HIAS-E-100

Fiscal multipliers in the most aged country:
Empirical evidence and theoretical interpretation
Hiroshi Morita
Hosei University, Institute of Comparative Economic Studies,4342 Aihara, Machida, Tokyo, 194-0298, Japan

September, 2020

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Fiscal multipliers in the most aged country:
Empirical evidence and theoretical interpretation∗
Hiroshi Morita†
Hosei University
September, 2020

Abstract
This study investigates how population aging impacts the eﬀectiveness of agovernment spending shock. We estimate a panel VAR model with prefectural
data in Japan, the world’s fastest aging country and reveal that a government
spending shock becomes less eﬀective as the aging rate increases. Subsequently, weconstruct a New Keynesian model with workers and retirees, which can replicate
our empirical findings. This highlights the role of the supply-side channel throughwhich workers facing a liquidity constraint can benefit from increased disposable
income, in generating the state-dependent eﬀect of the government spending shock.
Our theoretical finding may suggest that promoting labor market participation byelderly people could increase the eﬀectiveness of a government spending shock amid
a rapidly aging society.

JEL Classification Number: E62, C11, C23.
Key words: Population aging, Panel VAR model, New Keynesian model, Fiscal
policy.

∗This research was funded by Grant-in-Aid for Young Scientists 19K13652 and the Nomura Foundation. Errors, if any, are entirely the author’s responsibility.
†
Address: Hosei University, Institute of Comparative Economic Studies, 4342 Aihara, Machida, Tokyo
194-0298, Japan. E-mail address: morita-h@hosei.ac.jp1

1

Introduction

Population aging is a growing issue faced by many developed countries. From the economic perspective, it can cause long-term problems, such as low economic growth andfiscal unsustainability. Indeed, many studies revealed the relationship between population aging and such problems in the growth model framework (e.g., Gonzalez-Eiras andNiepelt, 2012; Nishiyama, 2015; Hansen and İmrohoroǧlu, 2016).
However, there is little research addressing this issue in the context of short-term
economic fluctuations. Specifically, it is not clear how the aging structure aﬀects the economic stimulus eﬀect of fiscal policy shock, although population aging is likely to makefiscal policy management diﬃcult.1 In this study, we aim to clarify whether the eﬀectiveness of a government spending shock on output changes depending on the economy’s
aging rate, both theoretically and empirically. This study contributes to the literatureby presenting a desirable labor market structure that allows the government to maintainthe eﬀectiveness of a government spending shock amid rapid population aging.
To this end, we estimate a panel VAR model that includes government spending,tax revenue, and output across Japan’s prefectures, as Japan is the world’s most rapidly
aging country. We illustrate that the progression of population aging is likely to dampen
the eﬀectiveness of a government spending shock measured by the fiscal multiplier. To
interpret the empirical findings theoretically, we build a New Keynesian (NK) modelwith heterogeneous agents, in which we categorize households into three types: Ricardian workers, Non-Ricardian workers, and retirees. Our model replicates the empiricalevidence well and carries the implication that the state-dependent eﬀect of a government spending shock on the aging rate is mainly caused by the supply-side channel, forlack of which retirees cannot benefit from the increased disposable income arising from agovernment spending shock.
This study closely relates to that by Basso and Rachedi (forthcoming), who estimate
1

Anderson et al. (2016) and Janiak and Santos-Monteiro (2016) are among the few to examine theconnection between demographic structure and fiscal policy.

2the fiscal multiplier in U.S. states depending on the aging structure and construct a lifecycle NK model to rationalize their empirical findings. Basso and Rachedi’s (forthcoming)
study is similar to ours in that we both reveal the negative eﬀect of population aging on thefiscal multiplier. In this sense, our empirical analysis strengthens the findings in Basso and
Rachedi (forthcoming), though our study diﬀers from theirs in several regards. First, wederive the fiscal multiplier from the VAR model, while Basso and Rachedi (forthcoming)
estimate a single equation to obtain a local fiscal multiplier in the manner of Nakamura
and Steinsson (2014). The advantage of the VAR model over the single-equation model isthat it provides the aggregate fiscal multiplier directly instead of having to derive the local
fiscal multiplier. The disadvantage is its lower statistical accuracy, as the loss of degrees
of freedom due to the inclusion of many endogenous variables leads to a wider error
band for the estimates. To cope with this shortcoming, we incorporate a hierarchicalstructure into our panel VAR model. This method improves the statistical accuracyof estimated coeﬃcients and thus underscores the diﬀerences across the demographic
structures. In addition, the VAR analysis gives us the impulse responses of tax revenue,
which we can use to calibrate the fiscal rule in the theoretical model. Second, regarding
the theoretical model, we build on Yoshino and Miyamoto (2017), whose model has
infinitely lived workers and retirees. In the absence of life-cycle agents, our model ismore tractable than the one developed in Basso and Rachedi (forthcoming). Despite
its simplicity, however, our theoretical model successfully traces the empirical responses
qualitatively and quantitatively. Finally, and most importantly, we attempt to identify
a source of state dependency in the eﬃcacy of a government spending shock by altering
the assumption of retirees’ behavior. The results of this theoretical analysis imply that
retirees’ lack of access to the labor market is more relevant than is the lack of access tothe financial market in generating the state-dependent eﬀect on the aging situation.
The remainder of this paper is organized as follows. Section 2 describes our data,explains the empirical model, and presents the empirical evidence and the robustness of
our main findings. In Section 3, we develop an NK model to rationalize our empirical3

findings. Section 4 examines the source of the state-dependent eﬀect of the government
spending shock by relaxing the assumption on retirees’ behavior. Finally, Section 5
concludes the paper.

2

Empirical analysis

2.1

Data

The benchmark model is a Blanchard and Perotti (2002)-type three-variable fiscal VAR
model that includes government spending (gt ), tax revenue (τt ), and output (xt ) in realterms. We obtained the data on prefectural government spending and output from theAnnual Report on Prefectural Accounts published by the Cabinet Oﬃce, Governmentof Japan. We retrieved the local tax revenue data for each prefecture from the Nikkei
NEEDS Financial Quest database. The local tax consists mainly of the municipal tax
and business tax collected by the local government, though this study does not cover all
types of taxes. Since the original tax revenues for each prefecture are published in nominal
terms, we deflate them by a prefecture-specific GDP deflator, which is also available in
the Annual Report on Prefectural Accounts. We define government spending as the sumof government consumption and public investment. All data are at the annual frequency
for the fiscal year. The sample period is from 1990 to 2014. 2
To examine whether the eﬃcacy of a government spending shock depends on the
demographic structure, we divide the prefectures into two groups, low- and high-aging
prefectures, depending on their aging ratio; that is, the ratio of the population over 65
to the total population. We construct this ratio using the population data publishedin Population Estimates by the Statistics Bureau of Japan, and then calculate its time
average for each prefecture. Finally, we select the top (worst) 12 prefectures whose aging
ratios are high (low).3 The aging ratios diﬀer across Japanese prefectures, as seen in
2

