IMES DISCUSSION PAPER SERIES

Missing Wage Inflation? Estimating the Natural Rate ofUnemployment in a Nonlinear DSGE Model
Yuto Iwasaki, Ichiro Muto, and Mototsugu Shintani
Discussion Paper No. 2018-E-8

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
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IMES Discussion Paper Series 2018-E-8
July 2018

Missing Wage Inflation? Estimating the Natural Rate ofUnemployment in a Nonlinear DSGE Model
Yuto Iwasaki*, Ichiro Muto**, and Mototsugu Shintani***
Abstract
During the recovery from the global financial crisis, most advanced economies haveexperienced a surprisingly weak response of wage inflation to the decline inunemployment. In this study, we investigate whether downward wage rigidity(DWR) is the source of the flattening wage Phillips curve and the lack of wage
inflation in the four advanced economies: Japan, the euro area, the UK, and the US.
Specifically, we apply Markov chain Monte Carlo methods with a particle filter toestimate a nonlinear New Keynesian dynamic stochastic general equilibrium model
incorporating asymmetric wage adjustment costs. This enables us to jointly estimate
the degree of DWR as well as the natural rate of unemployment, that is, the rate ofunemployment expected in the absence of (downward) wage rigidity. Our results
indicate that wage adjustment costs are highly asymmetric in Japan, the euro area,
and the UK, but not in the US. Especially, an L-shaped wage Phillips curve between
wage inflation and the unemployment gap clearly emerges in Japan, due to thepresence of DWR. As for the US, wage adjustment costs are large but symmetric,
which means that wages are inherently quite sticky both in an upward and downward
direction. Our results suggest that missing wage inflation in Japan, the euro area, andthe UK is attributable largely to DWR, but not in the US.
Keywords: downward wage rigidity; natural rate of unemployment; Phillips curve;
particle filter
JEL classification: E24, E31, E32* Director and Senior Economist, Institute for Monetary and Economic Studies, Bank of Japan(E-mail: yuuto.iwasaki@boj.or.jp)**Director and Senior Economist, Institute for Monetary and Economic Studies (currently, Headof Price Statistics Division, Research and Statistics Department), Bank of Japan (E-mail:ichirou.mutou@boj.or.jp)***Research Center for Advanced Science and Technology, University of Tokyo (E-mail:shintani@econ.rcast.u-tokyo.ac.jp)
The authors are grateful to Kosuke Aoki, Martin Bodenstein, James Bullard, Christopher Erceg,
Andrea Ferrero, Marvin Goodfriend, Hirokuni Iiboshi, Jinill Kim, Kevin Lansing, David LopezSalido, Sophocles Mavroeidis, Taisuke Nakata, James Nason, John Roberts, Kwanho Shin,Toshiaki Watanabe, Francesco Zanetti, seminar participants at Federal Reserve Board, Federal
Reserve Bank of St. Louis, RCAST Macro-economic Analysis Workshop at University of Tokyo,
Korea University, CEF 2018, and the staffs of the Bank of Japan for their helpful comments. The
views expressed in this paper are those of the authors and do not necessarily reflect the official
views of the Bank of Japan.

1Introduction

During the recovery from the global …nancial crisis and European debt crisis, advanced economies have experienced a surprisingly weak response of wage in‡ation
to the decline in unemployment. As shown in Figure 1, the unemployment rate inadvanced economies such as Japan, the euro area, the UK, and the US has fallen
substantially from post-crisis peak levels, but wage in‡ation in these economies has
not picked up markedly despite the clear improvement in labor market condition.1
Against this background, central bankers have expressed their concern over “missing wage in‡ation,” since this creates a weak in‡ationary environment in which thein‡ation rate persistently remains below the central bank’s target.2
There have been a wide range of discussions regarding the possible causes of thismissing wage in‡ation. Potential causes that have been mentioned include macroeconomic factors such as the weakness of trend productivity growth and in‡ation expectations, as well as recent structural changes in the labor market, such as changes
in labor force participation or the increase of part-time workers in total employment(see, for example, IMF (2017)). However, there are still few analyses trying to quantify the relative importance of the possible causes and the discussion on this issue
seems to be far from having reached a conclusion.
In this study, we investigate whether downward wage rigidity (DWR) is the sourceof the ‡attening wage Phillips curve and the lack of wage in‡ation in the four ad1

Figure 1 shows wage in‡ation in terms of increases in labor compensation per worker. However,the picture with regard to subdued wage in‡ation remains the same when increases in hourly wages
are used (Appendix Figure 1).
2

For example, in her remarks on the weak developments in wage growth in the post-recovery
phase, Yellen (2014) highlighted that “since wage movements have historically been sensitive totightness in the labor market, the recent behavior of both nominal and real wages point to weakerlabor market conditions than would be indicated by the current unemployment rate.” Similarly,
Constâncio (2017) observed that “[t]he fact that wages are not increasing more is an importantpuzzle in advanced economies. Higher wage increases were to be expected due to the strengtheningof the economic recovery and are necessary in order to normalise in‡ation.” Meanwhile, Haldane(2017) noted that “[a]s unemployment has fallen recently, this Phillips curve relationship would
have led us to expect wage growth to pick up. That, plainly, has not happened. Over recent years,
the Phillips curve relationship has been anything but strong and stable. And that same ‡atness
in the Phillips curve has been found in a number of other countries.” Finally, Cunli¤e (2017) also
expressed his concerns regarding the ‡attening or disappearance of the wage Phillips curve.

1vanced economies. The possibility that DWR was the reason for the subdued wage
in‡ation in recent years has been highlighted by Kuroda (2017), who observed that“wage reductions of full-time employees had been conducted only marginally in an
economic downturn, and wages barely rise even if the economy recovers and labormarket conditions tighten. It also can be said that past downward rigidity in wages
has led to the present upward rigidity.”3 DWR is an issue that has been widely discussed by Keynes (1936) and Tobin (1972), who identi…ed it as a prominent featureof macroeconomic dynamics in a low-in‡ation environment.4 As more recent studies
have shown, the wage Phillips curve becomes highly nonlinear and quite ‡at at low
rates of wage in‡ation in the presence of DWR (see e.g., Akerlof, Dickens and Perry(1996), Benigno and Ricci (2011), and Daly and Hobijn (2014)). The ‡at shape atlower wage in‡ation rates means that wage in‡ation is irresponsive not only to downside shocks but also to upside shocks in the labor market. The mechanism behindthe latter is clearly explained by Elsby (2009), who argues that wage increases are
irreversible to a certain degree. First, forward-looking …rms temper wage increases
as a precaution against future costly wage cuts. Second, even in the absence of
forward-looking behavior, DWR raises the level of wages that …rms inherit from thepast and makes it less necessary for …rms to raise wages as much to obtain their
3

Similarly, Yellen (2014) observed that “the sluggish pace of nominal and real wage growth inrecent years may re‡ect the phenomenon of ‘pent-up wage de‡ation.’ The evidence suggests thatmany …rms faced signi…cant constraints in lowering compensation during the recession and theearlier part of the recovery because of ‘downward nominal wage rigidity’– namely, an inability or
unwillingness on the part of …rms to cut nominal wages. To the extent that …rms faced limits in
reducing real and nominal wages when the labor market was exceptionally weak, they may …ndthat now they do not need to raise wages to attract quali…ed workers. As a result, wages mightrise relatively slowly as the labor market strengthens. If pent-up wage de‡ation is holding down
wage growth, the current very moderate wage growth could be a misleading signal of the degree ofremaining slack. Further, wages could begin to rise at a relatively more rapid pace once pent-up
wage de‡ation has been absorbed.”
4

Paul Krugman has also repeatedly stressed the importance of DWR in understanding in‡ation
dynamics, especially after the global …nancial crisis. For example, he stated that “[w]hat employers
learned during the long slump is that you can’t cut wages even when people are desperate forjobs; they also learned that extended periods in which you would cut wages if you could are a lot
more likely than they used to believe. This makes them reluctant to grant wage increases even ingood times, because they know they’ll be stuck with those wages if the economy turns bad again,”(Krugman (2018)).

2

desired wage level.5
The highly nonlinear nature of the wage Phillips curve especially at low levels of
wage in‡ation is easily discernible in scatter plots of wage in‡ation and the unemployment rate for the four advanced economies, as shown in Figure 2. The pattern isparticularly notable in Japan, where the nonlinear relationship between wage in‡ation and the unemployment rate is highly stable and the wage Phillips curve is nearlyvertical for high levels of wage in‡ation but becomes almost monotonically ‡atter for
lower levels of wage in‡ation. The patterns for the euro area and the UK are quite
similar, suggesting that in both economies the wage Phillips curve is nonlinear witha ‡at part for lower rates of wage in‡ation, especially since the 2000s. In contrast,
for the US, no clear relationship between wage in‡ation and unemployment can beobserved for the observation period as a whole. However, if we focus on data sincethe 1990s, there seems to be a downward-sloping wage Phillips curve with a ‡atter
slope at low rates of wage in‡ation.6
These observations suggest the presence of DWR in all four advanced economies.
However, based on this evidence only, it would be rash to conclude that DWR isan entrenched feature of these economies. This is particularly so since the unemployment rate is a relatively crude measure that does not necessarily provide an
accurate measure of labor market slack once the possibility that the natural rate ofunemployment is time-varying is taken into account. That is, as suggested by thenatural rate hypothesis, what matters for wage in‡ation is the deviation of the actual unemployment rate from the natural rate. This means that we need to correctly
identify developments in the natural rate of unemployment to assess the true pressure of labor market tightness on wage in‡ation. This means that the importance of5

Our theoretical model emphasizes the former mechanism in explaining the irresponsiveness ofwage in‡ation to upside shocks to the labor market.
6

The emergence of an inverse relationship between wage growth and the unemployment rate in
the US economy since the mid-1980s has been highlighted as the “return of wage Phillips curve”
by Galí (2011a), who derives and estimates a linear forward-looking structural relationship between wage growth and the unemployment rate, the so-called “New Keynesian wage Phillips curve(NKWPC).” Muto and Shintani (2018) compare wage in‡ation dynamics in the US and Japanthrough the lens of the NKWPC taking into account time variations in the parameters of theNKWPC.

3

DWR can be correctly assessed only by using an appropriate measure of the naturalrate of unemployment. This should be done through an empirical analysis using a
coherent framework that explicitly incorporates both DWR and the natural rate ofunemployment.
Against this background, the aim of this study is to examine the importance of
DWR in the recent weak wage in‡ation observed in the four advanced economies.
To this end, we apply Markov chain Monte Carlo (MCMC) methods with the particle …lter developed by Fernández-Villaverde and Rubio-Ramírez (2007) to estimate a nonlinear New Keynesian dynamic stochastic general equilibrium (DSGE)
model incorporating DWR for the four economies. Our model essentially follows
the models developed by Kim and Ruge-Murcia (2009, 2011) and Aruoba, Bocola,and Schorfheide (2017), which introduce DWR through an asymmetric wage adjustment cost function to produce downwardly rigid wage in‡ation dynamics.7 Usingan asymmetric wage adjustment cost function provides a reasonable setup, because
it does not preclude wage cuts, which are rare but observed in practice in microand macro-level data.8 However, since their models do not incorporate unemployment and assume that wage in‡ation is determined by a wage markup (the ratioof real wages to the marginal rate of substitution between consumption and labor),
we extend their models by introducing the link between the wage markup and theunemployment rate derived by Galí (2011a, b) based on the assumption of indivisible
labor input à la Hansen (1985).
In order to obtain an accurate measure of labor market slack, we calculate thenatural rate of unemployment fully taking DWR into account. In the New Keynesianframework, the natural rate of unemployment is de…ned as the hypothetical ratethat should be realized in the absence of wage rigidity. There exist some previous
studies which have attempted to estimate the natural rate of unemployment based
7

As explained below, we use an asymmetric wage adjustment cost function that is close to theone employed by Fahr and Smets (2010) rather than the linex adjustment cost function used by
Kim and Ruge-Murcia (2009, 2011) and Aruoba, Bocola, and Schorfeide (2017).
8

An alternative setup to incorporate DWR would be to introduce an inequality constraint thatstrictly requires the wage in‡ation rate not to be below zero. However, this kind of setup would
contradict the fact that wage cuts are observed in practice in the data.

