CIGS Working Paper Series No. 23-006E

Sources of Inequality and Business Cycles:
Evidence from the US and Japan

Masaru Inaba (Senshu University / The Canon Institute for Global Studies)
Kengo Nutahara (Senshu University / The Canon Institute for Global Studies)
Daichi Shirai (Tohoku Gakuin University / The Canon Institute for Global Studies)2023.4※Opinions expressed or implied in the CIGS Working Paper Series are solely those of theauthor, and do not necessarily represent the views of the CIGS or its sponsor.
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Sources of Inequality and Business Cycles:
Evidence from the US and Japan
Masaru Inaba∗Kengo Nutahara†

Senshu University and CIGS

Senshu University and CIGS

Daichi Shirai‡
Tohoku Gakuin University and CIGS
April 8, 2023∗

E-mail: imasaru@isc.senshu-u.ac.jp
E-mail: nutti@isc.senshu-u.ac.jp‡
E-mail: shirai.daichi@mail.tohoku-gakuin.ac.jp†1Abstract
The main objectives of this paper are twofold. The ﬁrst is identifying the sourcesof inequality and business cycle ﬂuctuations in the US and Japan. The second
is investigating the eﬀects of reducing inequality on business cycles. We develop
a tractable heterogeneous-agent business cycle model with unconstrained (U) andhand-to-mouth (HtM) households. We also introduce wedges, which imply various
types of distortions in economic activities, into the model following the businesscycle accounting approach and estimate them by the Bayesian method. We focus
on consumption inequality as inequality. We ﬁnd that, in the US, the labor market
distortions speciﬁc to the U households have signiﬁcant impacts on both businesscycles and consumption inequality, while the primary source of business cycles is the
distortion in aggregate productivity, and that of consumption inequality is the labormarket distortion speciﬁc to the HtM. In contrast, we ﬁnd that no common factors
signiﬁcantly impact both business cycles and consumption inequality in Japan. In
Japan, the key for business cycles is the distortion in aggregate productivity, and thatfor consumption inequality is the labor market distortions speciﬁc to U and HtMhouseholds. We also investigate the eﬀects of reductions in consumption inequality
on business cycle volatility through two types of experiments: (1) removing labormarket distortions speciﬁc to two types of households, which are primary sourcesof consumption inequality, and (2) redistribution policy. Removing the labor market
distortions increases output growth volatility in the US while it reduces in Japan.
Removing cyclical consumption inequality by redistribution policy reduces output
growth volatility in the US and increases in Japan. In contrast, reducing the levelof consumption inequality from the estimated steady-state levels increases output
volatility in both countries. However, the relation between the level of consumption
inequality and output growth volatility is not monotonic. If consumption inequality isquite severe, a reduction in consumption inequality reduces output growth volatility.
Keywords: Inequality; consumption inequality; business cycle accounting; wedges;
distortions; hand-to-mouth
JEL codes: E25; E32; E372

1Introduction

Inequality and business cycles are two important aspects of modern macroeconomics,
each with signiﬁcant implications for social welfare. However, the relation between thesetwo phenomena is complex. While inequality is often a consequence of macroeconomic
dynamics, it can also aﬀect the business cycle through various channels.
This paper seeks to accomplish two main research objectives. First, we aim to identify
the driving sources of inequality and business cycles in the US and Japan. In particular,
we investigate whether there are common factors that signiﬁcantly impact both inequality
and business cycles. Second, we explore the eﬀects of reducing inequality on businesscycles. Our focus is on whether reducing inequality increases or decreases business cycle
volatility.
To reach these research objectives, we develop a tractable heterogenous-agent businesscycle model with unconstrained (U) households and hand-to-mouth (HtM) households.
While U households are the standard representative permanent income households, HtM
households consume their entire disposable income in each period. Each household faces
idiosyncratic risks of being HtM from U and those of being U from HtM, as employed by
Bilbiie (2020, 2021), and Bilbiie, Primiceri, and Tambalotti (2022).
We also introduce, into the model, time-varying “wedges,” which resemble aggregate
productivity, distortionary taxes on investment and labor income, and government consumption, following the business cycle accounting (BCA) approach proposed by Chari,Kehoe, and McGrattan (2002, 2007). These wedges can be interpreted as distortions in
the economy; the eﬃciency wedge is for distortion in aggregate productivity, the investment wedge is for distortion in investment activity, the labor wedge is for labor market
distortions, and the government wedge is for distortion in the resource constraint. While
our model is relatively simple and describes only the real side, introducing these wedges
enables the model to cover various detailed models, including monetary ones. Because of
the heterogeneity among households, we introduce three types of labor wedges: neutral
to two types of households, U-speciﬁc and HtM-speciﬁc labor wedges.
In this paper, we focus on consumption inequality as inequality. This contrasts recent
studies by Bilbiie et al. (2022) and Bayer, Born, and Luetticke (2023), who focus on income3inequality and wealth distributions in the US to investigate the relation between inequality
and business cycles. As Attanasio and Pistaferri (2016) and Meyer and Sullivan (2023)
point out, consumption inequality is a good indicator of economic well-being because
social welfare is measured by consumption and leisure. Then, consumption inequality is
an important topic, as well as income inequality and wealth distribution. We construct
the data on consumption inequality by using the Consumer Expenditure Surveys of the
Bureau of Labor Statistics for the US and the Family Income and Expenditure Survey(Kakei Chosa) of the Ministry of Internal Aﬀairs and Communications for Japan.
By the Bayesian estimations of the model, we ﬁnd that, in the US, the U-speciﬁclabor wedge signiﬁcantly impacts both business cycles and consumption inequality, while
the primary source of business cycles is the eﬃciency wedge, and that of consumption
inequality is the HtM-speciﬁc labor wedge. On the other hand, we ﬁnd that no common
factors signiﬁcantly impact both business cycles and consumption inequality in Japan. In
Japan, the key for business cycles is the distortion in aggregate productivity, and that for
consumption inequality is the labor market distortions speciﬁc to U and HtM households.
The most signiﬁcant contributor to consumption inequality is the HtM in the US, while itis the U in Japan.
To investigate the relation between inequality and business cycles, we conduct two
counterfactual simulations and analyze the eﬀects of reducing consumption inequality on
output growth volatility. The ﬁrst is by removing the eﬀects of the U-speciﬁc and HtMspeciﬁc labor wedges, which are the primary driving sources of consumption inequality
in both countries. We ﬁnd that it increases the output growth volatility in the US, whereas
it reduces the output growth volatility in Japan. The second is by reducing consumption
inequality through redistribution policies. We ﬁnd that removing cyclical consumption
inequality reduces the output growth volatility in the US, whereas it increases in Japan.
We also ﬁnd that reducing the level of consumption inequality from the estimated steadystate levels increases the output growth volatility in both countries. However, the relation
between the level of consumption inequality and output growth volatility is not monotonic.
If consumption inequality is quite severe, a reduction in consumption inequality reduces
output growth volatility.

4

Related literature: Many researchers have studied the sources of business cycle ﬂuctuations. In particular, King and Rebelo (1999) and Hayashi and Prescott (2002) emphasize
the importance of aggregate productivity shock. Since the seminal work of Smets andWouters (2003, 2007), estimations of medium-scale DSGE models using the Bayesian
method have become popular in investigating the source of business cycles. Recent studies emphasize the importance of investment shocks as well as productivity shocks inbusiness cycles, including Justiniano, Primiceri, and Tambalotti (2010); Justiniano, Primiceri and Tambalotti (2011), and Kaihatsu and Kurozumi (2014a) for the US, and Sugo andUeda (2008), Hirose and Kurozumi (2012), Kaihatsu and Kurozumi (2014b), and Inaba,Nutahara, and Shirai (2022) for Japan.
The relation between inequality and business cycles by the estimation of a heterogeneousagent New-Keynesian (HANK) model has been studied by Bayer et al. (2023) and Bilbiie
et al. (2022). Bayer et al. (2023) emphasize that including data on inequality measures inobservations is important for the estimation results. Bilbiie et al. (2022) is closely related
to our paper since they also investigate the eﬀects of reductions in consumption inequality
on business cycles in their models. There are two main diﬀerences between their papers
and ours. The ﬁrst is on the models; they develop medium-scale HANK models withmany frictions and shocks, while our model is relatively simple with time-varying wedges
following the BCA approach. The second is that they use income inequality measures as
an observable variable in estimations, while we use the data on consumption inequality.
The paper by Berger, Bocola, and Dovis (2019) is also related to ours. They show thata class of HANK models can be equivalently represented as a representative-agent NK
model with a time-varying discount factor wedge, as in the BCA approach. By measuring
their wedges by micro data and estimating their model, they show the importance ofimperfect risk-sharing among households in business cycles.
While we employ a DSGE model to investigate the relation between business cycles andinequality, some researchers conduct time-series analyses by VAR models. De Giorgi andGambetti (2017) and Geiger, Mayer, and Scharler (2020) investigate the eﬀects of macro
shocks on consumption inequality in the US. Inui, Sudo and Yamada (2017) investigate
the impact of monetary policy shocks on inequality in Japan. Theophilopoulou (2022)
studies the eﬀects of uncertainty shocks on inequality in the UK.
5