The fiscal year in Japan starts on April 1 and ends the next March 31.
Because Japan has 47 prefectures, the top and bottom 12 prefectures are approximately equal to the
1st and 3rd quartiles.
34Figure 1, which displays the transition of the aging rate across prefectures using a box
plot. Therefore, the Japanese prefectural data are suitable to our purpose of determining
the eﬃcacy of a government spending shock according to the aging situation, in the sensethat the sizable heterogeneity among Japanese prefectures can highlight the diﬀerence inthe value of the aging rate between the low- and high-aging groups.
[Figure 1 about here.]
The high aging prefectures include Akita (AK), Yamagata (YG), Nagano (NA),
Wakayama (WA), Tottori (TT), Simane (SM), Yamaguchi (YA), Tokushima (TK), Ehime(EH), Kochi (KO), Oita (OI), and Kagoshima (KG). The low aging prefectures consistof Miyagi (MG), Ibaraki (IB), Tochigi (TC), Saitama (ST), Chiba (CB), Tokyo (TY),
Kanagawa (KN), Aichi (AI), Shiga (SI), Osaka (OS), Hyogo (HG), and Okinawa (OK).
As Figure 2 shows, the low-aging prefectures tend to be concentrated in metropolitan
areas such as Tokyo and Osaka, while high-aging prefectures exist across the country.
Here, we are not concerned about the location, but rather with whether or not these prefectures share common characteristics besides the demographic structure. To clarify thatthe state-dependent eﬀect of a government spending shock, if any, is caused by the aging
situation, it is necessary to prove that the aging rate in each prefecture is not related
to other possible economic factors. As a representative example reported in Auerbachand Gorodnichenko (2012), the eﬀectiveness of government spending shocks changes depending on economic conditions, such as a boom or recession. To this end, we computethe correlation between the aging rate and prefectural unemployment rate as a proxy foreconomic conditions. We find no significant correlation unless we include Okinawa prefecture. Okinawa, located in the south-most part of Japan, is an outlier with a considerably
low aging rate and high unemployment rate, making the entire correlation significantly
negative. Hence, we perform an additional estimation to check the robustness of the main
results by excluding Okinawa from the low-aging prefectures.
[Figure 2 about here.]5

2.2

Hierarchical panel VAR model with sign restrictions

We employ panel data on Japanese prefectures to examine whether the economic stimulus
eﬀect of increased government spending depends on the aging rate. As explained above,the sample period is from 1990 to 2014 with data at an annual frequency, and we divide
the prefectures into two groups of 12 depending on the aging rate. One notable feature ofthis study is that we adopt a hierarchical panel VAR model to address the short sample
period and the possible heterogeneity across the selected prefectures. As Canova andPappa (2007) and Pappa (2009) emphasize, the hierarchical structure improves the quality of the estimates by exploiting unit-specific and cross-sectional information eﬃciently
compared with the pooled estimators, which are likely to give us imprecise estimates insuch circumstances.
For each unit i, we can formulate the three-variable VAR(p) model as

yit = Bi0 + Bi1 yit−1 + · · · + Bip yit−p + uit ,(1)

where yit = [git , τit , xit ]′ is a 3 × 1 vector of endogenous variables, and uit is a 3 × 1][ ′′and βi =, · · · , yit−pvector of reduced-form residuals. We define Xit = Ik ⊗ 1, yit−1
]′
[ ′
′
′
) , and rewrite the model as), · · · , vec(BipBi0 , vec(Bi1yit = Xit βi + uit ,(2)

where the vec operator creates a column vector from Bis (s = 1, · · · , p) by stacking thecolumn vector of Bis , and ⊗ denotes the Kronecker product. We assume that we canrepresent the reduced-form residuals as a linear combination of structural shocks:

Ai uit = eit , eit ∼ N (0, Di )(3)

where eit is a vector of structural shocks that can interpret economically and are mutuallyindependent. Thus, the variance of each structural shock is located in a diagonal element
6

of Di . Note that the variances of structural shocks and contemporaneous matrix Ai areunit-specific, as are the VAR coeﬃcients. Suppose that Ai is a lower triangular matrix,
as in the form of




0 0 
 1



Ai = 
 a21i 1 0  .
a31i a32i 1(4)Then, we can transform (3) into

uit = Zit αi + eit ,(5)

where αi is a stacked vector of the lower triangular elements in Ai , and Zit takes the formof

 0 0 0 ,1
Zit = − 
0
0
u

 it


0 u1it u2it

(6)

where ulit indicates the l-th element from the top of the vector of the reduced-form residual
uit . In summary, we can estimate the unit-specific coeﬃcients βi and contemporaneous
relationship αi from (2) and (5).
For the unit-specific coeﬃcients βi and αi , we postulate the following hierarchicalstructure:βi = β̄ + νi , νi ∼ N (0, Σν ),(7)αi = ᾱ + ωi , ωi ∼ N (0, Σω ),(8)

implying that each unit i’s coeﬃcients share a common prior distribution. Namely, each7

unit i’s coeﬃcients are random variables drawn from the common distribution. Using
the Bayesian technique, we can calculate the conditional posterior distributions of βiand αi from the data and prior distributions (7) and (8). In addition, we can specify
the conditional posterior distributions of β̄ and ᾱ by regarding individual draws of βiand αi as observations. Therefore, this mutual use of information between unit-specificand cross-sectional cyclically improves the accuracy of both estimates in the hierarchicalmodel.
We identify the structural shocks using the sign restriction developed by Uhlig (2005).
We adopt this method because, in contrast to Blanchard and Perotti (2002), who use
quarterly data, it is reasonable to suppose that the fiscal variables respond to contemporaneous economic activity in annual data. The methodology, which imposes restrictions
on the shape of the impulse responses, identifies the shocks by allowing for simultaneous
responses among the variables. Therefore, we can elude the problem of endogeneity originating from the use of annual frequency data. The only shock of interest is government
spending for this study; nevertheless, we follow Peersman (2005) to identify a full set ofstructural shocks, which can help us identify government spending shocks. Furthermore,
Caldara and Kamps (2017) argue recently that the estimated impact of a government
spending shock on output is determined by the extent to which the unconditional covariance between them is allocated as a contribution of the government spending shock.
In this sense, it is preferable to identify shocks besides a government spending shock.
The structural shocks identified here are government spending, tax revenue, and businesscycle shocks.
Table 1 summarizes the sign restrictions employed in this study. Each row and column
correspond to a shock and variable, respectively. We set the sign conditions to be basicallyconsistent with the theoretical and empirical evidence documented in the existing literature. We impose all restrictions on the variables for two periods. We first assume thatthe government spending shock increases government spending for two years. Note that
we impose no sign restrictions on the other variables to allow the data to speak for itself,8

