CIGS Working Paper Series No. 20-008E

Secular Stagnation and Low Interest Rates
under the Fear of a Government Debt
CrisisKeiichiro Kobayashi (Keio University / The Canon Institute for Global Studies)
Kozo Ueda (Waseda University/ The Canon Institute for Global Studies)2020.11※Opinions expressed or implied in the CIGS Working Paper Series are solely those of theauthor, and do not necessarily represent the views of the CIGS or its sponsor.
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Secular Stagnation and Low Interest Rates
under the Fear of a Government Debt Crisis
Keiichiro Kobayashi∗

Kozo Ueda†

November 24, 2020

Abstract
In this study, we explain the driving forces behind the secular stagnation associated with a persistent decrease in interest rates by employing a model thatincorporates a crisis risk triggered by government debt accumulation. The modelshows that fear of large-scale capital taxation and capital misallocation in future
debt crises accounts for almost half the economic slowdown in Japan over the past
two decades. Over the same period, the government bond yield declines, because
a decrease in the expected returns on capital makes investing in government bonds
more attractive than investing in capital.

JEL Classification Numbers: E32, E62, G18, H12, H63
Keywords: default; government bond; capital levy; lost decades; bank run∗

Keio University, Canon Institute for Global Studies (CIGS), and Tokyo Foundation for Policy Research (E-mail: kobayasi@econ.keio.ac.jp).
†
Waseda University and CIGS (E-mail: kozo.ueda@waseda.jp). We would like to thank Kenneth
West, three anonymous referees, Tamon Asonuma, Gauti Eggertsson, Ippei Fujiwara, Masazumi Hattori, Fumio Hayashi, Selo Imrohoroğlu, Takatoshi Ito, Etsuro Shioji, and the conference and seminar
participants at the Bank of Japan, CIGS, Columbia University, IMF, Ministry of Finance Japan, andSummer Workshop on Economic Theory for their useful comments and suggestions. This research was
funded by the JSPS Grant-in-Aid for Scientific Research (Japan) (19H01491 for Ueda). All remaining
errors are our own.

1

1

INTRODUCTION

The Great Recession of the late 2000s raised concerns about future secular stagnation(Hansen 1939) that would cause the U.S. and European economies to stagnate persistently in the coming decades (Gordon 2012, Krugman 2013, Summers 2013). This effectis evident in Japan in the so-called “lost decades” following the collapse of the asset
market bubble around 1990. There are many possible reasons for secular stagnation;1
however, we focus on the fact that these concerns arose when government bonds outstanding (GBO) started to expand substantially in these countries. Moreover, it is equallyimportant to emphasize that despite this development the interest rate on government
bonds did not increase but actually decreased to historically lowest levels.
This study offers a new explanation for the factors that drive secular stagnation andlower government bond yields; the key factor being a government debt crisis. A complete
loss of market confidence in government debt forces the government to collect extremely
large tax revenues. The government reduces its debt by imposing a once-off increasein taxes on, for example, GBO (a partial default of government debt), capital stock(capital levy), and consumption. By contrast, taxes are low in normal periods, and thus
the government budget is not balanced. Consequently, government debt continues torise. A crisis occurs with an exogenous probability that increases with GBO. Using asimple neoclassical closed-economy model, we show that fear of a government debt crisis
causes a persistent economic slowdown in normal times.
The model works as follows. In normal periods, people anticipate that returns on
capital will decline during a crisis due to a heavy tax on capital stock and/or misallocations of capital caused by bank runs triggered by a partial default on GBO. This fear
increases the required return on capital and discourages capital investment. Government
debt accumulation intensifies this adverse effect because the probability of a government
debt crisis increases. Moreover, the tax distortion in a crisis increases with GBO because
the government imposes higher tax rates to meet its debt obligations. Consequently, a
growing risk of government debt crisis persistently depresses output.
The model also shows that fear of a capital levy and capital misallocation decreases
the government bond yield, which is consistent with the Japanese data. People expect
their capital to lose value in a crisis, whereas the government does not tax bond holdings
heavily. Consequently, people perceive capital investment as riskier than government
bond investment and choose to buy more bonds and less capital. This increase in demandfor bonds reduces the bond yield, despite the accumulation of GBO.
In our model, the key to the increase in the required return on capital is the tax on
1

Examples include a slowdown in innovations (Gordon 2012), demand shortfall (Summers 2013,
Eggertsson, Mehrotra, and Robbins 2019), and pessimism (Benigno and Fornaro 2018). Focusing on
Japan, Hayashi and Prescott (2002) note the effect of a decrease in the total factor productivity (TFP)
growth rate, and Caballero, Hoshi, and Kashyap (2008) emphasize the malfunctioning financial sector.

2

capital stock. However, it seems problematic to posit that increases in government debt
trigger widespread fear of a capital levy but not fear of a government bond default. To
quantify the model, we calibrate key parameter values associated with taxes by examining Japan’s economic crisis after World War II.2 We show that the post-war Japanese
government imposed a heavy wealth tax to avoid defaulting on their debt obligations.
Furthermore, our model incorporates bank runs that cause severe misallocation of capitalstock during a crisis, which reinforce the adverse GBO effect on capital investment. Our
simulation calibrated to Japan demonstrates that the expectation of heavy capital taxation and capital misallocation in a crisis accounts for approximately half of the output
slowdown in Japan during the first two decades of the 21st century.
We also examine the case of a preemptive tax hike, where the government introduces
a distortionary consumption tax in normal times to prevent a debt crisis. We find thata preemptive tax hike increases social welfare.
The remainder of this paper proceeds as follows. Section 2 provides several facts on
Japan’s secular stagnation and reviews related literature. Section 3 introduces the modeland specifies the equilibrium. Section 4 presents the results of the numerical simulation.
Section 5 presents several extensions of our analyses, and Section 6 concludes.

2

SOME BACKGROUND ON JAPAN AND A LITERATURE REVIEWIn this section, we explain the background of Japan’s secular stagnation and review the
related literature.

2.1

Background

Economic slowdowns coincide with debt increases in many economies. Figure 1 shows
the trend in real GDP per capita and government debt for Japan, the United States, 17
countries in the euro area, and Italy. The graph begins in 1975 for Japan and in 1992 forthe other countries, representing the 15-year periods before the financial crises in eachregion (i.e., 1990 and 2007, respectively, shown as vertical dashed lines). The thick solidline represents the logarithm of real GDP per capita, and the thin solid line represents
its linear trend (left axis). The lines with crosses and circles represent the ratio of grossand net government debt, respectively, to nominal GDP (right axis). The figure showsthat government debt increased after a financial crisis in all regions. Notably, Japan’s
gross government debt now exceeds 200% of its nominal GDP and net government debt
is around 120% of its nominal GDP. Together with the increase in GBO, real GDP
2

Kozlowski, Veldkamp, and Venkateswaran (2020) assume that learning from history leads to theformation of expectations on tail events.

3

decreased persistently compared with its trend. Hereafter, we provide further empiricalevidence related to Japan.
Although this government debt accumulation is commonly cited as the result rather
than the cause of the stagnation, Figures 2 and 3 show that increased government debt
does indeed cause anxiety in Japan. According to a household survey conducted by
Japan’s Cabinet Office, “Overview of the Public Opinion Survey on the Life of the People,” an increasing number of Japanese worry in their everyday lives and are concerned
about their prospects, with one-third indicating that the fiscal balance is one of thereasons for this.3
The left-hand panel of Figure 3 shows how often specific words appear in the morningand evening editions of the Nihon Keizai Shinbun, Japan’s financial newspaper, from 1981
to 2019. The frequency with which terms that suggest a government debt crisis—such as“fiscal failure” (zaisei hatan) or “fiscal crisis” (zaisei kiki )— appear has been increasing
since 1981. Moreover, combinations of “tax increase” (zozei ) and either “fiscal failure”
or “default” are increasingly common, which seems to suggest a growing fear of a tax
increase in the event of a government debt crisis. The euro crisis around 2010 caused
a sharp increase in these terms; however, we confirm a steady increase even when we
exclude the word “Europe.”
The right-hand panel of Figure 3 shows the time-series movements of the sovereign
credit default swap (CDS) spread for five-year Japanese government bonds from 2003 to
2019. Sovereign CDS contracts protect investors against sovereign default, which enables
us to calculate the probability of sovereign default numerically. Similar to the occurrenceof specific words in the left-hand panel, the euro crisis around 2010 contributed to theincrease in the CDS spread. If we exclude this episode, we find that the CDS spread in
the latter half of the 2010s is several percentage points higher than that in the mid-2000s.
To investigate the effect of government debt on the CDS spread more quantitatively, we
estimate the following equation using the ordinary least squares (OLS) method:
CDStJapan = c + α · bt + β · CDStItaly + εt ,where CDStJapan , bt , and CDStItaly represent the logarithm of the CDS spread for Japan,
logarithm of the debt ratio for Japan, and logarithm of the CDS spread for Italy, respectively, in month t. The observation period is from March 2003 to July 2017.4 The
3Health and natural disasters are the top two reasons, followed by concerns about public services.
This worry is likely to be related to the government debt accumulation because it prevents the government from providing sufficient public services in the future, such as pensions, medical services, andinvestment in infrastructure that is resilient to disasters.
4
Although the debt data are annual, we use monthly data to secure a sufficient number of observations
for estimation. We calculate the debt ratio as follows. First, we collect annual data on the net debt
to nominal GDP ratio for Japan (source: OECD, as in Figure 1). Second, we calculate the annual
net debt, BT in year T , by multiplying the debt-to-GDP ratio by the annual nominal GDP for Japan4

explanatory variable CDStItaly aims to capture the international spillover effect, particularly the euro crisis effect, on Japan’s CDS spread. The estimates of α and β are
1.00 and 0.61, respectively, with 95% confidence intervals of [0.37, 1.63] and [0.48, 0.73].
Thus, both coefficients are significantly positive; that is, the CDS spread for Japanese
government bonds is positively associated with both Japan’s net debt ratio and Italy’s
CDS spread. We confirm the robustness of the results using the monthly logarithm difference for all three variables in the equation. The estimates of α and β are 1.47 and
0.59, respectively, with 95% confidence intervals of [0.49, 2.44] and [0.45, 0.73]. These
results suggest that the elasticity of the CDS spread to debt is approximately one.
The probability of a government debt crisis for Japan is also studied by Morikawa(2016, 2017). By conducting surveys, he finds that Japanese consumers and firm managers believe that a debt crisis will occur by 2030 with a probability of approximately
24% (consumers) and 27% (firm managers), on average. However, the survey is open tointerpretation by respondents.
A noteworthy puzzle is that the price (yield) of government bonds is high (low). The
left-hand panel of Figure 4 shows the government bond yield in real terms for Japan,
defined as the nominal bond yield with five-year maturity minus the annual CPI inflation
rate in the following year.5 The figure shows that bond yields decreased, which hardly
suggests a mounting public default risk.
Simultaneously, the return on capital seems to be increasing. The right-hand panelof Figure 4 shows the increase in the credit spread, defined as the bank loan rate with
one-year maturity or longer minus the government bond yield with five-year maturity.
Data taken from the JIP Database 2015, compiled by the Research Institute of Economy,
Trade and Industry in Japan, suggest that the capital cost spread, a proxy for the credit
spread, has been increasing. Consequently, the ratio of investment to capital has been
decreasing.
(source: the Cabinet Office). Third, we set annual net debt in March (i.e., the end of the fiscal year) in
year T to BT and linearly interpolate the values for the other months to construct monthly data on net
debt Bt for month t (e.g., debt in September 2010 equals the average of debt in March 2010 and debt
in March 2011). Last, we divide Bt by the monthly CPI (excluding fresh food, source: the Statistical
Bureau) and the monthly Indices of Industrial Production (source: the Ministry of Economy, Trade andIndustry) to construct the debt ratio. It should be noted that we do not use the net debt to nominal
GDP ratio for the regression but only to construct the monthly debt ratio. Furthermore, the CDS data
are missing for several months during the observation period.
5
For 2019, we use the realized inflation rate from 2018 to 2019.