We employ a kind of the BCA approach proposed by Chari et al. (2002, 2007). We
extend the prototype model of BCA to a heterogenous-agent one and introduce additional
wedges. BCA has been applied to diﬀerent periods in various countries. The results
obtained by BCA vary depending on the country and the sample period. The eﬃciency
wedge is found as the most important driving force by Chari et al. (2002, 2007) for the US
during the Great Depression, by Kobayashi and Inaba (2006) and Otsu (2011) for Japan
during the 1990s, by Šustek (2011) for the US during 1959–2004, by Gerth and Otsu
(2018) for the European Great Recession, by Chakraborty and Otsu (2013) for Brazil andRussia in the late 2000s, by Cho and Doblas-Madrid (2013) for the East Asian Crisis,and by Brinca, Chari, Kehoe, and McGrattan (2016) for many OECD countries during
the Great Recession. The labor wedge is found as the primary driving source of businesscycles by Brinca et al. (2016) for France, the UK, Belgium, and New Zealand in the 1980s
and the US during the Great Recession. Chakraborty and Otsu (2013) and Brinca et al.
(2016) ﬁnd that the investment wedge plays a dominant role in China and India in the late
2000s and in Spain, Ireland, and Iceland during the Great Recession.
Introducing HtM households into the model has a long tradition in macroeconomics,
as done by Campbell and Mankiw (1989), Galı́, López-Salido and Vallés (2007), Bilbiie(2008), Kaplan and Violante (2014), and Bilbiie et al. (2022). On the empirical side,
Kaplan, Violante, and Weidner (2014) estimate the share of HtM households in the total
households and analyze demographic characteristics and portfolio composition for the
US, Canada, Australia, the UK, Germany, France, Italy, and Spain. Hara, Unayama, andWeidner (2016) estimate them for Japan.
Finally, our paper is related to the literature on measuring consumption inequality. As
pointed out by Attanasio and Pistaferri (2016) and Meyer and Sullivan (2023), consumption
inequality provides a better indicator of economic well-being because social welfare ismeasured by consumption and leisure. In the US, the measurement of consumption
inequality and the relation between consumption inequality and income inequality havebeen studied by many researchers, including Krueger and Perri (2006), Attanasio, Battistin,and Ichimura (2007), Blundell, Pistaferri, and Preston (2008), Heathcote, Perri, andViolante (2010), Attanasio and Pistaferri (2014), Attanasio, Hurst, and Pistaferri (2014),
Aguiar and Bils (2015), and Meyer and Sullivan (2023). In Japan, there is also an
6accumulation of empirical analyses of consumption inequality, including papers by Ohtake
and Saito (1998), Abe and Yamada (2009), Yamada (2012), and Inui et al. (2017). Lise,
Sudo, Suzuki, Yamada, and Yamada (2014) and Higa (2019) show the widening of
consumption inequality in recent years in Japan.
Structure of the rest of the paper: The remainder of the paper is organized as follows.
Section 2 introduces the model. Section 3 explains the data for estimations and ourestimation strategy. Section 4 explains the main estimation results. The driving sourcesof inequality and business cycles are shown. Section 5 examines the eﬀects of reductionsof consumption inequality on business cycle volatility through two types of counterfactual
simulations. Finally, Section 6 presents the conclusion.

2

The Model

2.1

Households

There are two types of households: Unconstrained (U) and Hand-to-mouth (HtM). The
total population is one, and the number of HtM households is θ.
The U households are the standard representative permanent income households. All
U households pool their income and consume the same amount CtU . They supply labor
LUt given the wage rate Wt . They also invest XtU , and their capital holding at the end of
period t is KtU . The rental rate of capital is Rt .
Each U households face an idiosyncratic risk of being HtM at the next period at
probability 1 − s, as employed by Bilbiie (2020, 2021), and Bilbiie et al. (2022). Thisrisk engages precautionary saving of U households. Then, the U households maximize
the following utility VtU
 U
H+ (1 − s)Vt+1,
VtU = u(CtU , LUt ) + βEt sVt+1(1)

where u(CtU , LUt ) is the instantaneous utility, and VtH is the utility of HtM.
The budget constraint of U households isU
CtU + (1 + τtX )XtU ≤ sRt Kt−1
+ (1 − τtL − τtLU )Wt LUt + TtU ,7(2)

where τtX , τtL , and τtLU denotes the investment wedge, the neutral labor wedge, and the
U-speciﬁc labor wedge, respectively. TtU is the lump-sum tax to the U households.
As explained by Chari et al. (2007), these wedges, which resemble distortionary taxes
on investment and labor supply activities, can be interpreted as reduced forms of distortionsin the economy. We consider the neutral labor wedge, which aﬀects both households andhousehold-type speciﬁc ones since the model has two households. All members of the U
household at period t − 1 have the claim of the rental rate of capital at the period t. Then,the income from capital holding is multiplied by s in the budget constraint.
The evolution of the capital stock is given byU
KtU = (1 − δ)Kt−1+ XtU .

(3)

HtM households consume their entire disposable income. As in U households, all
HtM households pool their income and consume the same amount CtH . They supply laborLH
t given the wage rate Wt . Unlike U households, HtM households cannot invest. Then,the budget constraint of HtM households is
CtH = (1 − s)1−θUH
Rt Kt−1+ (1 − τtL − τtLH )Wt LH
t + Tt ,θ

(4)where τtLH is the HtM-speciﬁc labor wedge, and TtH is the lump-sum tax to HtM households. The capital income comes from HtM households that were U households in theprevious period.
The ﬁrst-order necessary conditions of two households are given byX(1 + τtX )ΛUt = βEt sΛUt+1 (1 + τt+1)(1 − δ) + sRt+1+(1 − s)ΛH
t+1 (1 − s)1−θRt+1 },θ(5)

uL (CtU , LUt ) = ΛUt (1 − τtL − τtLU )Wt ,

(6)

H
L
LH
uL (CtH , LH
)Wt ,t ) = Λt (1 − τt − τt(7)where ΛUt and ΛHt are the marginal utilities. The instantaneous utility function is speciﬁedasu(Cti , Lit ) = log(Cti ) + ξ log(1 − Lit ),for i = U and H.
8

2.2

Firms

Firms are perfectly competitive. Firms produce output Yt using labor input Lt and capital
service Kt . The production function is given by
Yt = Ktα (Zt Lt )1−α ,(8)where α ∈ (0, 1) is the cost share of capital, and Zt is the eﬃciency wedge, which
resembles aggregate productivity.
The marginal productivity conditions are given by
Rt = αYt,
Kt

Wt = (1 − α)

2.3

(9)
Yt
.
Lt

(10)

Fiscal policy

The government spends Gt using lump-sum taxes of two households: TtU and TtH . Then,the government budget constraint is given by
Gt = (1 − θ)TtU + θTtH .

(11)

The government spending Gt is given by
G t = Zt g t ,(12)where gt is the government wedge. In the model, Gt is the government spending, but toﬁt the data, we consider Gt also includes the net exports in the empirical part following
Chari et al. (2007).
The lump-sum taxes are exogenous for households, and they are given by
TtU = τtX XtU + (τ L + τ LU )Wt LUt −TtH = (τ L + τ LH )Wt LH
t .

1
Gt ,
1−θ

(13)
(14)

In this setting, the government spending Gt is ﬁnanced by the lump-sum tax to the U
household. This setting works as a redistribution policy from the U (non-poor) to HtM(poor).
92.4

Aggregations and the market clearing conditions:

In our model, only U households invest and hold capital. Then, the aggregate investment
Xt and the aggregate capital, which is equal to capital service Kt at equilibrium, are givenby
Xt = (1 − θ)XtU ,

(15)

U
.
Kt = (1 − θ)Kt−1(16)

The aggregate consumption Ct and the aggregate labor Lt are the weighted sum of
those of U and HtM. Then, they are given by
Ct = (1 − θ)CtU + θCtH ,(17)

Lt = (1 − θ)LUt + θLH
t .

(18)

Finally, the resource constraint is given by
Yt = C t + X t + G t .

2.5

(19)

Exogenous shocks and equilibrium conditions

There are six exogenous wedges in the model. They follow the AR(1) processes:1
zt = (1 − ρz )z + ρz zt−1 + εzt ,(20)X
τtX = (1 − ρX )τ X + ρX τt−1
+ εX
t ,

(21)

L
τtL = (1 − ρL )τ L + ρL τt−1
+ εLt ,(22)

LU
τtLU = (1 − ρLU )τ LU + ρLU τt−1
+ εLU
t ,

(23)

LH
+ εLH
τtLH = (1 − ρLH )τ LH + ρLH τt−1
t ,

(24)

gt = (1 − ρg )g + ρg gt−1 + εgt ,(25)

1In the business cycle accounting literature, it is often that the wedges follow the following VAR(1)process. In this paper, we assume AR(1) process for simplicity of analysis. An extension to VAR(1)speciﬁcation of wedges is considered in Appendix E. In addition, Chari et al. (2007) deﬁne the eﬃciency
wedge is on the level of aggregate productivity, while we deﬁne the eﬃciency wedge as the growth rate ofaggregate productivity. This setting eliminates arbitrariness in taking the trend and increases the ﬁt of the
data.

10

where zt = Zt /Zt−1 , and z, τ X , τ L , τ LU , τ LH , and g are the steady-state values. For
j = z, X, L, LU , LH, and g, ρj is the persistent parameter and εjt is an i.i.d. structural
shock to a wedge.
The model is detrended by the eﬃciency wedge Zt . The detrended equilibrium
conditions are log-linearized around a balanced growth path. Appendix A shows the
log-linearized equilibrium system.

