The Society for Economic Studies The University of Kitakyushu
Working Paper Series No.2016-8 (accepted in March , 2017)

Tax Incidence in DSGE Model †Minoru Hayashida ‡・Ryoichi Nanba §・Masaya Yasuoka ¶・Hiroyuki Ono *Abstract
Our paper presents an examination of how an increase in consumption tax affects tax
incidence with Dynamic Stochastic General Equilibrium (DSGE) model. Results showthat an increase in consumption tax increases the price index level and wage level anddecreases consumption, output, and the real interest rate. In addition to these results,
our paper presents derivation of results related to inequality. Wage inequality between
low-skilled and high-skilled labor shrinks in the short run. However, in the medium run,wage inequality rises. The labor income share increases in the short and medium run.
However, the capital income share decreases.
Keywords: Consumption tax, DSGE model, Inequality, Tax incidence
JEL Classifications: E62, H22† We would like to thank Kazuki Hiraga and seminar participants for helpful comments. Research forthis paper was supported financially by JSPS KAKENHI Grant Numbers 26380367 and 26380253.
Nevertheless, any remaining errors are the authors' responsibility. This paper is written as the shortpaper version of Hayashida, Nanba, Yasuoka and Ono (2017) “DSGE moderuniyoru Sozeikichakuno
Bunseki” The Society for Economic Studies The University of Kitakyushu Working
Paper Series No.2016-6.
‡ The University of Kitakyushu§ Chubu Region Institute for Social and Economic Research¶ Corresponding to: School of Economics, Kwansei Gakuin University, 1-155 Uegahara, Ichiban-Cho,
Nishinomiya, Hyogo 662-8501 Japan, Tel.:+81-798-54-6993, E-mail: yasuoka@kwansei.ac.jp* Toyo University

11. Introduction
Our paper presents examination of the effects of an increase in a consumption tax torebuild fiscal systems with a Dynamic Stochastic General Equilibrium (DSGE) model. A
DSGE model can show tax incidence in a transition pass, which cannot be shown clearlyby an analytical model. Moreover, our manuscript uses Markov chain Monte Carlo
methods (MCMC) to estimate parameters appropriately.
In Japan, the ratio of public debt to gross domestic product (GDP) is greater than
200%. An increase in tax revenue seems necessary to reduce public debt needs. However,consumption taxes present difficulties such as regressivity, even if the effect of aconsumption tax on economic performance is less than that of an income tax.
Our paper sets a standard DSGE model including a consumption tax and two types oflabor (High-skill labor and Low-skill labor) to consider inequality. These analyses
examine the tax incidence of a consumption tax and derive the following results. First,
an increase in a consumption tax raises the price index level and reduces consumption
and output. Second, because of a decrease in an opportunity cost of leisure, the wage
level increases because an increase in a consumption tax reduces the labor supply. Third,
the real interest rate decreases because the marginal productivity of capital decreases
as a result of decreased labor input. Fourth, the wage inequality between low-skill andhigh-skill labor shrinks in the short run. However, wage inequality rises in the medium
term. Fifth, considering capital income and labor income shares, the labor income sharerises and the capital income share declines in the short and medium run.
Some related reports exist the relevant literature. Hayashi (1995) examines tax
incidence empirically. However, when considering tax incidence theoretically, it isnecessary to set household optimization. Heer and Trede (2003), Nishiyama andSmetters (2005), and Lehmus (2011) set the DSGE model with household optimizationand examine tax incidence for some economically development countries. Sakuragawa
and Hosono (2011) examine fiscal sustainability in the DSGE Model. However, thesestudies do not consider income inequality. The problems of consumption tax are
attributed to the income inequality because of its regressivity. Therefore, our paper sets
the DSGE model with labor of two types: highly skilled labor and low skilled labor. Hara,
Katayama and Kato (2014) set the DSGE model with labor of two types. However, they
do not examine the effect of tax reform on deriving the tax incidence. Doi (2010) derives
the tax incidence of corporation tax, but the inequality between laborers is notconsidered.
The motivation for the analyses presented in our manuscript is appropriate. The resultsof these analyses are expected to hold many contributions for this field. Moreover, the2parameters are appropriately set by Bayesian statistics. Recent reports of the literaturesuch as those by Smets and Wouters (2007) and Benchimol and Forcans (2012)
described studies using Bayesian statistics to estimate parameters. In the DSGE model,
few reports of the literature describe studies that use Bayesian statistics except for
Hirose and Naganuma (2010) and others.
The remainder of this paper comprises the following. Section 2 sets the model. Section
3 derives the equilibrium. Section 4 sets the parameters. Section 5 presents
examination of the results of the simulation.
2. ModelThere exist agents of three types: households, firms, and government.
2.1 Household
Households exist for an infinite time and obtain utility from consumption and themoney stock. The population size is assumed as unity and no population growth. Our
paper assumes the following CRRA form utility as
𝑢𝑡 =