5

2.2

Literature Review: Sovereign Default and Public Debt Overhang

Empirical studies report that economies tend to stagnate when government debt is high,6
referred to by Reinhart, Reinhart, and Rogoff (2012) as the public debt overhang. They
review 26 cases of high government debt in advanced countries, reporting that in 23 cases,
economic growth remained stagnant for more than a decade. Based on their finding of anonlinear correlation between higher debt and lower growth, they argue that an increasein government debt causes lower economic growth. Our study provides a theoreticalbasis for the public debt overhang phenomenon.
Aguiar, Amador, and Gopinath (2009), Aguiar and Amador (2011), Bocola (2016),
Balke (2018), Arellano, Bai, and Mihalache (2018, 2019), and Roldan (2019), among
others, propose models in which sovereign default risk dampens real economic activity
and causes a recession before a default occurs. The mechanism depends on the model.
In Aguiar, Amador, and Gopinath (2009) and Aguiar and Amador (2011), expectations
that the government will impose a tax on investment returns during a default dampens
capital formation before a default occurs. In Bocola (2016), banks hold government bonds
and loans to non-financial firms. The fear of sovereign default tightens banks’ funding
constraints and increases the stochastic discount factor, both of which make banks require
higher returns on the loans, dampening capital formation before a default. Balke (2018)
models a contraction in labor demand. The fear of sovereign default decreases the price ofgovernment bonds and hurts the balance sheets of banks that hold government bonds as
an asset. This decreases bank loans to non-financial firms, which in turn contracts labordemand. In Arellano, Bai, and Mihalache (2018), default risk decreases capital inflows,which prevents investment, particularly in nontraded sectors, whereas Arellano, Bai, andMihalache (2019) discuss the importance of both monetary and fiscal policy. Roldan(2019) constructs a model of incomplete markets to show that sovereign risk decreases
consumption demand. Our model has a similar mechanism to that of Aguiar, Amador,and Gopinath’s (2009), Aguiar and Amador’s (2011), and Bocola’s (2016) models. In ourmodel, as GBO increases, people are prepared for a higher capital levy during a crisis
and increase their stochastic discount factors. This causes banks to require a higher
return from capital investment and, in turn, decreases capital investment.
However, these standard sovereign default models often predict an increase in thegovernment bond yield as GBO increases. The most important departure from thesemodels is that we seek to explain the decrease in the government bond yield, despite
the GBO increase peculiar to episodes of secular stagnation (e.g., the lost decades of
6

See Reinhart and Rogoff (2010), Reinhart, Reinhart, and Rogoff (2012), Checherita-Westphal andRother (2012), and Baum, Checherita-Westphal, and Rother (2013) for the negative effect of government
debt on output. Barro and Sala-i-Martin (1995) show that government consumption has a negative effecton output. Fischer (1991) shows that a fiscal deficit has a negative effect on output.

6Japan). The key feature of the model is a capital levy. When people anticipate relatively
heavier capital taxation than government bond taxation during a crisis, investment ingovernment bonds becomes less risky as the probability of a crisis increases, whereas
capital investment becomes riskier. Thus, the government bond yield decreases, while
the credit spread increases, both of which are consistent with the data. Furthermore,
the increase in the credit spread helps explain the stagnation in real economic activity
quantitatively better.
It should be noted that the type of government debt crisis we describe in the model
does not necessarily refer to a large-scale sovereign default. The government repays mostof its debt by taxing private agents heavily (notably, a capital levy), which contributes
to a stable government bond price and an increasing credit spread. From a historicalperspective, Japan’s Ministry of Finance (MOF 1976) and Eichengreen (1989) study
capital levies.
Other differences from the early studies are that we incorporate domestic rather
than external debt using a closed-economy model. No investor in our model has accessto global financial markets, and thus, to a riskless world interest rate. This structure
helps to stabilize the price of government bonds, despite the increase in GBO. Another
difference is that a crisis occurs with an exogenous probability in our model, whereas inmany of the models mentioned above, the government’s default decision is endogenous.

2.3

Other Strands of the Literature

Regarding the low, stable government bond yield, Hoshi and Ito (2014) point out thatJapanese government bonds are held predominantly by Japanese institutional investors,
arguing that they have a strong home bias. However, their simulation results show thatgovernment debt will exceed private sector financial assets within 10 years, and thus theywarn of a potential fiscal crisis. Sakuragawa and Sakuragawa (2016) use the absence ofsafe assets to explain the low government bond yield. Caballero and Simsek (2017) pointout a secular increase in risk intolerance.
Our study is also related to the literature in the 1990s on the non-Keynesian effectof fiscal policy developed by Giavazzi and Pagano (1990), Alesina and Perotti (1996),
Alesina and Ardagna (1998), and Perotti (1999). Perotti (1999) shows theoretically andempirically that an increase in government debt has a contractionary effect on consumption when government debt is large. Alesina and Perotti (1996) and Alesina and Ardagna(1998) show that government expenditure cuts have a longer-lasting effect on improving the economy than do tax increases. Our study has similar implications, in that anincrease in government debt is contractionary.
The model we present here has its basis in the neoclassical model of a rare disaster,following the work of Rietz (1988), Barro (2006, 2009), Gabaix (2012), and Gourio(2012, 2013). Specifically, our model is a simplified version of Gourio’s (2013). However,7the “disaster” properties in our case are different because a debt crisis is an abrupt
redistribution of wealth, whereas a disaster destroys resources. Therefore, we refer toa crisis, rather than a disaster, which affects mainly the household (through taxes),
whereas a disaster in Gourio (2013) affects firms through changes in their productivity
and capital values.7

3

MODEL

The model economy consists of a representative household, firm, bank, and government.
For simplicity, we assume that, in normal times, the government makes transfers, butcollects no tax.8 Thus, government debt keeps increasing. When a crisis occurs, thegovernment imposes once-off taxes in order to repay its debt. The crisis probability isgiven exogenously and increases with GBO. We assume a closed economy because around
90% of government bonds are held by domestic investors in Japan. The asset market isincomplete, meaning that the crisis risk is not insured.

3.1

Crisis Risk

We define the crisis indicator xt as xt = 0 in normal times and as xt = 1 when agovernment debt crisis occurs. The variable xt is an exogenous sunspot shock to theeconomy. A government debt crisis occurs when debt holders lose confidence in thegovernment debt and rush to exchange it for real assets and goods, which is exogenous
to the government’s actions. More specifically, when xt = 0, the government exchanges
maturing GBOs for new GBOs. When xt = 1, (i) a sunspot shock changes people’s
preferences such that they demand real goods (as much as a portion ω of BtG + Gt )7

Kozlowski, Veldkamp, and Venkateswaran (2020) use Gourio’s model to analyze the exogenous
disaster risk of a financial crisis. Isore and Szczerbowicz (2017) later extend this work to the New
Keynesian model. Kunieda (2015) applies Barro’s (2006) model to the risk of a natural disaster.
8
In reality, there are at least four issues related to government policy in normal times. First, thegovernment may increase taxes to reduce its debt. Bohn (1998) and Lo and Rogoff (2015) report
empirical results that governments tend to improve their fiscal balance in response to an increase in theirdebt (see also Aguiar et al. 2015, Bianchi, Hatchondo, and Martinez 2018). Second, the government may
impose several distortionary taxes. Third, government spending is counter-cyclical and tax revenue is
pro-cyclical. Finally, Gt may not necessarily be a transfer, but government spending on goods. However,these tax and spending policies in normal times do not affect our main results qualitatively because we
focus on how changes in tax policy from normal times to a crisis event influence economic activity.
Moreover, with regard to the first issue, the political costs are large, given that very few countries
implement policies that are sufficiently conservative to eliminate the possibility of a crisis. The third
issue strengthens our results quantitatively because the risk of a government debt crisis causes a larger
decrease in output and investment when GBO increases more rapidly. The fourth issue may have
implications for the optimality of tax smoothing. However, this does not matter for our main results
because our main analysis is not normative.

8

rather than government bonds. (ii) The government chooses either to impose taxes, asmuch as ω(BtG + Gt ), or not to impose taxes. (iiia) If the government chooses the latter,then maturing GBO holders refuse to purchase new GBOs. This results in the default
and zero value for government bonds. The government suffers considerable nonpecuniary
disutility. (iiib) If the government chooses to impose taxes of ω(BtG +Gt ), then no default
occurs. The government does not suffer disutility, and thus, chooses this option. Notethat in this environment, maturing GBO holders have no incentive to deviate fromthe strategy of demanding that the government exchange the maturing bond for real
goods. That is, if only one individual stops demanding the exchange and the others
continue with their demand, then he/she can still receive real goods, which does notchange his/her utility. However, if all holders stop demanding the exchange, then thegovernment will not impose taxes, which makes them unable to receive real goods anddecreases their utility. Thus, deviating from the strategy never increases bondholders’utility, and possibly decreases their utility, meaning that the strategy of demanding theexchange is weakly dominant.
The government debt crisis in our model is essentially a rollover crisis, in that the
government does not need to repay all of its debt. However, even part of its debt is likely
to be sizable because the Japanese government relies on the issuance of new bonds foras much as 30% of its annual expenditure. Thus, when a crisis causes a loss of market
confidence in government debt, it is plausible that the government will be forced tocollect extremely large tax revenues. Indeed, this was the case in post-war Japan.
A crisis has two main consequences. First, it forces the government to raise a substantial amount in tax revenue. The government imposes a once-off tax only at the timeof the crisis, where τtC is the consumption tax rate, τtK is the tax rate on capital stock,and τtG is the tax rate on GBO. Note that the latter two taxes are wealth taxes, not taxes
on the net returns from holding these assets. The tax on GBO is essentially equivalent
to a partial default (a full default when τ G = 1).9 The tax on capital stock, also calleda capital levy, plays the most important role in explaining persistent recessions.
Second, when a crisis occurs, banks become insolvent. Because banks and other financial institutions hold government debt as a large part of their assets, it is straightforwardthat a decrease in the value of government debt will make them insolvent. Then, alldepositors rush to withdraw early, and banks are forced to sell both productive capitaland government bonds to finance the early withdrawals (bank runs), causing fire sales of
productive capital. These fire sales misallocate capital stock within the economy, leading to its inefficient use and decreasing its value. From the household’s perspective, thisreduction in capital value works as a capital levy.
The probability that a crisis occurs in period t + 1 is pt = p(BtG /zt ) = Pr(xt+1 =9

Although our model is a real model without an explicit role for the nominal variables, we caninterpret a partial default in our model as a debt reduction by inflation tax or seigniorage.