as we are interested in the quantitative and qualitative eﬀects of a government spending
shock. Subsequently, we assume that output responds negatively to an exogenous positive tax revenue shock; that is, an unexpected increase in tax revenue. Mountford andUhlig (2009) report a negative co-movement between tax revenue and output for the U.S.
and Kuttner and Posen (2002) report the same for Japan. We assume that a favorable
business cycle shock is a shock that increases output and tax revenue while decreasing
government spending. We consider an increase in tax revenue due to a business cycle
shock as an automatic response associated with an increase in the tax base. Conversely,
this type of shock describes the situation in which a discretionary increase in government
spending is implemented in response to a drop in tax revenue and output by a recession.
The signs imposed on tax revenue and output can separate tax revenue shocks from business cycle shocks, and we can identify government spending shocks and business cycle
shocks by the sign on the responses of government spending. However, we cannot distinguish between government spending shocks and tax revenue shocks suﬃciently using the
restrictions listed in Table 1 because there are no restrictions on government spending
after a tax revenue shock, and vice versa. Hence, we impose a size restriction such thatthe impact response of government spending to government spending shocks in absoluteterms is greater than that to tax revenue shocks. This condition assumes that the government spending shock best explains the forecast error variance in government spending
during the impact period. We can consider this final restriction as the simplest versionof Max Share identification proposed by Francis et al. (2014).
[Table 1 about here.]
In practice, identification using sign restrictions is implemented by an algorithm involving QR decomposition, as proposed by Rubio-Ramı́rez et al. (2010). After drawing
a sample of βi , Ai , and Di for each unit i, we generate a matrix Q such as W = QR
by the QR decomposition, where Q is an orthogonal matrix (i.e., QQ′ = I), R is anupper-triangular matrix, and W is a real square matrix. Moreover, we draw each column
9

of W randomly from N (0, I). Given the matrices of Ai , Di , and Q, we can construct
an alternative contemporaneous matrix Ãi and structural shocks ε̃it by transforming (3)intouit = A−1
i Di εit
1/2

′
= A−1
i Di QQ εit
1/2

(9)

= Ãi ε̃it ,′where Ãi = A−1
i Di Q, ε̃it = Q εit , and the new structural shock εt follows a standard
1/2

normal distribution. We repeatedly draw a matrix Q until the impulse response computedfrom βi and Ãi satisfy the sign restrictions in Table 1.
Following Canova and Pappa (2007) and Pappa (2009), we assume that the impulse
response function also has a hierarchical structure. We denote the stacked vector ofeach unit i’s impulse responses of the variables to the government spending shock as
ψi . Representing the maximum horizon of impulse responses as J, ψi is a 3J × 1 vector
storing the respective responses of government spending, tax revenue, and output untilhorizon J, in this order. Then, we impose the unit invariant specification on ψi , given as
ψi = ψ̄ + ηi , ηi ∼ N (0, Ση ),(10)

where ψ̄ is the “typical” responses according to the terminology in Pappa (2009), Ση isa diagonal matrix, and (Ση )j , j = 0, 1, · · · , J, which corresponds to the variance of the
responses at horizon j, is set to 0.2
exogenously, as in Canova and Pappa (2007).
j
Finally, we discuss the prior distributions adopted in this study. As discussed above,
the prior distributions for βi and αi are

βi ∼ N (β̄, Σν ) αi ∼ N (ᾱ, Σω ),(11)

as described in (7) and (8). Note that the hyperparameters in (11) also generate their10

conditional posterior distributions. While we assume diﬀuse priors on β̄ and ᾱ, we set
the priors for Σν and Σω to be an inverse Wishart distribution:Σν ∼ IW(100, 0.01I), Σν ∼ IW(100, 0.01I),(12)

where IW represents the inverse Wishart distribution. We assume that the prior distribution for the inverse of the l-th diagonal element of Di , denoted by δil , follows a gamma
distribution, as follows:( l )−1δi∼G(

100 0.01
,
2
2

)
,

(13)

where G also indicates a gamma distribution. Similar to β̄ and ᾱ, we set the prior for
impulse response ψ̄ to be diﬀuse priors.

2.3

Empirical results

All data in the VAR system are at the log level of the real per-capita values. The system
contains a constant and linear trend as a deterministic term and we set the lag length
to two as we use annual frequency data. We conduct the estimation using the Gibbs
sampler of the Bayesian Markov Chain Monte Carlo (MCMC) method, iterating 60, 000
draws of each parameter from the conditional posterior distributions and discarding thefirst 10, 000 draws as a burn-in. Then, we exploit every 10-th draw for the inference toalleviate the autocorrelation among each sequential draw. Moreover, we save only thedraws in which the roots of the VAR coeﬃcients are inside the unit circle to ensure thestationarity of the VAR system.
Figure 3 illustrates the responses of (a) government spending, (b) tax revenue, (c)output, and (d) the accumulated fiscal multiplier. We denote the median responses
obtained from the low- and high-aging prefectures by solid lines and dash-dotted lines
with circles, with the associated 68% credible intervals depicted by shaded areas and11dash-dotted lines. We normalize the scale of all responses to be interpreted as yens. For
example, an increase in government spending of 1 yen leads to an increase in output of
1.5 yen in the impact period in low-aging prefectures. We compute the J-period-ahead
accumulated fiscal multiplier as∑J

j=0 IRj (x)

x̄
× ,
ḡ
j=0 IRj (g)Accum. fiscal multiplierJ = ∑J

(14)where IRj (·) denotes the impulse response of the variable in the parentheses at horizon j,
and x̄ and ḡ represent the means of output and government spending in each group, taking
the average for both the time-series and cross-sectional dimensions. Fiscal multipliers
allow us to quantitatively compare the eﬀect of a government spending shock on output,
even if the dynamics of government spending diﬀer between the two groups.
[Figure 3 about here.]
We first observe from Figure 3 that output exhibits diﬀerent dynamics across thetwo groups to the same 1-yen increase in government spending. In particular, the impact response of output in low-aging prefectures is at least 1.5 times that in high-aging
prefectures, suggesting that the eﬀect of government spending shocks changes dependingon the demographic structure. When assessing the diﬀerence in the eﬀects across thegroups over the impact period, it is appropriate to examine the fiscal multiplier insteadof the output response because the path of government spending is also diﬀerent acrossthe two groups, as Figure 3 (a) shows. The fiscal multiplier, derived from (14), measures
how much output increases for 1 yen of government spending for each horizon basedon the cumulative response of each variable so far, allowing us to directly compare themagnitude of a government spending shock on output across the groups by unifying thesize of government spending. Figure 3(d) shows that the response of the multiplier in
low-aging prefectures is always located above the one in high-aging prefectures without
overlap between the confidence bands, demonstrating that population aging diminishes
the stimulus eﬀect of a government spending shock on output significantly. In addition,
12the fact that the size of the multiplier in the high-aging prefecture fluctuates around one
suggests that there are a little or no “multiplier eﬀects” under the condition of progressive
population aging. On the other hand, the multiplier in low-aging prefectures is larger
than those derived from the standard neoclassical model (e.g., Aiyagari et al., 1992; Baxter and King, 1993), matching those derived from the “ultra” NK model (e.g., Galı́ etal., 2007). This motivates us to build on the Galı́ et al. (2007)-type NK model in thetheoretical part below to rationalize our empirical findings.
Finally, we discuss the fiscal stance in Japan by examining the response of tax revenue
in Figure 3 (b). The path of the tax response also changes depending on the degree of
population aging, which traces that of output proportionally. Additionally, we shouldstress that the scales of the tax response are considerably modest compared with that ofgovernment spending, regardless of the situation. Although we do not cover all types oftax revenue, our estimates indicate that the increase in tax revenue is only about 10%of the increase in government spending, suggesting that government spending in Japanis mostly financed by debt issuance rather than tax increases. Prior studies often pointout this lack of fiscal discipline (Doi et al., 2011). We use this VAR result to calibrate
the structural parameters in the theoretical model below.
In sum, our empirical findings strongly support the state-dependent eﬀect of a government spending shock on the aging structure in Japan. More specifically, in line with
Basso and Rachedi (forthcoming), the fiscal multiplier in low-aging prefectures is significantly higher than that in high-aging prefectures. We next address the factors that cause
state dependency, as we observed in the empirical analysis. To answer this question, weconstruct a dynamic stochastic general equilibrium (DSGE) model in the next section toderive a structural interpretation of the empirical evidence.