1−𝜇

𝑐𝑡1−𝜃 𝑚𝑡
𝑙𝑡1+𝜅
+
−
.
1−𝜃 1−𝜇 1+𝜅

(1)

In that equation, ct, mt, and 𝑙𝑡 respectively denote consumption, the money stock, andlabor supply time. Each household has a unit of time. Then the leisure time is 1 − 𝑙𝑡 . 𝛽

is the discount rate (0 < 𝛽 < 1) and 𝜃, 𝜇, 𝜅 are parameters showing the level of relativerisk aversion. t represents time.

Each household supplies labor to obtain the labor income and to have capital stock to

obtain the capital income. However, money stock has no interest. Household allocatesthe income to money 𝑚𝑡 , bond 𝑏𝑡 , investment 𝐼𝑡 to have capital stock 𝐾𝑡 , andconsumption 𝑐𝑡 .

This model includes labor of two types: high-skill labor (Type H) and low-skill labor(Type L). Type H labor obtains wage rate 𝑤𝑡𝐻 . Type L labor obtains wage rate 𝑤𝑡𝐿 .

The budget constraint of type H can be shown as follows. Superscripts h and lrespectively denote type H and type L variables. Lack of a superscript denotes variables
that are applicable for both type H and type L.
𝑚𝑡 + 𝑏𝑡 + (1 + 𝜏𝑐 )𝑐𝑡ℎ + 𝐼𝑡=

1
�(1 + 𝑖𝑡 )𝑏𝑡−1 + 𝑚𝑡−1 � + 𝜑𝑡 + (1 − 𝜏)𝑎ℎ 𝑤𝑡ℎ 𝑙𝑡ℎ + 𝑟𝑡 𝐾𝑡−1 .
1 + 𝜋𝑡(2)

The budget constraint of type L is shown as follows. Type L labor allocates all labor
income to consumption: our manuscript assumes no saving about type L.
3

In that equation, 𝑎ℎ(1 + 𝜏𝑐 )𝑐𝑡𝑙 = (1 − 𝜏)𝑎𝑙 𝑤𝑡𝑙 𝑙𝑡𝑙 .

(3)

and 𝑎 respectively denote the labor productivity of type H and𝑙type L labor; 𝑎ℎ > 𝑎𝑙 > 0 is assumed. bt denotes the stock of no risk asset, which brings
about the interest. The nominal interest rate is 𝑖𝑡 . Households have capital stock andlend to firms to obtain capital income. The real interest rate is 𝑟𝑡 . Moreover, thehousehold of type H has firms and obtains monopolistic profit 𝜑𝑡 . 𝜋𝑡 denotes the
inflation rate, which is given as 1 + 𝜋𝑡 =

𝑝𝑡

𝑝𝑡−1

. 𝑝𝑡 denotes the price index level. 𝑤𝑡

represents the effective wage rate. Also, 𝜏𝑐 and 𝜏 respectively denote the tax rate of

consumption and labor income.

The capital stock in t period is formed as

𝐼𝑡
𝐾𝑡 = 𝐼𝑡 + (1 − δ)𝐾𝑡−1 − S �� 𝐼𝑡 .
𝐼𝑡−1(4)

We assume S ′ > 0, 𝑆(1) = S ′ (1) = 0. We can set the following Lagrange equation as

𝐿 = 𝐸0∞∞� 𝛽𝑡 �
𝑡=0

𝑐ℎ𝑡

1−𝜃

1−𝜃

+

1−𝜇

1+𝜅

𝑚𝑡
𝑙𝑡ℎ
−
�
1−𝜇 1+𝜅+𝐸0 � 𝜆𝑡 �𝑚𝑡 + 𝑏𝑡 + (1 + 𝜏𝑐 )𝑐𝑡ℎ + 𝐼𝑡 −𝑡=0

− 𝑟𝑡 𝐾𝑡−1 �∞1�(1 + 𝑖𝑡 )𝑏𝑡−1 + 𝑚𝑡−1 � − 𝜑𝑡 − (1 − 𝜏)𝑎 ℎ 𝑤𝑡ℎ 𝑙𝑡ℎ
1 + 𝜋𝑡(5)𝐼𝑡+𝐸0 � 𝛾𝑡 �𝐾𝑡 − 𝐼𝑡 − (1 − 𝛿)𝐾𝑡−1 + 𝑆 �� 𝐼𝑡 �,𝐼
𝑡−1
𝑡=0

where 𝜆𝑡 and 𝛾𝑡 denote Lagrange multipliers.

The optimization problem for a household of type H is reduced to the following

equations.