2.6

Interpretations of wedges

Our model is a tractable two-agent business cycle model with various exogenous wedges.
However, our simple model can cover various detailed models by adjusting wedges.
The eﬃciency wedge resembles aggregate productivity in the production function.
Then, the eﬃciency wedge would cover distortions from misallocation among the multisector model. In addition, Chari et al. (2007) show that equilibrium allocation generated
by an input-ﬁnancing friction model employed by Kiyotaki and Moore (1997) is coveredby their prototype model with the eﬃciency wedge.
The labor wedges would cover the distortions of labor market behaviors and distortionary labor income taxations. As in Equations (6) and (7), the labor wedge captures the
distortions on the consumption–leisure decision. Chari et al. (2007) show that their prototype model with the labor wedge covers a sticky-wage model à la Erceg, Henderson, andLevin (2000). In sticky-price models, the real marginal cost of ﬁrms ﬂuctuates, generating
the gap between the wage and the marginal product of labor. As shown by Šustek (2011),
the labor wedge would capture this gap. If these distortions are common among two types
of households, the neutral labor wedge works. If the distortions are speciﬁc to U or HtM,
U-speciﬁc or HtM-speciﬁc labor wedge works. In other words, our model is real, but it cancapture monetary models by adjusting the labor wedge. Wage markup shock is supposed
as an important source in business cycles as shown by Smets and Wouters (2007). Since
the wage markup generates the deviation of the real wage rate from the marginal productof labor, the labor wedge can capture its eﬀect.
The investment wedge covers the distortions of investment activities in addition to
distortionary investment taxation. As in Equation (5), the investment wedge captures the
distortions on the intertemporal decision. Chari et al. (2007) show that their prototype
11

model with the investment wedge can cover the models with ﬁnancial frictions à laCarlstrom and Fuerst (1997) and Bernanke, Gertler, and Gilchrist (1999). In sticky-pricemodels, the real marginal cost of ﬁrms ﬂuctuates, generating the gap between the rental rateof capital and the marginal product of capital. Then, the investment wedge would capture
this gap, as shown by Šustek (2011). Bilbiie et al. (2022) emphasize the importance ofrisk to switch from the U to the HtM (1 − s) as countercyclical in business cycles. Thisrisk distorts intertemporal decisions, then the investment wedge would be able to capture
it in our model.
Finally, as shown by Chari et al. (2007), the government wedge covers a model with anopen economy model with a sudden stop in addition to wasteful government consumption.
Then, while our model is a closed economy, our model can cover open economy models.

3

Data and Estimation Strategy

Data:

We use ﬁve observable variables. Four of them are quarterly macro series: growth

rates of real GDP per capita Yt , real consumption per capita Ct , real investment per capita
Xt , and labor supply Lt . The last one is inequality, which is annual. As a measure ofinequality, we focus on consumption inequality between HtM and aggregate consumption:
CtH /Ct . A smaller value of CtH /Ct means severe consumption inequality. If amounts ofconsumption are identical among households, then CtH /Ct is one.
According to the empirical facts, the fraction of HtM households is about 20%, andtheir income is lower than non-HtM. Then, as a proxy of CtH /Ct , we calculate the ratio ofconsumption of the lowest 20 % to that of all consumer units in the “Quintiles of income
before taxes” of the Consumer Expenditure Surveys (CEX) of BLS for the US. For Japan,
we use the Family Income and Expenditure Survey (Kakei Chosa) of the Ministry ofInternal Aﬀairs and Communications. This data is annual.
The sample period for the US is from 1984:Q1 to 2019:Q4, and that for Japan is from1980:Q2 to 2019:Q4. The availability of GDP and consumption inequality data determines
the sample period. Since our model is log-linearized, we stopped the sample period at
2019:Q4 to avoid the signiﬁcant ﬂuctuation caused by Covid-19. The data construction
details are in Appendix B.
12

Table 1: Fixed Parameter Values
Parameter

Value

β

Discount factor

0.99

α

Cost-share of capital

0.37

δ

Depreciation rate of capital

0.025

θ

Share of HtM HH

0.2

1−s

Transition rate from U to HtM

1 − 0.987

G
Y

Steady-state value of Gt /Yt

0.1645 (US)0.2439 (JPN)

Estimation strategy: The observation equations for macro variables (Yt , Ct , Xt ,and
Lt ) are100∆ log Yt = 100(z − 1) + (ŷt − ŷt−1 + ẑt ),100∆ log Ct = 100(z − 1) + (ĉt − ĉt−1 + ẑt ),100∆ log Xt = 100(z − 1) + (x̂Ut − x̂Ut−1 + ẑt ),100 log Lt = 100 log L + L̂t ,where z and L denote steady-state values, and v̂t is the log-deviation from its steady-statevalue: v̂t = log(vt ) − log(v) for v = c, z, y, xU , and L. The data of the consumption
inequality CtH /Ct is annual. We follow Pfeifer (2021) to model the observation equationof consumption inequality with this mixed frequency. It is given by
 H
 H
C
Ct
= 100 log+
100 log
Ct annual
C
 H

1
H
2 H
3 H
(ĉ
−
ĉ
)
+
z(ĉ
−
ĉ
)
+
z
(ĉ
−
ĉ
)
+
z
(ĉ
−
ĉ
)
,
t−3
t−2
t−1
t
t−3
t−2
t−1
t
1 + z + z2 + z3
where C H /C is the steady-state consumption inequality.2
Table 1 shows the ﬁxed parameter values. The model is speciﬁed to be quarterly. The
discount factor β is 0.99. The cost share of capital α is 0.37. The depreciation rate ofcapital δ is 0.025. These values are standard in the literature. The share of HtM households
2Appendix C explains the derivation of this observation equation.

13

Table 2: Prior Distribution
Parameter

Distribution

Mean

SD

τX

Steady-state investment wedge

Gamma

0.5

0.2

100(z − 1)

Steady-state output growthGamma0.4017 (US)

0.2

Gamma

0.3112 (JPN)

0.2

Gamma

0.5788 (US)

0.05

Gamma

0.7080 (JPN)

0.05

Gamma

0.1973 (US)

0.05

Gamma

0.2129 (JPN)

0.05

CH
C

L

Steady-state consumption inequality

Steady-state total labor supplyρzPersistence of eﬃciency wedge

Beta

0.5

0.2

ρx

Persistence of investment wedge

Beta

0.5

0.2

ρL

Persistence of neutral labor wedge

Beta

0.5

0.2

ρLU

Persistence of U-speciﬁc labor wedge

Beta

0.5

0.2

ρLH

Persistence of HtM-speciﬁc labor wedge Beta

0.5

0.2

ρg

Persistence of government wedge

Beta

0.5

0.2

σz

SD of eﬃciency wedge shock

IG

0.5

Inf

σx

SD of investment wedge shock

IG

0.5

Inf

σL

SD of neutral labor wedge shock

IG

0.5

Inf

σLU

SD of U-speciﬁc labor wedge shock

IG

0.5

Inf

σLH

SD of HtM-speciﬁc labor wedge shock

IG

0.5

Inf

σg

SD of government wedge shock

IG

0.5

Inf

θ is 0.2. The transition rate from U to HtM 1−s is 1−0.987. These two values are from thepaper by Bilbiie et al. (2022). Finally, the steady-state share of government consumption
in GDP G/Y is set as the data mean: 0.1645 for the US and 0.2439 for Japan.
The Bayesian estimation is employed by using Dynare. The prior distributions of theparameters are presented in Tables 2. For the steady-state investment wedge τ X , growthrate 100(z − 1), labor supply L, consumption inequality C H /C, the prior distribution is
the gamma. The means of the prior distribution except for τ X are set at the sample mean.
Since there is no prior information on τ X , we set the mean of τ X as 0.5 so that the shape14

of the posterior mode is close to the log-likelihood kernel in the mode check plots in
Dynare. For the persistence parameters of structural shocks, the prior distribution is thebeta distribution with a mean of 0.5 and a standard deviation of 0.2.
Following standard Bayesian likelihood approaches, we use the Kalman ﬁlter to evaluate the likelihood function of the log-linearized equilibrium system and the Metropolis–
Hastings algorithm to generate draws from the posterior distribution of the deep parameters.

4

Empirical Results

4.1

Posterior Estimates

The posterior estimates are presented in Table 3. Many posterior estimates are close
between the US and Japan. Judging by the credible intervals, τ X is higher in the US,
implying that the distortion in investment activity is higher in the US. In addition, C H /Cis higher in Japan than in the US, implying that consumption inequality is less severein Japan than in the US. The persistence parameter of the eﬃciency wedge is higher inJapan, while the persistence of the HtM labor wedge is much lower in Japan. The standard
deviations of eﬃciency, neutral labor, and government wedges are larger in Japan than in
the US.