4

on this concept. For example, Galí, Smets and Wouters (2012) develop a mediumscale sticky wage model with unemployment and present an estimate of the naturalrate of unemployment through a counterfactual exercise in which wage rigidity isabsent.9 Other studies presenting estimates of the natural rate of unemployment
are those by Gertler, Sala and Trigari (2008) and Sala, Söderström, and Trigari(2008), which employ search and matching models with staggered wage contracts.
However, wage rigidity in these studies is assumed to be symmetric in an upward anddownward direction. To the best of our knowledge, no previous studies calculate thenatural rate of unemployment in a New Keynesian DSGE model with downward wage
rigidity, despite its potential importance in understanding wage in‡ation dynamics ina low in‡ation environment. By employing the recently developed technique of using
an MCMC algorithm with particle …lter, our study contributes to the literature by
jointly estimating (i) the degree of downward wage rigidity as well as (ii) the naturalrate of unemployment within the coherent framework of a New Keynesian DSGE
model for four advanced economies.
The main …ndings of our analysis can be summarized as follows. First, wage
adjustment costs are highly asymmetric in Japan, the euro area, and the UK but notin the US. This result indicates that, except in the case of the US, it is extremely
important to take DWR into account to accurately measure the natural rate ofunemployment in the economies examined here. Second, in the case of Japan, an
L-shaped wage Phillips curve between wage in‡ation and the unemployment gap
emerges, once DWR is properly taken into account. This indicates that wage in‡ationin Japan is not responsive to labor market slack as long as the unemployment rate
is above the natural rate but then accelerates once the unemployment rate falls well
below the natural rate. Therefore, our results imply that the missing wage in‡ation
observed in Japan is not a permanent phenomenon and wage in‡ation is likely toreappear with further improvement in the labor market. Third, in the case of the US,
wage adjustment costs are large but symmetric, which means that wages are highly
9

They also use the setup proposed by Galí (2011a, b) to introduce unemployment into the New
Keynesian sticky-wage framework.

5

sticky both in an upward and downward direction. This suggests that wage in‡ation
has responded only sluggishly to the decline in unemployment because the wage
Phillips curve is inherently quite ‡at in the case of the US. These results suggestthat missing wage in‡ation in Japan, the euro area, and the UK can be largely
explained by DWR, but this is not the case for the US.
The rest of this study is organized as follows. Section 2 presents the model used
for our analysis. Section 3 explains the data and empirical methodologies. Section
4 presents the empirical results of the model and the natural rate of unemployment. Section 5 compares the estimated wage adjustment cost functions for thefour economies and examines whether missing wage in‡ation can be explained by
DWR. Section 6 discusses some caveats and possible extensions. Finally, Section 7
concludes.

2

Model

Our model essentially follows the New Keynesian sticky wage models developed by
Kim and Ruge-Murcia (2009, 2011) and Aruoba, Bocola, and Schorfheide (2017),
which introduce asymmetric wage adjustment costs to produce downwardly rigid
wage in‡ation dynamics. However, we modify and extend their models in the following respects. First, since their models do not incorporate unemployment, we extend
their models to introduce the unemployment rate, by adopting the setup of Galí(2011a, b) and Galí, Smets and Wouters (2012). This means that our model focuses on variations in labor input in the form of the extensive margin (employment)
rather than the intensive margin (hours).10 Second, we add several types of exogenous shocks - namely, wage markup shocks, discount factor shocks, and labor supply
shocks - to the model. Incorporating wage markup shocks is important to capture
time-variations in the natural rate of unemployment, while incorporating discountfactor shocks plays a key role in creating ‡uctuations in aggregate demand. Incorporating labor supply shocks is useful to prevent overestimating the importance of
10

To be consistent, wage is de…ned as the wage per worker rather than the wage per hour.

6

wage markup shocks.11 Third, in order to facilitate the identi…cation of asymmetry
in wage adjustment costs, we slightly change the speci…cation of the wage adjustment
cost function following Fahr and Smets (2010) and Abbritti and Fahr (2013).

2.1

Setup

2.1.1

Households

The economy is populated by a large number of identical households. Each household
consists of a continuum of family members indexed by a pair (j; k) 2 [0; 1][0; 1].

Households provide di¤erentiated types of labor services Nt (j) and the index j represents the type of labor service in which a household member is specialized. We
assume that labor input is indivisible in the sense that an individual works a …xed
number of hours or does not work at all. This means that all endogenous variations
in labor input take place at the extensive margin.
A household’s period utility is speci…ed as:
U (Ct ; fNt (j)g)(Ct =At )1
1

1

1

1

t

(Ct =At )1t

Z 1 Z Nt (j)1k dkdj000

Nt (j)1+
dj;
1+ 1

Z 1

1

(1)

where 1= is the intertemporal elasticity of substitution. The index k determines
the disutility from working, labor disutility iszero otherwise,tk

1

if the member is employed andt is an exogenous preference shifter andelasticity. The exogenous preference shifteris the Frisch labor supply

t , which is referred to as a “labor supply

shock,”follows a …rst order autoregressive process in log:
ln

t =

ln

11

t 1 + " ;t ;(2)

As highlighted by Galí, Smets and Wouters (2012), introducing the unemployment rate makes
it possible to distinguish between wage markup and labor supply shocks, which prevents the overestimation of the importance of wage markup shocks in the ‡uctuations of macroeconomic variables.

7

where " ;t is an iid white noise with variance

2 12

.

Household j obtains utility from consumption relative to a habit stock which is
given by the level of technology At .This assumption is introduced in order to ensurethat the economy ‡uctuates along a balanced growth path (Aruoba, Bocola, andSchorfheide (2017)). The utility function is additively separable and we assume fullrisk sharing within a household. This implies the same level of consumption for all
household members, which is represented by Ct .
The household seeks to maximize intertemporal utility:

Et

"1
X
s=0

where

s

#

dt+s U (Ct+s ; fNt+s (j)g) ;(3)is the constant component of the discount factor and dt is a discount factor

shock with law of motion:ln dt =

(4)d ln dt 1 + "d;t ;where "d;t is an iid normal random variable with mean zero and variance

2
d.

The

budget constraint is given by
Pt Ct + Bt (j) + Tt
= Wt (j) Nt (j) (1(5)

w ( w;t (j))) + Rt 1 Bt 1 (j) + Pt Dt (j) + Pt SCt (j) :where Pt is the price of …nal goods, Bt (j) is purchases of government bonds, Tt
is lump-sum taxes, Wt (j) is the nominal wage for type i labor,
wage adjustment cost (wherew;t (j)w ( w;t (j)) is theWt (j) =Wt 1 (j)), Rt is the risk-free nominalinterest rate on government bonds, Dt (j) is residual real pro…ts, and SCt (j) is thenet cash in‡ow from a set of state-contingent securities.
Household j maximizes lifetime utility (3) with respect to consumption Ct , bondholdings Bt , and nominal wage Wt (j). Let

t (j) be the Lagrange multiplier onbudget constraint (5) in the utility maximization problem for a household providing
type j labor services. The …rst order condition with respect to consumption is as
12

Following Galí, Smets and Wouters (2012),

is …xed to 0.999.

8

follows:
UC;t =(6)t (j)Pt ;where UC;t denotes the partial derivative of period utility with respect to consumptiont ;Nt (j))
= A1t
(UC;t = @U (C@C
t

Ct
At

), which is the same for all households.

Let Qt+sjt be the time t value of a unit of the consumption good in period t + s(Qt+sjt =UC;t+s
UC;t

=

At+s
At

1

Ct+s
Ct

). The …rst order condition with respect tobond holdings can then be expressed as follows:
dt+1
Rt
Qt+1jtdt
t+1

1 = Et(7):

Labor services provided by households are aggregated in the following constantelasticity of substitution form:

Nt =

Z 1

1

Nt (j)

1
w;t

1
w;t

dj

;

(8)

0

where the inverse demand elasticity

w;t is time-varying and evolves according to a

…rst order autoregressive process in log:
ln
where

w;t = (1w ) lnw is the steady state value ofwith mean zero and variance

2
w.

w +

w ln

w;t 1 + "w;t ;(9)

w;t and "w;t is an iid normal random variable

The value ofw;tdetermines workers’ market

power, and, as is explained later, it is the sole determinant of the wage markup in
this model. Therefore, we refer tow;t as a “wage markup shock”below.

Given input prices Wt (j) and output prices Wt , the optimal demand for eachtype of labor for the production that maximizes intermediate goods …rms’pro…ts is

Nt (j) =Wt (j)
Wt

9

1
w;t

Nt :

(10)

where the aggregate wage index is given by
Z 1

Wt =

w;t 1
w;t

Wt (j)

w;t
w;t 1

dj

(11)

:

0

Household j maximizes lifetime utility (3) by taking into account the labor demand schedule (10). The …rst order condition with respect to nominal wage Wt (j)is obtained as follows:
1

0 =UN;t (j)

w;t

+ Et

1
w;t

Wt (j)Wt+ t (j)

dt+1
dt

1
w;t

Wt (j)
Wt

1

Nt
Wt
1

Nt (1

w ( w;t (j)))

Nt+1

0
2
w ( w;t+1 (j)) w;t+1 (j);

w;t
1
w;t+1

Wt+1 (j)Wt+1

t+1 (j)0w ( w;t (j)) w;t (j)

) (1

(12)

where UN;t (j) denotes the partial derivative of period utility with respect to labort ;Nt (j))=
services (UN;t (j) = @U (C@Nt (j)2.1.2

1t Nt (j)).

Wage adjustment costs

The speci…cation of wage adjustment costs is a key element of our analysis. Following
Fahr and Smets (2010), we introduce the following asymmetric wage adjustment costfunction:w ( w;t (j)) =

where

w

1

w

w;t (j)

2

2

1

+

1

w

is the steady state value ofdegree of convexity andw;t (j)

exp

2
w

w

1

w

+

w;t (j)ww(13)w;t (j).

The parameterw

determines the

w the degree of asymmetry in wage adjustment costs.

Figure 3 illustrates the properties of wage adjustment cost function (13). If theasymmetry parameter is close to zero ( w ! 0), (13) becomes a Rotemberg-stylequadratic function (Rotemberg (1982)) where wage rigidity is solely determined by
the value of

w:

lim

w !0

w ( w;t (j)) =w

w;t (j)

2

w

10

2

1

:

1

1 ;

However, if

w takes a strictly positive value, the wage adjustment cost is asymmetricand wage cuts are more costly than wage increases. As shown in Figure 3, the increasein the asymmetry parameterw raises the cost of wage cuts, leaving the cost of wage

increases almost unchanged. Therefore, we can interpretw

as representing thedegree of wage rigidity generally seen in both an upward and a downward direction,
andw determines the asymmetry of wage rigidity.

This speci…cation is slightly di¤erent from the linex function introduced by Kim
and Ruge-Murcia (2009, 2011) and Aruoba, Bocola, and Schorfheide (2017) which is
given by:w ( w;t (j)) =w
2 (exp (w

w ( w;t (j)w )) +w ( w;t (j)

w)

1) :

(14)

Speci…cation (13) di¤ers from the linex form (14) in that the third derivative is givenby the single parameter

w , although it is given by

w in the case of (14). Becausew

we carry out Bayesian estimation with second-order approximation, this property of
speci…cation (13) is useful since it facilitates the identi…cation of parameters

w and

w.

2.1.3

Final goods production

We assume perfectly competitive …nal-goods-producing …rms that combine a continuum of intermediate goods indexed by i 2 [0; 1] using the technology
Yt =

Z 1

1

1

Yt (i)

1
p;t

p;t

di

;

(15)

0

where 1= p;t denotes the elasticity of demand for each intermediate good. Under theassumption of a perfectly competitive market, pro…t maximization and free entry
imply that the demand for intermediate goods is

Yt (i) =Pt (i)
Pt

11

1
p;t

Yt ;

(16)

where Pt (i) is the price of intermediate goods. The relationship between Pt (i) and
Pt is given by

Z 1

Pt =

p;t 1
p;t

Pt (i)

p;t
p;t 1

(17)

:

di

0

The inverse demand elasticity, referred to as the “price markup shock”below, evolves
according to a …rst order autoregressive process in log:
ln

p;t =

1

ln

p

p+

p ln

(18)

p;t 1 + "p;t ;where "p;t is an iid normal random variable with mean zero and variance
2.1.4

2
p.