𝑐𝑡ℎ

−𝜃

−𝜇

𝑚𝑡

𝛾

where 𝑞𝑡 = 𝜆𝑡 .
2.2 Firmsℎ= 𝛽𝐸𝑡 𝑐𝑡+1

= 𝑐𝑡ℎ

−𝜃

−𝜃 1 + 𝑖𝑡+1𝐸𝑡

1 + 𝜋𝑡+1

2 + 𝑖𝑡+1,
1 + 𝑖𝑡+1

𝐸𝑡 �𝑟𝑡+1 + 𝑞𝑡+1 (1 − 𝛿)� = 𝑞𝑡 𝐸𝑡,

1 + 𝑖𝑡+1,
1 + 𝜋𝑡+1

(6)
(7)
(8)

𝑡

This model includes firms of two types: firms that produce final goods by inputtingintermediate goods and firms that produce intermediate goods.
42.2.1 Final Goods Production
It is assumed that final goods are produced in a perfectly competitive market. Then, theproduct function is shown as presented below:
1

𝑌𝑡 = ��

0

𝜀
𝜀−1
𝜀−1
𝑌𝑗𝑡 𝜀 𝑑𝑗� .

(9)

The firm produces final good 𝑌𝑡 by inputting each intermediate good 𝑌𝑗𝑡 . We assume

that the intermediate goods firm is distributed between 0 and 1 and aggregateintermediate goods firm is unity. Then, the profit function 𝜋𝑡 is
𝑓

1

𝑓

(10)

𝜋𝑡 = 𝑝𝑡 𝑌𝑡 − � 𝑝𝑗𝑡 𝑌𝑗𝑡 𝑑𝑗.
0

Therein, 𝑝𝑗𝑡 denotes the price of j th intermediate goods. The demand function of the jth intermediate good is derived as shown below:
𝑝𝑗𝑡 −𝜀𝑌𝑗𝑡 = � � 𝑌𝑡 .
𝑝𝑡

(11)

2.2 Intermediate Goods Production

Intermediate goods are produced by inputting capital stock and labor. The productfunction is assumed as follows.
𝜁

𝑌𝑗𝑡 = 𝐾𝑗𝑡𝛼 �𝑁𝑗𝑡ℎ 𝑁𝑗𝑡𝑙

1−𝜁 1−𝛼

�

(12)

.

In that equation, 𝐾𝑗 denotes capital stock to produce j th intermediate goods. 𝑁𝑖𝑡ℎ and𝑁𝑗𝑡𝑙 respectively denote type H and type L labor input to produce j th intermediate goods.

Our paper assumes that the labor share of type H and type L are, respectively, ν, 1 − ν.
Then, each labor input is(13)

𝑁𝑗𝑡ℎ = 𝜈𝑎ℎ 𝑙𝑡ℎ .

(14)

𝑁𝑗𝑡𝑙 = (1 − 𝜈)𝑎𝑙 𝑙𝑡𝑙 .

Now, we can consider the following Lagrange equation of cost minimization as follows.
𝜁

𝑀 = 𝑤𝑗𝑡ℎ 𝑁𝑗𝑡ℎ + 𝑤𝑗𝑡𝑙 𝑁𝑗𝑡𝑙 + 𝑟𝑗𝑡 𝐾𝑗𝑡 + 𝜔𝑗𝑡 �𝑌𝑗𝑡 − 𝐾𝑗𝑡𝛼 �𝑁𝑗𝑡ℎ 𝑁𝑗𝑡𝑙

1−𝜁 1−𝛼

�

(15)

�,

Therein, 𝜔𝑗𝑡 denotes a Lagrange multiplier. We can obtain the following demand forthe input factor as shown below.

α

1−𝜁

𝑙
𝑎𝑙 (1 − 𝑣)𝑙𝑗𝑡
�
�
�
ℎ
ℎ 𝜁
𝑙 1−𝜁
𝑎ℎ 𝑣𝑙𝑗𝑡
�𝑎ℎ 𝑣𝑙𝑗𝑡
� �𝑎𝑙 (1 − 𝑣)𝑙𝑗𝑡
�

𝑤𝑗𝑡ℎ = ωjt (1 − α)𝜁 �

K jt

5

.

(16)

𝑤𝑗𝑡𝑙 = ωjt (1 − α)(1 − 𝜁) �

𝜁

K jt

1−𝜁

ℎ
𝑙
�𝑎ℎ 𝑣𝑙𝑗𝑡
� �𝑎𝑙 (1 − 𝑣)𝑙𝑗𝑡
�αℎ𝑎ℎ 𝑣𝑙𝑗𝑡

𝜁

� � 𝑙
𝑙 � .
𝑎 (1 − 𝑣)𝑙𝑗𝑡

𝐾𝑗𝑡

𝛼−1

𝑟𝑗𝑡 = 𝜔𝑗𝑡 𝛼 �
�
ℎ 𝜁
𝑙 1−𝜁
�𝑎ℎ 𝑣𝑙𝑗𝑡
� �𝑎𝑙 (1 − 𝑣)𝑙𝑗𝑡
�

.

