Forthcoming in the Japanese Economic Review

Fiscal Sustainability of Japan: A Dynamic Stochastic General Equilibrium Approach

Masaya Sakuragawa and Kaoru Hosono

Abstract
The purpose of this paper is to investigate the fiscal sustainability of Japan by applying a dynamic
stochastic general equilibrium (DSGE) model to the Japanese economy. By introducing
intermediation costs into the model, we succeed in explaining the observed relationship between
the interest and GDP growth rates, which is crucial in testing for sustainability. When the
projected real growth rate is 2.5 percent, the average real interest rate becomes 2.57 percent, andthe debt-to-GDP ratio gradually increases stochastically so that government debt is notsustainable. To recover sustainability, the primary surplus must be 0.2 percent of GDP.

JEL Classification Numbers: H68, G12 and E62.

Key Words: Fiscal Sustainability, Intermediation Cost, DSGE Model1. Introduction
The huge budget deficit in recent years has raised great concerns about the long-run
sustainability of Japanese fiscal stance. Since the beginning of the 1990s, government debt has
accumulated so much faster than GDP growth that the outstanding stock of debt as a proportion of
GDP amounts to about 1.7 as of 2006. This figure is by far the highest among OECD countries(other countries whose debt-to-GDP ratio exceeds unity are only Belgium and Greece). It is
obvious that the Japanese budget is in a deficit crisis. The government, responding to a possible
fiscal crisis, proposed a fiscal reform plan entitled “Basic Policies for Economic and Fiscal Policy
Management and Structural Reform” (Honebuto no Houshinn in Japanese) in 2006 and released aplan under which it will attain a primary surplus in fiscal year 2011.
At the same time, there is the puzzling fact that the real interest rates on government bonds
have remained relatively low and have not reacted sensitively to the recent increase in thedebt-to-GDP ratio. Figure 1 depicts the nominal interest rate of government bond and the nominal
GDP growth rate for the period 1981–2005. The interest rate was 5.0 percent on average over theperiod prior to the zero-interest-rate policy (i.e., 1981–1998), while the growth rate was 4.2
percent on average over the same period, with a gap of 0.8 percent1, which is far smaller than
predicted by the neoclassical growth theory. Small gaps between interest rates on government
bonds and the GDP growth rate are not specific to Japan. In fact, the average realized real rates ofreturn on government bonds in major OECD countries over the past 30 years have been smaller
than the real growth rate (e.g., Blanchard and Weil, 2001). In investigating the test for fiscal
sustainability, it is of extreme importance to understand the relationship between the two rates2.
There are two strands of approaches to explaining low interest rates for government bonds
from the viewpoint of the neoclassical growth theory. One approach is to introduce uncertainty
into the framework of a complete, frictionless economy. Abel et al. (1989) and Bohn (1995)

1

provide examples of stochastic growth models in which risk premium drives down the safeinterest rate, often below the economic growth rate. Their argument, along with that of Zilcha(1992), demonstrates that dynamic efficiency is preserved and that the no-Ponzi condition for theintertemporal government budget constraint holds even if the safe interest rate is below thegrowth rate. Another approach deviates from complete markets and assumes the presence offinancial intermediation costs (e.g., Bohn, 1999). Aiyagari and Gertler (1991) and Heaton andLucas (1996) introduce intermediation costs to resolve the Mehra–Prescott puzzle3.
In this paper, we attempt to resolve the issue of low interest rates by extending Bohn’s (1999)
model with financial intermediation costs to a stochastic environment. Specifically, we provide adynamic stochastic general equilibrium (DSGE) model of an exchange economy featured byintermediation costs and heterogeneous agents. Intermediation costs reduce the deposit interest
rate and hence the rate of return of the government bond, a perfect substitute for deposits, andcontribute to explaining the relatively low interest rate of government bond. We use this model toinvestigate whether the Japanese economy can avoid a possible fiscal crisis if the released reform
plan to attain a primary surplus in fiscal year 2011 is realized.
We did not take the approach on risk aversion. As Mehra and Prescott (1985) and Weil (1989)
argue, when the degree of relative risk aversion takes admissible values of greater than 0.5 but
less than 2, the magnitude of risk premium is at most 0.06–0.4 percent, which cannot account for
the relatively low interest rate of government bond. Given that the degree of relative risk aversion
reported by existing studies on the Japanese economy lies between 0.5 and 1, however, thisapproach does not appear to improve our results greatly. We think it essential to introduce
financial intermediation costs to match the data with the model.
Our results are summarized as follows. When the targeted growth rate is 2.5 percent, theaverage interest rate becomes 2.57 percent in the presence of significant financial intermediation2

costs. We report a small gap between the interest and growth rates, which is consistent with theactual data. If the targeted primary surplus is zero on average, the debt-to-GDP ratio gradually
increases from the initial level of 1.72 to 1.896 in 100 years, suggesting that the government debt
is not sustainable. Sustainability requires the primary surplus to be no less than 0.2 percent of
GDP. Lowering the GDP growth rate to 2 percent, 1 percent and 0 percent and assuming that the
government runs a zero primary surplus for each growth rate, we find that the average interest rateof government bonds decreases but the gap between the interest rate and growth rates widens.
Therefore, the government faces more difficulty in meeting the sustainability condition.
This paper belongs to the literature on methodology to test sustainability of fiscal deficits.
Hamilton and Flavin (1986) tested sustainability by examining the transversality condition in theintertemporal budget constraint of the government. Bohn (1998) proposed a simple and pragmatic
test for sustainability under which it is sufficient to ascertain whether the debt-to-GDP ratio
displays a mean-reversion property. In their famous paper entitled “Deficit Gamble,” Ball et al.
(1998) projected future growth rates and interest rates using past data and calculated theprobability under which the debt-to-GDP ratio enters a dangerous zone. What is common in theexisting literature is that all studies rely heavily on past data, and so their approaches are oflimited use for estimating the effect of possible fiscal reforms in periods of fiscal crisis.
This paper is also related to the empirical literature investigating fiscal sustainability in Japan.
Doi and Ihori (2003) take the intertemporal approach following Hamilton and Flavin (1986) andreach a pessimistic conclusion regarding the period of 1965–20004. A substantial body ofliterature examines the Bohn’s condition (for example, Ihori et al., 2007) and reports findings
against that condition when the sample includes the period since 1990.
Ihori et al. (2007), the study most similar to ours, simulates fiscal sustainability using africtionless growing economy of overlapping generations, and makes predictions more

3

pessimistic than ours. The primary reason behind their pessimistic results lies in the gap between
the interest and GDP growth rates that amounts to about 3 percentage points that are greater thanactual data and our study. Dekle (2005) simulates fiscal sustainability using a small-open
economy version of the growth model and makes a pessimistic projection based on his assumed
wide gap between the interest and TFP growth rates. Broda and Weinstein (2005) and Doi (2006)
provide some fiscal implications by applying several exogenous pairs of economic growth rates
and interest rates5.
This paper is organized as follows. Section 2 introduces the model used in calibration. Section
3 reports the results of this fiscal calibration. Section 4 concludes.