9

1|BtG /zt ), where BtG is the quantity of government bonds and zt is productivity (TFP)
given by log zt+1 = log zt + µ + σe et+1 , where et+1 ∼ N (0, 1). This probability depends
positively on GBO divided by the TFP. We assume the following logistic form:
pt = p(BtG /zt ) =1,
1 + (1/d0 − 1)exp(−d1 BtG /zt )(1)

where d0 represents the crisis probability in the next period when there is no government
debt today, and d1 represents the steepness of the crisis probability curve with respect
to GBO divided by the TFP.10
The crisis probability, given by equation (1), is exogenous. Abstracting away frommodeling the government’s decision, we discuss the equilibrium dynamics of macroeconomic variables by focusing on optimizations by private agents.11

3.2

Household

There exists a unit mass of identical households. A household has the nonseparable
lifetime utility Ut , defined as1−ψ
Ut1−ψ = (1 − β)(Ctν (1 − Nt )1−ν )1−ψ + βEt (Ut+1
),
10

(2)

Specifically, we assume that pt becomes one (a crisis occurs for sure) if the tax rate on either capitalor GBO during the crisis becomes one or greater.
11
A crisis is more likely to occur as GBO increases (Reinhart and Rogoff 2010). Arellano (2008)
constructs a small open-economy model, showing that a default is more likely to occur in a recession,
which is consistent with the data. She points out that this result is due to the incomplete asset market,
whereas models based on a complete asset market tend to predict the opposite result. D’Erasmo andMendoza (2016) show theoretically that a default on domestic government debt is more likely to occur
when GBO increases and the tax revenue decreases. See also Aguiar, Amador, and Gopinath (2009)
and Arellano, Bai, and Mihalache (2018, 2019). In contrast, the probability of a default is arbitrary inthe self-fulfilling crisis model of Cole and Kehoe (2000). They consider an optimal government policy inwhich government debt is rolled over every period. In this case, there is a risk of a self-fulfilling default
caused by a loss of confidence in the government. However, this does not deny the assumption that aself-fulfilling crisis is more likely to occur as GBO increases.
Though the sovereign debt in these models is external debt, similar reasoning would hold for thedomestic debt in our model and lead to an endogenous crisis probability that is increasing in GBO. We
can consider the following decision problem for the government on whether to impose heavy taxes torepay its debt now or in the future. Here, we assume that the sunspot shock on the preference occurs
exogenously with a constant probability. The shock makes private agents demand real goods rather
than government bonds, regardless of the government’s decision. Given that the tax distortion grows
as the debt increases, the government faces tension between taxing today or in the future. Then, thegovernment optimally taxes if its debt exceeds an endogenous threshold. Adding a preference shock on
the government due to the stochastic event of losing office (Aguiar, Amador, and Gopinath 2009) would
lead to an endogenous crisis probability that the government imposes taxes today, which is increasing
in GBO. The crisis probability would be the sum of the exogenous probability that a preference shockon households occurs and the endogenous probability that the government chooses taxation. However,providing such a model with an endogenous crisis probability is beyond the scope of this study.

10where β represents a discount factor, ψ represents the intertemporal elasticity of thesubstitution of consumption, and ν represents the utility weight on consumption. The
utility function takes the standard Cobb–Douglas form in consumption Ct and leisure
1 − Nt , where Nt is the labor supply, which is consistent with balanced growth.
Owing to the inefficiency described below, the household holds certificates of bankdeposit Dt rather than capital stock Kt and government bonds BtG directly.12 The rateof return on the bank deposit is RtD (xt ). The budget constraint for the household is(1 + xt τtC )Ct + xt Tt + Dt+1 ≤ RtD (xt )Dt + Gt + Wt Nt ,(3)

where Tt is the lump-sum tax, Gt is the lump-sum transfer from the government, and
Wt represents the real wage. We assume that a technical constraint exists for the government’s ability to impose a lump-sum tax such that Tt cannot exceed a certain limit,
given by a parameter ω T , which we define shortly. Alternatively, Tt can be interpreted
as a decrease in the lump-sum transfer during a crisis, representing that the government
makes some transfers at all times and that in the crisis the government could cut thistransfer by Tt . The household maximizes the lifetime utility (2) subject to the budget
constraint (3). This optimization problem implies thatD
Et [Mt+1 Rt+1
(xt+1 )] = 1,

(4)

where Mt+1 is the stochastic discount factor:
Mt+1 = β

3.3

1 + xt τtCC
1 + xt+1 τt+1



Ct+1
Ct

ν(1−ψ)−1 

1 − Nt+1
1 − Nt(1−ν)(1−ψ)
.

(5)

Firm

There is a unit mass of identical non-financial firms that face perfect competition. Its
production function is Yt = Ktα (zt Nt )1−α , where Yt represents output. The static profit
maximization isπ(Kt , zt ; Wt ) = max {Ktα (zt Nt )1−α + (1 − δ)Kt − RtK Kt − Wt Nt },Kt ,Nt ≥0(6)

1which yields the return on capital RtK = 1−δ +αYt /Kt and Nt = Kt zt1−α (1 − α)/Wt α ,where δ represents the capital depreciation rate.
The household cannot operate Kt efficiently for two reasons. First, the household
can use only λYt Kt for production, where λYt < 1. Second, (1 − λDt )Kt is destroyed ifD
the household directly holds and uses Kt , where λt < 1. In a crisis, the household
12

The household can hold government bonds BtG , but chooses not to do so because the rate of returnGis higher for the bank deposit; that is, Et [Mt+1 RtD ] ≥ Et [Mt+1 (1 − xt+1 τt+1)/qtG ], in equilibrium. We
provide the definitions of the variables below.

11uses Kt inefficiently because banks become insolvent, whereas the banks operate capitalKt efficiently in normal times. Consequently, the return on capital RtK (xt ) becomes arandom variable that depends on xt :Yt (1)+ (1 − δ)λDt ,KtYt (0)RtK (0) = α+ (1 − δ),Kt
Yt (1) = (λYt Kt )α (zt Nt )1−α ,RtK (1) = αα1−αYt (0) = (Kt ) (zt Nt )

3.4

.

(7)
(8)
(9)
(10)

Bank

There is a unit mass of one-period-lived banks. A bank is born, collects deposits, andinvests them in capital stock and government bonds at the end of period t to maximize
the return on the deposit. It then pays out all the return to depositors and ceases tooperate in period t + 1. The deposit contract can be contingent on zt+1 , but not on xt+1 .
D(0), which is the return on a deposit when xt+1 = 0:
The banks can offer only Rt+1
D
Rt+1
(0) =

G
K
(0)Kt+1 + Bt+1
Rt+1,
G
Kt+1 + qtG Bt+1

(11)

where qtG is the price of government bonds. The deposit contract is such that depositors
D(0)Dt+1 whenever they want during period t + 1, even before output
can withdraw Rt+1
13
is produced. When xt+1 = 0, they withdraw after output is produced in equilibrium.14
When xt+1 = 1, the banks adopt the first-come-first-served principle for depositors.
D
Thus, depositors can withdraw the full amount of their deposits Rt+1
(0)Dt+1 at any timein period t + 1, as long as the bank has assets remaining. However, they receive nothing
if the bank pays all of its assets out to early withdrawers. Thus, when xt+1 = 1, all13

We impose the incompleteness such that the deposit contract cannot specify the timing of withdrawal. Diamond and Dybvig (1983) provide a model that justifies such a contract as constrained
optimal. We impose incompleteness as a constraint rather than providing a Diamond–Dybvig typemodel to simplify our analysis.
14
If one depositor withdraws early and all the other depositors do not, then the depositor can withdraw
the full amount. If all depositors withdraw early, then the bank is forced to sell their asset (physicalcapital) to households at a discounted price because households are inefficient in operating capital.
Then, the depositors cannot withdraw the full amount of their deposits. Therefore, the depositors’
dominant strategy is to withdraw after output is produced, and all depositors choose to withdraw latein equilibrium. Rigorously speaking, there exists another equilibrium in which all depositors withdraw
early before output is produced when xt+1 = 0 because withdrawing early is the best response for adepositor given that all other depositors withdraw early. Following Allen and Gale (1998), we exclude
this equilibrium since it is Pareto inefficient, whereas the equilibrium where all depositors withdraw lateis Pareto efficient.

12

depositors try to withdraw before production and all capital is paid out to households.
As the households produce output inefficiently, the average return on a deposit decreases
to
K
K
G
G
(1 − τt+1)Rt+1(1)Kt+1 + (1 − τt+1
)Bt+1
D
.
(12)
Rt+1
(1) =
G
Kt+1 + qtG Bt+1
Given deposit Dt+1 , the bank’s problem isD(xt+1 )]Dt+1 =Et [Mt+1 Rt+1

G
G
K
K
}],
)Bt+1
(xt+1 )Kt+1 + (1 − xt+1 τt+1
)Rt+1
max Et [Mt+1 {(1 − xt+1 τt+1

G
Kt+1 ,Bt+1(13)G+ Kt+1 , where qtG is the price of
subject to the balance sheet constraint Dt+1 = qtG Bt+1
the government bonds. Then, the first-order condition with respect to Kt+1 implies thatK
K
Et [Mt+1 (1 − xt+1 τt+1
)Rt+1
(xt+1 )] =1
G
Et [Mt+1 (1 − xt+1 τt+1
)].
qtG

(14)

Furthermore, we assume that the inefficiency λYt at the crisis is determined endogenously by
λYt = F (ξt ),(15)

where F 0 (ξ) < 0 and ξt represents the ratio of nonperforming loans to assets:ξt ≡ 1 −(1 − τtK )RtK (1)Kt + (1 − τtG )BtG.
RtK (0)Kt + BtG(16)This assumption is based on the premise that production activity by the household
necessitates bank deposits as a means of payment, while the function of the means of
payment is disrupted if the ratio of nonperforming loans is large. In the next section, we
provide empirical evidence and calibrate parameter values for this specification.
We define RtG ≡ 1/qtG , and call RtG − 1 the government bond yield. The price ofgovernment bonds, qtG , is normally less than one because Mt+1 is less than one.

3.5

Government

The government spends through a lump-sum transfer Gt > 0, which has a constant ratio
to the ratio of TFP, zt , for all xt . Tax is zero if xt = 0 and non-negative if xt = 1. The
government budget constraint isG+ xt τtC Ct + xt τtK RtK Kt + xt τtG BtG + xt Tt = BtG + Gt .
qtG Bt+1

13(17)

To determine the tax rates, we assume a tax weight of ω i (i = C, K, G, T ), which isexogenous and satisfies
τtC Ct=ω C (BtG + Gt ),(18)

τtK RtK Kt=

ω K (BtG + Gt ),(19)

τtG BtG=ω G (BtG + Gt ),

(20)

Tt

=

ω T (BtG + Gt ),(21)

0 < ω ≡ ω C + ω K + ω G + ω T ≤ 1.

(22)

If ω C + ω K + ω G + ω T = 1, the government issues no new bonds; in other words, it owes
nothing after the crisis period.
Although we assume a tax rate of zero in non-crisis periods (xt = 0), this assumption
is not critical to our results as long as the government collects insufficient tax to coverits expenditure.