(17)

(18)

Considering the constant returns to scale product function and (16)--(18), total cost C isshown as follows.
𝐶 = 𝑤𝑗𝑡ℎ 𝑁𝑗𝑡ℎ + 𝑤𝑗𝑡𝑙 𝑁𝑗𝑡𝑙 + 𝑟𝑗𝑡 𝐾𝑗𝑡 = 𝜔𝑗𝑡 𝑌𝑗𝑡 .

(19)

Lagrange multiplier 𝜔𝑗𝑡 denotes the marginal cost. Considering (11) and (19), the profit

function of intermediate goods firm are shown as

𝑝𝑗𝑡 𝑝𝑗𝑡 −𝜀
𝑝𝑗𝑡 −𝜀� � 𝑌𝑡 − 𝜔𝑗𝑡 � � 𝑌𝑡 .
𝑝𝑡 𝑝𝑡
𝑝𝑡

𝜋𝑗𝑡 =

(20)

Profit maximization is reduced to the following equation.
𝜔𝑗𝑡 =2.3. Government𝜀 − 1 𝑝𝑗𝑡
.
𝜀 𝑝𝑡

(21)

The government levies a labor income tax and a consumption tax for households. The
tax revenue is assumed to be expended for non-productive government expenditure.
2.4. Sticky Price
Our paper presents consideration of sticky prices as shown by Calvo (1983). Calvo(1983) assumes that each firm in monopolistic competition cannot set optimal prices tomaximize the profit because of some reason. Then, the firm considers the probability of
not changing the price level in a future period and sets the price in the present period.
Some calculations engender the following New Keynesian Phillips Curve (NKPC)
considering homogeneous firms.
𝑙𝑛(1 + 𝜋𝑡 ) = 𝐸𝑡 𝑙𝑛(1 + 𝜋𝑡+1 ) +𝜌2𝜀+ 𝑙𝑛𝜔𝑡 �,
�𝑙𝑛
1−𝜌
𝜀−1

(22)

In that equation, ρ denotes the probability that the firm can change the price; 1 − ρdenotes the probability that the firm cannot change the price. Linearization of (22)engenders𝜋�𝑡 = 𝐸𝑡 𝜋�𝑡+1 +where 𝜋�𝑡 = 𝜋𝑡 − 𝜋 and 𝜔
�𝑡 =

𝜔𝑡 −𝜔
𝜔

𝜌2
𝜔
�,
1−𝜌 𝑡

(23)

. Actually, 𝜋 and 𝜔 denote the inflation rate and themarginal cost from the steady state. Hereinafter 𝑧̂𝑡 denotes the rate of change of 𝑧𝑡
6from the steady state. Furthermore, 𝑧̃𝑡 the difference of 𝑧𝑡 from the steady state.

2.5. Monetary Policy

Based on the Taylor rule, monetary policy is provided as𝚤̃𝑡 = 𝜒𝚤̃𝑡−1 + (1 − 𝜒){𝜙1𝐸𝑡 𝜋�𝑡+1 + 𝜙2 𝑦�𝑡 }.

(24)

The nominal interest rate in period t depends on the nominal interest rate in period𝑡 − 1, the expectation of inflation in period t+1, and the output gap in period t.

3. Equilibrium

This section presents derivation of the equilibrium and linear model. Considering
aggregate consumption as 𝑐𝑡 = 𝜈𝑐𝑡ℎ + (1 − 𝜈)𝑐𝑡𝑙 , we obtain 1
𝐶̂𝑡 =

where

𝜈𝑐 ℎ ℎ (1 − 𝜈)𝑐 𝑙 𝑙
𝑐̂ +
𝑐̂𝑡 ,
𝐶 𝑡
𝐶

(25)

1
1
ℎ
𝑐̂𝑡ℎ = 𝐸𝑡 𝑐̂𝑡+1− 𝐸𝑡 𝚤̃𝑡+1 + 𝐸𝑡 𝜋�𝑡+1 ,
𝜃
𝜃

(26)

𝑤
�𝑡ℎ = 𝜅𝑙̂𝑡ℎ + 𝜏̃ 𝑡 + 𝜃𝑐̂𝑡ℎ + 𝜏̃ 𝑐 ,𝑤
�𝑡𝑙 = 𝜅𝑙̂𝑡𝑙 + 𝜏̃ 𝑡 + 𝜃𝑐̂𝑡𝑙 + 𝜏̃ 𝑐 .