2. ModelWe construct a dynamic stochastic general equilibrium (DSGE) model of an exchange
economy with infinitely lived agents to investigate the sustainability of government debt. We
assume intermediation costs and heterogeneous agents to provide fiscal implications of low safeinterest rates6. The developed model is a stochastic version of Bohn (1999).
The model has an equal number of two types of infinitely lived agents, A and B. Type A has
income Yt A = Yt in even periods (zero otherwise) and type B has income Yt B = Yt in odd
periods (zero otherwise). The growth rate of income, xt +1 ≡ (Yt +1 Yt ) − 1 , is a random variable
that follows a Markov process according to a continuous and differentiable probability
distribution function G ( xt +1 xt ) with a positive density g (.) over the support [ x, x] , with amean µ and x > −1 .
To simplify the notation, let there be one representative agent of each type so that individual
income Yt denotes the aggregate income. There is no population growth. The model developed4here is an exchange economy without production so that xt +1 may be interpreted as the sum of
the growth rate of per capita income and the population growth rate. Both types have preferences
over consumption

∑

∞
t =0

β t u (Ct ) , where u (.) is strictly increasing and concave and 0< β <1.

Government spending is Gt = gYt , with 0 < g < 1 . The government finances its spending by
imposing lump-sum taxes Tt on the type that receives income, and by issuing debt.
There is a room for private lending and borrowing among agents. We introduce market
imperfections by supposing that private lenders must bear a proportional intermediation costκ > 0 per unit of deposit. One may interpret κ as the cost of monitoring or identifying aborrower or verifying credit. The cost is measured in terms of loss of effort so that the goodmarket is not influenced by the intermediation cost. The bank is compensated for the loss of effort
by income, but the income accruing to the bank is transferred to households in a lump sum fashion.
This assumption is an artifact to avoid complicating the model despite the presence of theintermediation cost. In a world of competitive intermediation, if a loan is made at the borrowing
rate Rt* , the after-cost return to the lender is 1 + Rt = (1 + Rt* ) (1 + κ ) . The key assumption of
the model is that loans to the government can be monitored without cost, so that the government
can borrow at the lending rate Rt .
It is useful to examine first an economy with no government. Let V ( Bt −1 , Yt , xt ) denote thevalue function of the agent who inherits the debt Bt −1 from period t − 1 and receives income~
Yt in period t , and let V (Wt −1 , Yt , xt ) denote the value function of the agent who held thedeposit Wt −1 in period t − 1 and does not receive income in period t 7. The agent who receives~

income Yt in period t maximizes V ( Bt −1 , Yt , xt ) = max u (Ct ) + β Et V (Wt , Yt +1 , xt +1 ) 8 subject
C t ,Wt

to the budget constraint Yt − Bt −1 (1 + Rt* ) = Ct + Wt , where Ct is consumption when agents5~receive income. For later reference, let Ct denote consumption when agents do not receive
income. The F.O.C. for consumption yields~u ' (Ct ) = βEt V1 (Wt , Yt +1 , xt +1 ) .

(1)

Using the envelope theorem, we obtain

V1 ( Bt −1 , Yt , xt ) = − u ' (Ct ) (1 + Rt* ) .

(2)

On the other hand, the agent who receives income in period t − 1 maximizes~~
V (Wt −1 , Yt , xt ) = maxu (Ct ) + βEt V ( Bt , Yt +1 , xt +1 ) subject to the budget constraint~
C t , Bt~
Wt −1 (1 + Rt ) + Bt = Ct . The F.O.C. for consumption yields~u ' (Ct ) + βEt V1 ( Bt , Yt +1 , xt +1 ) = 0 .

(3)Using the envelope theorem, we obtain~~
V1 (Wt −1 , Yt , xt ) = u ' (Ct ) (1 + Rt ) .

(4)

It follows from (1) and (4) that we derive the Euler equation for the agent who receives income inperiod t~u ' (Ct ) = β Et {u ' (Ct +1 ) (1 + Rt +1 )} .

(5)

It follows from (2) and (3) that we derive the Euler equation for the agent who receives income inperiod t –1

6

~u ' (Ct ) = β Et {u ' (Ct +1 ) (1 + Rt*+1 )} .

(6)

Hereafter we proceed with the analysis by assuming that agents have identical CRRA utility
functions, u (Ct ) = Ct1−α (1 − α ) , where α is the degree of the relative risk aversion and hence
the inverse of the elasticity of substitution of consumption between periods. Accordingly, (5) and(6) are rewritten, respectively, as~
Ct−+α1
1 = Et {β −α (1 + Rt +1 )} ,
Ct

(7)

C −α
1 = Et {β ~t−+α1 (1 + Rt*+1 )} .
Ct

(8)

and

~

In addition, market clearings in the good market are Ct + Ct = Yt , and~~Ct +1 + Ct +1 = Yt +1 = (1 + xt +1 )Yt . Let Ct Ct ≡ θ t denote the consumption ratio between when
agents receive income and do not. By limiting focus on an equilibrium in which θ t is constantthrough time ( θ t = θ t +1 = θ ), we obtain~~Ct +1 Ct = Ct +1 Ct = 1 + xt +1 .