3.6

Market Clearing

The goods market is cleared whenYt = Ct + It ,(23)

where investment It equals Kt+1 − (1 − δ)Kt .
The labor market is cleared when
(1 − α)3.7Yt
1 − ν (1 + τtC )Ct= Wt =.
Ntν
1 − Nt(24)

Equilibrium

In the economy, the state variables are {Kt , BtG , xt , zt }; however, as in Gourio (2013),
G
the equilibrium can be expressed using {kt ≡ Kt /zt , bGt ≡ Bt /zt , xt }. See Appendix A
for the calculation of the equilibrium.
In the model, the fear of a government debt crisis can cause persistent stagnation
beforehand when people share the expectation that a considerable distortion occurs at
the time of a crisis. In particular, two distortions are important. The first is a capital
levy, a one-time tax on all wealth holders with the goal of retiring government debt,
denoted by ω K or τtK . The second is capital-use inefficiency resulting from bank runs,
K
denoted by λYt and λD
t . Both distortions increase the required return to capital, Rt+1 (0),in a normal state and, in turn, dampen capital investment before a crisis occurs.

14

3.8

How the Crisis Risk Affects the Economy

Before describing our simulation, it is useful to discuss how the crisis risk affects the
economy in the model. The most important equation, equation (14), is related to thebank’s optimization problem with respect to investing in either capital or government
bonds. We can rewrite this asK(1 − pt )Et Mt+1 Rt+1
(0)





1
G
K
K
= G Et Mt+1 (1 − pt ) + pt (1 − τt+1
) − pt Et Mt+1 (1 − τt+1
)Rt+1
(1)
qt



1
G
)
= G Et Mt+1 (1 − pt ) + pt (1 − τt+1
qt


K
K
(1)
)Rt+1
− pt Et [Mt+1 ] Et (1 − τt+1

KK
− pt Covt Mt+1 , (1 − τt+1
)Rt+1
(1) .

(25)

K
This equation suggests that the return on capital, Rt+1(0), in normal times (xt+1 = 0)
increases as GBO increases through the following three factors. Note that a higher
K(0) decreases the desired level of capital, and in turn, investment, before a crisis
Rt+1
occurs through equation (7).
K
(0).
First and most important is that the expectation of a capital levy increases Rt+1
K
K
Unless ω = 0, τt+1 during a crisis is positive and increases as GBO increases. Thus, theKsecond term on the right-hand side of the equation increases, which increases Rt+1(0) onthe left-hand side of the equation. Banks require a higher return on capital investment.
K
Second, the expectation of a higher crisis risk increases Rt+1(0). As GBO increases,
K
K
K
pt increases. As long as Rt+1 (0) > (1 − τt+1 )Rt+1 (1); that is, when the return on capitalin normal times is higher than that during a crisis,15 an increase in pt leads to an increaseKin Rt+1(0) through the second term on the right-hand side of the equation.16
Third, banks require a higher risk premium. As GBO increases, the stochastic discount factor Mt+1 increases, while the post-tax return on capital decreases. This negativeK
comovement increases Rt+1(0) through the third term on the right-hand side of the equation.
This clarification is similar to that in Bocola (2016). Bocola (2016) decomposes theexcess return on bank loans to a liquidity premium and a risk premium. The former
K
K
K
)Rt+1
(1) is not restrictive because it holds when there exists
The condition that Rt+1(0) > (1 − τt+1
K
either a capital levy (ω > 0) or capital-use inefficiency (either λY < 1 or λD < 1).
16
K
The increase in Rt+1(0) is partially offset by the first term on the right-hand side of the equation
GKG
if τt+1 > 0; that is, ω G > 0. When both τt+1
and τt+1
are positive, the credit risk decreases returns
from investing in capital and government bonds. If the latter decrease is larger, then banks will chooseK(0). However, this is likely of
to invest in capital rather than government bonds, which decreases Rt+1
G
decreases (if Gt = 0, it is
low possibility, especially when GBO is large because as GBO increases, τt+1
K
constant), while τt+1 increases. Thus, the credit risk increasingly affects return on capital, rather thanreturn on government bonds, as GBO increases.
15

15

is similar to the first and second factors in our model while the latter corresponds tothe third factor in our model. However, the source of the loss of return during a crisis
is different. It is banks’ funding constraints in Bocola (2016), while it is a capital levy(ω K > 0) in our model. Moreover, there is no riskless asset in our model, which makes
it difficult to define a premium by comparing the return on capital to a riskless rate.
K
Finally, it should be noted that the above equation influences not only Rt+1(0) but
G
also qt . Thus, the three factors we discussed also influence the government bond yield
differently and the government bond yield may increase or decrease as GBO increases.
It likely decreases when the distortion associated with the capital stock increases. In
other words, banks prefer to hold government bonds rather than capital. This makes
the government bond yield lower due to the no-arbitrage condition between capital andgovernment bonds.
In Appendix B, we provide more rigorous analytical results using a simplified twoK
and Ktperiod version of the model. For example, we show that the derivatives of τt+1
with respect to GBO are non-negative and non-positive, respectively. The derivative of
qtG with respect to GBO can be both positive and negative depending on the relative
sizes of τ K and τ G .

4SIMULATION

4.1

Simulation Method

Time is measured in years. The equilibrium can be expressed using {kt = Kt /zt , bGt =G
Bt /zt , xt }. In what follows, we denote the variables divided by zt using lower-case lettersν(1−ψ)(e.g., yt = Yt /zt ), with the exception of ut , which is defined as Ut1−ψ /zt. Note thatG
the GBO, bt , does not explode, although there is no deterministic steady state. Equation(1) indicates that the crisis probability increases with the increase in bGt , provided d1 > 0,
G 17
which stabilizes bt .
Following Gourio (2013), we solve the model by employing a projection method. The
policy functions are approximated using two-dimensional Chebychev polynomials withrespect to kt and bGt , with degree five in each dimension.
17

G
For the numerical calculation, we set an upper bound on the value of bGt ; that is, b̄ = 0.85. Thissetting implies that the government imposes a lump-sum tax when necessary to keep bGt within theupper bound; hence, this satisfies the transversality condition. According to the simulation, the meanof yt is 0.39, and thus, b̄G implies that the highest debt-to-GDP ratio is approximately 2.2, whereas it
is currently around 1.2 (net).

16

4.2

Calibration

Table 1 shows the benchmark parameter values we use for the simulation. Many arestandard and based on Gourio (2013); others are chosen to fit Japan, and are based on
Sugo and Ueda (2008) and Hirose and Kurozumi (2012). The TFP trend growth rate,µ, is 0.0182, which is the actual annual growth rate of real GDP per capita from 1990 to2019 for Japan (see Figure 1).18 The standard deviation of the productivity shock σe is0.023, calibrated to match the standard deviation of the change in output (∆logYt ) forJapan. Government spending g = Gt /zt is chosen as 0.014, consistent with the speed ofJapan’s government debt accumulation.19
There are three main parameter categories, each of which needs careful calibration:
the tax policy during the crisis (ω i (i = C, K, G, T )), capital-use inefficiency during thecrisis (λYt and λDt ), and probability of the crisis (d0 and d1 ). We explain our calibration
strategy below.
4.2.1

Tax in a Crisis

As in Kozlowski, Veldkamp, and Venkateswaran (2020), we assume that expectations of
the tax policy during a crisis, ω i (i = C, K, G, T ), are formed by learning from history.
A crisis is a typical tail event, about which agents in the economy have no knowledge
besides historical episodes of similar events. Here, the capital levy ω K is particularly
important, so we review episodes in interwar Europe and post-war Japan.
Eichengreen (1989) notes that prominent British economists and policymakers debated the use of a capital levy in the 1920s when the UK government was suffering fromthe public debt overhang due to the war debt from World War I. This active debate about
the capital levy exemplifies the strength of the ex post temptation for policymakers to
introduce a one-time capital levy when government debt builds up.20 Similar debates
took place in Italy, Czechoslovakia, Austria, Hungary, Germany, and France. Although
Eichengreen concludes that the capital levies in those countries failed,21 if history repeats
18

As Gourio (2012, 2013) and Isore and Szczerbowicz (2017) argue, occasional disasters affect thecalculation of a trend from data. However, Japan did not experience a government debt crisis between
1975 and 2019, and thus, we do not exclude any sample observations when calculating the trend.
19
The mean of 0.39 for yt in our simulation implies that the ratio of government spending to GDP(g/yt ) is approximately 3%. Note that Gt in our model corresponds to government expenditure minus
tax in normal times. According to the OECD, the mean of Japan’s actual general government net
lending as a percentage of GDP is −5% for 1990–2019. Given the considerable size of the volatility forthis variable (the standard deviation is 3 percentage points), the size of g is comparable with the actual
size of the government deficit.
20
Eichengreen (1989) states that, “[i]n modern times, capital levies have come under consideration
following every period of major military expenditure and rapidly rising debt/income ratios. (...) Noneof these proposals was adopted. For examples where capital levies were actually implemented, we mustturn to the 20th century.”
21
Eichengreen (1989) finds that Italy and Czechoslovakia were the closest to success. A factor in the17itself or people learn from previous events, these episodes suggest that some people anticipate the capital levy when a crisis strikes. Moreover, post-World War II Japan is anoteworthy example of a successful implementation.
In Japan, the government debt inherited from the wartime period amounted to 267%of the national income in 1944, more than 99% of which was internal debt (MOF 1976,
Kawamura 2013). Although Eichengreen (1989) emphasizes that the absolute power of
the Supreme Commander of the Allied Powers that occupied Japan was crucial in the
successful implementation of the capital levy of 1946–47, the MOF (1976) and Kawamura(2013) show memorandums that prove that it was the MOF that decided to impose acapital levy to avoid an outright default of government debt. The Supreme Commander
actually recommended declaring a partial default. The capital levy, or wealth tax, taxed
all real and financial assets owned by Japanese residents, including land, houses, government bonds, bank deposits, and machinery. Tax rates varied progressively from 25%
to 90% depending on the taxpayer’s income class. The capital levy worked effectively to
reduce wealth inequality among the Japanese.22 “With important elements of democracy in suspension, the levy could be quickly and effectively implemented” (Eichengreen,1989) using the deposit blockade and withdrawing the legal tender status of old yen.
The package of these policies, which could not have been implemented in normal times,
helped the Japanese government seize domestic wealth efficiently.23
We use this post-war Japanese experience reported by the MOF (1976) to calibrate
the parameters of the tax weights ω i (i = K, C, G, T ). The government debt outstanding(B) was 215,867 million yen at the end of 1945, and the GBO and war indemnity (T G )were 199,454 million yen (Chapter 11 in the MOF 1976) and 16,413 million yen (Chapter7), respectively. The government owed war indemnity because it suspended payments ofsubsidies, loss compensation, wartime insurance, evacuation costs, and so on. This war
indemnity was virtually unpaid because 100% of the tax was imposed under the War
Indemnity Special Measures Law. The share of the unpaid debt constitutes ω G ; thus, we
obtain the fraction of the partial default as ω G = T G /B = 0.076. The capital levy (T K )failure of the capital levies in these countries was the democratic decision-making processes because
political resistance from property owners led to extreme delays and opportunities for capital flight.
22
According to the MOF (1976), the main aim of the capital levy was to avoid default, whereas theSupreme Commander was attempting to reduce inequality. Kawamura (2013) points out that although
the super rich class was taxed most heavily, the middle class paid the largest proportion of the total
revenue generated by the capital levy.
23
Saito (2017) points out that the exchange of old yen for new yen was an effective way for the government to seize private assets concealed on the black market. However, note that the tax revenue fromthe capital levy was less than the amount needed to restore the sustainability of government debt. By
contrast, Hattori and Oguro (2016) estimate that the seigniorage revenue resulting from hyperinflation
immediately after World War II was nearly 29% of the GDP. However, this large seigniorage revenue
was not intended by the government or MOF officials. Rather, the MOF was concerned about the riskof hyperinflation, as in Germany, which prompted it to introduce the capital levy (MOF, 1976).