4.2

Driving sources of inequality and business cycles

Table 4 shows the variance decompositions of the growth rates of output (100∆ log Yt ),consumption (100∆ log Ct ), and investment (100∆ log Xt ), labor supply (100 log Lt ), andconsumption inequality (ĉHt − ĉt ). We focus on consumption inequality generated in themodel (ĉHt − ĉt ) rather than the observation data since the observation data is annual.
In the US, the eﬃciency wedge plays the dominant role in output growth, accounting
for more than 70%. The U-speciﬁc labor wedge is the second, accounting for about 15%.
For consumption growth, the eﬃciency wedge is still the most signiﬁcant factor, and 50%
is accounted for. The U-speciﬁc and HtM-speciﬁc labor wedges also have substantial roles
in consumption growth; each wedge accounts for about 15%. In the case of investment
15

Table 3: Posterior Estimates

US
Mean

Japan

90% credible interval

Mean

90% credible interval

τX

1.1202[0.8730, 1.3724]

0.5673[0.3322,0.8041 ]100(z − 1)

0.3566[0.2280, 0.4808]

0.4342[0.2243,0.6475 ]CHC

0.5706[

0.5378, 0.6072]

0.7122[0.6790,0.7436]L

0.2000[0.1888, 0.2094]

0.2208[0.2041,

0.2364]ρz

0.0788[0.0421, 0.1158]

0.1922[0.1454,0.2412]

ρx

0.9493[0.9178, 0.9810]

0.9727[0.9551,

0.9918]

ρL

0.8359[0.6659, 0.9890]

0.9916[

0.9847,0.9997]

ρLU

0.9854[

0.9741, 0.9991]

0.9905[0.9829,0.9990]ρLH

0.8706[0.8055, 0.9369]

0.4500[0.2646,

0.6359]ρg

0.9885[0.9797, 0.9979]

0.9773[0.9636,

0.9916]σz

0.9835[0.8841, 1.0758]

1.6196[1.4745,1.7714]σx0.3863[0.3067, 0.4696]

0.5537[0.4586,

0.6420]σL

0.1907[0.1046, 0.2821]

0.6791[0.5066,0.8369]σLU

0.7115[0.6186, 0.8110]0.5893[0.3998,0.7649]σLH

3.4108[2.8603, 3.9387]

3.5238[2.9183,4.1320]σg1.5785[1.4204, 1.7267]

2.1127[1.9144,2.3123]growth, the investment wedge is the most signiﬁcant contributor, and the eﬃciency wedge
is the second. The total labor supply is mainly accounted for by the U-speciﬁc laborwedge (about 65%) and the government wedge (about 20%). Except for total labor
supply, the role of the eﬃciency wedge is crucial in these variables. On the other hand,consumption inequality is explained by household-speciﬁc labor wedges: about 75% by
the HtM-speciﬁc labor wedge and about 20% by the U-speciﬁc labor wedge. In contrast,
the eﬃciency wedge has almost no contribution to consumption inequality.
In Japan, the eﬃciency wedge is the main contributor to output, consumption, andinvestment growth, as in the US. Unlike the US, the role of the U-speciﬁc labor wedge is16

Table 4: Variance Decompositions
US
τL

τ LH

τ LU

τx

g

∆ log Yt

71.38 2.40

4.78

15.41

2.23

3.80

∆ log Ct

56.36 0.73

14.94

16.45

5.13

6.39

∆ log Xt

31.10 7.30

0.09

15.21

45.04

1.25

log Lt

3.52

1.18

3.47

65.79

1.79

24.24

ĉH
t − ĉt

0.08

0.01

75.63

23.67

0.04

0.57

z

Japan
z

τL

τ LH

τ LU

τx

g

∆ log Yt

81.11

5.94

1.72

3.31

1.98

5.93

∆ log Ct

55.47

8.15

16.66

4.20

4.60

10.92

∆ log Xt

48.03

6.72

3.50

4.18

36.46

1.11

log Lt10.10 43.47

0.33

21.04

2.26

22.81

ĉH
t − ĉt

1.45

44.91

49.85

0.32

3.28

0.19

Note: Inﬁnity-horizon forecast error variance decompositions are performed.

minor to them. The investment wedge signiﬁcantly aﬀects investment growth, as in the US.
Unlike the US, the neutral labor wedge accounts for about 40% of the total labor supply.
The U-speciﬁc labor wedge also signiﬁcantly aﬀects the total labor supply, accounting forabout 20%. Consumption inequality is explained by household-speciﬁc labor wedges, asin the US. However, Japan’s primary source of consumption inequality is the U-speciﬁclabor wedge, whereas it is the HtM-speciﬁc labor wedge in the US.
According to our results, household-speciﬁc labor wedges are essential for consumption inequality in the US and Japan. Bayer et al. (2023) and Bilbiie et al. (2022) ﬁnd
that markup shock is an important source of income inequality and wealth distribution
dynamics. Since the markup shock distorts the factor prices, including the wage, our laborwedges might capture a similar eﬀect. If so, our result implies that markup shock speciﬁc
to each household is crucial, not neutral to two types of households.
17

We focus on the impulse response functions to interpret the mechanism behind theresults of the variance decomposition. Figure 1 shows the estimated impulse responses to a
one standard deviation shock to the wedges on output growth (100∆ log Yt ), consumptionsof two households, and consumption inequality (ĉHt − ĉt ) in the US. For output growth,
the response to the eﬃciency wedge shock is the largest among shocks. This result is
consistent with our ﬁnding that the main driver for output growth is the eﬃciency wedge
in the variance decomposition. The second largest response is to the U-speciﬁc laborwedge shock, which is also in line with the variance decomposition.
On the other hand, the consumption responses to the eﬃciency wedge shock are very
close for the two types of households. Thus, only a slight consumption inequality is causedby the eﬃciency wedge shock. In the case of investment, neutral labor, and government
wedges, the diﬀerences in consumption responses between the two types of households arealso slight. The consumption responses diﬀer signiﬁcantly from the U-speciﬁc and HtMspeciﬁc labor wedge shocks; these signiﬁcantly contribute to consumption inequality in thevariance decomposition. Remarkably, the impact of the HtM-speciﬁc labor wedge is themost signiﬁcant to consumption inequality. In the variance decomposition, the U-speciﬁclabor wedge accounts for about 15% of output growth and about 20% of consumption
inequality. The impulse responses imply that the U-speciﬁc labor wedge shock decreases
output growth and increases consumption inequality (ĉHt − ĉt ). So then, erasing the eﬀects
of the U-speciﬁc labor wedge would positively impact the business cycles while widening
the consumption gap.
Figure 2 is the analogue of Figure 1 in Japan. As in the US, output growth responds
most signiﬁcantly to eﬃciency wedge shocks, and its magnitude is much larger than in the
US case. Unlike the US, the magnitudes of the neutral labor wedge and the government
wedge on output growth are almost the same as those of the U-speciﬁc labor wedge. On
the other hand, in the US, the magnitude of the U-speciﬁc labor wedge on output growth ismore signiﬁcant than those of the neutral labor and the government wedges. This diﬀerence
is consistent with the variance decomposition, which shows that the contributions of theneutral labor wedge and the government wedge to output growth are more signiﬁcant thanthat of the U-speciﬁc labor wedge in Japan.
In Japan, the consumption response to the HtM-speciﬁc labor wedge is less persistent18

Output growth

Consumption

0

0.06

-0.2

0.4

0
0

0.04
U
HtM

-0.4

0.2
10

20

30

40

Output growth

-0.6

0

10

20

30

0.02
40

30

40

-0.02

0

-0.04
10

20

30

40

Output growth

-0.2

0

10

-0.05

20

30

40

0

10

20

30

40

Inequality (CH/C t )
tConsumption00

HtM Labor Wedge

20

0

U
HtM

0

0

-0.05

-0.02
U
HtM

-0.1
0

0

10

20

30

40

-0.1

0

10

Output growth

20

30

40

Consumption

-0.04

0

0.5

10

20

30

40

Inequality (CH/C t )
t

U
HtM

0

0.4

-0.2

-0.2
-0.4
0

10

20

30

40

Output growth

0

10

20

-1
-2

0.2

10

20

30

40

Output growth

0
0

0

10

20

30

20

30

40

-0.2

10

19

20

30

20

30

40

Inequality (CH/C t )
t

-2
40

-3

0

10

20

30

40

Inequality (CH/C t )
t

0.06

U
HtM

0

10

-1

0.08

-0.1

10

0.3
0

0

Consumption

0

0.1

40

U
HtM

-0.1

0

30

Consumption

0

0

-0.2

10

Inequality (CH/C t )
t

-0.1

-0.4

0
0

Consumption

0.2

-0.05

Government Wedge

Inequality (CH/C t )
t

0.6

0

U Labor Wedge

Neutral Labor Wedge

Investment Wedge

Efficiency Wedge

Figure 1: Impulse Responses to a One Standard Deviation Shock to Wedges (1): US

40

0.04
0

10

20

30

40

U Labor Wedge

Neutral Labor Wedge

Investment Wedge

Efficiency Wedge

Figure 2: Impulse Responses to a One Standard Deviation Shock to Wedges (2): Japan