(9)Equation (9) implies that the growth rate of consumption when agents receive income is equal tothat when not, and to the growth rate of income. We use (9) to rewrite (7) as~ ~C −α C −α
1 = Et {β ~t−+α1 t−α (1 + Rt +1 )} = Et {β (1 + xt +1 ) −α θ α (1 + Rt +1 )} , (10)
Ct Ct

7

and, using (8), have~~Ct−α Et {βCt−+α1 (1 + Rt +1 )}Et {β Ct−+α1 (1 + Rt +1 )}=.
~ =Ct−α Et {βCt−+α1 (1 + Rt*+1 )} Et {β (θCt−+α1 )(1 + κ )(1 + Rt +1 )}Finally, we have

θ = (1 + κ )1 2α >1(11)

for any κ > 0 . In the presence of intermediation cost, agents fail to smooth consumption acrossperiods. They consume more when they receive income than when they do not.
We are now ready to examine the relation between the economic growth rate and the depositinterest rate. If there were no uncertainty in the model, (10) would be reduced to

1 = β (1 + xt +1 ) −α θ α (1 + Rt +1 ) . It follows from this equality and (11) that

1 + Rt +1 =(1 + xt +1 )α.
β 1+ κ

(12)

The deposit interest rate is higher than the growth rate in the neoclassical case of κ = 0 . However,if κ is positive, the deposit interest rate tends to be lower, and if κ is sufficiently high, it may
become lower than the growth rate. In any case, there exists a causal relationship between them sothat formulation of their relationship is crucial in investigating the test for fiscal sustainability.
Let us return to the analysis in a world with uncertainty. We assume Et {β (1 + xt +1 )1−α } < 1 toguarantee the finiteness of the expected utility. Consider an auxiliary variable vt +1 that is
introduced to measure the difference between Rt +1 and xt +1 , to satisfy

1 + Rt +1 = (1 + xt +1 )(1 + vt +1 ) . It follows from (10) that
1 = Et {β (1 + xt +1 )1−α θ α }(1 + vt +1 ) < θ α (1 + vt +1 ) . When there is an intermediation cost,8( θ > 1 ), vt +1 < 0 is feasible, so that the deposit interest rate may be lower than the growth rate if

κ is sufficiently high. Of course, for κ = 0 ( θ = 1 ), vt +1 is positive; hence the deposit interest
rate is always higher than the growth rate. We log-linearize (10) around the steady state andobtainRt +1 = R + α {log(1 + xt +1 ) − log(1 + µ )} ,(13)

with R = (1 + µ )α ( β 1 + κ ) − 1 . The deposit is a risky asset for which the rate of return ispositively dependent on the growth rate. The sensitivity of the deposit interest rate to the growthrate is dependent on the degree of relative risk aversion α .
Abel et al. (1989) and Bohn (1995) point out that the interest rate of a safe asset may be lower
than the economic growth rate in the presence of uncertainty. However, as Mehra and Prescott(1985) and Weil (1989) argue, when α takes admissible values of more than 0.5 but less than 2,
the magnitude of risk premium is at most 0.06–0.4 percent. Weil (1989) attempted to resolve therisk premium puzzle by proposing a preference in which the degree of relative risk aversion isunrelated to the elasticity of intertemporal substitution of consumption, but reported that the safeinterest rate exceeds the actually observed value for admissible parameter values. The findings in
the existing literature might suggest that intermediation cost may play an important role ofexplaining the lower interest rate of government bonds relative to the growth rate.
We are now ready to introduce a government bond in the model. Rigorous analysis may
require an investigation of both the case when deposits and government bond are both held andthe case when only the government bond is held. Bohn (1999) studies both cases in a
deterministic economy, and demonstrates that if the debt-to-income ratio Dt Yt is below acertain threshold both assets are held, while otherwise only the government bond is held
(Proposition 2 in Bohn 1999). Because the latter case is not practically feasible, we focus mainly

9

on the interesting and realistic case when both assets are held. Later in this section, we discuss
another case.
Let V ( Bt −1 , Dt −1 , Yt , xt ) denote the value function of the agent who inherits debt Bt −1 fromperiod t − 1 and the government bond Dt −1 , and receives income Yt in period t and let~
V (Wt −1 , Dt −1 , Yt , , xt ) denote the value function of the agent who held the deposit Wt −1 and thegovernment bond Dt −1 in period t − 1 and does not receive income in period t .
Agents who receive income Yt in period t maximize~
V ( Bt −1 , Dt −1 , Yt , xt ) = max u (Ct ) + βEt V (Wt , Dt , Yt +1, , xt +1 ) , subject to the budget constraint
Wt , Dt , C t

Yt + (1 + rt ) Dt −1 − Bt −1 (1 + Rt* ) = Ct + Wt + Dt + Tt . On the other hand, agents who do not~~receive income in period t maximize V (Wt −1 , Dt −1 , Yt , xt ) = max~ u (Ct ) ,Bt , Dt , C t+ β EtV ( Bt , Dt , Yt +1 , xt +1 ) subject to the budget constraint~(1 + rt ) Dt −1 + (1 + Rt )Wt −1 + Wt = Ct + Dt . This formulation allows for the possibility that any
agent holds the government bond at any period. Letting St ≡ Tt − Gt denote the primary surplus,
the budget constraint in period t is thenDt +1 = (1 + rt +1 )( Dt − S t ) .

(14)

~

Market clearings in the good market are rewritten as Ct + Ct = (1 − g )Yt , and~Ct +1 + Ct +1 = (1 − g )Yt +1 = (1 − g )(1 + xt +1 )Yt .
When private finance is available, agents have no incentive to purchase the government bondin the period when they do not receive income, so that agents purchase the government bond only10

in the period when they receive income. Equilibrium conditions are finally expressed as threeequations:~u ' (Ct ) = β Et u ' (Ct +1 ) (1 + Rt +1 ) ,(15)~
u ' (Ct ) = β Et u ' (Ct +1 ) (1 + rt +1 ) ,(16)~
u ' (Ct ) = β Et u ' (Ct +1 ) (1 + Rt*+1 ) .

(17)

and

Equations (15) and (16) jointly imply that the bank deposit and the government bond are perfect
substitutes for agents who are willing to hold these assets, so that without loss of generality wewill set Rt = rt . It turns out that the rate of return on the bond is determined to satisfy

1 = Et {β (1 + xt +1 ) −α 1 + κ (1 + rt +1 )} . It follows from (15) and (17) that Rt < Rt* and hencert < Rt* , which implies that the rate of return of the government bond rt +1 is below the loaninterest rate offered by the bank, Rt*+1 .
The intermediation cost leads to a reduction of not only the deposit interest rate but also therate of return on the government bond. Importantly, the lending rate to the government is lower
than that to individuals. The model’s feature in which only the government is able to lend toothers at no transaction cost allows the government to exploit “seigniorage.” When the economy
is financially “repressed”, the rate of return on the government bond tends to be low relative to thegrowth rate. If so, the government budget is likely to be sustainable even if the primary balance isnegative.