18

was 43,500 million yen (Chapter 7); thus, we obtain ω K = T K /B = 0.202. For ω C , we use
the fact that the government increased personal income tax and corporation tax to raise
revenues by T C = 3, 907 million yen in 1946 (Chapter 7). Considering that consumption
tax plays virtually the same role as income tax, we calculate ω C as T C /B = 0.018.
Finally, we calibrate ω T based on the decrease in government expenditure under thegovernment consolidation plan in 1946 (T T = 3, 398 million yen); we find no evidence ofan increase in lump-sum tax collection. Then, we obtain ω T = T T /B = 0.016.
4.2.2

Capital-Use Inefficiency in a Crisis

We calibrate the parameters associated with the capital-use efficiency of production λYt
and depreciation λD
t . In the former case, we use the database on systemic banking crises
from 1970 to 2017 constructed by Laeven and Valencia (2018). We estimate the following
equation using the OLS method:∆Yi= θ0 + θ1 log(1 + ξi ) + εi ,
Yi

(26)

where ∆Yi and ξi represent the output loss and the ratio of nonperforming loans to totalloans, respectively, for crisis episode i, and εi is a residual. The output loss is measured
as the cumulative sum of the differences between the actual and trend real GDP for
a period of four years after a crisis, divided by four (i.e., average annual decrease in
output). The coefficient θ0 indicates the size of the output loss in the event of a banking
crisis when there are no nonperforming loans, whereas the coefficient θ1 indicates theelasticity of the output loss to the ratio of nonperforming loans to total loans. We
use 29 of the 151 systemic banking crisis episodes by restricting our estimation to theOECD member countries when a crisis occurred. Figure 5 shows the scatter plot of theresults, where each circle indicates a systemic banking crisis episode. The estimates of θ0and θ1 are 0.059 and 0.202, respectively; their standard errors are 0.016 and 0.097, andtheir p-values are 0.0013 and 0.048, respectively. This result suggests that a significant
output loss occurs in the event of a banking crisis. Furthermore, as nonperforming loans
increase, the degree of the loss increases significantly at the 5% level.24 Combining thisresult with the production function (9) in a crisis, we formulate λYt as follows:
λYt = F (ξt ) = (1 − θ0 − θ1 log(1 + ξt ))1/α ,(27)

where we assume that the crisis does not affect labor Nt , which is a quantitatively good
approximation in our simulation.
D
D
We assume that λD
t is constant (λt = λ ) because we are unable to find informative
empirical evidence. We determine the parameter value using Japan’s Flow of Funds
24

When we use the following equation: ∆Yi /Yi = θ0 + θ1 ξi + εi , the estimates of θ0 and θ1 become
0.062 and 0.154, respectively; their standard errors are 0.015 and 0.079, and their p-values are 0.0005
and 0.064, respectively.

19

Accounts compiled by the Bank of Japan. When Japan’s systemic banking crisis occurredin 1997 (recorded in Laeven and Valencia 2018), the liability of non-financial private
corporations decreased by 6.074%.25 Considering that this corresponds to a change inthe private firms’ real firm value, we obtain λD = 1 − 0.0607.
4.2.3

Probability of a Crisis

We need to calibrate two parameters associated with the probability of a crisis, d0 and
d1 , in equation (1). For domestic debt defaults, empirical data are scarce, as shown
by Reinhart and Rogoff (2010). However, as we discussed in the Introduction, several
sources of data indicate a mounting concern about a government debt crisis in Japan.
Of these, we employ the CDS spread as a target variable to calibrate d0 and d1 , mainly
because it can be linked numerically to the probability of a crisis. Based on our model,
the CDS spread for the five-year maturity can be priced as follows:26
" j
!#
5
X
Y
Mt+k (xt+k = 0)(1 − pt+k−1 )
CDS
Et
j=1

= (1 − R)

k=1
5
X
j=1

"
Et

j−1
Y

!
Mt+k (xt+k = 0)(1 − pt+k−1 )#· Mt+j (xt+j = 1)pt+j−1 ,

(28)

k=1

where pt and R represent the arrival rate of a crisis in period t + 1 and the government
bond recovery rate during a crisis, respectively. The left-hand side of the equation
represents the present value of the premiums paid by the protection buyer contingent
upon a crisis event not occurring. The right-hand side of the equation represents thepresent value of the payment by the protection seller contingent on a crisis event. Notethat the stochastic discount factor Mt+1 (xt+1 ) changes depending on whether a crisis
occurs in t + 1 (i.e., xt+1 = 1) or not (i.e., xt+1 = 0). Because ω G represents the degreeof partial default, the recovery rate of government bonds R equals 1 − ω G .
Using this formula and the observed CDS values, we calibrate d0 and d1 . Specifically,
we target the following two variables. The first target variable is the level of the CDS
spread when the debt ratio is low. Because CDS data are available only from 2003 andthe global financial crisis around 2008 led to a surge in the CDS spread, we use theaverage value for 2003–2006, which is 0.008 (i.e. 0.8 percentage points). The second
target is the elasticity of the CDS spread to the debt ratio. As we stated in Section 2.1,
the estimate of the elasticity is approximately one.
Based on the model, we calculate the level and elasticity of the CDS spread for the
given values of d0 and d1 . To this end, we first generate the path of the CDS spread as well25

At the end of March 1997, private bank borrowings and equity were 400 and 375 trillion yen,respectively. At the end of March 1998, they had decreased to 389 and 339 trillion yen, respectively.
26
Refer to Pan and Singleton (2008) and Longstaff et al. (2011).

20

as the debt ratio.27 Second, we calculate the average CDS spread when the simulated
debt ratio is low, which we compare with the first target variable above.28 Third, we
regress the logarithm of the simulated CDS spread by the logarithm of the simulated
debt ratio using the OLS method to obtain the elasticity of the CDS spread.
We find that the combination of d0 = 0.025 and d1 = 4 is the best fit to the data.
This makes the level and elasticity of the CDS spread 0.008 and 0.96, respectively. The
surveys by Morikawa (2016, 2017) show that the average crisis probability from 2016 to2030 is around 0.02 (∼ 1 − (1 − 0.25)1/15 years ). While this value is comparable with thevalue of d0 (i.e., the crisis probability when the debt ratio is low), the former is much
lower than the crisis probability simulated for the same period (i.e., when the debt ratio
is above 1.2); that is, 0.19. In contrast, Hoshi and Ito (2014) imply that there is a fairly
high crisis probability because the amount of government debt will exceed that of private
sector financial assets within 10 years. Our model simulation shows that the probabilityof a crisis in 2024, 10 years after their study, is 0.29. This suggests that our benchmark
calibration lies between those of Morikawa (2016, 2017) and Hoshi and Ito (2014).

4.3

Simulation Results Based on Standard Sovereign Default
ModelsIn this subsection, we investigate the extent to which a standard model of sovereigndefault can or cannot explain Japan’s secular stagnation. Our model becomes equivalent
to a standard model when the government imposes tax only or heavily on GBO during
a crisis. Specifically, we set ω G = 0.2 and ω C = ω K = ω T = 0, which implies that the
government declares a partial default for 20% of its debt.
Figure 6 shows the simulation results based on this model (the solid line) and thedata for Japan from 1990 to 2019 (the solid line with circles). For the simulation, we
assume that normal times, xt = 0, continue from t = 1 to 35 years. Log productivity,
logzt , grows with the speed of µ and no shock (et = 0). We set the initial value of bGt
to 0.05 to match the actual value in 1990. The initial value of kt is set as k0 = kSS e0.11 ,where kSS is the steady-state value of capital stock. This is calculated as the mean of
the time-series paths of kt obtained using the method in Subsection 4.4.1. The valueof e0.11 comes from the actual log deviation of the capital stock from its linear trend in
1990, based on the JIP 2015 Database.
For comparison, we plot the actual paths for Japan from 1990 to 2019 as the solid
27

See Section 4.4.1 for the detailed simulation procedure. In the CDS spread simulation, we assume
that the future productivity shock, et+1 , is zero for simplicity. In other words, whether a crisis occurs
or not is the only uncertain event.
28
To be precise, we calculate the average when the simulated debt ratio falls within the range of theactual debt ratio for 2003–2006 (i.e., between 0.66 and 0.73). The average value hardly changes when
we calculate the average CDS spread when the simulated debt ratio is below the 2006 level (i.e., 0.73).

21

line with circles. Here, real GDP and investment per capita are shown as deviations fromthe linear trends of the respective variables from 1975 to 2019 (see Figure 1). Output(yt ) and investment (it ) based on the model are detrended by zt , and then demeaned sothat their average from 1990 to 2019 equals that in the data for the same period.
The figure shows that as the debt-to-GDP ratio (bG /y) increases, output (y) decreases,
which is consistent with the data. Output decreases because capital-use efficiency in acrisis (λYt ) deteriorates as banks hold nonperforming assets if a crisis occurs. However,the size of the output decrease based on the simulation is smaller, and investment (it ) is
more or less unchanged. The more important simulation results are that the government
bond yield (RG − 1) increases while the credit spread (RK − RG ) decreases, both ofwhich are inconsistent with what we observe in the data. The gap in the government
bond yield between the model and data widens in the 21st century. As the government
bond yield increases, the CDS spread and the probability of a crisis increase, both at anaccelerated pace.29
These simulation results emerge because investment in government bonds becomesriskier as the probability of a crisis increases, while investment in capital becomes relatively less risky. Thus, the government bond yield increases and the credit spreaddecreases, while investment does not decrease much. Introducing a capital levy (i.e.,ω K > 0) is a simple and powerful way to improve the model fit, as we see in the next
subsection.

4.4

Simulation Results Based on the Benchmark Model

Now we assume the tax weights of ω K = 0.202, ω C = 0.018, ω G = 0.076, and ω T = 0.016
that are calibrated in Section 4.2.1.
4.4.1

Moments of Variables: Does the Model Have a Good Fit?

In what follows, we discuss two sets of simulation results. The first simulation checks
the fit of our model with the data by comparing the key first and second moments ofvariables, such as the mean of the debt-to-output ratio, the standard deviation of thechange in consumption, the correlation coefficient between the change in output and thedebt-to-output ratio in the previous period, and the correlation coefficient between thechange in investment and the credit spread.
Using the model, we calculate the time-series paths of key economic variables. We
generate the time-series path of TFP (zt ) for t = 1, 2, · · · , 45 years. The initial values ofthe state variables are bG
0 = 0.05 for GBO, which we set to be constant with the value
29

The size of ω G that we assume here may be modest because in principle we can consider the caseof a full default (i.e., ω G = 1). We find that a higher value of ω G accelerates changes in bG /y, RG − 1,
the CDS spread, and the probability of a crisis, so that the probability becomes one before 2019.