2.4

Robustness check: Excluding the outlier

Before we construct a theoretical model that can replicate the state-dependent eﬀectof a government spending shock on the aging ratio observed in the data, we perform a13

robustness check of the main empirical findings. As mentioned above, Okinawa prefecture,
which is a low-aging prefecture, is an outlier in terms of the aging rate and unemployment
rate; it has the lowest aging rate and the highest unemployment rate in Japan. This
strict negative relationship generates a significant negative correlation between the aging
rate and the unemployment rate for all prefectural data. This is problematic because
the selected prefectures are likely to be categorized by economic conditions rather than
aging, meaning that it remains possible that the state-dependency is caused by factors
other than population aging. Fortunately, however, this correlation is insignificant forthe data except for Okinawa prefecture. Hence, we now re-estimate our empirical model
for low-aging prefectures without Okinawa’s data.
In Figure 4, we plot the median responses of the variables derived from the sample
excluding Okinawa prefecture (denoted by dashed lines), together with the median responses and credible intervals derived from the benchmark estimates. At a glance, wefind that the benchmark findings are robust because the responses derived from the robustness check almost follow the ones from the benchmark, leading us to conclude that
population aging, not economic conditions, is a main factor that influences the eﬀectiveness of a government spending shock in our present analysis. Based on the empiricalfindings so far, we build a DSGE model in which the aging ratio aﬀects the stimulus
eﬀects of a government spending shock on output.
[Figure 4 about here.]

3

Theoretical model

To rationalize the empirical evidence using a theoretical model, we construct a simple
NK model that contains a retiree as the counterpart of an elderly person in the empiricalanalysis. Our theoretical model is similar to that in Yoshino and Miyamoto (2017), except
for the presence of non-Ricardian households and wage unions, which are keys to replicate
the large fiscal multipliers observed in low-aging regions.
14

3.1

Households

We divide households into three types: Ricardian workers, Non-Ricardian workers, andretirees. The diﬀerence between workers and retirees is whether the households provide
labor services. While workers earn wages as compensation for labor supply, retirees
receive social security benefits from the government. We assume that a fraction ζ of the
population consists of retirees, and the remaining 1 − ζ consists of workers. In addition,
we categorize workers as Ricardian or and Non-Ricardian. Non-Ricardians cannot accessthe bond market, but Ricardians can. In other words, Ricardians can smooth theirconsumption by bond holdings, while Non-Ricardians face liquidity constraints and thus
consume all their disposable income each period. Among the workers, we assume that thefraction ξ consists of non-Ricardians and the remaining population consists of Ricardians.
We also assume that retirees are unable to access the bond market and thus behave as
Non-Ricardians.

3.1.1

Ricardian workers

Ricardian workers derive utility from consumption cR
w,t and government spending gt , while
receiving disutility from labor supply hw,t . The lifetime utility function is

E0

∞ (
∑
t=0

1
1+ρ


)t 

1
1 − σ

[{

η−1
η

η−1

η
ωcR
+ (1 − ω)gt
w,t

η ]1−σ
} η−1



−


h1+µ
w,t
1 + µ

,(15)

where the parameters ρ ≥ 0, σ ≥ 0, and µ ≥ 0 represent the rate of time preference,risk aversion, and the inverse of the Frisch labor elasticity, respectively. In addition,
the parameter η is the elasticity of substitution between consumption and government
spending, and the weight ω ∈ (0, 1) governs the eﬀect of government spending on utility.
We follow Galı́ et al. (2007) and assume an imperfectly competitive labor market, where
the labor unions set real wages, firms determine labor demand under the given wages, andworkers supply their labor to fulfill the labor demand from firms. Moreover, we assume
that labor demand is the same across all types of workers (hence omitting the superscript15

R from the labor supply). Therefore, Ricardian workers choose only consumption to
maximize (15) subject to the budget constraint

Pt cR
w,t +

R
Bw,t
R
R
= Pt wt hw,t + Bw,t−1+ Pt d R
w,t − Pt τw,t ,
Rt

(16)

R
where Pt is the final good price, Bw,tis the nominal government bond, wt is real wages,RdR
w,t is the real dividend Ricardian workers receive, τw,t is the lump-sum tax imposed on

Ricardian workers, and Rt is the gross nominal interest rate. We can solve the first-orderconditions of the Ricardian workers’ problem as

λR
t =

(
) 1−ση
η−1
η−1
η−1
− η1
η
R η
ωcw,t
+ (1 − ω)gt
,
ωcR
w,t

λR
t =

1
Rt R
Et
λ ,
1 + ρ Πt+1 t+1

(17)

(18)

where Πt+1 ≡ Pt+1 /Pt is the gross inflation rate and λR
t is the Lagrange multiplier on the
Ricardian workers’ budget constraint, representing the marginal utility of consumption.

3.1.2

Non-Ricardian workers

The period utility function of non-Ricardian workers is also
1
1−σ

[{
ωcN
w,t

η−1
η

η−1
η

+ (1 − ω)gt

η ]1−σ
} η−1

h1+µ
w,t
−
,
1+µ

(19)

where cNw,t is the consumption of non-Ricardian workers. We assume that non-Ricardian
workers simply consume all their current disposable income every period because theylack access to the bond market. Therefore, we can represent their consumption asNcN
w,t = wt hw,t − τw,t ,

16

(20)

N
where τw,t
is the lump-sum tax paid by non-Ricardian workers, which is set to equate thesteady-state consumption between Ricardian and non-Ricardian workers.