Intermediate goods production

Intermediate good i is produced by a monopolist who has access to the following
production technology:(19)

Yt (i) = At Nt (i) :where At represents the productivity level, which is exogenous and common to all…rms and follows(20)

ln At = ln + ln At 1 + ln zt ;where ln zt =z ln zt 1 + "z;t and "z;t is an iid normal random variable with mean

zero and variance2
z . Intermediate goods producers buy labor services at a nominal

wage of Wt . Moreover, they face a Rotemberg-style quadratic adjustment cost withrespect to price changes:p ( p;t (i)) =where

Pt (i) =Pt 1 (i) andp;t (i)pp;t (i)

2

p

2

1

(21)

;

p is the steady state value of

13

The Rotemberg-style adjustment cost is a speci…c form (the case of
more general function, which allows upward or downward price rigidity:p ( p;t(i) ) =

1

p

2

p;t (j)
p

2

1

+

1
2
p

p;t (j)

exp

p

12

p

1

+

13
p;t (i).

p ! 0) of the followingp;t (j)
p

p

1

1 :

Given the demand for intermediate goods (16), each …rm maximizes pro…ts bychoosing its output price and labor input based on the following equation:

Et

"1
X

s

Qt+sjts=0

Pt+s (i)(1Pt+sp;t+s ( p;t (i))) Yt+s (i)Wt+s
Nt+s (i)
Pt+s

#

;

(22)

where Qt+sjt is the time t value of a unit of the consumption good in period t + s,
which is exogenous to intermediate goods producers.
The …rst order condition with respect to intermediate goods price Pt (i) is derived
as follows:
1

1

p;t
Pt (i)
00 =Yt + A t N t 1
p ( t (i))
p ( t (i)) t (i)
Pt
p;t
dt+1
1
+ Et
Qt+1jtAt+1 Nt+1 0w ( t+1 (i)) 2t+1 (i);
dt
t+1

t

where

(23)

t is the Lagrange multiplier for the demand for intermediate goods (16).

The …rst order condition with respect to labor input Nt (i) is given byt =Pt (i)(1Pt

2.2

Aggregate relationships

2.2.1

Unemployment ratep ( t (i)))

Wt 1:
P t At(24)

Next, we derive the relationship between the wage markup and the unemployment
rate, following Galí (2011a, b). Consider household member (j; k). Taking nominal
wage Wt (j) and price level Pt as given, an individual …nds it optimal to provide
labor services if and only if
Wt (j)
Pt

1

k
t At
(Ct =At );(25)

Many preceding studies, such as Kim and Ruge-Murcia (2009, 2011), Fahr and Smets (2010), andAbbritti and Fahn (2013), focus on the case of symmetric price adjustment costs ( p ! 0), basedon empirical analyses using micro data (such as Altissimo, Ehrmann and Smets (2006) and Chenet al. (2008)). Following these studies, we also use a Rotemberg-style quadratic price adjustment
cost.

13

that is, the real wage is above his or her disutility from working, expressed in termsof the marginal utility of consumption. Thus, the marginal supplier of type j labor,
which is denoted by Lt (j), is given by1

Lt (j)t At(Ct =At )Wt (j)
=
Pt

(26)

:

De…ne the aggregate labor force Lt as follows:
Lt =

Z 1

(27)

Lt (j) dj:

0

In a symmetric equilibrium, the aggregate labor supply condition is given by thefollowing equation:
1

Wt
=
Pt
Wage markup

Ltt At(Ct =At )

(28)

:

W
t , which is de…ned as the ratio of the real wage to the averagemarginal rate of substitution, is given by the following equation:
1

Wt
W
t =
Pt

Nt
t At
(Ct =At )

! 1

:

(29)

Combining these equations,
W
t =

Lt
Nt

1

:

(30)

De…ne the unemployment rate ut as the deviation of labor input from labor
supply (ut

Lt Nt
).
Nt

Then the relationship between the wage markup and theunemployment rate is as follows:1W
t = (1 + ut ) :

14

(31)

2.2.2

Wage in‡ation dynamics

In a symmetric equilibrium, wages and utilities are the same for all households(Wt (j) = Wt ,w;t (j)

w;t ,

=

t (j)

t,

=

UN;t (j) = UN;t , and UC;t (j) = UC;t ).

Then, from (6), (12), and (29), aggregate wage in‡ation dynamics can be obtained
as follows:
0 =

1

1

w;t

W
t

+ Et

+

1

1

(1

w ( w;t ))

w;t

dt+1
Nt+1 1Qt+1jtdt
Nt t+10w ( w;t ) w;t02w ( w;t+1 ) w;t+1

(32)

:

This represents a nonlinear and forward-looking structural relationship between
aggregate wage in‡ation

w;t and wage markup

Wt .

In the special case where wages are perfectly ‡exible ( w =0
w = 0), the above

equation becomes
0=
where

W
t

1

1

w;t

W
t

+ 1

1

(33)

;

w;t

denotes the natural level of the wage markup (“natural wage markup”

hereafter) that should prevail in a ‡exible-wage economy. Therefore, the natural
wage markup is solely determined byw;t (i.e.

Wt= 1 1w;t ). Because (31) holds

even in a ‡exible wage economy, the relationship between the natural wage markup
Wnt and the natural rate of unemployment ut , which is the unemployment rate thatshould prevail when wages are rigid, is given by the following equation:Wn 1
t = (1 + ut ) :(34)

From (33) and (34), the natural rate of unemployment can be expressed in terms
of

w;t :

unt =

1
1

1:

(35)

w;t

Therefore, the natural rate of unemployment rate is solely determined by wage
markup shockw;t .

By subtracting (33) from (32) and substituting

15

W
t and

W
t using (31) and (34),we obtain the following structural relationship between aggregate wage in‡ation andthe unemployment rate:
0 =

1

1

(1 + ut )

w;t

+ Et

(1 + unt )Nt+1 1
dt+1
Qt+1jtdt
Nt t+11

1

1w ( w;t ) +w;t02w ( w;t+1 ) w;t+10w ( w;t ) w;t

(36)

:

This can be viewed as a nonlinear version of the New Keynesian wage Phillips curve(NKWPC) in which the current wage in‡ation ratew;t is related to the unemploy-ment rate (actual rate ut and natural rate unt ) and the expectation for one-periodahead wage in‡ation Et w;t+1 . Similar to the linearized version of the NKWPC derived by Galí (2011a, b), the unemployment gap, namely, the deviation of the actual
unemployment rate ut from the natural rate unt , is the fundamental determinant ofwage in‡ation.14
2.2.3

Price in‡ation dynamics

Aggregate price in‡ation dynamics are obtained by combining intermediate goods…rms’…rst order conditions (23) and (24), and by assuming a symmetric equilibrium
where all intermediate goods producers choose the same prices and labor inputs(Pt (i) = Pt and Nt (i) = Nt ):
0=

Wt
+ 1
p;t Pt At

1

1

(1

p ( t ))p;t0p( t) t+

Et

dt+1
Yt+1
Qt+1jtdt
Yt

0p ( t+1 ) t+1(37)
This can be viewed as a nonlinear version of the New Keynesian Phillips curve(NKPC) in which current price in‡ationWt
t is related to markup ratio Pt At

and theforward-looking expectation for one-period ahead price in‡ation Et t+1 .
14

When the wage adjustment cost function is quadratic, the linearized version of the NKWPC
can be derived as the same functional form as those derived by of Galí (2011a, b), which assumes
Calvo-style symmetric wage rigidity (for the derivation of the linearized NKWPC under quadratic
wage adjustment costs, see Born and Pfeifer (2016)).

16:

2.2.4

Monetary and …scal policy

Following Aruoba, Bocola, and Schorfheide (2017), monetary policy is governed byan interest rate feedback rule:
Rt = Rt1

R

Rt R1 e"R;t ;(38)

where "R;t is a monetary policy shock given by an iid normal random variable withmean zero and variance2
R and Rt is the target rate of the nominal interest rate. Rt

is given by
Yt
Yt 1

1

t

Rt = r ppwhere r is the steady state real interest rate (r =

2

(39)
) and

is the target in‡ation

rate.
The …scal authority consumes a fraction

t of aggregate output Yt , where[0; 1] follows an exogenous process. De…ne gt = 1=(1ln gt = 1g

ln g +t ). We assume that(40)g ln gt 1 + "g;t ;where "g;t is an iid normal random variable with mean zero and variance
2.2.5

t 2

2
g.

Aggregate resource constraint

Intermediate goods producers’total dividend payments to households are given by
Dt = (1p ( p;t )) Yt

Wt
Nt ;
Pt

(41)

Combining the household budget constraint and the government budget constraint,
we obtain the following aggregate resource constraint:
Pt Ct +t Pt Yt = (1p ( p;t )) Pt Yt

17

w ( w;t )Wt Nt :

(42)

2.3

Model Solution

The dynamics of aggregate variables in this model are governed by eleven equations
determining the endogenous variables ((5), (7), (13), (19), (21), (35), (36), (37), (38),(39) and (42)) and six equations describing the shock processes ((2), (4), (9), (18),(20) and (40)). Since a closed-form solution is not available, we solve the rational
expectation equilibrium using the perturbation method. It should be noted that ourmodel involves a strong nonlinearity generated by the asymmetric wage adjustment
costs. For this reason, we employ a second-order approximation of the equilibrium
equation and the monetary policy function, instead of a …rst-order approximation,and solve the local dynamic equations around the steady state. The second-order
perturbation method has also been used to solve a similar model with asymmetric price and wage adjustment costs by Kim and Ruge-Murcia (2009) and Aruoba,
Bocola, and Schorfheide (2017).15

2.4

Wage adjustment costs and wage in‡ation dynamics

Before conducting our empirical analysis, we examine the relationship between thewage adjustment cost and the wage Phillips curve. Speci…cally, we show how theasymmetry of the wage adjustment cost function, captured by the parameterw,a¤ects wage in‡ation dynamics and eventually the shape of the wage Phillips curve.
Because the model is forward-looking and nonlinear, we conduct numerical simulations based on a calibrated model, rather than deriving the relationship analytically.16
Figure 4 presents the impulse responses of the unemployment rate and wage in‡ation to exogenous shocks (productivity and monetary policy shocks) under alternativevalues of

w . In the case of

w = 0, which corresponds to the case of Rotemberg-stylequadratic wage adjustment costs, the responses to positive and negative shocks are
symmetric. However, if

w has a large positive value (we set= 5000 in the …gure),

meaning that wages are downwardly rigid, the responses to positive and negative
15
See Schmitt-Grohe and Uribe (2004), and Kim et al. (2008) for the implementation and advantages of the second-order perturbation method.
16

The parameters of the model are based on the prior distribution presented later in Section 3.3.

18

shocks are asymmetric. The decline of wage in‡ation in response to a negative shockis limited and, as a result, the rise of the unemployment rate is much larger, than in
the case of symmetric adjustment costs.
Figure 5 shows the relationship between the asymmetry parameterw and theshape of the wage Phillips curve based on stochastic simulations using the calibrated
model.17 When wage adjustment costs are quadratic ( w = 0), the wage Phillips
curve is almost linear. However, as the asymmetry parameterw increases, the wage

Phillips curve becomes highly nonlinear. It is almost linear and steep at high levelsof wage in‡ation but is nearly ‡at at low levels of wage in‡ation. This shows clearly
that in our model DWR bends the wage Phillips curve.

3

Empirical approach

3.1

Data

The data we use for our empirical analysis are presented in Table 1, while Figure 6
shows developments in the key variables. We use (i) GDP per capita, (ii) consumption per capita, (iii) price in‡ation, (iv) wage in‡ation, (v) employment per capita,(vi) the unemployment rate, and (vii) the policy interest rate as observable variables.
As for wage in‡ation, we use the nominal wage per worker (rather than per hour) for
consistency with the model. For the policy interest rates we use the shadow interest
rates estimated by Ueno (2017) for Japan and by Krippner (2015) for the other three
economies, in order to capture the e¤ects of unconventional monetary policies. The
observation period is from 1970Q2 to 2017Q3.

3.2

Estimation method

We estimate our nonlinear DSGE model with asymmetric wage adjustment costs
using a Bayesian approach. Since our model is nonlinear, we cannot use a Kalman
17
In the stochastic simulation, in order to check the shape of the wage Phillips curve, we add
exogenous shocks except for labor supply shocks and wage markup shocks, since these two types ofshocks are the factors shifting the wage Phillips curve.