1.5

Consumption

00.2

Inequality (CH/C t )
t

1
-0.5

0.1
U
HtM

0.5
0
0

0

-1
10

20

30

40

-0.2

0

10

Output growth

20

30

40

Consumption

0

20

30

40

Output growth

20

30

40

-0.05

-0.2
10

10

0

U
HtM

0

0

0
0

Inequality (CH/C t )
t

0.2

-0.1

0

10

20

30

40

Consumption

-0.3

-0.1

0

0
U
HtM

10

20

30

40

Inequality (CH/C t )
t

-0.02

-0.4

-0.2

-0.04
-0.4

0

0

10

20

30

40

Output growth

-0.5

20

30

40

-0.06

0

10

20

30

40

Inequality (CH/C t )
tConsumption
U
HtM

0.36
0.34
0.32

-0.2

0.3

0.1

10

20

30

40

Output growth

-0.4

-1

-0.1

-2
0

0.4

10

20

30

0

10

40

Output growth

20

30

40

Consumption00

Government Wedge

10

0

-0.2

-0.2

0

0.2

-0.1

0

HtM Labor Wedge

Output growth

0

0

10

20

30

40

Inequality (CH/C t )
t

-1

U
HtM

-2
0

10

20

30

40

Consumption

0

-3

0

0.2

10

20

30

40

Inequality (CH/C t )
t

0.15
0.2

-0.2
U
HtM

0
0

10

20

30

40

-0.4

0

10

20

20

30

0.1
40

0.05
0

10

20

30

40

than in the US. Then, the gap in consumption between the two types of households
disappears quickly in Japan. In the US, on the other hand, the HtM-speciﬁc labor wedge
has a long-lasting impact on consumption inequality. This diﬀerence in the shapes of these
impulse responses can be interpreted as creating the fact that the U-speciﬁc labor wedge
explains more consumption inequality in Japan, whereas the HtM-speciﬁc labor wedge isthe major contributor to consumption inequality in the US.
What is the central message of these results? In the US, the labor market distortions speciﬁc to the U households signiﬁcantly impact business cycles and consumption
inequality, while the primary source of business cycles is the distortion in aggregate
productivity (i.e., the eﬃciency wedge), and that of consumption inequality is the labor market distortion speciﬁc to the HtM household, the poor. Then, the labor market
distortions speciﬁc to the U households are essential to understand the relation between
business cycles and consumption inequality in the US. On the other hand, we ﬁnd thatno common factors signiﬁcantly impact both business cycles and consumption inequality
in Japan. In business cycles, the distortion in aggregate productivity is dominant, andthe labor market distortions are less critical. The key to understanding consumption inequality is the U households: non-poor households. In both countries, the eﬀects of labormarket distortions common to the two household types (i.e., the neutral labor wedge) areinsigniﬁcant except for aggregate labor supply. In particular, the eﬀects on output growthand consumption inequality are minor.
We report the variance decomposition of the aggregate labor wedge if we consider arepresentative agent economy in Appendix D.

5Does a Reduction in Inequality Increase or Reduce theVolatility of Business Cycles?

This section analyzes how reducing consumption inequality would aﬀect the businesscycle based on the estimates in Section 4. Here, we consider the following two types ofcounterfactual simulations. The ﬁrst is by eliminating the eﬀects of the U-speciﬁc and
HtM-speciﬁc labor wedges, which are the main drivers of consumption inequality in thevariance decomposition. Since each labor wedge implies distortions in the labor market,
21

Table 5: Output Growth Volatilities by Reducing Consumption Inequality (1): Removing
the eﬀects of the U-speciﬁc and the HtM-speciﬁc Labor Wedges
Data

(1)

(2)

(3)

US 0.583

0.643

0.720

0.783

JP

1.017

1.003

1.006

1.023

Note: Standard deviations of 100∆ log Yt are reported.
(1) τtLH is constant at the estimated steady-state level.
(2) τtLU is constant at the estimated steady-state level.
(3) τtLH and τtLU are constant at their estimated steady-state levels.

we can analyze what happens to the business cycle if the government removes thosedistortions and reduces consumption inequality. The second is by reducing consumption
inequality through redistribution policies by the government. In this approach, we examine
the eﬀects of government policies while the market distortions represented by the wedges
are left. Bilbiie et al. (2022) also employ a similar method to our second strategy.

5.1

Eliminating the eﬀects of the U-speciﬁc and the HtM-speciﬁclabor wedges

In this experiment, we perform a counterfactual experiment in which we ﬁx the values of
either the U-speciﬁc or the HtM-speciﬁc labor wedges or both at their steady-state values
while the other wedges are the same as the estimated ones.
Table 5 shows the standard deviations of output growth. Case (1) is where the HtMspeciﬁc labor wedge τtLH is constant at the estimated steady-state level. Case (2) is where
the U-speciﬁc labor wedge τtLU is constant at the estimated steady-state level. Case (3)is where the HtM-speciﬁc and U-speciﬁc labor wedges are constant at their estimatedsteady-state levels.
In the US, eliminating the eﬀects of the U-speciﬁc and the HtM-speciﬁc labor wedges
increases output growth volatility. In contrast, it reduces the volatility of output growth in

22

Japan. This diﬀerence comes from the correlations between wedges and output growth.
In the US, the U-speciﬁc and the HtM labor wedges are positively correlated with output
growth: corr(τtLH , ∆ log Yt ) = 0.25 and corr(τtLU , ∆ log Yt ) = 0.12. In contrast, inJapan, they are negatively correlated with output growth: corr(τtLH , ∆ log Yt ) = −0.12
and corr(τtLU , ∆ log Yt ) = −0.26. In other words, the labor market distortions speciﬁc tothese households are severe in a boom in the US, while they are severe in a recession in
Japan. These wedges have adverse eﬀects on output growth. Thus, removing the eﬀects
of these wedges increases the volatility of the output growth in the US, whereas it reduces
in Japan.
The magnitudes of removing the eﬀects of these wedges are quantitatively small inJapan. This would be because the contributions of τtLH and τtLU are small in the variancedecomposition. On the other hand, in the US, the U-speciﬁc labor wedge is the second
contributor to output growth and removing wedges signiﬁcantly impacts output growthvolatility.

5.2

Redistribution policy

In the second experiment, we change the lump-sum taxes to U and HtM, Equations (13) and(14), to achieve constant CtH /Ct . To implement counterfactual simulations, we replace
the HtM budget constraint in the equilibrium system with the equation where CtH /Ct isconstant.
Table 6 shows the standard deviations of output growth. Case (1) is where CtH /Ct isconstant at the estimated steady-state level. Case (2) is where CtH /Ct is constant at its
maximum steady-state level. In the US, CtH /Ct = 0.744, and in Japan, CtH /Ct = 0.884.
Our numerical simulations imply that, in our model, CtH /Ct = 1 cannot be achieved by
the redistribution policy. This result would be because the lump-sum transfer to HtM
becomes enormous. The income eﬀect from this transfer makes HtM reduce its labor
supply to zero, and the assumption of interim solutions does not hold. Case (3) is of themodel without HtM: the case of the number of HtM θ is zero, and the risk of being of
1 − s is zero. While Case (3) is not achieved by redistribution policy, we investigate it as
the case of perfect consumption equality.
Case (1) shows that in the US, eliminating ﬂuctuations of consumption inequality
23

Table 6: Output Growth Volatilities by Reducing Consumption Inequality (2): Redistribution Policy
Data

(1)

(2)

(3)

US 0.583

0.547

0.589

0.641

JP

1.153

1.231

1.002

1.023

Note: Standard deviations of 100∆ log Yt are reported.
(1) CtH /Ct is constant at the estimated steady-state level.
(2) CtH /Ct is constant at its maximum level. (US: 0.744, Japan: 0.884)(3) Model without HtM (θ = 0, s = 1): Perfect consumption equality

reduces output growth volatility, whereas it increases output growth volatility in Japan.
The US case is consistent with the result of Bilbiie et al. (2022), who also ﬁnd thateliminating ﬂuctuations of consumption inequality reduces output growth volatility. Our
results show that it is diﬀerent in Japan. Case (2) shows that reducing the steady-state
consumption inequality (increasing in CtH /Ct ) from the estimated level increases output
growth volatilities in both countries. This result is in contrast to that of Bilbiie et al.
(2022). In their model, reducing steady-state consumption inequality decreases output
growth volatilities. Case (3) shows that output growth volatility becomes more volatile if
HtM does not exist in the US, whereas it becomes less volatile in Japan. The intuitions of
this diﬀerence would be similar to those of our ﬁrst experiment by removing the eﬀects
of τtLH . The eﬀects of removing the HtM households entail those of removing the HtMspeciﬁc labor wedge. In Table 5, removing the eﬀects of τtLH increases output growthvolatility in the US, whereas it reduces output growth volatility in Japan. As explained in
the previous subsection, it comes from the correlation of τtLH and output growth.
Comparing Cases (1) and (2) in Table 6, an increase in CtH /Ct from the estimatedsteady-state level increases the volatility of output growth both in the US and Japan. Is thisa general property? Figure 3 shows the volatility of output growth under various values of
CtH /Ct . According to this ﬁgure, the relation between CtH /Ct and output growth volatility
is not monotonic. If CtH /Ct is quite low and consumption inequality is severe, increasing
CtH /Ct reduces output growth volatility. However, if consumption inequality is mild to24

Figure 3: Output Growth Volatilities and CtH /Ct under Redistribution Policy
US

log Y
Std. of 100

t

0.6
0.59

0.57

0.58

0.56
0.55
0.54
0

0.5

1

H

Ct /CtJapan

Std. of 100

log Y

t

1.2
1.15
1.1
1.05
1
0

0.5

1

H

Ct /CtNote: In the US, the minimum standard deviation of output growth is 0.533 at CtH /Ct =0.275. In Japan, the minimum is 0.974 at CtH /Ct = 0.09.

some extent, increasing CtH /Ct increases the volatility of output growth. In the US, theminimum standard deviation of output growth is 0.533 at CtH /Ct = 0.275. In Japan, theminimum is 0.974 at CtH /Ct = 0.09.
In Appendix E, we consider the case where the wedges follow the VAR(1) process.
Even if the wedges are VAR(1) and the shocks are correlated, the main implication of thissection doesn’t change.