(28)

Labor supply is given as𝑐̂𝑡𝑙 = 𝑤
�𝑡𝑙 + 𝑙̂𝑡𝑙 − 𝜏̃ − 𝜏̃ 𝑐 .

(27)

(29)

Tobin’s q dynamics follow as
𝐸𝑡 𝑞�𝑡+1 =

1
1+𝑖
𝑟
�
�𝑞�𝑡 + 𝐸𝑡 (𝚤̃𝑡+1 − 𝜋�𝑡+1 )� − 𝑟̂𝑡 �.
1−𝛿 1+𝜋𝑞(30)

The investment dynamics can be expressed as
𝐼̂𝑡 =

1+𝑖
1+𝑖
1+𝑖
𝐼̂ +
𝐸 𝐼̂ +𝑞� .
2 + 𝑖 + 𝜋 𝑡−1 2 + 𝑖 + 𝜋 𝑡 𝑡+1 (2 + 𝑖 + 𝜋)𝑆 ′′ (1) 𝑡(31)

The expression of the capital stock dynamics is�𝑡 = 𝛿𝐼̂𝑡 + (1 − 𝛿)𝐾�𝑡−1 .
𝐾(32)

Market clearing conditions of final goods market are shown as follows.
𝑌�𝑡 =

𝐶
𝐼
𝐶̂ + 𝐼̂ .
𝑌 𝑡 𝑌 𝑡

The rates of change of factor prices are shown as follows,�𝑡 − (𝛼𝜁 + 1 − 𝜁)𝑙̂𝑡ℎ + (1 − 𝛼)(1 − 𝜁)𝑙̂𝑡𝑙 ,𝑤
�𝑡ℎ = 𝜔�𝑡 + 𝛼𝐾
1

(33)
(34)

We consider 𝜏̃ = 𝜏̃ 𝑐 =0 for type L labor because the government provides a redistribution policy toexempt income tax burden and provides benefit for low income family for regressivity of theconsumption tax in OECD countries.

7�𝑡 + (1 − 𝛼)𝜁𝑙̂𝑡ℎ − (𝛼(1 − 𝜁) + 𝜁)𝑙̂𝑡𝑙 ,𝑤
�𝑡𝑙 = 𝜔�𝑡 + 𝛼𝐾�𝑡 + (1 − 𝛼)𝛽𝑙̂𝑡ℎ + (1 − 𝛼)(1 − 𝜁)𝑙̂𝑡𝑙 .
𝑟̂𝑡 = 𝜔�𝑡 + (𝛼 − 1)𝐾

(35)
(36)

Linearization of the product function is shown below.

�𝑡 + (1 − 𝛼)𝜁𝑙̂𝑡ℎ + (1 − 𝛼)(1 − 𝜁)𝑙̂𝑡𝑙 .
𝑌�𝑡 = 𝛼𝐾

(37)

4. Parameter Setting

Data that we use for estimation are GDP, consumption, investment, the wage rate, theinflation rate, and the interest rate from the first quarter of 1993 to the second quarterin 2016 in Japan. 2 Results of estimation of parameters are shown as follows.
[Insert Table 1 around here.][Insert Table 2 around here.][Insert Table 3 around here.]
5. Results and Conclusions
In this section, we examine how the consumption tax affects variables such as
consumption. We consider the tax reform of an increase in the consumption tax from 5%
to 8% and simulate a permanent shock of an increase in the consumption tax. The
following figure presents the effects of an impulse shock of an increase in the
consumption tax.
[Insert Fig. 1 around here.]
An increase in the consumption tax reduces the aggregate output, consumption,
investment, and others. This is an intuitive result. An increase in the consumption tax
raises consumer prices and reduces demand for goods.
The main aim of this paper is to examine tax incidence. Our paper presents
consideration of the following index to examine tax incidence as shown below. This
index refers to Feldstein (1974) and Doi (2010), as

2

With the HP Filter, we detrend the data of GDP, consumption, investment, and wage rate. We referto the total cash salary of the “Monthly Labour Survey (more than 30 persons)” in Japan for the wage
rate. We consider that the wage of type H is the index of permanent employees and that the wage oftype L is the index of part-time employees. We de-mean the data of the inflation rate and the interest
rate. To remove the effect of the bubble economy, we use data from 1993.