11

We briefly discuss the economy without private finance. If Dt Yt grows beyond a certain
threshold, the economy reaches a state in which agents hold only the government bond and there
is no private lending or borrowing. Agents first purchase the government bond only in the period
when they receive income and realize imperfect consumption smoothing, but as the debt-to-GDP
ratio reaches a threshold, agents begin to purchase the bond also in the period when they do notreceive income, and they realize perfect consumption smoothing. See Bohn (1999, Proposition 2)
for details. The rate of return on the bond is, as in the frictionless economy, determined to satisfy

1 = Et {β (1 + xt +1 ) −α (1 + rt +1 )} .
If the government adjusts the level of taxation to maintain the average debt-to-income ratio

Dt Yt below a constant level, its budget will be sustainable, irrespective of the relation between
the average growth rate and the average rate of return of the government bond (e.g., Bohn 1998).
On the other hand, if the debt-to-income ratio grows beyond a certain threshold, in some finite
period the economy reaches a state in which only the government bond is held and there is no
private lending and borrowing. In the exchange economy, the endowment grows irrespective ofthe level of private lending so that fiscal sustainability might coexist with an economy without
private lending. However, it is difficult to imagine that the economy without private lending
would be successful. Therefore, in the following section we take the debt-to-income ratio, Dt Yt ,being below a given constant in the long term as a criterion of fiscal sustainability.
We comment on the relation between fiscal sustainability and the no-Ponzi condition when thegovernment implements a policy to stabilize the debt-to-income ratio in every period to satisfy

Dt = dYt , with a constant d . Then we derive the discounted value of future outstanding debt as

lim EtN →∞Dt + N∏ (1 + r )
N

i =1

t +i

[

= dYt lim Et β 1 + κ (1 + xt +1 )1−α
N →∞

12

],
N

which should be zero at the limit. Unlike the standard case of κ = 0 , the condition for thebounded expected utility Et {β (1 + xt +1 )1−α } < 1 does not guarantee that the no-Ponzi condition
will hold. The government may be able to run a Ponzi scheme. If the government were to run a
Ponzi game by rolling over debt, it would have to forsake the stable debt-to-income policy,
reaching in a finite period a state in which private lending disappeared. Then the no-Ponzi
condition would be recovered if the stable debt-to-income policy were adopted.

3. Calibration

3.1 Methodology
To simulate the model, we need dynamic equations for the GDP growth rate, the interest rateon government bond and the debt-to-GDP ratio. We assume that the GDP growth rate follows an
AR(1) process:xt +1 = µ (1 − ρ ) + ρxt + ε t +1 ,(18)

where ε t +1 is a random shock following an i.i.d. normal distribution with the standard deviationof σ 9.We have the interest rate on the government bond from Equation (13) with Rt replaced
by rt :

rt +1 = r + α [ln(1 + xt +1 ) − ln(1 + µ )] ,(19)

where r = R is used. We drive the debt-to-GDP ratio from (14) as

dt +1 =

(1 + rt +1 )(dt − st ) ,(1 + xt +1 )(20)

where d t and st are debt and primary surplus divided by GDPt , respectively,

13

Under some alternative fiscal policy rules for the process of st , we simulate the modelrecursively by combining Equations (18), (19), and (20), starting with d 0 . Specifically, we takethe following steps.
Step 1. We generate a series of random shocks ε t +1 for 100 time periods.
Step 2. We construct xt +1 by substituting xt and ε t +1 into Equation (18) recursively withthe starting value of x0 = µ .
Step 3. We construct rt +1 by substituting xt +1 into Equation (19).
Step 4. We construct d t +1 by substituting rt +1 , xt +1 , and st under a particular fiscal policy
rule recursively into Equation (20) with a starting value of d 0 .
Step 5. We repeat Steps 1–4 10,000 times to obtain the distribution of d t +1 , and particularly
its mean value.

3.2 Parameters
First, we select the parameters in the utility function. We set the annual discount factor β at
1/1.02=0.9803, a figure often applied to the Japanese economy10. The inverse of the elasticity ofintertemporal substitution, α , plays a key role in the model, relating the interest rate to the GDP
growth rate. To set α , we regressed the nominal government bond yield r̂r on the nominal
GDP growth rate x̂t using OLS. We chose a sample period from 1981 to 2005 to exclude theperiod with a regulated interest rate. The estimation result is as follows:rˆt = 0.020 + 0.668 xˆt(0.004) (0.089), Adj. R 2 = 0.696 ,(21)
where the numbers in the parentheses are standard errors. Adj. R 2 denotes the coefficient ofdetermination adjusted for degrees of freedom. Based on Equation (21), we set α at 0.668, or

14

equivalently the elasticity of intertemporal substitution at 1.497. The goodness of fit is reasonably
high. The 95 percent confidence interval for the coefficient on x̂t implies that the elasticity of
substitution lies between 1.17 and 2.07, which is consistent with much of the recent micro andmacro evidence for the Japanese economy, reporting the elasticity in the range between one andtwo11.
Next, we choose the parameters of technology. We set the average real GDP growth rate µ
to 0.025 that is the average growth rate over the period of fiscal years 1981 to 2004. The mid-term
economic outlook published by the Japanese government predicts that the growth rate will
gradually accelerate from fiscal year 2006, reaching 2.5 percent in fiscal year 2011 (Cabinet
Office, 2007)12. The parameters in the autoregressive process of the real GDP mainly affect theprobability that the debt-to-GDP ratio decreases from the initial level. We estimated the AR(1)process, applying OLS to the annual real GDP growth rate data over the period of fiscal years
1981 to 2004:

xt = 0.014 + 0.451xt −1(0.006) (0.195), Adj. R 2 = 0.166 ,(22)

where xt is the real GDP growth rate. The estimated value of ρ is positive and significant,consistent with the model’s assumption that the growth rate of income follows a Markov process.
We set ρ to 0.451 and σ to 0.01917, where the latter is the root MSE of the above regression.
Third, we set the financial intermediation cost, κ , to be the interest margin between bank loans
and deposits. We calculate the average interest margins of domestic banks by taking differences
between the average loan interest rate (loan interest income divided by loans outstanding) and theaverage deposit interest rate (deposit interest expenses divided by deposits outstanding) reported
in their financial statements13. We obtain 1.88 percent for 1975–79, 2.28 percent for 1980–89,
2.08 percent for 1990–99, and 2.15 percent for 2000–06. The average interest margins for15

1980–2006, when financial liberalization was promoted, is 2.17 percent. We set κ at 0.022.
Although the interest margin may reflect a default risk, we interpret the interest margin as theintermediation cost, because the intermediation cost should include the costs of defaults14.