6

Concluding Remarks

In this paper, we have investigated the sources of inequality and business cycles in the US
and Japan, using a tractable heterogeneous-agent business cycle model with U and HtM
25households. Following the BCA approach, we introduced six “wedges,” interpreted as
distortions in the economy: eﬃciency, neutral labor, U-speciﬁc labor, HtM-speciﬁc labor,
investment, and government wedges. While our model is simple, including these wedges
enables the model to cover various detailed models. We have highlighted consumption
inequality as inequality.
We have found that, in the US, the U-speciﬁc labor wedge signiﬁcantly impacts bothbusiness cycles and consumption inequality, while the primary source of business cycles
is the eﬃciency wedge, and that of consumption inequality is the HtM-speciﬁc laborwedge. On the other hand, we have found that no common factors signiﬁcantly impact
both business cycles and consumption inequality in Japan. In Japan, the key for businesscycles is the distortion in aggregate productivity, and that for consumption inequality is
the labor market distortions speciﬁc to U and HtM households. The most signiﬁcant
contributor to consumption inequality is the HtM (poor) in the US, whereas it is the U(non-poor) in Japan.
We have also investigated the relation between inequality and business cycles. Especially we have conducted two types of counterfactual simulations to analyze the eﬀects of
the reduction in consumption inequality on output growth volatility. The ﬁrst is by removing the eﬀects of the U-speciﬁc and HtM-speciﬁc labor wedges, which are the primarydriving sources of consumption inequality. We have found that it increases the output
growth volatility in the US, whereas it reduces the output growth volatility in Japan. The
second is by reducing consumption inequality through redistribution policies. We havefound that removing cyclical consumption inequality reduces the output growth volatility
in the US, whereas it increases in Japan. We also have found that reducing the level ofconsumption inequality from the estimated steady-state levels increases the output growthvolatility in both countries. However, the relation between the level of consumption inequality and output growth volatility is not monotonic. If consumption inequality is quite
severe, a reduction in consumption inequality reduces output growth volatility.
Our model is tractable and simple, and there are remaining future tasks. Nonetheless,we have found a complex relation between inequality and the business cycle, and our work
has a certain contribution to this ﬁeld.

26

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33

A

Equilibrium System

The equilibrium system of the model is given by
ΛUt =

1
,
CtU




X
)(1 − δ) + sRt+1(1 + τtX )ΛUt = βEt sΛUt+1 (1 + τt+12
+ΛH
t+1 (1 − s)1−θRt+1 },
θ1= ΛUt (1 − τtL − τtLU )Wt ,1 − LUt
1
ΛH
,
t =
CtH
1
L
LH
= ΛH
)Wt ,
t (1 − τt − τt
1 − LHt
1−θ
URt Kt−1,
CtH = Wt LH
t + (1 − s)θU
KtU = (1 − δ)Kt−1+ XtU ,

Yt = [Kt ]α (Zt Lt )1−α ,Rt = αYt,
Kt

Wt = (1 − α)

Yt
,
Lt

Yt = C t + X t + G t ,U
Kt = (1 − θ)Kt−1,

Xt = (1 − θ)XtU ,Ct = (1 − θ)CtU + θCtH ,Lt = (1 − θ)LUt + θLHt .
Let the detrended variables as
Ct
,
Zt
Xt
xt =
,
Zt

ct =

CtU
,
Zt
XU
xUt = t ,
Zt

cUt =

CtH
,
Zt
Kt
kt =
,
Zt

cH
t =

H
λH
t = Λ t Zt ,ktU =and zt = Zt /Zt−1 .

34

KtU
,
Zt

H
λH
t = Λ t Zt ,

yt =

Yt
,
Zt

wt =

Wt
,
Zt

Then, the detrended equilibrium system is given by
λUt =

1
,
cUt


λU 
X
)(1 − δ) + sRt+1(1 + τtX )λUt = βEt {s t+1 (1 + τt+1
zt+1
λH
1−θ
+ t+1 (1 − s)2
Rt+1 },zt+1
θ1= λUt (1 − τtL − τtLU )wt ,ξ
1 − LUt
1
L
LH
ξ
= λH
)wt ,
t (1 − τt − τt
1 − LHtU
1 − θ kt−1
H
R
,
cH
=
w
L
+
(1
−
s)
t
t t
t
θ
zt
1
λH
t = H,
ctkU
ktU = (1 − δ) t−1 + xUt ,zt
yt = (kt )α L1−α,tyt
Rt = α ,ktyt
wt = (1 − α) ,Ltyt = ct + xt + g t ,kU
kt = (1 − θ) t−1 ,zt
xt = (1 − θ)xUt ,ct = (1 − θ)cUt + θcH
t ,Lt = (1 − θ)LUt + θLHt .
At a balanced growth path, the equilibrium system becomes
λU =

1
,
cU

(1 + τ X )λU = β {s

 λH
1−θ
λU 
(1 + τ X )(1 − δ) + sR +(1 − s)2
R },
z
z
θ

1
= λU (1 − τ L − τ LU )w,
1 − LU1= λH (1 − τ L − τ LH )w,
ξ
1 − LHξ

35

cH = wLH + (1 − s)

1 − θ kUR ,θz1
λH = H ,
c


1−δ U
1−
k = xU ,zy = k α L1−α ,
y
R=α ,
k
y
w = (1 − α) ,L
y = c + x + g,
k = (1 − θ)kU,zx = (1 − θ)xU ,c = (1 − θ)cU + θcH ,L = (1 − θ)LU + θLH .
If cH /c and L are known, this balanced growth path can be computed as follows. First,
if cH /c is known, then
cU
λH
=
λU
cH
 H −1
=

c
c

−θ

1−θ

.

R can be calculated by the Euler equation asR=(1 + τ X ) − βs(1 + τ X )(1 − δ)
z 

 .
β(1−s)2
βs2
1−θ
λH
+
z
z
λU
θ

Given the value of R, the following values are calculated as
α
k
= ,
y
R

 
x
1−δ
k
= 1−
z,
y
z
y
c
x g
=1− − ,yy yα  1−α
y
k
=
,
L
y
y
.
w = (1 − α)L36

If L is known, then
y=

y

L,
 L
c
c=
y,
y
 
k
k=
y,
y
kz
kU =
.
1−θ
Finally, since cH /c is given, then

 H
c
c,
c =
c
H

U

cH − (1 − s) 1−θ
R kz
θ
L =
,
w
L − θLH,
LU =1−θ
c − θcH
cU =
.
1−θ
H

The ratios of the labor disutility parameter ξ and labor wedge are given byξ= λU (1 − LU )w,(1 − τ L − τ LU )ξ
(1 − τ L − τ LH )= λH (1 − LH )w.

Let v̂t denote the log-deviation from its steady-state: v̂t = log(vt ) − log(v). In thecase of investment and labor wedges, τ̃tJ is deﬁned as the diﬀerence from its steady-state:τ̃tJ = τtJ − τ J for J = X, L, LU , and LH.
Then, the log-linearized equilibrium system is given by
λ̂Ut = −ĉUt ,βs(1 − δ)τ̃tX + λ̂Ut , =X
Et λ̂Ut+1 − Et ẑt+1 + Et τ̃t+1

z
βs2 R 
U
+
E
λ̂
+
E
R̂
−
E
ẑ
t t+1
t t+1
t t+1
z(1 + τ X )βs(1 − δ)
βs2 R
H
+ 1−
−
E
λ̂
−
E
ẑ
+
E
R̂
t t+1
t t+1
t t+1 ,z
z(1 + τ X )LU
bU − τ̃ L − τ̃ LU + ŵt ,L̂Ut = λ
t
t
t
U
1−L
LH
bH − τ̃ L − τ̃ LH + ŵt ,L̂H = λttt
1 − LH t
37

wLH
ĉH
=
(ŵt + L̂H
t
t )+
H



c

H
λ̂H
t = −ĉt ,1−δ Uk̂tU =(k̂t−1 − ẑt ) +
z

wLH
1− H
c



h

i
U
R̂t + k̂t−1− ẑt ,


1−δ U
1−
x̂t ,
z

U
ŷt = α(k̂t−1− ẑt ) + (1 − α)L̂t ,U+ ẑt ,
R̂t = ŷt − k̂t−1ŵt = ŷt − L̂t ,c(1 − θ)xU U gŷt = ĉt +x̂t + ĝt ,
y
y
y


H
H
θc
θc H
ĉt = 1 −ĉUt +
ĉ ,
c
c t


θLH H
θLH
L̂Ut +L̂ .
L̂t = 1 −L
L t
The evolution of wedges is given byẑt = ρz ẑt−1 + εzt ,X
τ̃tX = ρX τ̃t−1
+ εX
t ,
L
τ̃tL = ρL τ̃t−1+ εLt ,LU
τ̃tLU = ρLU τ̃t−1
+ εLU
t ,
LH
τ̃tLH = ρLH vt−1
+ εLH
t ,

ĝt = ρg ĝt−1 + εgt .

B Data Appendix
The data sources are summarized in Table A1.