22

until 1990, and k0 = 0.91, which is the midpoint of the range of k in our numericalcomputation. The simulated key economic variables include the debt-to-output ratio(bGspreads (RtG − 1 andt /yt ), the change in log output (∆logYt ) , and the interest-rate DK(0) −RtG ). The(0) −RtG ). In addition, we simulate the deposit spread (Et Rt+1
Et Rt+1
moments are calculated by excluding the crisis periods because Japan has not experienced
a government debt crisis for more than half a century. For the same reason, we calculateDK(xt+1 ) in the normal state (xt+1 = 0). We set(xt+1 ) and Rt+1
the interest rates Rt+1
xt = 0 for t = 1, 2, · · · , 45; agents in the model are prepared for the risk of a crisis
according to equation (1). We repeat this calculation 50 times.
We tabulate the actual moment values for Japan from 1975 to 2019, except for thecredit and deposit spreads, which run from 1993 to 2019 and 1988 to 2019, respectively.
We express government debt in net rather than gross terms for BtG , except for the figures
shown in parentheses. The spread data are the same as those reported in the left-handpanel of Figure 4. The deposit rate is the interest rate on the five-year fixed-term deposit.
Table 2 compares the moments in the actual data with those based on the benchmark
model. Although we do not target moments other than σ(∆logYt ), the fit of the model isgood. The mean and standard deviation are of the same order as those in the data. The
main caveat seems to be that in the benchmark model, the mean simulated government
bond yield, (RtG − 1), is 4.4 percentage points, which is 2.5 percentage points higher than
the actual mean. In other words, the actual interest rate is lower than that suggested by
the actual TFP growth rate for Japan. In reality, government bonds act as a medium
of exchange, like money, in that they serve as collateral in short-term transactions in
the interbank market. This should add a liquidity premium to the price of government
bonds, decreasing the government bond yield. One of the major causes of the discrepancy
between the simulated and actual bond yields may be that our model does not consider
such a liquidity premium.
Comparing the correlation coefficients, we find that the benchmark model successfully
explains the negative coefficients between output growth and both the ratio of GBO(model −0.24 and data −0.30) and the credit spread (model −0.19 and data −0.15).
Furthermore, output growth is positively correlated with the government bond yield(model 0.35 and data 0.28). Investment growth has a similar correlation coefficient withthe credit spread and the government bond yield, although that between investment
growth and the ratio of GBO is almost zero in the data. According to the model, thedeposit spread is negatively correlated with output and investment growth; however, thisis not the case in the data.
4.4.2

Time-series Path: Does the Model Explain the Persistent Stagnation?

The second simulation is similar to that in Section 4.3. We generate the time-series
paths of the economic variables by assuming that the crisis indicator xt has an exogenous
23

path. Then, we discuss whether our model can explain Japan’s persistent stagnation.
We assume that normal times (xt = 0) continue from t = 1 to 39 years, then a crisis(xt = 1) occurs at t = 40, followed by xt = 0 thereafter.30
Figure 7 shows that the simulated path of the government debt-to-GDP ratio (bGt /yt ;
solid line) is similar to the actual path (solid line with circles). This is not surprisingbecause we choose government expenditure g so that the two paths match.
Our model accounts for nearly half the decrease in output (yt ) during the first two
decades of the 21st century. Output decreases between 2000 and 2019 as capital investment (it ) slows down in this period. The model accounts for actual investment during
2000–2019 fairly well. Investment decreases because agents become increasingly cautious
about a future crisis, for two reasons. First, the crisis probability increases with GBO.
Second, given the crisis probability, agents anticipate a greater capital tax rate during
a crisis (τtK ) as GBO increases because the government will need greater tax revenue torepay its increased debt. Furthermore, banks’ losses during a crisis (ξt ) increase as GBO
increases, which worsens capital-use efficiency (λYt ). The higher capital levy and lower K
capital-use efficiency both lead to an increase in the credit spread (Et Rt+1(0) − RtG )in normal periods, which further discourages capital investment as GBO accumulates.31
Last, but not least, the model shows that the bond yield (RtG − 1) decreases rather
than increases with GBO. Although a tax on government bonds is essentially a default,
the parameter value of ω G is low in our benchmark simulation. Thus, agents invest
more of their savings in government bonds and less in capital. In other words, Japanesepeople do not anticipate a large-scale government bond default, although they do fear
tax increases when a crisis strikes. This pattern is consistent with the Japanese data.
The decrease in the bond yield in the model is similar to that in the data. However,there are caveats. The level of RtG − 1 based on the model is several percentage points
higher than that in the data, as we discussed in Section 4.4.1. Moreover, in the model,
30

The agents in our model do not know this predetermined event; instead, they form expectations on
the crisis probability based on equation (1). Thus, the simulated paths do not change until year t = 39,
as long as the crisis occurs at t = 40 or later. The longer the delay before the crisis, the larger will be
the crisis effect because the government introduces larger tax increases.
31
In the model, we assume, for simplicity, that a crisis ends in one period, unless xt happens to be
one in two or more consecutive periods. Our results remain quantitatively similar, even if we assume
that a crisis continues for more than one period, as long as the distortionary taxes during that crisis
have a similar discounted present value. This is a type of the Ricardian equivalence, in that the timing
or duration of a tax imposition does not affect the ex ante slowdown in investment and output, as longas the total tax distortion remains the same.
Output based on the model increases during the 1990s when GBO is low. This is driven by an increasein consumption. In the 1990s, the probability of a crisis is low and, even if a crisis occurs, the resulting
distortion caused by the capital levy and capital-use inefficiency is small. As a result, the householdchooses to consume in the current period rather than save. In reality, the decrease in actual output
during the 1990s might have been caused by factors outside our model, such as the collapse of theasset-price bubble in the early 1990s and the subsequent decade-long period of financial distress.

24

 K

the increase in the credit spread (Et Rt+1(0) − RtG ) is more sluggish in the first two
decades and steeper in the subsequent decade than the data suggest.

4.5

Differing Expectations of Tax Scenarios in the Crisis Period

In the benchmark model, we assumed tax weights of ω K = 0.202, ω C = 0.018, ω G =0.076, and ω T = 0.016 as the government tax policy following a government debt crisis. In
this subsection, we investigate how the simulation results change when Japanese people
have differing expectations of tax scenarios at the point of a crisis. This exercise helps
clarify the underlying transmission mechanism caused by the taxes. We illustrate theresults in Figures 7 and 8.
We begin by considering a first-best case in which the government imposes a lumpsum tax (Tt ) only. The size of the tax collection is the same as that in the benchmark
case, namely ω T = 0.312. Because banks remain solvent, bank runs do not occur, andcapital use is efficient (i.e., λY = λD = 1). This hypothetical case partly reflects adecrease in government transfers (e.g., a decrease in pension obligations or medical care)during a crisis. “All T tax” in Figure 7 indicate almost no change in real economic
activity as GBO increases. Slight changes occur because the capital stock starts fromk0 = kSS e0.11 . Note that the growth rate of the debt ratio (bGt /yt ) slows down because
output no longer decreases.
In order to investigate the effects of various taxes independently, we set the tax weightiω (i = K, C, G) at zero, one by one, and simulate the time-series paths of the economic
variables. By the size of the decrease in ω i , the value of ω T increases to maintain the
total amount of tax collected during a crisis. The results are shown in Figure 8. When
we eliminate the capital levy during the crisis (“No K tax” in the figure), the decreases in
output and investment are greatly mitigated, and the credit spread decreases, rather thanincreases. In this scenario, capital use during the crisis remains inefficient (λY , λD < 1),which contributes to a decrease in investment. However, the figure suggests that thecontribution of capital-use inefficiency is quantitatively much smaller than that of thecapital levy. The capital-use efficiency for output (λYt ) deteriorates only slightly because
the ratio of banks’ nonperforming loans does not increase much without the capital levy.
“No G tax” in Figure 8 shows that eliminating the tax on GBO during a crisis
causes little change, particularly before 2010. However, the loss of fear of a government
bond default decreases the government bond yield, which slows the accumulation of thedebt ratio. This reduces the crisis probability compared with that of the benchmark
model, which mitigates the output stagnation after 2010. “No C tax” in the figure
shows that eliminating the consumption tax during a crisis does not alter our simulation
results considerably. In our calibration, both ω C and ω G are small, and thus, they havequantitatively limited effects.
There are many alternatives to the tax policies considered above. One of the most
25

 K

realistic options is an income tax imposed on the return on capital (e.g., Et Rt+1(0) − 1in our model), return on government bonds (e.g., RtG − 1), or labor income (e.g., Wt Nt ).
However, such taxes on incomes (flow variables) are insufficient to repay government
debt because their revenue cannot exceed the income. Thus, expectations of income
taxes have small quantitative effects on pre-crisis output. Qualitatively, the effect of acapital income tax is similar to that of a capital levy, and the effect of a tax on theinterest on government bonds is similar to that on GBO. The effect of a labor income
tax is similar to that of a consumption tax, both qualitatively and quantitatively.

5

FURTHER ANALYSES

5.1

A Smaller Loss of Capital-Use Efficiency

In our model, Japan’s persistent stagnation is caused by two main expectations: thecapital levy and capital-use inefficiency at the time of a crisis. In Figure 9, we investigate
the latter effect in more detail by assuming a smaller loss of capital-use efficiency. To doso, we first assume no additional capital depreciation during the crisis (i.e., λD = 1), andwe keep the production inefficiency (λYt ) at the same value as the benchmark. The dottedline shows that the pace of the government bond yield decrease and credit spread increase
becomes slower. Moreover, the size of investment decrease declines, which suggests thatthe capital-use inefficiency reinforces the adverse effect of GBO on capital investment.
We observe an increase, rather than a decrease, in output one period after the crisis
occurs because in this case, the crisis does not depreciate capital.
Second, we assume that θ1 is zero. Although we obtained a positive estimate for theelasticity of output loss to the ratio of nonperforming loans to total loans, it was not
significant at the 1% level. To check the robustness of our results, we simulate the modelby setting θ1 = 0. The simulated paths, shown as solid lines, are almost the same as
those based on the benchmark model.

5.2

Low and Constant Crisis Probability

We examine the robustness of our results to changes in the parameter associated with thecrisis probability, particularly, d1 . We assume d1 = 0, meaning that the crisis probability
is unchanged (constant at d0 ) even if the debt ratio changes. Figure 10 shows thesimulation results. Since agents face a smaller, constant crisis risk, the increase in thedebt ratio leaves key economic variables such as output, investment, bond yield, credit
spread, and CDS spread, almost unaffected.

26

5.3

Foreign Direct Investment

Although a closed-economy model serves as a good starting point to describe the caseof Japan because around 90% of GBO is held by domestic investors, domestic investors
have the opportunity to invest abroad. Thus, it is valuable to consider implications forforeign direct investment under the fear of the government debt crisis.
In Appendix B, we construct a simple two-period model, in which private agents caninvest in not only government bonds and capital at home, but also in capital abroad.
In the model, the benefit of foreign direct investment is that it ensures a riskless returnirrespective of a crisis. The government does not impose tax on this return. However,there is a cost, represented by a convex cost function for foreign direct investment. In
this environment, we show analytically that foreign direct investment increases as thegovernment debt increases as long as ω K > 0. Because investing in capital at homebecomes riskier as the debt increases, private agents increase investment overseas.
Figure 11 shows the actual changes in foreign direct investment from Japan abroad(i.e., outward) in the solid line and from abroad to Japan (i.e., inward) in the dashedline. It shows a secular increase in outward investment from 1990 to 2019 consistent withthe model prediction. By contrast, inward investment is almost unchanged and remains
low. Although factors besides taxes and crisis risk influence foreign direct investment,
the fear of large-scale taxation on capital and misallocations of capital in future debt
crises may be a factor that discourages inward investment.