3.3 Fiscal policy rules
Bohn (1998) proposes a fiscal policy under which the government increases a primary surplus
when the debt-to-GDP ratio increases, and vice versa. While he claims that the US experience
was consistent with this rule, the Japanese government does not seem to have followed it, at least
during the past 30 years15. Rather, as is shown by Figure 2, we observe a positive correlation
between the primary balance and the GDP growth rate. This observation will motivate us to guessthat the Japanese primary balance is an increasing function of the GDP growth rate16. Regressing
the primary balance as a proportion of nominal GDP to the growth rate of nominal GDP over theperiod of fiscal years 1981–2004, we obtained the following result:st = −0.053 + 0.997 xˆt(0.006) (0.131), Adj. R 2 = 0.713 .

(23)

Given the high goodness of fit, we can safely say that the primary balance actually depends
heavily on the nominal GDP growth rate.
We used the nominal GDP growth rate and not the real one because the government is likely todetermine expenditures considering tax revenues that are strongly affected by nominal GDP. In
fact, the nominal GDP growth rate provides a better fit than the real GDP growth rate17. This
policy rule is not necessarily inconsistent with our theoretical model defined in real terms. Basedon Equation (23), we assume the following fiscal policy rule:st = −η + 1.0 xt ,

(24)

16

where the intercept η is a free parameter that is intended to capture the government’s fiscal
stance. Equation (24) implies that the primary balance as a proportion of GDP increases by 1
percent if the GDP growth rate increases by 1 percent. We set η to 0.025 so that the primarybalance becomes zero if the growth rate is 2.5 percent, considering the government’s goal for a
fiscal surplus in fiscal year 2011. The figure of 0.025 is higher than the constant term of Equation(23), –0.053, by 2.8 percent, reflecting the government’s new policy rule, which will differ fromits past behavior. It is the advantage of calibration that we can incorporate a change in rules into
calculations, as well as utilize past variables. By increasing η we also investigate whether thesustainability condition is met for each value of η .
The initial value of the debt-to-GDP ratio is set at 1.72, the actual gross debt (net ofgovernment short-term securities) of the general government as a proportion of GDP at the end of
200518.
As a criterion for fiscal sustainability, we set the threshold value of the debt-to-GDP ratio at
this initial value. It is reasonable to judge fiscal sustainability based on whether the future
expected value of the debt-to-GDP ratio does not exceed its initial value. Our criterion of fiscal
sustainability coincides with the definition of Blanchard et al. (1990), which is that the ratio ofdebt-to-GDP eventually converges back to its initial level over the finite horizon. However, thevalue of 1.72 is quite high, from the perspectives of both Japanese history and cross-country
comparisons. It should be noted that if economic conditions change, such as the savings rate
and/or the long-run growth rate, the target level of 1.72 may become too high to be a critical value
for sustainability19.
Finally, we comment on the definition of government debt. Some argue that instead of grossdebt, net debt of government financial assets should be used. Doi (2006), however, points out that
financial assets held by social security funds should be subtracted from government debt only if

17

we forecast government expenditures taking into consideration future social security outlays.
Because we do not forecast future social security outlays, we do not subtract financial assets ofsocial security funds from the government bond. Doi (2006) also points out that the decision ofwhether financial assets held by the central or local governments should be subtracted fromgovernment assets depends on political judgments of whether those assets will be used to repay
debt. We understand that government financial assets are not used to repay debt.
Table 1 summarizes the parameters that we use for the baseline calibration.

3.4 Baseline results
Column 1 in Table 2 shows the baseline result under the assumption that the primary balance
averages zero. Hereafter we evaluate all variables in real term. The mean interest rate on
government debt is 2.57 percent, slightly higher than the average GDP growth rate. Figure 3.A
shows that the average debt-to-GDP ratio gradually increases from the initial level of 1.72,
reaching 1.896 in 100 years. The probability that the debt-to-GDP ratio reaches a level below theinitial value in 100 years is 39.7 percent. The government debt is not sustainable based on our
criterion.
We investigate how much primary surplus is necessary to make government debt sustainable.
We find that a primary surplus of 0.1 percent of GDP raises the average debt-to-GDP ratio to1.791 in 100 years, but a primary surplus of 0.2 percent of GDP decreases the averagedebt-to-GDP ratio, reaching 1.681 in 100 years20. In the latter case, the expected debt-to-GDP
ratio decreases, although the probability that the debt-to-GDP ratio increases from the initial levelis still as high as 44.4 percent. A primary surplus of 0.2 percent of GDP makes the government
debt sustainable.
We calculate the primary surplus required for decreasing the debt-to-GDP ratio with aprobability of 95 percent. In Figure 3.B, increasing the average primary surplus further, we find18

that if the targeted primary surplus is 1 percent of GDP, the debt-to-GDP ratio decreases from theinitial level with a probability of 94.4 percent. If the targeted primary surplus is 1.1 percent ofGDP, the debt-to-GDP ratio decreases with a probability of 96.3 percent. If the government is toachieve a primary surplus of 1 percent of GDP by raising consumption taxes, it must raise the tax
rate by about 2.5 percentage points given that its expenditure as a proportion of GDP is constant21.