19

…lter to obtain exact likelihood functions. Instead, we use a bootstrap particle …lterto evaluate the likelihood function. This strategy is the same as that employed by
Fernández-Villaverde and Rubio-Ramírez (2007) to estimate a DSGE model withstochastic volatility. To use the bootstrap particle …lter, we need to assume thepresence of the measurement error in the measurement equation of the nonlinear
state-space representation. We follow Aruoba, Bocola, and Schorfheide (2017) andassume the measurement error follows an iid normal distribution with mean zero anda variance of 10% of the sample variances of GDP growth, in‡ation, interest rates, andnominal wage growth. The posterior distribution of parameters is evaluated using
a single-block MCMC/Metropolis-Hastings algorithm.18 We calculate the marginal
likelihood using the modi…ed harmonic mean estimator proposed by Geweke (1999).

3.3

Bayesian priors

The prior distributions for the Bayesian estimations are presented in Table 2. The
priors are mostly based on previous studies. We set the same values for the four
economies with a few exceptions.19 With respect to the risk-aversion parameterand the Frisch labor supply elasticity , we use gamma distributions with the samemeans and standard deviations as Aruoba, Bocola, and Schorfheide (2017). Following
many previous studies, the constant component of discount factor

is set to 0:99

and 1=g is …xed to 0:85.
The prior for wage rigidity

w is set as a gamma distribution with a range from 0

to 40, which covers the calibrated values set by Fahr and Smets (2010) and Abbritti
and Fahr (2013), which are 32 and 37:6, respectively. The degree of asymmetry ofwage rigidity

w , which is the central focus of our analysis, potentially takes both

positive and negative values, re‡ecting downward and upward wage rigidity. To take18

See Herbst and Schorfheide (2015) and Fernández-Villaverde, Rubio-Ramírez, and Schorfheide(2016) for details of this nonlinear DSGE estimation approach.
19

For the UK, under the same prior distributions as for the other economies, the natural rate ofunemployment coincides with the actual unemployment rate for almost all periods, which seems
quite unreasonable. Therefore, we use slightly di¤erent prior values for some parameters ( 2 = 0:5,
p = 0:25; w = 0:15), while for the remaining parameters we use the same values as for the othereconomies.

20

into account that the value and distribution of

w are highly uncertain, we employ

a uniform distribution as the prior with respect to
As for price rigidityw.

p , we use a gamma distribution with a range from 0 to 60,

which covers the calibrated values set by Fahr and Smets (2010) and Abbritti andFahr (2013), which are 45 and 60, respectively. Regarding the responsiveness of themonetary policy rule, we use prior means of

1 = 1:5 and

2 = 0:2, which are thesame values as those used by Aruoba, Bocola, and Schorfheide (2017). Taking theobserved high persistence of the nominal interest rate into account, we set the priormean for the parameter of interest rate smoothing to20
R = 0:80.

The priors for the steady state values of the inverse of the demand elasticityof labor and the inverse of the demand elasticity of intermediate goods ( w andp , respectively) are set as gamma distributions with 0:10 as the mean 0:05 as thestandard deviation.21 As for the prior distribution of long-run output growth ( ), we
use a gamma distribution with 2.0 as the mean and 1.0 as the standard deviation.
We calculate sample averages for the steady state price in‡ation ( p ).22 The steadystate value of wage in‡ation ( w ) is equal to the sum ofp and. Finally, the samepriors (persistence and variance) are set for the exogenous shock processes of mostof the shocks.23
20

This value is higher than the prior mean (0.50) used by Aruoba, Bocola, and Schorfheide (2017),
but is close to their posterior mean (0.81).
21

Aruoba, Bocola, and Schorfheide (2017) use a …xed value of 0:10 for boththan estimating them.
22
The sample mean offor the US.

w andp ratherp is.2:3% for Japan, 3:8% for the euro area, 5:2% for the UK, and 3:9%

23

Based on the empirical results of Aruoba, Bocola, and Schorfheide (2017), we set the priormean for the persistence of productivity shocks to z = 0:2, which is smaller than the persistenceof other exogenous shocks.

21

4

Estimation results for each economy

4.1

Japan

For each economy, we carry out Bayesian estimations based on two models re‡ecting alternative assumptions regarding the degree of asymmetry of wage adjustment
costs

w.

In the …rst model (the “asymmetric model”), we assume that wage ad-justment costs are potentially asymmetric and estimate the value ofw based on adi¤used prior. In the second model (the “symmetric model”), we assume that wage
adjustment costs are symmetric and impose a value of zero for

w a priori. We thencompare the estimation results of the two models to examine the importance of wage
adjustment cost asymmetry.
The posterior distributions for the case of Japan are presented in Table 3. The log
marginal likelihood for the asymmetric model is

1; 778:0, while it is1; 821:6 forthe symmetric model. Thus, the …t of the asymmetric model is signi…cantly better
than that of the symmetric model. In the asymmetric model, the posterior mean of
the parameter for the asymmetry of wage rigidity

w , which is the most importantparameter considered here, is 3; 560. The 90% con…dence interval ranges from 2; 367
to 4; 609, which means thatw is signi…cantly positive.

The posterior mean of thedegree of convexity in wage adjustment costs ( w ) is 29:4, which is larger than thatin the symmetric model (18:1). Figure 7 depicts the estimated wage adjustment costfunctions based on the two models. The …gure clearly shows that in the asymmetricmodel wage cuts are more costly than wage increases.
Turning to other parameters, based on the asymmetric model, our estimates of
the risk-aversion parameterand Frisch labor supply elasticity

are 5:87 and 0:20,

respectively. These values are not far from the estimates obtained by Aruoba, Bocola,and Schorfheide (2017) for the US economy from the mid-1980s onward, which are
4:10 and 0:10, respectively. The posterior mean of price rigidity
is smaller than the estimated wage rigidity
24

24
w.

p is 12:4, which

The parameters of the monetary

Let i be the so-called Calvo parameter, namely, the probability that prices or wages (i = por w) are unchanged for a given period. Then, as shown in Table 1 in Kahn (2005), the following22policy rule ( R = 0:89,

1 = 2:05 and

2 = 0:82) are in line with the values reported

in the literature on DSGE models. These structural parameters (except forw ) andthe exogenous shock processes (persistence and standard deviations of exogenous
shocks) are also estimated in the symmetric model, and they are somewhat - but not
substantially - di¤erent from those in the asymmetric model.
Next, we consider the natural rate of unemployment and the unemployment gap
in Japan, which are presented in Figure 8. The left panel of the …gure shows thenatural rate of unemployment estimated based on the two alternative models. The
bold (red) line depicts the estimated natural rate of unemployment in the asymmetric model, while the thin solid (blue) line depicts that in the symmetric model.
The two lines trace a similar path until the late-1990s but then diverge substantially.
Furthermore, the natural rate of unemployment in the asymmetric model was substantially lower than the actual unemployment rate (broken black line) from the late
1990s until around 2014, which more or less coincides with the period of de‡ationin Japan, while the natural rate in the symmetric model has closely followed theactual rate from the early 1990s onward. As a result, as shown in the right panelof Figure 8, the unemployment gap (i.e., the deviation of the actual unemployment
rate from the natural rate) obtained based on the asymmetric model took persistently large positive values for a period of almost 15 years from the end of 1990s
until the mid-2010s. This is a stark contrast from the symmetric model, in which
the unemployment gap was only marginally positive during this period. Since themarginal likelihood strongly prefers the asymmetric model, these results imply thatthe asymmetry of wage adjustment costs has played quite an important role in determining the tightness of labor market conditions during Japan’s so-called lost decades.
That being said, since the mid-2010s, the unemployment gap has shrunk and turned
relationship holds between the degree of convexity and the Calvo parameter:i =(1i (1= i

1)

i )(1

i

)

:

Based on the estimation results of the asymmetric model, we obtain p = 0:489 and w = 0:805
for the Calvo parameters. Therefore, the estimation results imply that in Japan prices are more‡exible than wages.

23

negative, implying that tightness in Japan’s labor market has increased in recent
years.
Next, Figure 9 shows the wage Phillips curve by plotting wage in‡ation and theunemployment gap estimated based on the asymmetric model and the symmetricmodel. We …nd that the shape of the wage Phillips curve for Japan di¤ers dependingon the model. In the symmetric model, the wage Phillips curve is almost linear. In
contrast, in the asymmetric model, the wage Phillips curve is L-shaped: the slope
is quite steep where the unemployment gap is negative, but almost ‡at where theunemployment gap is positive. Because the asymmetric model is statistically superior
to the symmetric model, we judge that the L-shaped wage Phillips curve provides abetter description of Japan’s wage in‡ation dynamics.

4.2

The euro area

The posterior distributions for the case of the euro area are presented in Table 4.
The log marginal likelihood is

1; 493:4 for the asymmetric model and

1; 524:9

for the symmetric model, which means that, as for Japan, the asymmetric model
provides a much better …t than the symmetric model. In the asymmetric model, theposterior mean of the asymmetry parameterw , is substantially positive (4; 508).

The posterior mean of the convexity parameterw is 51:7. Figure 10 shows the wage

adjustment cost function for the euro area. As in Japan, wage adjustment costs arehighly asymmetric, with the cost of wage cuts being considerably larger than thecost of wage increases. The other parameters do not di¤er substantially from thosefor Japan except for the convexity parameter of price adjustment costs
56:3 and hence slightly larger than the value ofp , which is

w . This implies that prices are more‡exible than wages.25
Figure 11 depicts the natural rate of unemployment in the euro area. The two
estimated series deviate substantially especially in the period before the early 1980s,
the period from the mid- to late-1990s, and after the global …nancial crisis. This
indicates that DWR had a sizable impact on the actual unemployment rate in these25

The Calvo parameters are

p = 0:741 andw = 0:899.

24

three periods. This impact is particularly pronounced from around the mid-1990s
onward, with the natural rate of unemployment in the asymmetric model being much
lower than the actual rate. As a result, the unemployment gap has been persistently
positive since the …rst half of the 1990s, reaching a peak of more than 5% after the
onset of the European debt crisis. We also …nd that the unemployment gap in thesymmetric model turned negative at the end of the observation period (2017Q3), but
the gap in the asymmetric model is still large and positive.
Finally, Figure 12 shows the wage Phillips curve for the euro area. While the wage
Phillips curve in the symmetric model is quite linear, that in the asymmetric modelis fairly nonlinear, since its slope is quite ‡at for very low levels of wage in‡ation.
Because the log marginal likelihood is higher in the asymmetric model, we judge
that, as in the case of Japan, the nonlinear wage Phillips curve in the asymmetricmodel provides better description of wage in‡ation dynamics in the euro area.

4.3

The UK

The posterior distributions for UK are presented in Table 5. The log marginal likelihood is

2; 029:4 for the asymmetric model and2; 047:5 for the symmetric model.

Therefore, as for Japan and the euro area, the asymmetric model is statistically superior to the symmetric model. In the asymmetric model, the posterior mean of theasymmetry parameterw is 1; 101, while that of the convexity parameterw is 20:1.

These values are smaller than those of Japan and the euro area. Figure 13 indicatesthat the wage adjustment cost function for the UK is fairly asymmetric, meaningthat DWR is important. The convexity parameter of price adjustment cost
35:7, which is larger than the value of

p is

26
w.

Figure 14 displays the natural rate of unemployment in the UK. The two estimated series deviate persistently, indicating that DWR had a sizable impact on
developments in the unemployment rate in the UK. Speci…cally, the asymmetric
26
Nevertheless, in terms of the Calvo parameters, prices are more ‡exible than wages ( p = 0:593
and w = 0:874), because w is far greater than p . This result is consistent with the …ndings
obtained by Faccini, Millard and Zanetti (2011) who also report that for the UK the Calvo parameter
for wages is larger than that for prices.

25

model shows a large and positive unemployment gap in the early-1980s and again
in the early-1990s, when actual unemployment rose sharply. In the wake of the
global …nancial crisis, the actual unemployment rose, but the natural rate also rose
sharply. As a result, the unemployment gap was around 0% even during the crisis
periods. Meanwhile, in the recovery since the crisis, the actual unemployment rate
has declined substantially and fallen below the natural rate. Finally, Figure 15 shows
the wage Phillips curve for the UK. The wage Phillips curve is not as clear as forJapan and the euro area. However, it seems more nonlinear in the asymmetric model
than in the symmetric model, since the ‡at part at low wage in‡ation is much longer
in the asymmetric model.