5.4

WelfareUsing the model, we calculate the loss in welfare caused by the risk of crises. We
generate time-series paths for TFP (zt ) and the crisis indicator (xt ) randomly for 1000
years, discard the first one-third of the samples, and take the mean. A welfare loss canarise from not only the expectation of a crisis before it occurs, but also the distortion
at the time of the crisis. Thus, we calculate the welfare loss by including crisis periods,
unlike the simulation conducted in Section 4.4. Table 3 reports the simulation results
based on the benchmark model and the various tax scenario models that we describedin Section 4.5. Output (yt ) and welfare (Ut , in terms of consumption equivalence) areindicated as differences from the values in the benchmark model.32
If the tax is not distortionary (“All T tax”), both output and welfare improve by 19%
and 7%, respectively. These large improvements are the result of a non-distortionary tax
and capital-use efficiency. Of the distortionary taxes, the capital levy is the most impor32

Welfare in terms of consumption equivalence is measured as follows. We calculate lifetime utility
in the case in question and the benchmark case. We then determine the permanent percentage increasein consumption needed in the benchmark case such that households will be indifferent between the twocases. This permanent increase is the consumption equivalence.

27

tant, contributing to the decreases in output and welfare by 11% and 1%, respectively.
The tax on GBO plays the second most important role, decreasing output and welfare
by 5% and 1%, respectively.

5.5

Permanent Distortionary Tax to Prevent a Crisis

Should a government raise tax rates in normal times to avoid a crisis? Because a crisis
event causes not only a depression at the time but also stagnation beforehand, it may bebetter to introduce higher tax rates preemptively. To answer this question, we consider amodel in which the government always aims to maintain a bounded range of government
debt. Specifically, we assume the following tax policy:τtC = max 0, min(τtC∗ , 0.3) ,b¯G G µ bGt +gC∗τt = 1 − G
qt e
.
ct
bt + g(29)¯G
The government imposes a consumption tax only to maintain bG
t+1 around its target b ,where the maximum tax rate for τtC is 30%, and the target b¯G is 0.3.33
Table 3 presents the simulation results. The bottom row shows that the debt ratioG(b /y) stabilizes at 70%, and that output increases by 11%. Consequently, welfare
increases by 6% in the consumption unit. Therefore, preemptive tax increases improvesocial welfare.34

5.6

Final Thoughts on Government Tax Policy

Given the results on welfare, one may think that the Japanese government will (or should)
not adopt a capital levy, but should instead use a less distortionary tax on government
bonds or a partial default if a debt crisis occurs. Moreover, one may wonder why theJapanese government does not raise taxes now, before a crisis occurs. In the final partof this study, we discuss several issues related to government policy, although our modelcannot address these questions formally because it does not endogenize government actions.
With regard to the tax policy during a crisis, three points are worth making. First,
as we noted in Section 4.2.1, the post-war Japanese government did not resort to a lump33

Because the future TFP shock et+1 is unknown, the government cannot perfectly stabilize bG
t+1 atUG
its target. In our numerical calculation, we set an upper bound bU for bG
,
where
b

b̄
,
and
imposetU
a lump-sum tax only when the consumption tax alone cannot maintain bG
t+1 ≤ b .
34
An alternative approach is to calculate lifetime utility at t, conditional on the state in which thedebt-to-GDP ratio, bG
t−1 /yt−1 , is as high as it is today. We confirm that this hardly changes our results.
Because discount factor β in our model is close to one, the household is concerned about the long-run
state of the economy, rather than its short-run transition, even if a high debt ratio induces a very hightemporary consumption tax rate.

28

sum tax or a partial default as main means, although it decreased expenditure. Instead,the government imposed a heavy capital levy. Second, a capital levy has political appeal
because it reduces wealth inequality, as demonstrated in post-war Japan (Eichengreen1989). In contrast, a lump-sum tax hurts the poor relatively more than it does the rich.
Third, as noted by Eichengreen (1989) as well as Chamley (1986) and Chari, Christiano,
and Kehoe (1994), a once-off capital levy has no distortionary effect on economic activity
ex post facto, although this point applies to a lump-sum tax and a tax on government
bonds as well.
With regard to the tax policy before a crisis, a possible reason that the Japanese
government does not try to decrease its debt today by raising the tax rate may be its
short-time perspective. Parliament elections for the upper and lower houses occur atleast every three years. Thus, if the government does not believe a crisis will occur inthe near future, it has little incentive to raise taxes because doing so hurts utility inthe short run. Indeed, Prime Minister Shinzo Abe postponed a promised increase in the
consumption tax rate from 8% to 10% twice, in 2014 and 2016.

6

CONCLUDING REMARKS

We analyzed an economy at risk of a government debt crisis and provided a new perspective to explain the secular stagnation and low interest rates that occur. We demonstrated
that most of the persistent slowdown can be accounted for by the increasing fear of acapital levy and capital misallocation in a future debt crisis associated with increasing
government debt.
Because our framework is simple, it can be extended and enriched in numerous ways.
One possibility is to introduce nominal variables (e.g., Aguiar et al. 2015). The nominal
version of our model would be useful for analyzing price dynamics such as hyperinflation,and the implications for monetary policy in the event of secular stagnation with deflation.
Importantly, inflation is essentially equivalent to a tax on GBO (i.e., a default) because
inflation decreases the real value of nominal bonds directly, whereas it does not affect thereal value of capital much. Thus, we believe our analysis of a tax on GBO approximatesthe effects of high inflation in a nominal model. Second, our model can be extended
to incorporate uncertainty during a crisis. A crisis usually entails a large degree of
uncertainty in the market and government responses, which may quantitatively alter
our results. Finally, it would be worth extending our model to include heterogeneous
agents. As such, when a crisis occurs, taxes on capital stock and government bonds caninfluence the holdings of both assets, which plays an important role in the self-insuranceof heterogeneous households.
These extensions may help us explore whether the increasing risk of a crisis causes
other economic difficulties in addition to the persistent stagnation and low interest rates
29

demonstrated in this study.

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Economies, 22, 476–502.
[60] Summers, Lawrence (2013) “Why Stagnation Might Prove to be the New Normal,”
The Financial Times (December 16).

34

APPENDIX A

MODEL DETAILSWe denote the variables divided by zt by their lower case letters (e.g., yt ≡ Yt /zt ), withν(1−ψ)
the exception of ut ≡ Ut1−ψ /zt. In summary, we have 16 equations for 16 unknown
GCK
endogenous variables, {ct , kt , Nt , yt , Mt+1 , ut , xt , λYt , qtG , RtK , RtD , bGt , τt , τt , τt , tt }:yt
1 − ν (1 + xt τtC )ct=,Ntν
1 − NtD
1 = Et Mt+1 Rt+1
(xt+1 ) ,(1 − α)D
Rt+1(xt+1 ) =

(33)

yt
,
kt

(34)

α
yt = xt λYt + (1 − xt ) (kt )α Nt1−α ,(35)

1 + xt τtCC
1 + xt+1 τt+1e

(ν(1−ψ)−1)(µ+σe et+1 )ν(1−ψ)

ut = (1 − β)ct

(32)

1
G
E [Mt+1 (1 − xt+1 τt+1
)],
G t
qt

RtK = (1 − δ)(xt λD + 1 − xt ) + α

Mt+1 = β

(31)

K
K
G
G
(1 − xt+1 τt+1
)Rt+1
(xt+1 )Kt+1 + (1 − xt+1 τt+1
)Bt+1
,
G
Kt+1 + qtG Bt+1

K
K
Et [Mt+1 (1 − xt+1 τt+1
)Rt+1
(xt+1 )] =



(30)



ct+1
ct

ν(1−ψ)−1 

1 − Nt+1
1 − Nt(1−ν)(1−ψ)
(36)
,


(1 − Nt )(1−ν)(1−ψ) + βEt eν(1−ψ)(µ+σe et+1 ) ut+1 ,yt = ct + kt+1 eµ+σe et+1 − (1 − δ)(xt λD + 1 − xt )kt ,Pr(xt+1 = 1|bGt ) =1
1 + (1/d0 − 1)exp(−d1 bG
t )

(37)
(38)
(39)

µ+σe et+1
qtG bG+ xt τtC ct + xt τtK kt
t+1 eG+xt τtG bG
t + xt tt = bt + g,

(40)

τtK RtK Kt = ω K (bG
t + g),(41)

τtC ct = ω C (bG
t + g),(42)G G
τtG bGt = ω (bt + g),(43)

tt = ω T (bG
t + g),(44)

where λYt = F (ξt ) and et follows N (0, 1).

35

APPENDIX B

B.1

ANALYSIS USING A SIMPLIFIED
TWO-PERIOD MODEL

Setup

In this section, we provide an overview of the model by presenting the analytical resultsof a simplified two-period version of the model, in which the banking sector is abstracted
away and the household directly holds government bonds and capital stock. Time runs
from period 0 to 1. In period 0, the crisis does not occur. In period 1, the crisis
occurs if x = 1 and does not occur if x = 0. For simplicity of exposition, we assume
full depreciation (i.e., δ = 1) and no consumption tax during a crisis (i.e., τ C = 0).
Household utility is expressed simply as log consumption with fixed labor supply (i.e.,
N = 1). The probability that a crisis occurs is p(B0 ), which is increasing in B0 at the
beginning of period 0.
The government budget constraint in period 1 is
B + G = xτ K R1K K + xτ G B + xT + (1 − x)T̃ ,where we assume that the debt B is repaid only by the lump-sum tax T̃ if the crisis does
not occur (i.e., x = 0).
In period 0, the representative household is given the initial asset holdings, K0 and
B0 . The household solves the following problem in period 0:
U0 =max (1 − β)logC0 + βE0 logC1 ,C0 ,C1 ,K,B

subject to the budget constraint:
C0 + qB + K ≤ R0K K0 + B0 + W0 + G,

(45)

and
C1 ≤ (1 − xτ K )R1K (x)K + (1 − xτ G )B + W (x) − xT − (1 − x)T̃ .
The firm is modeled as before, producing goods using capital and employment. The
goods market is cleared as in equation (23).
If a crisis occurs, the government imposes taxes asτ K R1K (x = 1)K = ω K (B + G),
G

(46)

G

(47)

T

(48)

τ B = ω (B + G),
T = ω (B + G),

where ω K + ω G + ω T = 1 (0 ≤ ω K , ω G , ω T ≤ 1). If a crisis does not occur, the lump-sum
tax T̃ covers all necessary payments such that T̃ = B + G.
36

The probability of a crisis ispt = p(B0 ) =1.
1 + (1/d0 − 1)exp(−d1 B0 )(49)

The CDS spread is priced as
CDS · M1 (x = 0)(1 − p(B0 )) = (1 − R)M1 (x = 1)p(B0 ),(50)

where M1 (x) is the stochastic discount factor.