8

𝐽ℎ =

𝑤ℎ𝑡 𝑁𝑖𝑡ℎ,𝑤ℎ𝑡 𝑁𝑖𝑡ℎ + 𝑤𝑙𝑡 𝑁𝑖𝑡𝑙 + 𝑟𝑡 𝐾𝑡−1

𝐽𝐾 =𝑟𝑡 𝐾𝑡−1.
ℎ ℎ
𝑤𝑡 𝑁𝑖𝑡 + 𝑤𝑙𝑡 𝑁𝑖𝑡𝑙 + 𝑟𝑡 𝐾𝑡−1

𝐽𝑙 =𝑤𝑙𝑡 𝑁𝑖𝑡𝑙,𝑤ℎ𝑡 𝑁𝑖𝑡ℎ + 𝑤𝑙𝑡 𝑁𝑖𝑡𝑙 + 𝑟𝑡 𝐾𝑡−1

(38)
(39)
(40)

where 𝐽𝐾 denotes the share of capital income. 𝐽ℎ and 𝐽𝑙 respectively denote the shareof labor income of type H and that of type L. The results of an impulse shock ofconsumption tax are shown as the following figures.

[Insert Fig. 2 around here.][Insert Fig. 3 around here.]Fig. 2 and Fig. 3 respectively show the rate of change from the variables in a steadystate. As shown in Fig. 2, an increase in the consumption tax reduces the share ofcapital income and raises the share of labor income. Therefore, although regressivity
exists in the consumption tax, the tax burden brought about by regressivity is weakened
because the shares of labor income of both type H and type L raises.
As shown in Fig. 3, the rate of increase in wage rate of type L is larger than that oftype H in the short run. This result shows that the low income households are better off,compared with high income households. However, in the medium run, such is not the
case.

9

References
Benchimol, Jonathan & Fourçans, André 2012. "Money and risk in a DSGE framework:
A Bayesian application to the Eurozone," Journal of Macroeconomics, Elsevier, vol. 34(1),pages 95-111.
Calvo, Guillermo A. 1983. "Staggered prices in a utility-maximizing framework,"
Journal of Monetary Economics, Elsevier, vol. 12(3), pages 383-398, September.
Doi,

Takero2010.

“Houjinzeino

Kichakunikansuru

Dougakutekibunnseki-Kansonamoderuniyoru Bunseki-(in Japanese)” RIETI D.P.S.
10-J-034.
Eguchi, Masataka 2011. “Dougakutekiippankinkouniokeru Zaiseiseisakuno Bunseki (inJapanese)” The Mitsubishi Economic Research Institute.
Feldstein, Martin S. 1974. "Tax Incidence in a Growing Economy with Variable Factor
Supply," Quarterly Journal of Economics, vol. 88(4), pages 551-573.
Hara, Naoko & Katayama, Munechika & Kato, Ryo 2014. "Rising Skill Premium?: The
Roles of Capital-Skill Complementarity and Sectoral Shifts in a Two-Sector Economy,"
Bank of Japan Working Paper Series 14-E-9, Bank of Japan.
Hayashi,Hiroaki

1995.

“Sozeiseisakuno

Keiryoubunnseki(inJapanese)”

Nihonhyoronsha.
Heer, Burkhard & Trede, Mark 2003. "Efficiency and distribution effects of arevenue-neutral income tax reform," Journal of Macroeconomics, Elsevier, vol. 25(1),pages 87-107, March.
Hirose, Yasuo & Naganuma, Saori 2010. "Structural Estimation of the Output Gap: A
Bayesian DSGE Approach," Economic Inquiry, Western Economic Association
International, vol. 48(4), pages 864-879, October.
Lehmus, Markku 2011. "Labor or consumption taxes? An application with a dynamic
general equilibrium model with heterogeneous agents," Economic Modelling, Elsevier,vol. 28(4), pages 1984-1992, July.
10

Nishiyama, Shinichi & Smetters, Kent 2005. "Consumption Taxes and Economic
Efficiency with Idiosyncratic Wage Shocks," Journal of Political Economy, University of
Chicago Press, vol. 113(5), pages 1088-1115, October.
Sakuragawa, Masaya & Hosono, Kaoru 2011. "Fiscal sustainability in Japan," Journalof the Japanese and International Economies, Elsevier, vol. 25(4), pages 434-446.
Smets, Frank & Wouters, Rafael 2007. "Shocks and Frictions in US Business Cycles: A
Bayesian DSGE Approach," American Economic Review, American Economic
Association, vol. 97(3), pages 586-606, June.

11Appendix
Derivation of (6)--(8)First-order conditions of Lagrange equation (5) are shown below.
𝛽 𝑡 𝑐𝑡ℎ−𝜃ℎ
𝛽 𝑡+1 𝐸𝑡 𝑐𝑡+1
−𝜇

𝑤𝑡𝑙 =

(A.2)

+ 𝜆𝑡+1 (1 + 𝜏𝑐 ) = 0.

𝛽 𝑡 𝑚𝑡 + 𝜆𝑡 − 𝐸𝑡
𝑤𝑡ℎ =

(A.1)

+ 𝜆𝑡 (1 + 𝜏𝑐 ) = 0.