B.1

US DataGDP, consumption, investment, and government spending are all obtained in nominal
values and deﬂated using the GDP deﬂator. These data are seasonally adjusted quarterly
series. All per capita variables are obtained by dividing by labor force population. We
use Personal Consumption Expenditures as consumption. Investment is a series of Gross38

Table A1: Data Sources
Variables

Sources (FRED codes)

Quarterly series

US: NIPA tables
Japan: ESRI, Quarterly Estimates of the GDP

Labor force population

US: CNP16OVJapan: MIC, Labour Force Survey

Average hours worked per worker US: PRS85006023
Japan: Ministry of Health, Labour and Welfare,Monthly Labor Survey
The employed persons

US: CE16OVJapan: MIC, Labour Force Survey

Consumption inequality

US: Consumer Expenditure Survey (See below)
Japan: MIC, Family Income and Expenditure Survey

Variables for consumption inequality for the US

Sources (FRED codes)

Number of people in consumer units: all

CXU980010LB0101M

Number of people in consumer units: lowest 20%

CXU980010LB0102M

Total average annual expenditures: all

CXUTOTALEXPLB0101M

Total average annual expenditures: lowest 20%

CXUTOTALEXPLB0102M

Expenditures: vehicle insurance: all

CXU500110LB0101M

Expenditures: vehicle insurance: lowest 20%

CXU500110LB0102M

Expenditures: Health Insurance: all

CXUHLTHINSRLB0101M

Expenditures: Health Insurance: lowest 20%

CXUHLTHINSRLB0102M

Expenditures: personal insurance and pensions: all

CXUINSPENSNLB0101M

Expenditures: personal insurance and pensions: lowest 20%

CXUINSPENSNLB0102M

Notes: For the US, all data are taken from the FRED Database available through theFederal Reserve Bank of St. Louis. FRED codes are reported in the last column.

Private Domestic Investment. The government spending is deﬁned as the sum of Government Consumption Expenditures and Gross Investment, and Net Exports of Goods and39Services. The reason for including net exports of goods and services as an item of government expenditure is to assume a closed economy model following Chari et al. (2007).
The GDP deﬂator is from Implicit Price Deﬂators for Gross Domestic Product.
Labor supply Lt , following Hayashi and Prescott (2002) and Kobayashi and Inaba(2006), is constructed byLt =Weekly average hours worked per employed person × Employed person.
24 × 7 × Labor force

For the data of the consumption inequality CtH /Ct , we calculate the ratio of consumption of the lowest 20 % to that of all consumer units in the “Quintiles of income before
taxes.” We use average monthly consumption expenditures per household by quintile inannual income before taxes. We use adult-equivalent consumption expenditures as the observables for CtH and Ct . The adult-equivalent consumption expenditure of households is
deﬁned by dividing the average consumption expenditure by the square root of the numberof household members according to the square root scale, which is also used by OECD. To
make the deﬁnitions consistent with consumption in the model, we deduct “expendituresfor vehicle insurance,” “health insurance,” and “personal insurance and pensions” fromaverage annual expenditures.

B.2

Japanese Data

The data for Japan are constructed basically in the same way as for the US data. We use
Private Consumption as consumption. Investment is deﬁned as Gross Capital Formationby the private sector, which consists of Private Residential Investment, Private Nonresidential Investment, and Change in Private Inventories. The government spending isdeﬁned as the sum of Government Consumption, Public Investment, Changes in Public
Inventries, and Net Exports. The Economic and Social Research Institute (ESRI) releases
a series on the 2008 SNA basis for years from 1994 to 2022. In addition, the ESRI provides
a series on the 2008 SNA basis for years from 1980 to 1993 as provisional estimates. To
connect the data for the entire period, we splice the nominal variables and GDP deﬂators
using the overlapping years.
The aggregate labor Lt , following Hayashi and Prescott (2002) and Kobayashi and40Inaba (2006), is constructed by
Lt =Monthly average hours worked per employed person × Employed person.
24 × 7 × 4 × Labor force

For the consumption inequality CtH /Ct , we use average monthly consumption expenditures per household by quintile in annual income before taxes for working householdswith two or more members. We also use adult-equivalent consumption expenditures aswell as US data. The Ministry of Internal Aﬀairs and Communications (MIC) releases
consistent series for periods 1863–1975, 1976–2007, and 2008–2021, respectively. To
connect the data for the entire period, we splice data using the overlapping years.

C

Derivation of the Observation Equation of Consumption Inequality

The available data on consumption inequality is annual. Suppose the period t is the 4thHH++ Ct−2quarter of a year. The annual consumption of HtM (HtM C) is given by Ct−3
H
+ CtH . The aggregate consumption (AC) is given by Ct−3 + Ct−2 + Ct−1 + Ct . The
Ct−1

consumption inequality is HtM C/AC.
First, dividing the both numerator and denominator by Zt , it is obtainedH
H
C H + Ct−2
+ Ct−1
+ CtH
HtM C
= t−3
AC
Ct−3 + Ct−2 + Ct−1 + Ct

=

H
H
H
Ct−3
Ct−2
Ct−1
CtH
Zt−3
Zt−2
Zt−1
Zt−2
Zt−1
Zt−1
×
×
×
+
×
×
+
×
+
Zt−3
Zt−2
Zt−1
Zt
Zt−2
Zt−1
Zt
Zt−1
Zt
Zt
Ct−3
Ct−2
Ct−1
Zt−3
Zt−2
Zt−1
Zt−2
Zt−1
Zt−1
Ct
× Zt−2 × Zt−1 × Zt + Zt−2 × Zt−1 × Zt + Zt−1 × Zt + Zt
Zt−3

cH
cH
cH
t−3
t−2
t−1
+
+
+ cH
t
z
zt−1 zt
zt−1 zt
zt= t−2ct−3
ct−2
ct−1+ zt−1 zt + zt + ct
zt−2 zt−1 zt
HHH
cH
t−3 + ct−2 zt−2 + ct−1 zt−1 zt−2 + ct zt zt−1 zt−2=

ct−3 + ct−2 zt−2 + ct−1 zt−1 zt−2 + ct zt zt−1 zt−2.

Log-linearizing this equation yields
 \ 
HtM C
d
\C − AC.
= HtM
AC

Since the log-linearization of the numerator and denominator imply
 H

1
H
2 H
3 H
\C =
HtM
ĉ
+
z(ĉ
+
ẑ
)
+
z
(ĉ
+
ẑ
+
ẑ
)
+
z
(ĉ
+
ẑ
+
ẑ
+
ẑ
)
,
t−2
t−1
t−2
t
t−1
t−2
t−3
t−2
t−1
t
1 + z + z2 + z31d=ĉt−3 + z(ĉt−2 + ẑt−2 ) + z 2 (ĉt−1 + ẑt−1 + ẑt−2 ) + z 3 (ĉt + ẑt + ẑt−1 + ẑt−2 ) ,AC
2
3
1+z+z +z
41

then, the annual consumption inequality is given by
 \ 
 H

HtM C
1
2 H
3 H
=
(ĉt−3 − ĉt−3 ) + z(ĉHt−2 − ĉt−2 ) + z (ĉt−1 − ĉt−1 ) + z (ĉt − ĉt ) .
2
3
AC
1+z+z +z

D

Sources of Aggregate Labor Wedge

The importance of the labor wedge in the business cycle is pointed out by many researchers,
including Chari et al. (2002, 2007), Kobayashi and Inaba (2006), Shimer (2009), andKarabarbounis (2014). Our model has heterogeneous households and multiple laborwedges, the neutral, U-speciﬁc, and HtM-speciﬁc labor wedges. What is the source of theaggregate labor wedge in our model?
The aggregate labor wedge is deﬁned byξYtCt= (Aggregate Labor Wedge)t × (1 − α) .
1 − Lt
Lt

This deﬁnition is consistent with the representative agent model where the instantaneous
utility function of the representative household is speciﬁed asu(Ct , Lt ) = log(Ct ) + ξ log(1 − Lt ),and the production function is Cobb-Douglas, as employed by Chari et al. (2002) andChari et al. (2007).
Table A2 shows the results of the variance decomposition of the growth rate of theaggregate labor wedge. Since our observable variables in the estimation include GDP andconsumption growth rates, we focus on the growth rate of the aggregate labor wedge.
The results from the baseline model tell us as follows. In the US, 60% of the aggregate
labor wedge is accounted for by the U-speciﬁc labor wedge, and the HtM-speciﬁc laborwedge accounts for about 30%. The contribution of the neutral labor wedge is minor. In
Japan, the neutral and the HtM-speciﬁc labor wedges each account for about 40%, and the
U-speciﬁc labor wedge accounts for about 20%. In both countries, the contributions ofeﬃciency, investment, and government wedges are small.
The contribution of the HtM-speciﬁc labor wedge in the US is close to that of Japan.
The diﬀerence is that in the US, the contribution of the U-speciﬁc labor wedge is signiﬁcant,42

Table A2: Variance Decompositions on Aggregate Labor Wedge
z

τL

τ LH

τ LU

τx

g

US

0.01

6.12

33.25

60.61

0.01

0.01

Japan

0.01

40.86

37.15

21.95

0.01

0.02

Note: Inﬁnity-horizon forecast error variance decompositions are performed.

and that of the neutral labor wedge is very small. In Japan, on the contrary, the U-speciﬁclabor wedge and the neutral labor wedge signiﬁcantly contribute.
These results imply that to understand the aggregate labor wedge in the US, the labormarket distortions speciﬁc to HtM (poor) and U (non-poor) are important, and distortions
related to the overall labor market are insigniﬁcant. In contrast, in Japan, in addition
to household-speciﬁc distortions, we should focus on the distortions in the overall labormarket.