4.4

The US

The posterior distributions for the US are presented in Table 6. In the asymmetricmodel, the posterior mean of the asymmetric parameterw is positive and statisti-cally signi…cant. However, the mean value ( w = 579) is smaller than those obtained
for the other economies and the standard deviation is quite large.27 We also …ndthat the convexity parameter in the wage adjustment cost function is much larger( w = 242 in the symmetric model) than those for the other economies. Figure 16
displays the shape of the wage adjustment cost function for the US. Since the valueofw plays a dominant role, the wage adjustment cost function seems to be almost

quadratic even in the asymmetric model, which indicates that wage adjustment costs
are large but symmetric. This means that, in the case of the US, wages are highly
sticky both in an upward and downward direction. In other words, the asymmetry
parameter

w does not play a substantial role in wage dynamics, even though it is27

These results are in line with those reported by Aruoba, Bocola, and Schorfheide (2017), who
similarly …nd that the asymmetry parameter of the wage adjustment cost function is statisticallysigni…cant but small in size. Some studies using micro-data argue that the degree of DWR in the
US has increased since the global …nancial crisis (e.g., Fallick, Lettau, and Wascher (2016)). To
examine how the degree of DWR has changed in the US, we re-estimate the model using data forthe period the global …nancial crisis only (1970Q2-2007Q4). As shown in Appendix Table 1 andAppendix Figure 2, we …nd that the asymmetry parameter w is not statistically signi…cant. Thissuggests that the degree of DWR has increased somewhat in the post-crisis period, although it isstill quite small compared to the other three economies.

26

statistically signi…cant. Consequently, the log marginal likelihood isasymmetric model and1; 634:9 for the1; 627:2 for the symmetric model. Therefore, in contrast to

the other three economies, for the US the …t is better in the case of the symmetricthan the asymmetric model.
Turning to other parameters, in the symmetric model, the risk-aversion parameteris 3:37 and the Frisch labor supply elasticity

is 0:19 - values that are in line withprevious studies (such as Aruoba, Bocola, and Schorfheide (2017)). The convexity
parameter of price adjustment costp is 69:8, which is lower than the value ofw.

This implies that prices are more ‡exible than wages.28 The parameters of themonetary policy rule are

R = 0:75,

1 = 1:06 and

2 = 1:06. As for the propertiesof exogenous shocks, we …nd that the standard deviation of labor supply shock isquite large, compared to other economies.
Figure 17 shows the natural rate of unemployment in the US. Both of the estimated series - based on the symmetric and the asymmetric model - move very close
together. As a result, developments in the unemployment gap are almost the samein the two models, meaning that DWR has not had a substantial impact on thetightness of labor market conditions in the US.29 Variations in the natural rate ofunemployment are generally smaller than those in the actual unemployment rate.
Following the global …nancial crisis, the natural rate of unemployment jumped, but
this jump was less than 2 percentage points and thus only partially accounts for thelarge increase in the actual unemployment rate of more than 5 percentage points.
Finally, Figure 18 plots wage in‡ation/ unemployment gap observations based on the
two alternative models. Because the estimated series of the natural rate of unemployment are very close in these two models, no substantial di¤erence between thetwo panels can be observed. Furthermore, both panels indicate that, in the case of
the US, there is no clear relationship between wage in‡ation and the unemployment
gap.
28

The Calvo parameters are calculated as

p = 0:772 and29

w = 0:913.

The natural rate of unemployment in the asymmetric model is highly correlated with that
estimated by the Congressional Budget O¢ ce (the correlation coe¢ cient for the period from 1970Q2
to 2017Q3 is 0.96).

27

5

Downward wage rigidity and missing wage in‡ation

5.1

Comparisons of wage adjustment costs

The previous section provided estimation results of the model and the natural rateof unemployment for individual economies. As mentioned, the wage adjustment costfunction, which is the key focus of our study, is characterized by two parameters: thedegree of convexity ( w ) and the degree of asymmetry ( w ). In order to compare
the nature of wage rigidity in the four economies, Figure 19 present the shapes of
the respective wage adjustment cost functions.
In the case of Japan, wage adjustment costs are highly asymmetric. This meansthat DWR is very important, since the cost of wage cuts is very high compared to thecost of wage increase. Next, the shape of the adjustment cost function for the euroarea is also highly asymmetric, which indicates the importance of DWR in the euroarea. We …nd that wages are generally more rigid in the euro area than in Japan,
since adjustment costs are larger for the euro area both in an upward and downward
directions. In the case of the UK, wage adjustment costs are relatively small, meaningthat wages are relatively ‡exible. At the same time, however, the adjustment costfunction is also asymmetric, although the degree of asymmetry is somewhat smaller
than in Japan and the euro area. In contrast, the wage adjustment cost function for
the US is symmetric and adjustment costs are quite large both in an upward anddownward direction. The results thus indicate that wages are generally very sticky
in the case of the US, re‡ecting the very large value of convexity parameterw.

Let us examine whether these results are consistent with the …ndings of previous
studies. Unfortunately, there are relatively few studies that conduct a comprehensive
cross-country analysis of wage rigidity or downward rigidity. This is mainly due tothe fact that it is not easy to assemble comparable data sets across countries. In
particular, analyses using micro-data have to contend with di¤erences in the typeof workers included and the de…nition of wages, and the methods used for detecting

28

downward rigidity often di¤er across studies. There is also the problem that DWR
cannot be detected if the observation period is short. Furthermore, many existingstudies judge DWR based solely on the skewness of the distribution of microdata.
Because such analysis does not involve identi…cation of shocks hitting to the economy,
it is unable to quantify the in‡uence of DWR on aggregate wage in‡ation dynamics.
The studies mentioned below are subject to these issues.
Holden and Wulfsberg (2007), using industry-level wage data for 19 OECD countries for the period 1973-1999, …nd that DWR is weak in the US and the UK, while itis strong in some euro area countries (such as Italy, Greece, Portugal, and Spain).30
Another cross-country study is that by Dickens et al. (2007), who report that DWR
is relatively weak in the UK but is strong in some euro area countries (such as Germany, Greece, Italy, and Portugal).31 Meanwhile, Knoppik and Beissinger (2005)
compare DWR using micro-data for European households covering the period 19942001. They …nd that the degree of DWR for the UK is smaller than that for the
euro area. As for Japan, Kimura and Ueda (2001) and Kuroda and Yamamoto(2003a, 2003b) present empirical results for the emergence of DWR in the 1990s,
although they also …nd that DWR is not observed between the late 1990s and theearly 2000s. Yamamoto (2007) also shows that, even in the 2000s, DWR is foundin regular wages for full-time workers.32 With respect to the degree of wage rigidity (not DWR), which is captured by the parameterw in this analysis, Muto andShintani (2018) use aggregate data for Japan and the US to estimate the NKWPC,
30

They report that strong DWR is typically observed in countries where the unemployment rate islow, employment protection legislation is strong and labor union density is high. However, Dickens
et al. (2007) report that these measures do not necessarily explain the degree of DWR. These
results indicate that evidence on the relationship between DWR and the characteristics of labormarket institutions is still inconclusive.
31
This study reports the results of the International Wage Flexibility Project (IWFP). The observation period varies across countries. For example, it is 1970-1997 for the US, but 1976-2000 for
the UK.
32

Yamamoto (2008) explains that “wage stickiness should have increased after collapse of the
bubble economy in Japan, due to the existence of downward nominal wage rigidity under the low
in‡ation environment. Before collapse of the bubble economy when in‡ation rate was relatively
higher, downward rigidity of nominal wage was not a binding constraint in wage setting. When thein‡ation rate got lower and even negative after collapse of the bubble economy, however, nominal
wage cut became necessary to reduce real wages. Then, downward nominal wage rigidity started
to serve as a binding constraint.” This interpretation is consistent with our empirical …ndings.

29which assumes Calvo-style symmetric wage rigidity. They report that the slope of
the NKWPC is much steeper in Japan than in the US, which suggests that wages
are less sticky in Japan than in the US.33 López-Villavicencio and Saglio (2017) also
estimate the NKWPC using data for 15 OECD countries covering the period from
1985Q1 to 2014Q3. Their regression results suggest that wage rigidity is quite low
in Japan and very high in the US, while wage rigidity in European countries lies
between that Japan and the US. While these empirical studies do not necessarily
provide comprehensive analyses of wage rigidity or downward rigidity, our results
can be regarded as roughly consistent with previous studies.

5.2

Wage Phillips curve and missing wage in‡ation

Our remaining task is to investigate whether missing wage in‡ation observed in
the four advanced economies can be explained by DWR. To do so, we present recent
developments in wage in‡ation and the unemployment gap in Figure 20. Speci…cally,
we show how these two variables have evolved since unemployment rates peaked in
the wake of the crisis. In the case of Japan, the unemployment rate peaked in 2009Q3.
Subsequently, the unemployment gap declined substantially, but wage in‡ation has
remained almost unchanged and has stayed at around 0%, because the wage Phillips
curve has been quite ‡at due to the asymmetric wage adjustment costs. This meansthat the missing wage in‡ation observed in Japan’s economy very likely is caused by
DWR.
In the euro area, the situation is almost the same as in Japan. The unemployment
rate reached its peak in 2013Q2. The unemployment gap subsequently declined
substantially, re‡ecting the recovery from the European debt crisis. However, as inJapan, wage in‡ation has been unresponsive to the decline in unemployment, since
the wage Phillips curve has been almost ‡at due to DWR. In the case of the UK,
the unemployment rate peaked in 2011Q3 and the unemployment gap has declined
markedly since then. During this period, wage in‡ation rate has been somewhat
33

They estimate the NKWPC using data for Japan and the US for the period from 1970Q1 to
2013Q2. They also provide micro-level evidence (Barattieri, Basu, and Gottschalk (2014) for the
US and the “Survey on Wage Increases” for Japan) supporting this interpretation.

30

volatile, compared to Japan and the euro area. However, wage in‡ation and theunemployment gap can be regarded as evolving roughly in line with the ‡at part of
the wage Phillips curve.
Taken together, the empirical results indicate that DWR has been an importantsource of missing wage in‡ation in Japan, the euro area and the UK. We …nd thatin these economies wage in‡ation is not responsive to the degree of labor market
slack as long as the unemployment rate is above the natural rate but accelerates
when the unemployment rate falls well below the natural rate. The most recent datafor 2017Q3 shows that the unemployment gap is approaching the kink of the wage
Phillips curve. This suggests that wage in‡ation is likely to reappear with further
improvement in the labor market.
However, the situation seems to be quite di¤erent in the US. Since the crisis,
the unemployment gap has declined almost monotonically but wage in‡ation has‡uctuated considerably. This raises the fundamental question whether wage in‡ation
dynamics in the US are actually governed by the wage Phillips curve. Our estimation
results indicate that the estimated variance of labor supply shocks is particularly large
in the US. This raises the possibility that labor supply shocks play an important rolein shifting the location of the wage Phillips curve in the case of the US economy. To
examine this hypothesis, we investigate the shape of the wage Phillips curve throughstochastic simulations using the model estimated with the US data. In Figure 21, we
present the results of two alternative simulations. In the …rst simulation we add all
exogenous shocks. In the second simulation we add exogenous shocks except for thelabor supply shock. In the …rst simulation, the relationship between wage in‡ationand the unemployment gap is quite unclear. However, in the second simulation, therelationship is almost linear. These experiments suggest that there actually exists a
linear wage Phillips curve in the US once the distortionary impact of labor supply
shocks is eliminated. In other words, in the case of the US, wage in‡ation has
responded only sluggishly to the decline in the unemployment rate because the wage
Phillips curve is linear but inherently quite ‡at.34 This means that the mechanism
34

We have also checked the shape of wage Phillips curve by using the employment cost index

31

behind the missing wage in‡ation observed in the US is not explained by DWR.35

6

Caveats and possible extensions

Our analysis contributes to the literature by providing new empirical …ndings on
the shape of the wage Phillips curve and the natural rate of unemployment in thefour advanced economies considered here based on a recently developed particle …lter
technique for estimating nonlinear DSGE models. However, there are some caveats
and possible extensions.
First, our analysis is based on the sticky wage model developed by Galí (2011a,b) and Galí, Smets and Wouters (2012) to incorporate unemployment. A great
advantage of the model for our empirical analysis is its simplicity, since the estimationof a nonlinear DSGE model using Bayesian MCMC with a particle …lter is quite acomputationally demanding task. That being said, a natural extension would be toincorporate search and matching frictions into our model, since the current version of
the model is too simple to examine variations in the natural rate of unemployment,in that the natural rate of unemployment is solely determined by exogenous wage
markup shocks. In this respect, Abbritti and Fahr (2013) and Abo-Zaid (2013)
have already presented sticky wage models with search and matching frictions aswell as asymmetric wage adjustment costs. A potentially fruitful avenue would beto estimate these nonlinear models with search and matching frictions and thereby
identify the path of the natural rate of unemployment in the four economies.
Second, labor market slack in our model is de…ned in terms of the unemployment
gap. However, an interesting extension would be to incorporate broader measures oflabor market slack into the model. One possible extension in this direction would be
published by the Bureau of Labor Statistics, which is free from the in‡uence of employment shifts
among occupations and industries. Consequently, we have found that there exists a stable andvery ‡at wage Phillips curve in the US since the beginning of the 2000s. This also suggests that
employment shifts are quite important shifting factor of the wage Phillips curve in the US.
35
While we have highlighted the potential role of labor supply shocks in shifting the wage Phillips
curve, other factors a¤ecting the shape of the wage Phillips curve, such as a shift in in‡ation
expectations may also play a role especially in the earlier periods, such as the 1970s and the early
1980s.