3.1.3

Retirees

Retirees consume social security benefits in each period. Thus, we can describe their
behavior as

cr,t = sr ,(21)

where cr,t is the retiree’s consumption and sr is the retiree’s social security benefit. Here,
we assume that the social security benefit is time-invariant. Similar to the lump-sum
tax imposed on workers, we set the steady-state values of the social security benefit for
retirees to be equal to the steady-state consumption of all types of households.

3.2

Firms

The production sector consists of two types of firms: monopolistically competitive intermediate goods firms whose goods are diﬀerentiated by firm and a perfectly competitive
final good producer who transforms intermediate goods into final goods using constantelasticity of substitution technology.

3.2.1

The final good firm

A perfectly competitive final good firm bundles a continuum of diﬀerentiated intermediate
goods yj,t , j ∈ [0, 1] into final good yt using constant returns technology:
(∫ 1
yt =

εp −1
εp

yj,t djp) εpε−1

,

(22)

0

where εp is the degree of substitution between the diﬀerent intermediate goods. Given
the final good price Pt and the prices for intermediate goods Pj,t , the profit maximization17

problem for the final good firm yields the following demand function for intermediate
goods:(
yj,t =

Pj,t
Pt

)−εp
yt .

(23)

Substituting (23) into (22), we obtain the final good pricing rule:
(∫ 1
Pt =

1−ε
Pj,t p dj

1
) 1−ε
p

.

(24)

0

3.2.2

Intermediate goods firms

We assume a linear function for the production technology of the intermediate goods
firms:

yj,t = hj,t ,(25)

where hj,t is the labor hired by firm j. Then, the marginal costs mct of the firm are equal
to the real wage:

mct = wt .

(26)

We assume that intermediate goods firms set the price for their goods to maximize their∗be the optimal price that the
profit under Calvo (1983)-type price stickiness. Let Pj,tfirm can choose in period t; then, we can write the profit maximization problem as
)i
∞ (
∑
[ ∗
]
θ
max
Et
Pj,t − Pt+i mct+i yj,t+i ,
∗
Pj,t
1+ρ
i=0

(27)

subject to the demand function (23). Here, the parameter θ denotes the probability thatthe price cannot be reset at period t. This maximization problem yields the following

18

optimal price:

∗
Pj,t
=

εp
εp − 1Et∑∞ ( θ )i

Pt+i yj,t+i mct+i.
∑∞ ( θ )i
Et i=0 1+ρ yj,t+i
i=0

1+ρ

(28)

Because all intermediate goods firms that can reoptimize their price in period t set thesame markup εp /(εp − 1), the optimal price is the same for all 1 − θ firms that can adjust∗
their prices (i.e., Pj,t
= Pt∗ ). On the other hand, we assume that the remaining θ fractionof the firms keep their prices set at period t − 1. Therefore, under the final good pricing
rule (24), the evolution of the aggregate price is
1−εp

Pt

3.3

= θPt−1 p + (1 − θ)(Pt∗ )1−εp .
1−ε

(29)

Wage union

As mentioned above, we follow Galı́ et al. (2007) and introduce an imperfectly competitive labor structure, where labor unions exist for each type of diﬀerentiated labor input
indexed by l ∈ [0, 1] and they determine real wages collectively. In addition, we assume
that the Ricardian and Non-Ricardian workers are uniformly distributed across unions.
A perfectly competitive labor bundler provides eﬀective labor hj,t to intermediate good
firm j by bundling the diﬀerentiated labor input hj,t (l) according to(∫ 1
hj,t (l)hj,t =

εw −1εw) εwεw−1
dl

,

(30)

0

where εw is the degree of substitution between diﬀerent labor inputs. Parallel to the
problem for the final good producer, the labor bundler’s optimal problem yields the labordemand function:(ht (l) =wt (l)
wt

19

)−εw
ht .

(31)

The labor union l sets the real wage to maximize the following objective function:N(1 − ξ)λRt [wt (l)ht (l)] + ξλt [wt (l)ht (l)] −ht (l)1+µ
1+µ

(32)

subject to (31). Note that the labor income for each type of worker is weighted byN
4
their marginal utility of consumption, denoted by λRt and λt , respectively. Solving thisproblem, we obtain the optimal wage schedule:(N(1 − ξ)λRt + ξλt
hµt

)
wt =

εw
,
εw − 1(33)

where we omit the l index in symmetric equilibrium. This equation states that the real
wage is set by adding the markup εw /(εw − 1) to the (weighted sum of) the marginal rateof substitution.

3.4

Fiscal and monetary authorities

The government budget constraint in nominal terms is
Bt= Pt gt + Bt−1 + Pt s,
Rt

Pt τ t +(34)

where τt is the aggregate lump-sum tax. By defining τ̂t ≡ (τt − τ )/y, ĝt ≡ (gt − g)/y, andb̂t ≡ (bt − b)/y, we adopt the following tax rule:τ̂t = ϕb b̂t−1 + ϕg ĝt ,(35)

where any variables without a time subscript indicate the steady-state values of thecorresponding variables. This form of tax rule allows for debt financing. We assume that
4

From the utility function (19), the marginal utility of consumption for Non-Ricardians is
(
λN
t =

ωcN
w,t

η−1
η

η−1
η

+ (1 − ω)gt

taking the same form as for Ricardians.

20) 1−ση
η−1

ωcN
w,t

− η1

,

the dynamics of government spending follow the AR(1) process:ĝt = ρg ĝt + ugt ,(36)

where 0 < ρg < 1 and ugt is an exogenous innovation on government spending withugt ∼ N (0, σg2 ).
Finally, we assume that the monetary policy follows a simple Taylor rule:
Rt
=
R

3.5

(

Πt
Π

) ϕπ
.

(37)

Aggregation

Three types of households exist in the model economy: Ricardian workers, Non-Ricardian
workers, and retirees. The shares of these households are (1 − ξ)(1 − ζ), ξ(1 − ζ), and ζ,
respectively. Therefore, the aggregate consumption ct isN
ct = (1 − ξ)(1 − ζ)cR
w,t + ξ(1 − ζ)cw,t + ζcr,t .

(38)

Only workers provide the labor input, and it is the same across the types of workers.
Hence, the aggregate labor input ht is
ht = (1 − ζ)hw,t .

(39)

Similarly, workers pay a lump-sum tax. Thus, we have{}RNτt = (1 − ζ) (1 − λ)τw,t+ λτw,t
,

21

(40)

where the lump-sum tax diﬀers depending on the type of worker. In contrast, social
security benefits are distributed to the retiree:s = ζsr .

(41)

Finally, only Ricardians receive dividends and hold government bonds, and thus we represent the aggregate dividends and bonds as

3.6

dt = (1 − λ)(1 − ζ)dR
w,t ,

(42)

bt = (1 − λ)(1 − ζ)bR
w,t .