−𝜃

𝜆𝑡+1
= 0.
1 + 𝜋𝑡+1(1 + 𝜏𝑐 )𝑙𝑡ℎ𝜅

(1 − 𝜏)𝑎ℎ 𝑐𝑡ℎ(1 + 𝜏𝑐 )𝑙𝑡𝑙

(1 − 𝜏)𝑎𝑙 𝑐𝑡𝑙

−𝜃

𝜅

−𝜃

.

(A.3)
(A.4)

.

(A.5)
2

𝐼
1 + 𝜋𝑡+1 ′�𝐼𝑡+1
�� 𝑡+1 �
1 = 𝑞𝑡 �1 − 𝑆 �
� − 𝑆′ ����� + 𝐸𝑡 𝑞𝑡+1
𝑆 𝐼𝑡 𝐼𝑡 .
𝐼𝑡−1
𝐼𝑡−1 𝐼𝑡−1
1 + 𝑖𝑡+1

𝐼𝑡

𝐼𝑡

𝐼𝑡

−𝜆𝑡 𝑟𝑡 − 𝛾𝑡 (1 − 𝛿) + 𝛾𝑡−1 = 0.
𝜆𝑡 − 𝐸𝑡 𝜆𝑡+1

1 + 𝑖𝑡+1= 0.
1 + 𝜋𝑡+1

𝜕𝐿
= 0.
𝜕𝜆𝑡
𝜕𝐿
= 0.
𝜕𝛾𝑡

(A.6)
(A.7)
(A.8)
(A.9)
(A.10)

With (A.1), (A.2), and (A.8), we obtain the consumption Euler’s equation (6). With (A.1),(A.3), and (A.8), we obtain the marginal rate of substitution about money stock andconsumption (7). With (A.7) and (A.8), we obtain the equation of nominal interest rate,
real interest rate, and inflation rate (8).
Derivation of (22) and (23)The optimal price 𝑝𝑡∗ is set as follows because of (21) if the firms can set the price tomaximize their profit.

𝑙𝑛𝑝𝑡∗ = ln𝜀+ 𝑙𝑛𝜔𝑡 + 𝑙𝑛𝑝𝑡 .
𝜀−1

(B.1)

Based on Calvo (1983), we consider sticky prices in this model economy. It is assumed
that the share of 𝜌 can change the price in t period and share of 1-𝜌 cannot change theprice in period t. Then, the price that the firm cannot change is equal to the price inperiod t-1. Then, the firms set the following price 𝑥𝑡 if there exists uncertainty about
price setting, as

ln𝑥𝑡 = 𝜌𝑙𝑛𝑝𝑡∗ + 𝜌(1 − 𝜌)𝐸𝑡 𝑙𝑛𝑝𝑡∗ + ⋯
12

(B.2)

= 𝜌𝑙𝑛𝑝𝑡∗ + (1 − 𝜌)𝐸𝑡 𝑙𝑛𝑥𝑡+1 .

Defining ln∆𝑥𝑡+1 = 𝑙𝑛𝑥𝑡+1 − 𝑙𝑛𝑥𝑡 and substitute (B.1) into (B.2), we obtain the following
equation:

𝐸𝑡 𝑙𝑛∆𝑥𝑡+1 = 𝜌𝐸𝑡 𝑙𝑛𝑥𝑡+1 − 𝜌 �𝑙𝑛

𝜀+ 𝑙𝑛𝜔𝑡 + 𝑙𝑛𝑝𝑡 �.
𝜀−1

(B.3)

However, the price index in period t, 𝑝𝑡 , is given as the weighted average of price 𝑥𝑡

that the firms can set, and price 𝑝𝑡−1 that the firms cannot change in period t. Then,the price index in period t is given as

𝑙𝑛𝑝𝑡 = 𝜌𝑙𝑛𝑥𝑡 + (1 − 𝜌)𝑙𝑛𝑝𝑡−1 .

(B.4)

𝜌𝑙𝑛𝑥𝑡 = 𝑙𝑛(1 + 𝜋𝑡 ) + 𝜌𝑙𝑛𝑝𝑡−1 .

(B.5)

𝜌𝐸𝑡 𝑙𝑛∆𝑥𝑡+1 = 𝐸𝑡 𝑙𝑛(1 + 𝜋𝑡+1 ) − (1 − 𝜌)𝑙𝑛(1 + 𝜋𝑡 ).

(B.7)

𝑝

Considering 1 + 𝜋𝑡 = 𝑝 𝑡 , we obtain the following equation.
𝑡−1

𝜌𝐸𝑡 𝑙𝑛𝑥𝑡 = 𝐸𝑡 𝑙𝑛(1 + 𝜋𝑡+1 ) + 𝜌𝑙𝑛𝑝𝑡 .