3.5 Results under alternative parameters
In this section, we make several alternative assumptions to examine changes in the baseline
results. Table 3 shows the results.
In Case 1, we set the financial intermediation cost, κ , to zero. The average interest rate of thegovernment bond is 3.7 percent, 1.2 percentage points higher than the GDP growth rate. The realinterest rate is so high that the average debt-to-GDP ratio reaches 5.599. With an almost 100
percent probability, the debt-to-GDP ratio increases from the initial level. To stabilize thedebt-to-GDP ratio below the initial level, the primary surplus as a proportion of GDP must be 2.1
percent. How can we interpret this result? The average nominal interest rate of long-term
government bonds is 5.0 percent over the period 1981–1998, the period prior to thezero-interest-rate policy, while the average nominal GDP growth rate over the same period is 4.2
percent, 0.8 percentage points lower than the interest rate (see Figure 1). However, considering
the fact that the average funding cost of the government is no more than the long-term interest rate,
it does not seem likely that the interest rate of government bond is 1.2 percentage points higher
than the GDP growth rate.
In Case 2, we set the inverse of the elasticity of intertemporal substitution, α , at 1, which isoften assumed in the literature of dynamic stochastic general equilibrium analysis. The averageinterest rate of government bond is 3.42 percent, higher than the baseline result of 2.57 percent.
The debt-to-GDP ratio increases rapidly and reaches 4.204 in 100 years. The government debt is19

not sustainable. This case does not seem likely either, considering the past relationship between
the interest and growth rates.
We have so far assumed that the Japanese economy achieves a GDP growth rate of 2.5 percent.
Anticipating possible declines in labor force and technological progress due to the future
population aging, the growth rate may be lower than 2.5 percent. To consider this possibility, we
set the GDP growth rates at 2 percent, 1 percent, and 0 percent in that order. We also assume thatthe government achieves a zero primary balance under each growth rate. The results are shown inCases 3A, 3B and 3C, respectively. The average interest rates of government bonds decrease to
2.24 percent, 1.57 percent, and 0.9 percent as the growth rates decrease to 2 percent, 1 percent,and 0 percent, respectively. The gaps between the interest rate and the growth rate widen to 0.24
percent, 0.57 percent, and 0.9 percent, while it is 0.07 percent in the benchmark case. The
difference in the results comes from the fact that we set α to 0.668, put differently, the elasticity
of intertemporal substitution to be larger than unity. The debt-to-GDP ratios increase from theinitial level, reaching 2.206, 3.081 and 4.288 in 100 years, which suggests that the government
debt is not sustainable in any case. The primary surpluses necessary to reduce the expecteddebt-to-GDP ratio from the initial value are 0.5 percent, 1 percent and 1.6 percent of GDP,
respectively. The primary surpluses necessary to reduce the debt-to-GDP ratio from the initialvalue with a probability of 95 percent are 1.3 percent, 1.9 percent and 2.5 percent of GDP,
respectively. As the GDP growth rate declines, the gap between the interest and growth rates
widens and the primary surplus necessary for the sustainability condition increases. Ihori et al.
(2007) make a more pessimistic simulation under the assumptions that the rate of technologicalprogress is 0 percent and that the labor force declines, as we explain in detail in the next section.
Finally, we consider the possibility that the fiscal policy is featured by a primary surplus that is
a nonlinear function of the GDP growth rate. When the GDP growth rate is low, primary balances

20

may worsen extremely because of expansive fiscal policy measures. It may be difficult to restore
primary balances by increasing taxes or decreasing government expenditure, which would causesignificant redistribution. Progressive income tax may also cause a nonlinear relationshipbetween primary surplus and GDP growth rate. Regressing the primary surplus as a proportion of
nominal GDP on the nominal GDP growth rate and its squared values, we obtained the following
concave equation:st = -0.054 + 1.479 xˆ t - 7.321xˆ t2(0.006) (0.343) ( 4.846), Adj. R 2 = 0.729.

(25)

We employ the above fiscal policy rule by adjusting the constant term so that the primary balance
becomes zero under a 2.5 percent real GDP growth rate.
Case 4 shows that the probability that the debt-to-GDP ratio decreases from the initial level in
100 years is 22.2 percent, lower than the baseline result (39.7 percent). This result is obtained
from our assumption that primary balance deteriorates more than proportionately when the real
GDP growth rate declines.

3.6 Comparisons with previous studies
In this subsection, we compare our calibration results with previous studies on thesustainability of Japanese government debt. Ihori et al. (2007), directly comparable to ours,
calculate the interest rate from the growth model of overlapping generations. They project a
primary balance that is necessary to maintain the debt-to-GDP ratio. Their estimated primarybalance (3.9 percent of GDP on average) is greater than ours primarily because the gap between
the interest and growth rates exceeds almost 3 percentage points, which is much larger than our
result. The main reasons for this difference are as follows. First, they do not consider financial

21

intermediation costs. Second, they set the inverse of the elasticity of intertemporal substitution ata higher value than ours. Third, they assume a lower GDP growth rate than ours22.
Dekle (2005) examines the small-open version of the growth model and finds that the
government debt is not sustainable unless the primary surplus increases substantially to almost 5
percent of GDP over the next decade or so. His projection assumes a wide gap between theinterest rate (6 percent) and the TFP growth rate (1.2 percent).
Broda and Weinstein (2005) and Doi (2006) make simulations using exogenous pairs of
economic growth rates and interest rates. Broda and Weinstein (2005) project that the
government revenues as a proportion to GDP must increase moderately23. Doi (2006) points outthat more government revenues are necessary than Broda and Weinstein’s results if financial
assets held by the central and local governments are not subtracted from their debt and if we
consider the recent financial deterioration. Note that the simulation results of Broda andWeinstein (2005) and Doi (2006) are difficult to compare with our results directly because bothof them ignore general equilibrium consideration between growth and interest rates.
Oguro (2006) applies the approach of “deficit gamble” (Ball et al., 1998) to the Japanese
economy, finding that if the primary deficit is zero, the probability that the debt-to-GDP ratio will
fall below its initial value in 100 years is 49.4 percent, higher than our baseline result (39.7%).
His methodology is to estimate the joint distribution of the interest rate and the GDP growth rateover the period 1966–2005 and to project the estimated debt-to-GDP ratio to the future. Notably,
the sample period includes the high-growth era when the interest rate is lower than the average
growth rate. His optimistic prediction appears to come from the chosen sample period.