(43)

Market clearing

Equilibrium in the labor market requires
∫ 1
ht =

hj,t dj,(44)

0

implying that the supply of labor input equals its demand. The final goods market
clearing condition is

y t = ct + g t(45)

because we assume a no capital closed economy in this study.

3.7

Calibration

To quantify the dynamic responses to a government spending shock, we calibrate themodel to the Japanese economy for the sample period from 1990 to 2014. Table 2 summarizes the calibrated values for the parameters. The purpose of this exercise is to exam22

ine theoretically whether the eﬀect of a government spending shock changes dependingon the fraction of retirees, as it does in the empirical analysis. We derive the rational
expectations equilibrium from a system of stochastic diﬀerence equations composed ofthe log-linearized equilibrium conditions, which we provide in Appendix A.1.
We set the model period to one year to match the frequency in the empirical analysis.
First, we set the rate of time preference to 0.01 such that the annual real interest rateequals 1%, which approximately corresponds to the time average of the ex-post realinterest rate for the sample period.5 Based on the sample average in the data, we set thesteady-state values for the debt-to-output ratio at γb = 1.5. Similarly, the steady-stateshare of government spending in output is γg = 0.3 from the data, implying that theshare of consumption in the output of γc = 0.7.
Regarding the parameters in the utility function, we set the risk aversion to σ = 1.5
and the inverse of labor supply elasticity to µ = 1. We choose these parameters as they
are consistent with the micro evidence documented by Abe et al. (2007) and Kuroda
and Yamamoto (2008), respectively. Moreover, we follow Bouakez and Rebei (2007)
and set ω = 0.8 and η = 0.3 such that private consumption and government spending
are complements. Okubo (2003) also provides evidence supporting the complementarity
between them in Japan. As for price stickiness, we set the Calvo parameter to θ = 0.55
to accord with Watanabe and Watanabe’s (2018) finding that the annual rate of pricechanges for 55% of items that constitute the Japanese CPI is almost zero. The fractionof Non-Ricardians is ξ = 0.13, following Hara et al. (2016), who calculates the shareof hand-to-mouth households, including wealthy hand-to-mouth, in Japan using national
household survey data.
Following Galı́ et al. (2007), we calibrate the parameters in the tax rule and dynamics
of a government spending shock to be consistent with our VAR evidence in Figure 3.
Since the dynamics of the fiscal variables seem to diﬀer depending on the demographic
structure, we set some parameter values for each case. Specifically, we set the elasticity of5

We calculate the ex-post real interest rate by subtracting the inflation rate (CPI; all items) from theuncollateralized overnight call rate.

23

tax revenue to government spending ϕg to 0.08 in the low-aging cases and 0.04 in the highaging cases to match the respective tax responses during the impact period. On the other
hand, our empirical result demonstrates that tax revenues do not increase suﬃciently inresponse to an increase in government spending in both cases, implying that fiscal policy
in Japan might be unsustainable. Doi et al. (2011) also point out this implication in
their estimate of the Japanese tax rule using a two-state Markov switching model. Doi etal. (2011) find an insignificant coeﬃcient estimate on the debt-to-GDP ratio regardlessthe state, suggesting that the fiscal stance in Japan is “active” according to Leeper’s(1991) terminology. However, we must ensure fiscal sustainability in the model economy
to determine a unique equilibrium. Thus, we unify ϕb = 0.01 in both aging cases, which isthe smallest value that guarantees a unique equilibrium. Similarly, we set the persistenceof government spending to ρg = 0.6 in the low-aging state and ρg = 0.75 in the high-aging
state such that the theoretical responses of government spending at five periods after theshock correspond to the empirical responses. Finally, we follow Miyazawa (2011) and set
ϕπ = 1.2, which also satisfies the Taylor principle.
[Table 2 about here.]

3.8

Theoretical responses

We calculate the impulse responses in two aging cases by setting ζ = 0.2 in a low-aging
economy and ζ = 0.35 in a high-aging economy, which are approximately equivalent
to the aging rate in the lowest and highest prefectures in 2014. Figure 5 displays the
dynamic responses of (a) government spending, (b) tax revenues, (c) output, and (d)the calculated cumulative fiscal multiplier to a positive government spending shock. In
line with the empirically determined responses shown in Figure 3, we plot the responses
in low- and high-aging economies using solid lines and dashed-dotted lines with circles,
respectively. Note that we normalize the scale of all responses in Figure 5 by steadystate output because we define the theoretical responses of government spending and24

tax revenue as a deviation from the steady state and normalized by steady-state output
as in Galı́ et al. (2007), allowing us to directly compare the theoretical and empiricalresponses.
[Figure 5 about here.]
Overall, the theoretical responses closely follow the empirical responses in quantitativeand qualitative terms, although we cannot reproduce some hump-shaped responses dueto the simplicity of the model. Particularly, it is worth noting that our model perfectly
replicates the positional relationships between the responses such that the fiscal multiplierin a low-aging economy exceeds that in a high aging economy. We focus on the output
responses to elucidate the mechanism by which our model generates the state dependencyof the fiscal multiplier on the aging ratio. A glance at Figure 5 (c) reveals that in both
aging cases, a one-unit increase in government spending leads to a more than one-unit
increase in output at the impact period. From the market-clearing condition in (45), we
can attribute the gap in the responses between output and government spending to anincrease in consumption. The presence of non-Ricardian workers and wage unions causes
an increase in consumption, in contrast to a standard DSGE model, in which private
consumption responds negatively to a positive government spending shock due to a negative wealth eﬀect. As with Galı́ et al. (2007), consumption by Non-Ricardians greatly
increases owing to the postponement of a tax increase via debt financing and an increase
in real wages set by wage unions. A further important point is that, unlike Galı́ et al.
(2007), consumption by Ricardians also responds positively to an increase in government
spending thanks to a complementarity between them. Accordingly, while retirees cannot change their consumption at all, both types of workers increase consumption after apositive government spending shock, resulting in an increase in output. As the share of
retirees declines, the positive movement of workers’ consumption is increasingly reflectedin the aggregate consumption. Hence, it follows that the stimulus eﬀect of a government
spending shock on output becomes higher in a low-aging-rate economy. This is a key
25mechanism for generating the state dependency observed in our theoretical model. Additionally, the reversal of output responses after period 2 is due to the diﬀerence in thepersistence of government spending. As in the empirical findings, the fiscal multipliers
in low-aging economies continue to lie above those in high-aging economies, even though
the position of the output responses is reversed.
Moreover, the degree of population aging influences the size of the fiscal multiplierthrough our specific labor market structure. Note that output is equivalent to labor
supply because we adopt no capital and linear production function assumption in ourmodel, and real wages move in the same direction as labor supply by (32). Then, an
increase in output causes a rise in both labor supply and real wages, leading to a risein current disposable income and consumption by Non-Ricardians. Therefore, the labormarket channel, through which the consumption by Non-Ricardians expands cyclically,
plays an additional role in enhancing the eﬀect of a government spending shock on output
when the share of retirees is low.
The channels that generate the state dependency are similar to those in Basso and
Rachedi (forthcoming) in that the results emphasize the positive responses of consumption
and labor supply by a specific type of household to a government spending shock. In
Basso and Rachedi’s (forthcoming) model, young workers who have a high labor supply
elasticity and high marginal propensity to consume (as they face liquidity constraints)
are a major factor that alters the fiscal multipliers depending on demographics. In termsof liquidity constraints and a high marginal propensity to consume, young workers could
be regarded as a counterpart to the non-Ricardian workers who greatly increase theirconsumption and labor supply in response to a government spending shock. Similar tothe behavior of non-Ricardian workers in this study, young workers boost their labor
supply and consumption in response to an increase in government spending, leading toa significant increase in aggregate output when the share of young workers is high. The
negative wealth eﬀect in our model does not increase the labor supply due to the existenceof labor unions. Except for the origin of the increased labor supply, our theoretical model
26