(B.6)

Then, the following equation is obtained.

With (B.4), (B.6), and (B.7), we obtain (22). Linearization at approximation of the steadystate is reduced to (23).

13

parameter
θ
δ
α
ρ
χ
φ1
φ2
S''(1)
ξκ
standard error: NKPC shock
standard error: technology shock
standard error: preference shock
standard error: monetary policy shock
standard error: high skill labor shock
standard error: low skill labor shock
standard error: regulation cost of investment shock
AR(1) Coefficient: NKPC shock
AR(1) Coefficient: technology shock
AR(1) Coefficient: preference shock
AR(1) Coefficient: monetary policy shock
AR(1) Coefficient: labor shock
AR(1) Coefficient: regulation cost of investment shock

initialminimum
1
0.05
0.33
0.25
0.7
2
0.2
0.13
0.9
2
1.5
1.5
1.5
1.5
1.5
1.5
1.5
0.5
0.5
0.5
0.5
0.5
0.5

Table 1: Prior Distribution of Parameters(Based on Eguchi (2011), we set S’’(1)=0.13.)

14

0
0.01
0.23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

maximum distribution
10 normal distribution
0.1 uniform distribution
0.43 uniform distribution
0.9999 normal distribution
0.9999 β distribution
10 normal distribution
10 normal distribution
10 normal distribution
0.9999 β distribution
10 normal distribution
10 inverse γ distribution
10 inverse γ distribution
10 inverse γ distribution
10 inverse γ distribution
10 inverse γ distribution
10 inverse γ distribution
10 inverse γ distribution
0.9999 β distribution
0.9999 β distribution
0.9999 β distribution
0.9999 β distribution
0.9999 β distribution
0.9999 β distribution

average
1
0.25
0.6
2
0.2
0.13
0.5
2
1.5
1.5
1.5
1.5
1.5
1.5
1.5
0.6
0.6
0.6
0.6
0.6
0.6

paramerer content
consumption output ratio at the steady state
C/Yinvestment output ratio at the steady state
I/Ynominal interest rate at the steady statei
inflation rate at the steady stateπ
Tobin's q at the steady stateqreal interest rate at the steady stater
consumption of type H ratio
Ch/Clconsumption of type L ratio
C /CTable 2: Calibrated Parameters

15

value
0.8
0.2
(1+π)/β-101q(1/β+δ-1)
0.9
0.1

parameter
θ
δ
α
ρ
χ
φ1
φ2
S''(1)
ξκ
standard error: NKPC shock
standard error: technology shock
standard error: preference shock
standard error: monetary policy shock
standard error: high skill labor shock
standard error: low skill labor shock
standard error: regulation cost of investment shock
AR(1) Coefficient: NKPC shock
AR(1) Coefficient: technology shock
AR(1) Coefficient: preference shock
AR(1) Coefficient: monetary policy shock
AR(1) Coefficient: labor shock
AR(1) Coefficient: regulation cost of investment shock

Table 3: Posterior Distribution of Parameters

16

confidence interval
average
0.37
0.21
0.289
0.1
0.073
0.088
0.247
0.23
0.268
0.214
0.179
0.252
0.575
0.469
0.679
8.697
7.286
10
0.357
0.159
0.545
7.189
5.054
9.889
0.869
0.746
0.997
0.087
0
0.167
7.513
4.977
9.999
0.551
0.426
0.68
2.121
1.465
2.774
0.371
0.269
0.464
0.97
0.844
1.087
0.715
0.609
0.822
4.585
3.926
5.21
0.006
0
0.013
0.821
0.71
0.931
0.949
0.916
0.886
0.804
0.992
0.901
0.389
0.283
0.492
0.201
0.057
0.337

Fig. 1 Impulse Response of an Increase in Consumption Tax.
(Y_hat, C_hat, C_h_hat, C_l_hat, I_hat, K_hat, NI_tilda, RI_tilda, W_h_hat, W_l_hat,
PAI_tilda, L_h_hat, L_l_hat and Q_h_hat respectively denote� , 𝚤̃, 𝑟̃ , 𝑤
� ℎ, 𝑤
� 𝑙 , 𝜋�, 𝑙̂ℎ , 𝑙̂𝑙 , 𝑞�ℎ )𝑦�, 𝐶̂ , 𝑐̂ ℎ , 𝑐̂ 𝑙 , 𝐼̂, 𝐾

17

Fig. 2 Impulse Response of an Increase in Consumption Tax for 𝑗ℎ , 𝑗𝑙 , 𝑗𝑘 .

18

Fig. 3: Impulse Response of an Increase in Consumption Tax for 𝑤ℎ , 𝑤𝑙 .

19