32

to include the intensive margin (endogenous hours) in the model. Another issue israised by Erceg and Levin (2014), who incorporate households’decision with regard
to labor market participation into their model, assuming that labor market exit
and reentry decisions are associated with signi…cant adjustment costs. Under thissetup, the labor force participation gap, which is de…ned as the deviation of thelabor force participation rate from its potential path implied by demographic andstructural factors, enters the Phillips curve in addition to the unemployment gap.
Their study suggests that the reason why in‡ation in the US has been subdued
in recent years despite the notable decline in the unemployment rate, is that thelabor force participation gap widened markedly after the …nancial crisis. Our model
could be extended by incorporating the labor force participation gap to examine therobustness of our …ndings.
Third, although our sticky wage model assumes di¤erent types of labor services, itdoes not fully take into account the full variety of workers found in the labor market,
such as full-time and part-time workers, incumbent workers and newly hired workers,
skilled workers and unskilled workers, and any combination of these characteristics.
As highlighted in a number of studies (e.g., IMF (2017)), changes in the compositionof the workforce in terms of these characteristics are an important reason for the
recent subdued wage in‡ation in advanced economies. An important task therefore
is to develop a model which takes such changes in the workforce composition into
account and reassess the importance of DWR as a reason for the recent subdued
wage in‡ation.36
Finally, to capture the stance of monetary policy, we used estimates of shadowinterest rates obtained in empirical studies on yield curves in our model. Because
central banks in the four economies have employed an array of unconventional policies, we think that employing shadow interest rates provides a useful way to measure36

As shown in Appendix Figure 3, changes in the composition of the workforce in terms of full-timeand part-time workers have persistently restrained aggregate wage in‡ation in Japan. Similarly, inother economies, too, increases in the share of part-time workers have made a negative contribution
to aggregate wage in‡ation. This means that, if the depressing e¤ect of changes in the workforce
composition were excluded, aggregate wage in‡ation would be higher than it actually was. Thisimplies in turn that DWR might be even more important than our results suggest.

33

the stance of monetary policy. However, it would be possible to explicitly introduce
the zero lower bound (ZLB) or the e¤ective lower bound (ELB) into the model andestimate the model using a Bayesian method with a particle …lter, as some recent
studies have tried to do (e.g., Hirose and Sunakawa (2016), Iiboshi and Shintani(2017), Iiboshi, Shintani and Ueda (2017), and Aruoba, Bocola, and Schorfheide(2018)). Examining how explicitly introducing the ZLB or ELB a¤ects (or does
not a¤ect) the structural parameters, particularly the degree of (downward) wage
rigidity, is an issue we hope to tackle in the future.37;38

7

Conclusion

Wage in‡ation in advanced economies has been generally subdued in the aftermathof the global …nancial crisis. Against this background, the present study examined
whether this phenomenon can be explained by downward wage rigidity (DWR). The
results indicate that DWR likely is an important source of the ‡attening wage Phillips
curve and the lack of wage in‡ation in Japan, the euro area, and the UK.
Especially in the case of Japan, an L-shaped wage Phillips curve between wage
in‡ation and the unemployment gap emerges once the presence of DWR is properly
taken into account. This indicates that wage in‡ation is not responsive to the degreeof labor market slack as long as the unemployment rate is above the natural rate but
accelerates when the unemployment rate falls well below the natural rate. Therefore,
our results imply that the missing wage in‡ation observed in Japan is not a permanent
phenomenon and wage in‡ation is likely to reappear with further improvement in thelabor market.
However, our results also indicate that DWR is not necessarily the reason for37

To examine how sensitive our results are to the use of alternative data for policy interest rates,
we re-estimate the model and the natural rate of unemployment for the case of Japan using theactually observed call rate rather than the shadow interest rate. Using the call rate does notsubstantially alter the estimation results on the wage adjustment cost function and the natural rateof unemployment (see Appendix Table 2 and Appendix Figures 4 to 6).
38
Using a time-varying structural VAR model to identify shocks, Debortoli, Galí and Gambetti(2018) provide empirical evidence suggesting that the ZLB constraint is irrelevant for the dynamic
response of US macroeconomic variables to such shocks.

34the recent missing wage in‡ation in all advanced economies, since it has no notable
impact on the wage in‡ation dynamics in the US. In the case of the US, wage adjustment costs are large but symmetric, meaning that wages are highly sticky both in an
upward and downward direction. This suggests that wage in‡ation has responded
only sluggishly to the decline in the unemployment rate because the wage Phillips
curve is inherently quite ‡at in the case of the US.

35

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41

Table 1: Data Sources
Series1

OutputJapan

Cabinet Office, SNA: Real
GDP

Euro area
ECB, Area-wide Model
database: GDP;
Eurostat, Real Gross
Domestic Product for Euro
area (19 countries, retrieved
from FRED)
ECB, Area-wide Model
database, Individual
Consumption Expenditure;
OECD, Private Final
Consumption Expenditure
for the Euro Area(retrieved
from FRED)
ECB, Area-wide Model
database: HICP- All items
Excluding Energy;
Eurostat, Harmonized Index
of Consumer Prices: Overall
Index Excluding Energy forEuro area (19 countries,
retrieved from FRED)UKUSBank of England, Three
U.S. BEA, Gross Domestic
Centuries of Macroeconomic
Product: Real GDP
Data: Real GDP (retrieved(retrieved from FRED)
from FRED)Bank of England, Three
U.S. BEA, Personal income
Centuries of Macroeconomic
and outlays: Real Personal
Data: Real Consumption
Consumption Expenditure
Expenditure (retrieved from(retrived from FRED)
FRED)

Consumption

Cabinet Office, SNA: Real
Private consumption

CPI inflation

Ministry of Internal Affairs
and Communication,
Consumer Price Index: CPI
less food and energy

4

Wage per head

Calculated by deviding
Compensation of employees
by employment;Monthly Labor Survey; Total
ECB, Area-wide Model
cash earning
database: Compensation of
employees; OECD,
Quarterly National Account5

Employment

Labor Force Survey:
Employed person

ECB, Area-wide Model
database; OECD, Quarterly
National Account

Office for National
Statistics, Labor Market
OECD, Employed
Statistics: Number of People populationin Employment

6

Unemployment rate

Labor Force Survey:
Unemployment rate

ECB, Area-wide Model
database; OECD,
Harmonized Unemployment
Rate

Office for National
U.S. Bureau of Labor
Statistics, Labor Force
Statistics, Civilian
Survey: ILO Unemployment
Unemployment Rate
rate

7

Short-term interest
rateBank of Japan, Overnight call ECB, Area-wide Modelrate
database: Euribor 3-month

Bank of England, Bank ofEngland Policy Rate
(retrieved from FRED)

Board of Governors of theFederal Reserve System:
Effective Federal Funds Rate
(retrieved from FRED)

8

Shadow rate

Ueno (2017)

Krippner (2015)

Krippner (2015)

Krippner (2015)

Population

Ministry of Internal Affairs
and Communication,
Populaton estimatesEurostat

Office for National
Statistics, Population
estimates (Annual data isconverted to quarterly data)

U.S. Bureau of Economic
Analysis, Personal income
and outlays:
Population(retrived from
FRED)

2

3

9

42

OECD, CPI: All items
excluding food and energy
for UK

U.S. Bureau of Labor
Statistics, Consumer Price
Index for All Urban
Consumers: All items lessfood and energy(retrieved from FRED)

Calculated by deviding
Calculated by deviding
Compensation of employees Compensation of employees
by employment; OECD,by employment; OECD,
Compensation of Employees Compensation of Employees

Table 2: Prior distributions for Bayesian estimations

Parameters

Description

Distribution

Para1

Para2

𝜏

Inverse of the intertemporal elasticity of substitution

Gamma

2.00

1.00

𝜈

Frisch labor supply elasticity

Gamma

0.50

0.25

𝜙𝑝

Convexity in price adj. cost function

Gamma

30

15

𝜙𝑤

Convexity in wage adj. cost function

Gamma

20

10

𝜓𝑤

Asymmetricity in wage adj. cost function

Uniform−1000010000

𝜌𝑟Interest rate smoothing

Beta

0.80

0.20

𝜓1

Inflation coefficient in policy rule

Gamma

1.50

0.20

𝜓2

Output coefficient in policy rule

Gamma

0.20

0.10

𝜆̅ 𝑝

Inverse of demand elasticity of final goods

Gamma

0.10

0.05

𝜆̅ 𝑤

Inverse of demand elasticity of labor

Gamma

0.10

0.05

𝛾

Average technology growth

Gamma

2.00

1.00

𝜌𝑔

Autocorrelation of government spending shocks

Beta

0.50

0.20

𝜌𝑧

Autocorrelation of productivity shocks

Beta

0.20

0.10

𝜌𝑝

Autocorrelation of price markup shocks

Beta

0.50

0.20

𝜌𝑤

Autocorrelation of wage markup shocks

Beta

0.50

0.20

𝜌𝑑

Autocorrelation of discount factor shocks

Beta

0.50

0.20

𝜎𝑟

Std. dev. of monetary policy shocks

Inv.gamma

0.50

2.00

𝜎𝑔

Std. dev. of government spending shocks

Inv.gamma

0.50

2.00

𝜎𝑧

Std. dev. of productivity shocks

Inv.gamma

0.50

2.00

𝜎𝑝

Std. dev. of price markup shocks

Inv.gamma

0.50

2.00

𝜎𝑤

Std. dev. of wage markup shocks

Inv.gamma

0.50

2.00

𝜎𝜒

Std. dev. of labor supply shocks

Inv.gamma

0.50

2.00

𝜎𝑑

Std. dev. of discount factor shocks

Inv.gamma

0.50

2.00

Notes: Para(1) and Para(2) denote the means and the standard deviations for Beta, Gamma, and Normal
2
distributions; 𝑠 and 𝜈 for the Inverse Gamma distribution, where 𝑝𝐼𝐺 (𝜎|𝜈, 𝑠) ∝ 𝜎 −𝜈−1 𝑒 −𝜈𝑠 /2𝜎 ;For Uniform distribution, each denotes the upper and lower bound respectively.