4. Conclusions
To answer the question of whether government debt is sustainable in Japan, we havecalibrated a dynamic stochastic general equilibrium (DSGE) model. By incorporating financial

22

intermediation costs into the theoretical model, we have succeeded in explaining the actual
relationship between the interest rate and the GDP growth rate observed for the Japanese
economy in some consistent manner. Our approach has also made it possible to assume somecombinations of the interest rate and the economic growth rate on theoretical grounds. Our
approach will remain an effective analytical tool even when the future potential growth ratechanges. Our calibration has also enabled us to show the variability of our forecasts.
In investigating the fiscal sustainability by calibrating a DSGE model, it is extremely
important to specify the relationship between the interest rate on government bond and the GDP
growth rate in a manner that is consistent with the actual data. With parameter sets consistent with
the Japanese economy, it makes little difference whether government bonds are treated as safe or
risky assets, while it is essential to introduce financial intermediation costs to calibrate the modelwith realistic interest rates.
Our results can be summarized as follows. When the growth rate is 2.5 percent, the averageinterest rate is 2.57 percent under the assumption that financial intermediation costs remain at thesame level as in the past. If the targeted primary surplus averages zero, the debt-to-GDP ratio
gradually increases from the initial level of 1.72 to 1.896 in 100 years, suggesting that the
government debt is not sustainable. To make it sustainable, the targeted primary surplus must be
0.2 percent of GDP. Lowering the GDP growth rate to 2, 1, and 0 percent and assuming that the
government runs a zero primary surplus for each growth rate, we find that the average interest rate
for government bond decreases but the gap between the interest rate and the growth rate widens,
so that the government faces greater difficulty in meeting the sustainability condition.
It should be noted that our calibration results depend on several assumptions. First, we assume
that government outlays as a proportion to GDP are constant. If we consider an increase in social
security expenses associated with population aging, we may say that government revenues

23necessary to meet the sustainability condition will exceed our estimate. Next, we have argued thefiscal sustainability issues under the criterion that the debt-to-GDP ratio decreases from its 2005
level, 1.72. If we consider a decrease in savings rate, which will be inevitable with population
aging, we may have to set a more stringent criterion than 1.72. Furthermore, we do not consider
the possibility that the interest rate on government bonds incorporates the risk of an increasing
debt-to-GDP ratio. If we consider this possibility, the primary balance necessary to meet thesustainability condition may be higher than our estimate. The effect of the risk of fiscal
insolvency on the interest rate depends on how we model the economy after the fiscal insolvency,
which is an important direction for future research. Finally, it should be noted that our model is an
exchange economy with growth rates being exogenous. Generally, the accumulation of
government debt may deter private capital accumulation and have a negative impact on economic
growth.
We must not hastily conclude that a reduction of financial intermediation costs due to financial
innovation will raise the interest rate and tighten the government budget. We have employed an
exchange economy model without production to present a simple model that consistently explains
the actual relationship between the interest rate and the GDP growth rate. The next task is toextend the model to an endogenous growth model with production and to calibrate it to theJapanese economy. In the endogenous growth model, a reduction of financial intermediation
costs may possibly raise the economic growth rate, resulting in an increase in tax revenues and a
primary surplus.
Our model is a standard asset-pricing model incorporating financial intermediation costs. To
incorporate various other extensions that have been tried to explain the safe asset return, such as
the decoupling of the degree of relative risk aversion and the elasticity of intertemporal
substitution (Weil, 1989, among others), is left for future studies.

24

Acknowledgement
We thank two anonymous referees, Yasushi Iwamoto, Toshihiro Ihori, Takatoshi Ito, Akira
Okamoto, Masao Ogaki, Etsuro Shioji, Makoto Saito, Toru Nakazato, Tomoaki Yamada,Tsutomu Watanabe and seminar participants at the 9th Macroeconomics Conference, Hitotsubashi
University, Fiscal Studies Group at the Institute of Statistical Research for their valuablecomments. Sakuragawa thanks Kikawada Foundation for financial support. Remaining errors are
ours.

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29

Table 1. Parameters
Preference

Technology

Discount Inverse of
Rate
Elasticity ofIntertemporal
Substitution

Mean
GrowthRate

SerialCorrelation

Standard
Deviationβ
1/1.02

μ
0.025

ρ
0.451

σ
0.01917

α
0.668

30

Fiscal Policy Rule
Financial
Initial Condition
Intermediation
Intermediation Coefficient Constant Debt / GDP
Term
Cost
on Growth
Rate
κ
0.022

γ
1

η
0.025

d0
1.72

Table 2. Calibration Results
Average primary surplus-to-GDP Ratio (%)
Average real interest rate (%)
Average debt-to-GDP ratio in 100 years
Probability that debt-to-GDP ratio in 100 years is less than its initial value (%)
Note: We adjust η to realize each average primary surplus-to-GDP ratio.

31

0
2.57
1.896
39.7

0.1
2.57
1.791
47.0

0.2
2.57
1.681
55.6

0.5
2.57
1.362
76.0

1
2.57
0.852
94.4

1.1
2.57
0.737
96.3

Table 3. Calibration Results under Alternative Assumptions
Case

Parameters Changed
A. In case average primary surplus is zero
Average real interest rate (%)
Average debt-to-GDP ratio in 100 years
Probability that debt-to-GDP ratio is less than its initial value (%)
B. Primary surplus-to-GDP ratio necessary to make average debt-to-GDP
ratio in 100 years less than its initial value (%)
C. Primary surplus-to-GDP ratio necessary to make the probability thatdebt-to-GDP ratio in 100 years less than its initial value 95% or more (%)

1
2
3A
3B
3C
4
Benchmark Zero Financial
Unitary
2% GDP
1% GDP
0% GDP
Primary
Intermediatio Elasticity of Growth Rate Grwoth Rate Grwoth Rate Surplus as an CostIntertemporal
Quadratic
Substitution
Function of
GDP Grwoth
R
κ=0
α=1
μ=0.02μ=0.01
μ=0.0
2.57
1.896
39.7

3.70
5.599
0.0

3.42
4.204
0.0

2.24
2.206
22.6

1.57
3.081
3.9

0.90
4.288
0.3

0.2

2.1

1.6

0.5

1.0

1.6

1.1

3.0

2.1

1.3

1.9

2.5

Note: We adjust η to realize each average primary surplus-to-GDP ratio.
For the fiscal policy rule of case (4), see the text.

32

2.57
2.226
22.2

Figure 1

Yields to Government Bond and Nominal GDP Growth Rate%33

Figure 2

Primary Balance as a Proportion to GDP and Nominal GDP Growth Rate

34

Calibration Results (Government Debt-to-GDP ratio)

Figure 3

A. (Benchmark Case)： Primary balance is zero on average.
Debt/GDP

3.5

3

2.5

2

1.5

1

0.5

0

20

40

60
Year

80

100

120

Real line is the average. The vertical lines show the range within two times standard
errors.