shares almost the same mechanism as the one in Basso and Rachedi (forthcoming).
Consequently, it follows from our theoretical exercise that population aging reduces
fiscal policy eﬀectiveness in terms of both the demand and supply sides. From the demandside, a rise in the share of retirees, which is equivalent to decreasing the share of workers,
decreases the positive response of aggregate consumption because retirees’ consumption
is always unchanged, resulting in a lower output response. From the supply side, retirees
have no labor supply by definition, so they cannot benefit from an increase in disposable
income when wage unions set a higher wage in line with the rise in labor demand involvedin the surges of government demand. Consequently, the stimulus eﬀect on consumption
becomes more modest as population aging progresses. To summarize, we can theoreticallyinterpret the state-dependent eﬀect of a government spending shock on the aging rate
observed in the empirical analysis as stemming from the behavior of retirees, whose
consumption and labor supply do not react to government spending shocks at all.

4

Discussion

As retirees have no access to the labor and financial markets, a government spending
shock has no eﬀect on their behavior in the benchmark theoretical model. However,these two non-access assumptions are essential factors to characterize retirees in the model
economy, though it is not necessarily reasonable in practice. In particular, the Japaneseoﬃcial household surveys, such as the Family Income and Expenditure Survey reportthat elderly people in Japan have abundant financial assets, indicating the possibility
that retirees can access the financial market and thus smooth their consumption. In
this section, we examine how our theoretical outcome changes when retirees can accessthe financial market. We should stress that this analysis aims not only to bring themodel economy closer to the real economy, but also to clarify whether the aforementioned
demand- or supply-side factor contributes more to generating the state dependency of agovernment spending shock. Since retirees still have no access to the labor market and27

thus cannot benefit from an increase in disposable income in the model we develop in thissection, comparing the results from this model and the benchmark will highlight the roleof demand-side factors; that is, consumption fluctuations. Hence, we can conclude thatdemand-side factors (i.e., no consumption smoothing) are more relevant than supply-side
factors (i.e., no labor supply) are in generating the state-dependent eﬀect of a government
spending shock if it disappears in the current model. Conversely, if it does not disappear,then it follows that the supply-side channel, through which workers’ consumption expands
cyclically, is the essential factor generating the state dependency.
We refer to retirees who can access the bond market and smooth their consumption
by bond holdings as Ricardian retirees. Since retirees do not work, though they are
Ricardians, their lifetime utility function is[{η ]1−σ
} η−1
)t 
∞ (

η−1
∑
η−1
1
1
η
η
E0
+
(1
−
ω)g
,
ωcR
t
r,t

1 − σ
1
+
ρ
t=0

(46)

where cRr,t denotes the consumption of Ricardian retirees. This lifetime utility function is
maximized subject to the budget constraint

Pt cR
r,t +

R
Br,t
R
R
= Br,t−1+ Pt d R
r,t + Pt sr .
Rt

(47)

Ricardian retirees receive social security benefits, dividends, and repayments from bondholdings carried over from an earlier period, which they allocate to consumption andpurchase of new bonds. Although we assume that the social security benefit is timeinvariant, unlike in the benchmark model, Ricardian retirees can now change their consumption intertemporally through bond holdings. The optimization problem yields themarginal utility of consumption and the Euler equation for Ricardian retirees with thesame form as the Ricardian workers’ equations in (17) and (18). Namely, it follows that
the Ricardians’ optimal intertemporal consumption condition is the same regardless ofwhether they are workers or retirees, leading to the same responses of consumption to a28

government spending shock in both types of Ricardians.6
Figure 6 shows the impulse responses derived from the model in which retirees behave
as Ricardians. The introduction of Ricardian retirees slightly magnifies the responses of
output and the fiscal multiplier compared to Figure 5 because of the complementarity
between consumption and government spending by which Ricardian retirees increase theirconsumption in response to a government spending shock. However, the gap remains
in the responses across the diﬀerent aging ratios, even if we allow retirees to accessthe financial market, suggesting that the demand-side factor (retirees’ lack of access tothe financial market) alone does not explain the state-dependent eﬀect of a government
spending shock on the share of retirees. Therefore, we deduce from our theoretical exercisethat the supply-side factor; that is, retirees’ lack of access to the labor market, plays
an important role in generating the situation in which population aging diminishes theeﬀectiveness of a government spending shock. In other words, we can attribute thestate-dependent eﬀect of a government spending shock mainly to retirees’ receiving nobenefit from higher wages triggered by an increase in government spending as they supply
no labor. Importantly, this theoretical finding have the practical implication that thestimulus eﬀect of a government spending shock is likely to increase if elderly people areencouraged to participate in the labor market.
[Figure 6 about here.]6

In a practical manner, the introduction of Ricardian retirees modifies the system of the log-linearized
rational expectation equilibrium described in Appendix A.1 as follows. First, we add an equation toindicate that the deviation of Ricardian retirees’ consumption is equal to that of Ricardian workers:RĉRr,t = ĉw,t .

Second, the aggregate consumption equation becomesNRĉt = (1 − ξ)(1 − ζ)ĉR
w,t + ξ(1 − ζ)ĉw,t + ζĉr,t ,such that Ricardian workers’ consumption now fluctuates in response to the shock.

29

5Conclusion

In this study, we examined how population aging aﬀects the economic stimulus eﬀect ofa government spending shock. Specifically, we explored whether the eﬀectiveness of thegovernment spending shock on output changes depending on the demographic structure.
Given the relatively small amount of research investigating the relationship between population aging and fiscal policy eﬀectiveness in terms of short-run business fluctuations, thepresent study addresses this issue from both the theoretical and empirical perspectives.
We first reveal empirically that the government spending multiplier would diminish as
population aging progresses. Then, we develop an NK-DSGE model with heterogeneous
agents to explain the empirical evidence. The theoretical analysis suggests that a sourceof the state dependency is retirees’ lack of access to the labor market, due to which retirees
cannot benefit from the increase in disposable income caused by a positive government
spending shock. Therefore, our findings oﬀer a possible implication that the eﬃcacy ofa government spending shock on economic activity could be improved by creating anenvironment in which elderly people could join the workforce easily. Given the limited
scale of the fiscal stimulus package caused by an excess accumulation of sovereign debt
due to the ongoing corona pandemic, coupled with the continually aging the populationin many developed countries, the findings of this study provide implications for the labormarket structure to enhance the eﬀectiveness of a government spending shock. However,
future research could aim to determine the type of labor market system that would makeelderly people’s labor participation more flexible. This task is left to future work.