43

Table 3: Posterior distributions of the parameters for Japan

Posteriors
Asymmetric model

Symmetric model

Mean

90% interval

Mean

90% interval

𝜏

5.87

[5.02, 6.81]

5.05[4.75, 5.54]

𝜈

0.20

[0.18, 0.22]

0.16[0.15, 0.16]

𝜙𝑝

12.4

[6.89, 18.6]

31.8[29.7, 33.8]

𝜙𝑤

29.4

[22.5, 34.2]

18.1[14.8, 21.9]

𝜓𝑤

3560

[2367, 4609]

-

-

𝜌𝑟

0.89

[0.87, 0.91]

0.86[0.85, 0.88]

𝜓1

2.05

[1.94, 2.18]

1.85[1.80, 1.91]

𝜓2

0.82

[0.29, 1.40]

1.36[1.23, 1.50]

𝜆̅𝑝

0.13

[0.06, 0.20]

0.13[0.10, 0.16]

𝜆̅𝑤

0.41

[0.36, 0.47]

0.30[0.25, 0.33]

𝛾

2.05

[1.94, 2.18]

2.06[1.91, 2.18]

𝜌𝑔

0.98

[0.97, 0.99]

0.99[0.99, 0.99]

𝜌𝑧

0.75

[0.66, 0.83]

0.80[0.78, 0.83]

𝜌𝑝

0.41

[0.13, 0.81]

0.67[0.44, 0.95]

𝜌𝑤

0.71

[0.70, 0.72]

0.74[0.73, 0.76]

𝜌𝑑

0.96

[0.96, 0.97]

0.97[0.97, 0.98]

100𝜎𝑟

0.27

[0.24, 0.30]

0.30[0.27, 0.33]

100𝜎𝑔

0.21

[0.15, 0.26]

0.31[0.30, 0.33]

100𝜎𝑧

0.26

[0.19, 0.32]

0.26[0.23, 0.29]

100𝜎𝑝

0.52

[0.27, 0.83]

0.34[0.28, 0.42]

100𝜎𝑤

0.54

[0.27, 0.96]

4.00[3.16, 4.84]

100𝜎𝜒

0.50

[0.26, 0.98]

0.51[0.25, 1.03]

100𝜎𝑑

3.57

[3.10, 4.10]

3.01[2.54, 3.33]

Log marginal likelihood-1778.0-1821.6

44

Table 4: Posterior distributions of the parameters for the euro area

Posteriors
Asymmetric model

Symmetric model

Mean

90% interval

Mean

90% interval

𝜏

4.81

[4.17, 5.24]

4.16[3.84, 4.78]

𝜈

0.23

[0.21, 0.26]

0.23[0.13, 0.34]

𝜙𝑝

56.3

[52.4, 60.0]

49.7[40.1, 53.7]

𝜙𝑤

51.7

[46.2, 54.2]

53.9[50.7, 59.2]

𝜓𝑤

4508

[3802, 6306]

-

-

𝜌𝑟

0.86

[0.84, 0.88]

0.87[0.85, 0.90]

𝜓1

2.05

[1.77, 2.23]

1.88[1.68, 2.19]

𝜓2

1.37

[0.79, 1.99]

1.91[1.58, 2.21]

𝜆̅𝑝

0.16

[0.09, 0.29]

0.17[0.08, 0.26]

𝜆̅𝑤

0.61

[0.50, 0.74]

0.69[0.54, 0.77]

γ

2.12

[1.60, 2.81]

2.29[2,05, 2.48]

𝜌𝑔

0.89

[0.85, 0.93]

0.94[0.87, 0.97]

𝜌𝑧

0.77

[0.71, 0.86]

0.80[0.72, 0.84]

𝜌𝑝

0.43

[0.21, 0.71]

0.52[0.23, 0.85]

𝜌𝑤

0.91

[0.55, 0.99]

0.98[0.94, 0.99]

𝜌𝑑

0.98

[0.97, 0.98]

0.99[0.97, 0.99]

100𝜎𝑟

0.24

[0.22, 0.26]

0.22[0.19, 0.26]

100𝜎𝑔

0.22

[0.19, 0.24]

0.19[0.15, 0.20]

100𝜎𝑧

0.21

[0.19, 0.23]

0.18[0.17, 0.20]

100𝜎𝑝

0.57

[0.29, 0.81]

0.91[0.59, 1.21]

100𝜎𝑤

0.67

[0.29, 0.96]

0.62[0.41, 0.94]

100𝜎𝜒

1.17

[0.96, 1.55]

1.57[1.07, 2.05]

100𝜎𝑑

4.56

[4.02, 4.92]

6.93[5.48, 7.69]

Log marginal likelihood-1493.4-1524.945

Table 5: Posterior distributions of the parameters for the UK

Posteriors
Asymmetric model

Symmetric model

Mean

90% interval

Mean

90% interval

𝜏

2.62

[2.32, 2.86]

2.61[2.36, 2.83]

𝜈

0.34

[0.24, 0.49]

0.32[0.12, 0.45]

𝜙𝑝

35.7

[24.7, 45.5]

48.1[36.1, 68.2]

𝜙𝑤

20.1

[17.9, 22.4]

21.2[20.4, 22.2]

𝜓𝑤

1101

[849, 1289]

-

-

𝜌𝑟

0.58

[0.44, 0.70]

0.51[0.40, 0.60]

𝜓1

1.20

[1.18, 1.23]

1.12[1.10, 1.14]

𝜓2

1.38

[1.26, 1.50]

1.22[1.02, 1.36]

𝜆̅𝑝

0.09

[0.06, 0.12]

0.05[0.04, 0.08]

𝜆̅𝑤

0.72

[0.56, 0.84]

0.77[0.64, 0.93]

𝛾

2.36

[2.19, 2.48]

2.43[2.28, 2.58]

𝜌𝑔

0.79

[0.75, 0.84]

0.86[0.82, 0.89]

𝜌𝑧

0.76

[0.67, 0.82]

0.78[0.72, 0.84]

𝜌𝑝

0.51

[0.19, 0.72]

0.63[0.32, 0.86]

𝜌𝑤

0.22

[0.14, 0.31]

0.28[0.15, 0.32]

𝜌𝑑

0.99

[0.98, 0.99]

0.99[0.99, 0.99]

100𝜎𝑟

0.55

[0.39, 0.78]

0.60[0.49, 0.72]

100𝜎𝑔

0.49

[0.41, 0.55]

0.43[0.38, 0.45]

100𝜎𝑧

0.27

[0.25, 0.30]

0.27[0.26, 0.28]

100𝜎𝑝

0.98

[0.28, 2.21]

0.17[0.11, 1.40]

100𝜎𝑤

0.46

[0.24, 0.74]

0.54[0.27, 0.98]

100𝜎𝜒

1.19

[1.03, 1.39]

1.36[1.22, 1.45]

100𝜎𝑑

1.71

[1.52, 1.90]

2.04[1.79, 2.25]

Log marginal likelihood-2029.4-2047.5

46

Table 6: Posterior distributions of the parameters for the US

Posteriors
Asymmetric model

Symmetric model

Mean

90% interval

Mean

90% interval

𝜏

3.64

[3.07, 4.01]

3.37[3.05, 3.75]

𝜈

0.10

[0.09, 0.12]

0.19[0.17, 0.27]

𝜙𝑝

73.8

[67.4, 79.9]

69.8[64.4, 77.5]

𝜙𝑤

225

[219, 229]

242

[222, 257]

𝜓𝑤

579

[99, 845]

-

-

𝜌𝑟

0.77

[0.74, 0.79]

0.75[0.72, 0.77]

𝜓1

1.08

[1.05, 1.11]

1.06[1.04, 1.07]

𝜓2

0.98

[0.74, 0.79]

1.06[0.65, 1.27]

𝜆̅𝑝

0.12

[0.07, 0.17]

0.17[0.11, 0.24]

𝜆̅𝑤

0.50

[0.45, 0.55]

0.31[0.26, 0.37]

γ

0.71

[0.53, 1.04]

1.21[0.97, 1.45]

𝜌𝑔

0.98

[0.97, 0.98]

0.97[0.96, 0.98]

𝜌𝑧

0.53

[0.44, 0.63]

0.57[0.50, 0.66]

𝜌𝑝

0.41

[0.19, 0.71]

0.51[0.20, 0.82]

𝜌𝑤

0.44

[0.17, 0.98]

0.39[0.12, 0.85]

𝜌𝑑

0.81

[0.78, 0.84]

0.80[0.76, 0.85]

100𝜎𝑟

0.34

[0.28, 0.38]

0.37[0.32, 0.43]

100𝜎𝑔

0.22

[0.20, 0.25]

0.19[0.22, 0.26]

100𝜎𝑧

0.48

[0.31, 0.69]

0.24[0.21, 0.27]

100𝜎𝑝

0.22

[0.20, 0.25]

0.53[0.29, 0.78]

100𝜎𝑤

0.57

[0.29, 1.17]

0.42[0.25, 0.77]

100𝜎𝜒

4.93

[4.55, 5.33]

2.80[2.57, 4.03]

100𝜎𝑑

1.41

[1.20, 1.67]

1.12[1.00, 1.43]

Log marginal likelihood-1634.9-1627.2

47

Figure 1: Wage inflation and unemployment rate (time-series)<Euro area><Japan>y/y % chg.

40%

y/y % chg.

%

20

8

15

Wage inflation
30

Unemployment rate (right)

15

6

10

10

20
4

5

10
2

0
-10

70

75

80

85

90

95

00

05

10

15

5

0
-5

0

70

75

80

85

<UK>
y/y % chg.

%

30
20
10
0

85

90

95

10

y/y % chg.

15

0

00

05

10

15

%

10

10

12

8

10

6

8

4

6

2

4

2

0

2

0

-2

4

80

05

12

6

75

00

12

8

70

95

<US>

40

-10

90

48

70

75

80

85

90

95

00

05

10

15

14

0

Figure 2: Wage inflation and unemployment rate (scatter plot)<Euro area><Japan>wage, y/y % chg.

40

wage, y/y % chg.

20

35

1970s

30

1980s

25

15

1990s

20

2000s

15

10

2010s

10
5

5

0
-5
-10

0

1

2

3

0

4
5
6
unemployment rate, %

0

2

4

6

wage, y/y % chg.

wage, y/y % chg.

12

30

10

25

8

20

6

15

4

10

2

5

0

0
-5

10
12
14
unemployment rate, %

<US>

<UK>
35

8

0

2

4

6

8

10

12

-2

14

unemployment rate, %

2

4

6

8

10

12

14

unemployment rate, %49

Figure 3: Wage adjustment cost function

wage adjustment cost

0.006

= 5000= 1000
= 500
=0

0.005
0.004
0.003
0.002
0.001
0.000

Notes:

-6

-4

-2

0

wage inflation (%, annual)2

4

is 20. The horizontal axis depicts deviations from steady-state wage inflation.

50

6Figure 4: Impulse responses of wage inflation and unemployment
Productivity shock<Asymmetric model><Symmetric model>

Wage inflation
0.010

0.010

Positive shock

0.005

0.005

Negative shock

0.000

0.000

-0.005-0.005-0.0100Quarter48-0.01012

0Quarter

4

8

12

0

4

8

12

4

8

12

4

8

12

Unemployment rate
0.00500.00500.00250.00250.00000.0000-0.0025-0.0025-0.0050

0Quarter48-0.0050

12Quarter

Monetary policy shock
Wage inflation
0.03

0.03
0.02

0.02

0.01

0.01

0.00

0.00

-0.01

-0.01

-0.02

-0.02

-0.03

0

Quarter

4

8

-0.03

12

0

Quarter

Unemployment rate
0.02

0.02

0.01

0.01

0.00

0.00
-0.01

-0.01

-0.02

-0.02
0

Quarter

4

8

12

51

0

Quarter

Figure 5: Wage adjustment cost and the wage Phillips curve<𝜓𝑤 = 0><𝜓𝑤 = 500>

<𝜓𝑤 = 1000>

wage inflation, %wage inflation, %wage inflation, %

20

20

20

15

15

15

10

10

10

5

5

5

0

0

0

-5

-5

-5

-10

-5

0

unemployment gap, %

5

-10

-5

0

unemployment gap, %

52

5

-10

-5

0

unemployment gap, %5

Figure 6: Data used for estimation<Consumption per capita><GDP per capita>
y/y % chg.

10

Japan
Euro area
UK
US

5

5

0

0

-5

-5

-10

-10
70

75

80

85

90

95

00

05

10

y/y % chg.

10

15

70

80

85

90

95

00

05

10

15

<Wage inflation><Price inflation>y/y % chg.

25

75

y/y % chg.

40

20

30

15

20

10

10

5

0

0
-5

70

75

80

85

90

95

00

05

10

15

-10

70

75

80

<Employment per capita>

90

95

00

05

10

15

<Unemployment rate>

y/y % chg.

6

85

%

15

3

10

0
5

-3
-6

0
70

75

80

85

90

95

00

05

10

15

70

75

80

85

90

95

00

05

10

15

<Policy interest rate>
%

20
15
10
5
0
-5
-10

70

75

80

85

90

95

00

05

10

15

Note: All of the series other than the unemployment rate and the policy interest rate are displayed on a year-on-year basis.
However, we use a quarter-on-quarter change rather than a year-on-year change of a variable in our estimation.