E The Case Where Wedges Follow the VAR(1) Process
In this appendix, we consider the case where the wedges follow the VAR(1) process. Let
st and εt denote the vector of wedges and shocks such that st = [zt , τtx , τtL , τtLH , τtLU , gt ]′g ′LUand εt = [εzt , εxt , εLt , εLHt , εt , εt ] . Then, the wedge st evolves according the following

VAR(1) process:st = V st−1 + εt .
We specify the VAR(1) coeﬃcient matrix V asρzρzx ρzL ρzLH ρzLU 0




 ρxz
ρx ρxL ρxLH ρxLU 0 




 ρLz
ρLx ρL
0
0
0
.
V =


ρLHz ρLHx 0
ρLH
0
0




 ρLU z ρLU x 00
ρLU 0 


0
0
0
0
0
ρg

43

Table A3: Posterior Estimates: Model with VAR(1) Wedge

US
Mean

Japan

90% credible interval

Mean

90% credible interval

τX

1.0389[0.7818,1.2861]0.5543[0.2809,0.8225]100(z − 1)

0.2800[0.0994,0.4489]0.2407[0.0411,0.4105]CHC

0.5804[0.5661,0.5952]0.7200[0.7034,0.7400]L

0.1961[0.1916,0.2003]0.2093[0.2016,

0.2169]ρz

0.1427[0.0395,0.2445]0.0665[0.0106,

0.1140]ρx

0.9087[0.8406,

0.9826]0.6248[0.4030,0.8248]ρL

0.8107[0.7123,0.9193]0.9804[0.9661,0.9955]

ρLU

0.8884[0.8126,

0.9737]0.8041[0.6143,0.9783]

ρLH

0.8507[0.7850,0.9235]0.7485[0.6161,0.8833]ρg

0.9764[0.9622,0.9919]0.9733[0.9602,0.9880]ρzx-0.5914[-0.8777,-0.3441]0.0822[-0.3071,

0.4243]

ρzL

0.0490[-0.1201,

0.2088]-0.0681[-0.0990,-0.0381 ]ρzLU-0.2559[-0.3847,-0.1196]-0.0747[-0.3050,0.1734]

ρzLH-0.0156[-0.0337,

0.0028]0.0445[0.0073,0.0841]ρxz

0.0977[0.0288,0.1640]0.1005[0.0387,0.1627]

ρxL-0.0061[-0.0402,0.0260]0.0223[-0.0079,

0.0588]

ρxLU

0.0038[-0.0524,0.0620]0.2605[0.1061,0.4073]

ρxLH-0.0061[-0.0148,0.0026]0.0184[-0.0024,0.0401]

ρLz

0.0627[-0.0878,

0.2137]-0.0759[-0.2455,

0.0780]ρLx-0.7016[-0.9516,-0.4581]-0.0765[-0.1684,

0.0072]

ρLU z-0.1900[-0.3513,-0.0193]-0.1493[-0.3218,-0.0080 ]ρLU x-0.0833[-0.2947,0.1281]0.2442[-0.0100,

0.4720]ρLHz

0.1475[-0.2473,

0.5421]0.3146[0.0051,0.6729]

ρLHx-0.3004[-0.6817,0.0638]-0.1731[-0.4369,

0.0965]44

Table A4: Posterior Estimates: Model with VAR(1) Wedge

US
Mean

Japan

90% credible interval

Mean

90% credible interval

σz1.0271[0.9190,1.1436]1.5209[1.3648,1.6604]σx

0.3118[0.1942,0.4316]0.2334[0.1447,0.3162]σL

0.1650[0.1066,0.2205]0.8228[0.6217,1.0033]

σLU

0.5163[0.4131,0.6192]0.5781[

0.3383,0.8873]σLH

3.1857[2.7068,3.6515]2.4231[1.7768,3.0337]σg1.5924[

1.4374,1.7500]2.1094[1.9104,2.2958]σzx

0.2200[ -0.0212,

0.4669]0.1131[-0.3257,

0.5522]σzL

0.1431[ -0.3372,

0.6034]0.2642[0.0285,0.5052]σzLU0.3694[0.1783,0.5766]0.0701[-0.2902,0.4388]σzLH0.4943[0.3628,0.6303]-0.1710[-0.4059,

0.0554]σxL-0.2775 [-0.7804,

0.2092]0.0074[-0.4341,0.4468]σxLU-0.2530 [-0.5362,-0.0004]

0.0567[-0.3981,0.5092]σxLH0.0616-0.1755,

0.2922]0.1039[-0.3402,0.5520][

Following Chari et al. (2007), we assume that the government wedge is independentof other wedges. We also assume that each of the three labor wedges is independent of
the other two. In this speciﬁcation, we allow the correlations among shocks. We assume
that the government wedge shock is independent of other wedge shocks and that each of
the three labor wedge shocks is independent of the other two.
In the estimation, we set the prior distribution of the oﬀ-diagonal elements of V ,ρij , is Normal distribution with mean zero, and the standard deviation is 0.3. The priordistribution of the correlations between wedge shocks i and j, σij , is the generalized Beta
distribution with support [-1,1], mean zero, and the standard deviation is 0.3. The priordistributions of other parameters are the same as in Table 2.
The posterior estimates are in Tables A3 and A4. σij is the correlation between wedge
i and j. Judging by the credible intervals, the eﬃciency wedge shock positively correlates
45

Table A5: Output Growth Volatilities by Reducing Consumption Inequality in Model with
VAR(1) Wedges (1): Removing the eﬀects of U-speciﬁc and HtM-speciﬁc Labor Wedges
Data

(1)

(2)

(3)

US 0.583

0.642

0.614

0.673

JP

0.954

1.007

0.939

1.023

Note: Standard deviations of 100∆ log Yt are reported.
(1) τtLH is constant at the estimated steady-state level.
(2) τtLU is constant at the estimated steady-state level.
(3) τtLH and τtLU are constant at their estimated steady-state levels.

with U-speciﬁc and HtM-speciﬁc labor wedge shock in the US. In Japan, the eﬃciency
wedge positively correlates with the neutral labor wedge.
In this speciﬁcation of the wedges, the variance decomposition is not appropriate forinvestigating the driving sources of business cycles and inequality since the shocks arecorrelated. In addition, since a wedge shock aﬀects other wedges through the VAR(1)process, it is unclear the relation between the importance of a wedge shock and theimportance of the wedge itself. Then, we focus on the eﬀects of reducing consumption
inequality on business cycle volatility as in Section 5.
Table A5 is the analogue of Table 5. This table shows how removing U-speciﬁcand HtM-speciﬁc labor wedges aﬀects output growth volatility. Case (1) is where the
HtM-speciﬁc labor wedge τtLH is constant at the estimated steady-state level. Case (2) iswhere the U-speciﬁc labor wedge τtLU is constant at the estimated steady-state level. Case(3) is where the HtM-speciﬁc and U-speciﬁc labor wedges are constant at their estimatedsteady-state levels. As in Table 5, ﬁxing the levels of U-speciﬁc and HtM-speciﬁc laborwedges increases output growth volatility, whereas it decreases in Japan. This diﬀerence
comes from the correlations between wedges and output growth, as in the AR(1) case.
In the US, the U-speciﬁc and the HtM labor wedges are positively correlated with output
growth: corr(τtLH , ∆ log Yt ) = 0.25 and corr(τtLU , ∆ log Yt ) = 0.08. In contrast, inJapan, they are negatively correlated with output growth: corr(τtLH , ∆ log Yt ) = −0.03
and corr(τtLU , ∆ log Yt ) = −0.14.
46

Table A6: Output Growth Volatilities by Reducing Consumption Inequality in Model with
VAR(1) Wedges (2): Redistribution Policy
Data

(1)

(2)

(3)

US 0.583

0.530

0.561

0.674

JP

1.183

1.246

0.985

1.023

Note: Standard deviations of 100∆ log Yt are reported.
(1) CtH /Ct is constant at the estimated steady-state level.
(2) CtH /Ct is constant at its maximum level. (US: 0.752, Japan: 0.886)(3) Model without HtM (θ = 0, s = 1): Perfect consumption equality

Table A6 is the analogue of Table 6. This table shows the eﬀects of the redistribution
policy on output growth volatility. Case (1) is where CtH /Ct is constant at the estimatedsteady-state level. Case (2) is where CtH /Ct is constant at its maximum steady-state level.
In the US, CtH /Ct = 0.752, and in Japan, CtH /Ct = 0.886. Case (1) shows that in theUS, eliminating ﬂuctuations of consumption inequality reduces output growth volatility,
whereas it increases output growth volatility in Japan. Case (2) shows that reducing
the steady-state consumption inequality (increasing in CtH /Ct ) from the estimated level
increases output growth volatilities in both countries. Case (3) shows that output growthvolatility becomes more volatile if HtM does not exist in the US, whereas it becomes lessvolatile in Japan. These results are consistent with those of Table 6.
Figure A1 is the analogue of Figure 3. As in Figure 3, the relation between CtH /Ct
and output growth volatility is not monotonic. If CtH /Ct is quite low and consumption
inequality is severe, increasing CtH /Ct reduces output growth volatility. However, if
consumption inequality is mild to some extent, increasing CtH /Ct increases the volatilityof output growth. In the US, the minimum standard deviation of output growth is 0.5241
at CtH /Ct = 0.3286. In Japan, the minimum is 1.056 at CtH /Ct = 0.1073.
These results imply that the main implication of Section 5 is robust to the case where
the wedges follow the VAR(1) process and the shocks are correlated.

47

Figure A1: Output Growth Volatilities and CtH /Ct under Redistribution Policy
US

log Y

0.58

Std. of 100

t

0.59

0.56

0.57

0.55
0.54
0.53
0

0.2

0.4

0.6

0.8

0.6

0.8

/Ct
CH
t

log Y

1.2

Std. of 100

t

Japan

1.15

1.1

0

0.2

0.4
H

Ct /CtNote: In the US, the minimum standard deviation of output growth is 0.5241 at
CtH /Ct = 0.3286. In Japan, the minimum is 1.056 at CtH /Ct = 0.1073.

48