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34

A

AppendixA.1

Log-linearized model

Except for the fiscal variables in the text (i.e., τ̂t , ĝt , and b̂t ), we denote the log-deviation ofa variable from its steady-state as x̂t = ln xt −ln x. Then, we summarize the log-linearized
equilibrium conditions as follows:Rλ̂Rt = Θc ĉw,t + Θg ĝt

Marginal utility of consumptionRλ̂R
t = Et λ̂t+1 + r̂t − Et π̂t+1

Euler equation

γc ĉNw,t = ŵt + ĥt − τ̂t

Non-Ricardians’ consumption
Aggregate consumption
Wage schedule
New Keynesian Phillips Curve
Government budget constraintNĉt = (1 − ξ)(1 − ζ)ĉR
w,t + ξ(1 − ζ)ĉw,tΘcĉt − Θg ĝt + µĥtŵt = −1−ζ1
(1 − θ)(1 + ρ − θ)π̂t =Et π̂t+1 +ŵt1+ρθ(1 + ρ)b̂t = R(ĝt − τ̂t + b̂t−1 ) + γb (r̂t − Rπ̂t )Tax ruleτ̂t = ϕb b̂t−1 + ϕg ĝtTaylor ruler̂t = ϕπ π̂tMarket clearingŷt = γc ĉt + ĝt

Production functionŷt = ĥt

Government spendingĝt = ρg ĝt + ugt

where γc = c/y, γg = g/y, γb = b/y, and the associated coeﬃcients are given by[]1
1 − ση η−1ωc η −
,
Θc ≡
Γη
η

Θg ≡

1
1 − ση
(1 − ω)g − η y,
Γη

35

(48)

(49)

(
Γ = ωcA.2

η−1η

+ (1 − ω)g

η
η−1

)
.

(50)

Steady-state values

Given the steady-state value of labor supply h = h̄, we immediately obtainy = h̄,(51)

from the production function. Then, the steady-state values of consumption and government spending are

c = γc h̄,(52)

g = γg h̄,(53)from the market-clearing condition.
On the other hand, the first-order condition of the Ricardians’ problem yields thegross nominal interest rate:

R=1+ρ

(54)

in the zero-inflation steady state. Given the rate of time preference exogenously, we candetermine the gross nominal interest rate.

36

30

percent

25

20

15

10
1990

1995

2000

2005

2010

Figure 1: Time profile of the distribution of the aging rate among Japan’s prefectures
Notes: This figure shows the distribution of the aging rate among Japan’s prefectures for theperiod from 1990 to 2014 using a box plot. The tops and bottoms of each box are the 25thand 75th percentiles; the line in the box is the sample median; the lines extending from aboveand below each box are the ends of the interquartile ranges to the furthest observations thatare less than 1.5 times the interquartile range from the top or bottom of the box; and outliersof more than 1.5 times the interquartile range away from the top or bottom of the box, aredisplayed with +.

37

Figure 2:
Notes: This figures show the selected low and high aging prefectures, respectively. The low(high) aging prefectures are painted in the left-hand (right-hand) side of the map.

38

Figure 3: Empirical impulse responses to a government spending shock
Notes: This figure shows the empirical impulse responses of (a) government spending, (b) tax
revenue, (c) output, and (d) the fiscal multipliers to a positive government spending shock forthe two groups. The median responses obtained from the low- and high-aging prefectures aredenoted by solid lines and dash-dotted lines with circles, respectively. In addition, the 68%
credible intervals are depicted by the shaded areas and dash-dotted lines.

39

Figure 4: Empirical responses to a government spending shock (excluding the outlier,
Okinawa)
Notes: This figure shows the empirical responses to a positive government spending shock in
low-aging prefectures except for Okinawa prefecture.

40

(a) Government spending

1
0.8

0.15

0.6

0.1

0.4

0.05

0.2

0

0

0

1

2

3

4

5

-0.05

(c) Output

2.5(b) Tax revenue

0.2

0

1

2

3

4

5

(d) Accum. fiscal multiplier

3

2
2

1.5
1

1

0.5
0
0

1

2

3

4

0

5

0

1

2

3

4

5

Figure 5: Theoretical responses to a government spending shock
Notes: This figure shows the theoretical responses of (a) government spending, (b) tax
revenue, (c) output, and (d) the fiscal multiplier to a positive government spending shock.
The responses in low- and high-aging economies are plotted by solid lines and dashed-dottedlines with circles, respectively, as with the empirical responses.

41

(a) Government spending

1
0.8

0.15

0.6

0.1

0.4

0.05

0.2

0

0

0

1

2

3

4

5

-0.05

(c) Output

2.5(b) Tax revenue

0.2

0

1

2

3

4

5

(d) Accum. fiscal multiplier

3

2
2

1.5
1

1

0.5
0
0

1

2

3

4

0

5

0

1

2

3

4

5

Figure 6: Theoretical responses to a government spending shock (including Ricardian
retirees)
Notes: These figures show the theoretical responses derived from a model in which retirees canbehave as Ricardians.

42

Table 1: Sign restrictions
Gov. spending
Tax revenue
Business cycle

g
>0
<0

τ

x

>0
>0

<0
>0

Notes: This table summarizes the sign restrictions imposed on the impulse response functions.
The sign restrictions, if any, are imposed on each variable for two periods. A blank spaceindicates no restrictions.

43

Table 2: Calibration parameters
Parameters

Description

ρ
σ
µ
ω
η
θ
λ
ϕb
ϕg
ρg
ϕπ
γc
γg
γb

Time preference rate
Relative risk aversion
Inverse of labor supply elasticity
Weight attached to private consumption
Elasticity of substitution
Calvo parameter
Fraction of Non-Ricardians
Tax elasticity of bonds
Tax elasticity of government spending
Persistence of government spending shock
Taylor rule coeﬃcients for inflation
Share of consumption in output
Share of government spending in output
Debt-to-output ratio

Value
Low High
0.01
1.5
1.0
0.8
0.3
0.55
0.13
0.01
0.08 0.04
0.60 0.75
1.2
0.7
0.3
1.5

Notes: This table summarizes the values of the calibrated parameters in our theoretical model
based on the existing literature and our empirical results from the prior section. Note thatseveral fiscal policy parameters change depending on the demographic stricture.

44