35

B. Primary surplus is 1.0% of GDP on average.
Debt/GDP

2.5

2

1.5

1

0.5

0

-0.5

0

20

40

60
Year

80

100

120

Real line is the average. The vertical lines show the range within two times standard
errors.

36

Footnotes

1

Taking the averages over the whole sample period of 1981–2005, we find that the gap between thenominal interest rate on the government bond and the nominal GDP growth rate is 1.0 percentage
point. However, considering the possibility that the nominal interest rate remains at a higher level thanthe equilibrium rate under the zero-interest-rate policy, we believe that a gap of 1.0 percentage point is
not appropriate as a benchmark argument for fiscal sustainability.
2

At the beginning of 2005, politicians and government officials had a heated debate on “growth rateversus interest rates.” When investigating fiscal sustainability, it is of extreme importance tounderstand the relationship between the interest rate on government bonds and GDP growth rate. The
government released reports for fiscal sustainability for possible several pairs of economic growth
rates and interest rates, and provided a consensus that if the pair of high economic growth rates andlow interest rates is realized, the likelihood of fiscal sustainability increases. The government,
however, did not explain which pair is most consistent with the potential of the Japanese economy.
3

Bohn (1999) demonstrates that debt-financing strategies that exploit low interest rates lead todifferent welfare implications between the two approaches. Blanchard and Weil (2001) investigate
various models of the two approaches, and show that the feasibility of Ponzi games varies with models,
not necessarily depending on the average rate of return on bonds versus the growth rate.
4

Ihori et al. (2002) also apply the approach of Hamilton and Flavin and report that the sustainability

condition is not satisfied for the period of 1957–99, when the gap between the interest rate and the
GDP growth rate was 4 percent or more.
5

Broda and Weinstein (2005) report an optimistic projection for the sustainability, whereas Doi(2006) provides a pessimistic one. The difference in their results lies in the difference in whether
government assets are included or not.
6

Aiyagari and Gertler (1991) and Heaton and Lucas (1996) construct models with intermediation

costs and heterogeneous agents to resolve the Mehra–Prescott puzzle.
7

Note that it is easily possible to verify that rational agents never borrow in the period when theyreceive income and never lend in the period when they do not that we skip this part in the formulation.
8

Rigorously, we may formulate this asx ~V ( Bt −1 , Yt , xt ) = max u (Ct ) + β ∫ V (Wt , Yt +1 , xt +1 )g ( xt +1 xt )dxt +1 .
C t ,Wt

9x

The assumption of normal distribution may raise concern about the possibility of negative gross

growth rates. In practice, however, this is not the case. We simulated the model 10,000 times over 100
periods and found that the minimum value of the gross growth rate was 0.9143, far above zero, given
the values of µ and σ assumed as below.

37

10

For example, Sugo and Ueda (2007) set the quarterly discount factor at 0.995. Fujiwara et al. (2005)

set the annual discount factor at 0.99.
11

Though some of the studies in the 1990s report the elasticity of intertemporal substitution below 1(e.g., Kaneko, 1991; Akagi, 1995; and Uemura, 1997), more recent studies report it as being between 1
and 2. Abe and Yamada (2005) analyze household micro data, obtaining the estimated elasticity of
1.809. Hatano and Yamada (2006), using macro data, obtained the estimated elasticity above 1 in mostof the specifications (see also Hamori, 1996; Okubo, 2003; Fuse, 2004; Iiboshi et al., 2006; and Sugoand Ueda, 2007).
12

See the “transition-to-the-new-growing-economy scenario” in Cabinet Office (2007).

13

Domestic banks include city banks, long-term credit banks, trust banks, first-tier and second-tierregional banks.
14

Though we do not specify the intermediation cost, we can regard it as, for example, the expectedjudicial cost for forfeiting borrowers’ assets in case of defaults, which depends on the probability ofdefault. The costly-state-verification model posits that the loan interest rate is equal to the depositinterest rate plus the verification cost multiplied by the default probability (e.g., Gale and Hellwig,
1985, and Williamson, 1986).
15

Doi and Ihori (2002), using Japan’s data over the period of 1965–2000, provide evidence that

Bohn’s rule did not hold in Japan. Ito et al. (2006) also show that Bohn’s rule did not hold after 1975.
16

One may think that we should use the GDP gap, i.e., the difference between potential and actual

GDP, instead of the GDP growth rate to capture the role of a fiscal policy as a built-in stabilizer.
However, we choose the growth rate because most government expenditure, including social security
expenditure, does not depend on the growth rate, while tax revenues depend on it strongly. If we used
the GDP gap, we would implicitly assume that the government will increase taxes to finance primary
deficits caused by the decline in the potential growth rate (e.g., an increase in social security
expenditures arising with aging), which would be politically difficult and unrealistic due to its
significant redistribution effect among various interest groups (Alesina and Drazen, 1991).
17

Regressing the primary surplus as a proportion of GDP on the real GDP growth rate, we get

st = −0.039 + 0.710 xt(0.012) (0.362), Adj. R = 0.110
2

Following this estimation result, we assume that the fiscal policy follows the rulest = 0.7 xt − 0.0175 , which implies that the primary surplus is zero in case of the 2.5 percent growthrate. The calibration result under this policy rule was similar to that in Table 2, Column 1 (Theexpected government debt-to-GDP ratio is 1.893 in 100 years, and the probability that thedebt-to-GDP ratio in 100 years is below its initial value is 37.0 percent).

38

18

Total debt (excluding short-term government bonds) of the general government at the end of 2005

is 861705.7 billion yen (Flow-of-fund statistics by Bank of Japan).
19

Ball et al. (1998) analyze the U.S. fiscal sustainability using 1.0 or 1.5 as the critical value of thegovernment debt as a proportion of GDP.
20

In the following calibrations, we change η by 0.001.

21

The consumption tax rate was 5 percent and consumption tax revenues were 2.0 percent of nominal

GDP in fiscal year 2006.
22

They set the inverse of the elasticity of intertemporal substitution to 2.5. In their baseline scenario,

the annual rate of TFP growth is zero from 2003 on and the labor force declines.
23

In their “Case 2,” for example, the government revenues as a proportion to GDP must increase by

2.7 percentage points. In “Case 2”, the forecast for the population is based on Faruqee and Mühleisen(2001), the GDP growth rate is assumed to be 2 percent, the interest rate assumed to be 4 percent, andboth the per-capita government transfers to older generations and the per-capita government
expenditures to young generations increase at the same rate as GDP.

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