53

Figure 7: Wage adjustment cost function for Japan

wage adjustment cost
0.010

0.008

Asymmetric model
Symmetric model

0.006

0.004

0.002

0.000

-6

-4

-2

0

2

wage inflation (%, annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

54

4

6

Figure 8: Natural rate of unemployment in Japan<Unemployment gap>

<Natural rate of unemployment>
%

6

%

3
2

5

1

4

0

3

-1
2

Asymmetric model
Symmetric model1

Actual unemployment rate
0

70

75

80

85

90

95

00

05

10

-2

Asymmetric model-3

Symmetric model

-4

15

70

75

80

85

90

95

00

05

10

15

Figure 9: Wage Phillips curve for Japan<Asymmetric model><Symmetric model>

wage inflation, q/q % chg., annualized

30
25

25

20

20

15

15

10

10

5

5

0

0

-5

-5

-10

-4

-3

-2

-1

0

1

2

wage inflation, q/q % chg., annualized

30

-10

3

unemployment gap, %

55

-4

-3

-2

-1
0
1
unemployment gap, %2

3

Figure 10: Wage adjustment cost function for the euro area

wage adjustment cost
0.010

0.008

Asymmetric model
Symmetric model

0.006

0.004

0.002

0.000

-6

-4

-2

0

2

wage inflation (%, annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

56

4

6

Figure 11: Natural rate of unemployment in the euro area

<Natural rate of unemployment><Unemployment gap>

%

14

6

12

4

10

2

8

0

6

-2
Asymmetric model

4

Symmetric model

2
0

%

8

Actual unemployment rate
70

75

80

85

90

95

00

05

10

-4

Asymmetric model-6

Symmetric model

-8

15

70

75

80

85

90

95

00

05

10

15

Figure 12: Wage Phillips curve for the euro area

25

<Asymmetric model><Symmetric model>

Wage inflation, q/q % chg., annualized

Wage inflation, q/q % chg., annualized

25

20

20

15

15

10

10

5

5

0

0

-5

-8

-6

-4

-2
0
2
unemployment gap, %

4

-5

6

-8

-6

-4

-2

0

2

unemployment gap, %

57

4

6

Figure 13: Wage adjustment cost function for the UK

wage adjustment cost
0.010

0.008

Asymmetric model
Symmetric model

0.006

0.004

0.002

0.000

-6

-4

-2

0

2

4

wage inflation (%, annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

58

6

Figure 14: Natural rate of unemployment in the UK<Natural rate of unemployment>
14<Unemployment gap>

%

3

12

2

10

1
0

8

-1

6

-2

4

Asymmetric model

2

Symmetric model

0

%

4

-3

Actual unemployment rate
70

75

80

85

90

95

00

05

10

-4

Asymmetric model-5

Symmetric model

-6
15

70

75

80

85

90

95

00

05

10

15

Figure 15: Wage Phillips curve for the UK<Asymmetric model><Symmetric model>

wage inflation, q/q % chg., annualized

40
30

30

20

20

10

10

0

0

-10

-6

-4

-2

0

2

wage inflation, q/q % chg., annualized

40

-10

4

-6

-4

-2

0

2

unemployment gap, %unemployment gap, %59

4

Figure 16: Wage adjustment cost function for the US

wage adjustment cost
0.010

0.008

Asymmetric model
Symmetric model

0.006

0.004

0.002

0.000

-6

-4

-2

0

2

4

wage inflation (%, annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

60

6

Figure 17: Natural rate of unemployment in the US<Natural rate of unemployment>
12<Unemployment gap>

%

5

%

Asymmetric model

4

10

Symmetric model

3

8

2

6

1
0

4

Asymmetric model

0-1

Symmetric model2-2

Actual unemployment rate
70

75

80

85

90

95

00

05

10

-3

15

70

75

80

85

90

95

00

05

10

15

Figure 18: Wage Phillips curve for the US<Asymmetric model><Symmetric model>

wage inflation, q/q % chg., annualized

wage inflation, q/q % chg., annualized

15

15

10

10

5

5

0

0

-5

-5

-10

-4

-3

-2

-1

0

1

2

3

-10

unemployment gap, %

-4

-3

-2

-1

0

1

unemployment gap, %

61

2

3

Figure 19: Comparison of wage adjustment cost functions

wage adjustment cost
0.010

US
0.008

Euro Area
Japan

0.006

UK

0.004

0.002

0.000

-6

-4

-2

0

2

4

wage inflation (% annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

62

6

Figure 20: Recent development of wage inflation and unemployment gap<Euro area><Japan>wage inflation, q/q % chg., annualized

30
25

20

20

15

15
10

10

2009/Q3

5

2013/Q2

5

0
-5
-10

wage inflation, q/q % chg., annualized25-4

2017/Q3

0

-2

-5

-3

-1

0

1

2

3

unemployment gap, %2017/Q3
-8

-6

-4

2

4

6

<US>

wage inflation, q/q % chg., annualized

wage inflation, q/q % chg., annualized

15

2009/Q4

30

10

20

5

10

0

0
-10

0

unemployment gap, %

<UK>
40

-2

2017/Q3-6-4-5

2011/Q3-2

02

2017/Q3-104unemployment gap, %

-4

-3

-2

-1

0

1

unemployment gap, %

63

2

3

Figure 21: Stochastic simulation of wage Phillips curve for the US<Excluding labor supply shocks><All shocks>
wage inflation, q/q % chg., annualized

15

15

10

10

5

5

0

0

-5

-6

-4

-2

0

2

-5

4

wage inflation, q/q % chg., annualized

-6

-4

-2

0

2

unemployment gap, %unemployment gap, %Notes: To obtain simulated series, we use the posterior mean for the symmetric model in Table 6 and randomly
generated shocks.

64

4Appendix Table 1: Posterior distributions of the parameters for the US

Posteriors
Full-sample

Sub-sample(1970Q2-2017Q3)(1970Q-2007Q4)

Mean

90% interval

Mean

90% interval

𝜏

3.64

[3.07, 4.01]

3.08[2.70, 3.72]

𝜈

0.10

[0.09, 0.12]

0.24[0.20, 0.28]

𝜙𝑝

73.8

[67.4, 79.9]

82.9[79.2, 87.7]

𝜙𝑤

225

[219, 229]

212

[206, 222]

𝜓𝑤

579

[99, 845]-60.3[-117, 19.7]

𝜌𝑟

0.77

[0.74, 0.79]

0.71[0.67, 0.76]

𝜓1

1.08

[1.05, 1.11]

1.06[1.04, 1.08]

𝜓2

0.98

[0.74, 0.79]

0.40[0.20, 0.60]

𝜆̅𝑝

0.12

[0.07, 0.17]

0.08[0.06, 0.12]

𝜆̅𝑤

0.50

[0.45, 0.55]

0.25[0.22, 0.30]

γ

0.71

[0.53, 1.04]

0.96[0.97, 1.23]

𝜌𝑔

0.98

[0.97, 0.98]

0.97[0.97, 0.98]

𝜌𝑧

0.53

[0.44, 0.63]

0.41[0.30, 0.53]

𝜌𝑝

0.41

[0.19, 0.71]

0.50[0.12, 0.84]

𝜌𝑤

0.44

[0.17, 0.98]

0.41[0.14, 0.61]

𝜌𝑑

0.81

[0.78, 0.84]

0.80[0.79, 0.83]

100𝜎𝑟

0.34

[0.28, 0.38]

0.37[0.32, 0.43]

100𝜎𝑔

0.22

[0.20, 0.25]

0.24[0.22, 0.27]

100𝜎𝑧

0.48

[0.31, 0.69]

0.28[0.25, 0.32]

100𝜎𝑝

0.22

[0.20, 0.25]

0.67[0.36, 1.23]

100𝜎𝑤

0.57

[0.29, 1.17]

0.33[0.28, 0.37]

100𝜎𝜒

4.93

[4.55, 5.33]

2.41[1.90, 3.01]

100𝜎𝑑

1.41

[1.20, 1.67]

0.96[0.79, 1.23]

65

Appendix Table 2: Posterior distributions of the parameters for Japan

Posteriors
Using shadow rate

Using call rate

Mean

90% interval

Mean

90% interval

𝜏

5.87

[5.02, 6.81]

5.57[4.71, 6.00]

𝜈

0.20

[0.18, 0.22]

0.18[0.16, 0.20]

𝜙𝑝

12.4

[6.89, 18.6]

29.5[22.5, 35.3]

𝜙𝑤

29.4

[22.5, 34.2]

23.7[16.5, 30.3]

𝜓𝑤

3560

[2367, 4609]

2554[1240,3567]

𝜌𝑟

0.89

[0.87, 0.91]

0.86[0.84, 0.89]

𝜓1

2.05

[1.94, 2.18]

1.86[1.76, 1.97]

𝜓2

0.82

[0.29, 1.40]

0.79[0.58, 0.97]

𝜆̅𝑝

0.13

[0.06, 0.20]

0.13[0.08, 0.17]

𝜆̅𝑤

0.41

[0.36, 0.47]

0.38[0.31, 0.46]

𝛾

2.05

[1.94, 2.18]

2.06[1.91, 2.18]

𝜌𝑔

0.98

[0.97, 0.99]

0.99[0.99, 0.99]

𝜌𝑧

0.75

[0.66, 0.83]

0.72[0.67, 0.76]

𝜌𝑝

0.41

[0.13, 0.81]

0.47[0.13, 0.74]

𝜌𝑤

0.71

[0.70, 0.72]

0.71[0.69, 0.73]

𝜌𝑑

0.96

[0.96, 0.97]

0.97[0.96, 0.98]

100𝜎𝑟

0.27

[0.24, 0.30]

0.28[0.24, 0.31]

100𝜎𝑔

0.21

[0.15, 0.26]

0.27[0.19, 0.30]

100𝜎𝑧

0.26

[0.19, 0.32]

0.33[0.30, 0.36]

100𝜎𝑝

0.52

[0.27, 0.83]

0.52[0.28, 0.84]

100𝜎𝑤

0.54

[0.27, 0.96]

0.95[0.27, 3.51]

100𝜎𝜒

0.50

[0.26, 0.98]

0.48[0.26, 0.96]

100𝜎𝑑

3.57

[3.10, 4.10]

3.23[2.77, 3.68]

66

Appendix Figure 1: Hourly wage growth and unemployment rate (scatter plot)<Euro area><Japan>wage, y/y % chg.

40

wage, y/y % chg.

20

35

1970s

30

1980s

25

15

1990s

20

2000s

15

10

2010s

10
5

5

0
-5
-10

0

1

2

3

0

4
5
6
unemployment rate, %

0

2

4

6

wage, y/y % chg.

wage, y/y % chg.

12

30

10

25

8

20

6

15

4

10

2

5

0

0
-5

10
12
14
unemployment rate, %

<US>

<UK>
35

8

0

2

4

6

8

10

12

-2

14

unemployment rate, %

2

4

6

8

10

12

14

unemployment rate, %Notes:Figures for Japan and US are those for the overall sectors. Figures for Euro area and UK are thosefor the maufacturing sector.
Sources: Ministry of Health, Labour and Welfare; OECD

67

Appendix Figure 2: Wage adjustment cost function in the US (Full-sample vs Sub-sample)

wage adjustment cost
0.010

0.008

Full-sample
Sub-Sample

0.006

0.004

0.002

0.000

-6

-4

-2

0

2

wage inflation (%, annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

68

4

6

Appendix Figure 3: Decomposition of Japan's wage inflation (full-time vs part-time workers)

y/y % chg

Contribution of the share of part-time employees

4

Contribution of part-time employees
Contribution of full-time employees

2

Total Cash Earnings

0

-2

-4

-6

94

96

98

00

02

04

06

08

10

12

14

Source: Ministry of Health, Labour and Welfare.
Note: Q1 = March-May, Q2 = June-August, Q3 = September-November, Q4 = December-February.

69

16Appedix Figure 4: Wage adjustment cost function for the Japan (call rate)

wage adjustment cost
0.010

0.008

Using shadow rate
Using call rate

0.006

0.004

0.002

0.000

-6

-4

-2

0

2

wage inflation (%, annual)Notes: The horizontal axis depicts deviations from steady-state wage inflation.

70

4

6

Appendix Figure 5: Natural rate of unemployment in Japan (call rate)<Unemployment gap>

<Natural rate of unemployment>
%

6

%

3
2

5

1

4

0

3

-1
2

Using shadow rate
Using call rate

1

Actual unemployment rate
0

70

75

80

85

90

95

00

05

10

-2

Using shadow rate

-3

Using call rate

-4

15

70

75

80

85

90

95

00

05

10

15

Appendix Figure 6: Wage Phillips curve for Japan (call rate)<Using shadow rate><Using call rate>

wage inflation, q/q % chg., annualized

30

25

25

20

20

15

15

10

10

5

5

0

0

-5

-5

-10

-4

-3

-2

-1

0

1

2

wage inflation, q/q % chg., annualized

30

-10

3

unemployment gap, %

71

-4

-3

-2
-1
0
1
unemployment gap, %23