B.2

Equilibrium

Because of (45), we can rewrite the period 0 problem asmax(1 − β)log[R0K K0 + B0 + W0 + G − qB − K]
K,B

+βE0 log[(1 − xτK

)R1K (x)K + (1 − xτ G )B + W (x) − xT − (1 − x)T̃ ].

The first-order conditions with respect to K and B imply
1(1 − τ K )RK (1)RK (0)(1 − β)= β p(B0 )+ (1 − p(B0 )),C0C1 (1)
C1 (0)q1 − τG1(1 − β)= β p(B0 )+ (1 − p(B0 )).
C0C1 (1)
C1 (0)(51)(52)

Now, we can calculate the equilibrium by combining the solutions to the householdproblem and the firm’s problem. We have the following equilibrium values:
C0 = z 1−α K0α ,C1 (0) = C1 (1) = z 1−α K α ,W (0) = W (1) = (1 − α)z 1−α K α , z 1−α= αz 1−α K α−1 .
RK (0) = RK (1) = α
K
Then, we can derive the following equilibrium conditions from (51) and (52):
1−βK = αz 1−α K0α p(B0 )(1 − τ K ) + 1 − p(B0 ) ,
β

1−β α
K q = K0α p(B0 )(1 − τ G ) + 1 − p(B0 ) .
β

(53)
(54)

The equilibrium value of B is the government budget in period 0; that is,
qB = G + B0 ,

37

(55)

where the tax rates areτ K = min[1, ω K (B + G)/(αz 1−α K α )],(56)

τ G = min[1, ω G (B + G)/B].

(57)

Conditions (53), (54), and (55) determine the equilibrium with respect to K, q, and
B. Note that from (53) and (54), we can obtain the arbitrage condition between investingin capital and investing in government bonds:p(B0 )(1 − τ K ) + 1 − p(B) RK = p(B0 )(1 − τ G ) + 1 − p(B) /q.

(58)

The CDS spread is simply written as
CDS = (1 − R)

B.3

p(B0 ).
1 − p(B0 )(59)

Implications

Specifically, suppose (1 − β)/β = αz 1−α K0α−1 > 1, z = 1, and G = 0. We denote the0derivative of y with respect to B0 by y .
000PROPOSITION 1 The signs of B and q are ambiguous. Specifically, q < 0 if andonly if000
p (B0 )τ G
p (B0 )τ K + p(B0 )τ K
>
α.
(60)
p(B0 )(1 − τ G ) + 1 − p(B0 )p(B0 )(1 − τ K ) + 1 − p(B0 )We provide the proofs of this and the following propositions in the next subsection.
This proposition states that an increase in B0 does not necessarily lead to a decreasein the price of government bonds, q (i.e., an increase in the government bond yield).
When the capital levy (τ K ) is less than the size of the government bond default (τ G ), q
tends to decrease as B0 increases. However, when the capital levy (τ K ) is greater than
the size of the government bond default (τ G ), q tends to increase rather than decrease as
B0 increases. This occurs because the household chooses to invest in government bonds
rather than capital. Moreover, when the elasticity of the crisis probability to debt is zero0(i.e., p (B0 ) = 0), the above inequality is likely violated, and thus, q tends to increase
rather than decrease as B0 increases.
0
Furthermore, this proposition shows that somewhat surprisingly, the sign of B is notnecessarily positive. In other words, an increase in B0 does not necessarily lead to anincrease in debt in the next period B. When the capital levy is relatively more important
0
than the size of the government bond default, q becomes positive, which helps decrease
the amount of government bonds issued in the next period.
0
However, the case of B < 0 appears to be a little extreme. Although the Japanese0
economy has been experiencing an increase in q as B increases (i.e., q > 0), the amount38

0of debt, B, has not decreased (i.e., B > 0). Thus, in the following analysis, we consider0
the case in which B > 0.
The following proposition shows that a slowdown of real economic activity occurs if
ω K > 0.
0

0PROPOSITION 2 Suppose B > 0 and αp(B0 ) < 1 − p(B0 ). Then, K ≤ 0 and
00τ K ≥ 0. The strict inequality of K < 0 holds if ω K > 0 and either τ K < 1 or00
p (B0 ) > 0. The strict inequality of τ K > 0 holds if ω K > 0 and τ K < 1.
If the government imposes a capital levy during a crisis (i.e., ω K > 0), an increase in0
B0 leads to an increase in the capital levy rate (i.e., τ K > 0) unless it already reaches0
the maximum rate (i.e. one). This decreases investment in capital K (i.e., K < 0),which causes real economic activity to slow down. The decrease in investment in capital00
is also caused by an increase in the crisis probability (i.e., p (B0 ) > 0), and thus, K is
negative even if the capital levy rate is already one.
Propositions 1 and 2 suggest that the model can explain both a larger slowdown inreal economic activity and a decrease in the government bond yield as government debt
increases if people expect a relatively larger ω K than ω G during a crisis.
Finally, the next proposition provides a sharp prediction about the elasticity of the
CDS spread to B0 .
PROPOSITION 3 The elasticity of the CDS spread to B0 equals d1 .

B.4

Model Extension: Incorporating Foreign Direct Investment

We extend the two-period model by allowing the household to invest in not only government bonds and capital at home, but also capital abroad (denoted by F ). The household
solves the following problem in period 0:
U0 = max (1 − β)logC0 + βE0 logC1 ,C0 ,K,Bsubject to the budget constraint:
C0 + qB + K + Φ(F ) ≤ R0K K0 + B0 + W0 + G,
andC1 ≤ (1 − xτ K )R1K (x)K + (1 − xτ G )B + RF + W (x) − xT − (1 − x)T̃ .
Foreign direct investment ensures a riskless return, given by R, irrespective of a crisis.
The government does not impose a tax on this return. However, it is costly, as indicated
000
by the convex function Φ(F ), where Φ(F ) ≥ 0, Φ (F ) > 0, and Φ (F ) > 0 with Φ(0) = 0.
Then, we can obtain the following proposition illustrating that foreign direct investment F increases as B0 increases.
39

0

0

PROPOSITION 4 Suppose B > 0 and αp(B0 ) < 1 − p(B0 ). Then, F ≥ 0. The strict
0
inequality holds if ω K > 0 and either τ K < 1 or p (B0 ) > 0.

B.5

Proof of Propositions

B.5.1

Proof of Proposition 1

Differentiating (53) to (57) with respect to B0 yields
n 0
o
1−β 0
0
K = αz 1−α K0α p (B0 )(−τ K ) − p(B0 )τ K ,
β
n 0
o
1−β
α−1 0
α 0
α
G
G0
(αK K q + K q ) = K0 p (B0 )(−τ ) − p(B0 )τ
,
β
0

0

q B + qB = 1,000

τ K = ω K B /(αz 1−α K α ) − αω K (B + G)/(αz 1−α K α+1 )K ,00

τ G = −ω G G/B 2 B .
Then, under the assumption of (1 − β)/β = αz 1−α K0α−1 > 1, z = 1, and G = 0, theequilibrium condition is summarized as
K/K0 = p(B0 )(1 − τ K ) + 1 − p(B0 ),(1 − β)/β · (K/K0 )α q = p(B0 )(1 − τ G ) + 1 − p(B0 ),qB = B0 ,τ K = min[1, ω K B/(αK α )],τ G = ωG,
0

0

0

K /K0 = p (B0 )(−τ K ) − p(B0 )τ K ,
0

(61)
0

0

(1 − β)/β · (K/K0 )α (α(K /K)q + q ) = p (B0 )(−τ G ),00q B + qB = 1,000

τ K = ω K B /(αK α ) − αω K B/(αK α+1 )K ,0

τ G = 0.
0With respect to q , we can obtain000(1 − β)/β · (K/K0 )α q = −p (B0 )τ G − (1 − β)/β · (K/K0 )α α(K /K)q0

K K0
= −p (B0 )τ − {(1 − β)/β · (K/K0 ) q}αK0 KoGp(B0 )(1 − τ ) + 1 − p(B0 ) n 00
KK0= −p (B0 )τ G +
α
p
(B
)τ
+
p(B
)τ
.
0
0
p(B0 )(1 − τ K ) + 1 − p(B0 )
0

G

α

0

This result suggests that q < 0 if and only if000

p (B0 )τ G
p (B0 )τ K + p(B0 )τ K>
α.
p(B0 )(1 − τ G ) + 1 − p(B0 )p(B0 )(1 − τ K ) + 1 − p(B0 )
40

0

0

0

0

0

With respect to B , it is clear that B > 0 if q < 0 because q B + qB = 1. However,00
B can be negative when q is positive. To see this, suppose a special case, in whichτ G = 0 and τ K = 1. Then, we have00q = αp (B0 )q/(1 − p(B0 )) > 0
and0

qB = 1 − αB0d1.
1 + (1/d0 − 1)exp(−d1 B0 )which can be negative if either B0 , d0 or d1 is sufficiently large.
B.5.2

Proof of Proposition 2

In equation (61), we have000

K /K0 = p (B0 )(−τ K ) − p(B0 )τ K ,
τ

K0

K

0

α

K

= ω B /(αK ) − αω B/(αK

(62)
α+1

0

)K ,

which leads to
n 0
o
0
0
0
τ K = ω K B /(αK α ) − αω K B/(αK α+1 )K0 p (B0 )(−τ K ) − p(B0 )τ K ,000(1 − ατ K p(B0 )K0 /K)τ K = ω K B /(αK α ) + αω K B/(αK α+1 )K0 p (B0 )τ K ≥ 0.
00Note that 0 < p ≤ 1, p ≥ 0, and 0 ≤ τ K ≤ 1. The coefficient on τ K on the left-handside of the equation is positive becauseατ K p(B0 )p(B0 )(1 − τ K ) + 1 − p(B0 )p(B0 )≥1 − α1 − p(B0 )1 − ατ K p(B0 )K0 /K = 1 −>0.
The first line is derived from the first equation of (61), and in the second line, equality
holds when τ K = 1. The third line comes from the assumption in this proposition, which
requires that the probability of a crisis is not too high. Given α = 1/3, this assumption
holds when p(B0 ) < 3/4, and thus, this assumption is not restrictive. Therefore, we
00
obtain τ K ≥ 0, where the strict inequality holds if ω K > 0 and τ K < 1. Substituting τ K
0into equation (62) yields K ≤ 0, where the strict inequality holds if ω K > 0 and either0τ K < 1 or p > 0.

41

B.5.3

Proof of Proposition 3

Using
1
,
1 + (1/d0 − 1)exp(−d1 B0 )(1/d0 − 1)d1 exp(−d1 B0 )0p (B0 ) => 0,
{1 + (1/d0 − 1)exp(−d1 B0 )}2
pt = p(B0 ) =we obtain the derivative of the CDS spread as000

p (B0 )
p (B0 )CDS=+
CDS
p(B0 )
1 − p(B0 )(1/d0 − 1)d1 exp(−d1 B0 )d1=+
1 + (1/d0 − 1)exp(−d1 B0 ) 1 + (1/d0 − 1)exp(−d1 B0 )= d1 .
B.5.4

Proof of Proposition 4

The first-order conditions with respect to K and B are the same as before. The first-ordercondition with respect to F is given by
0RΦ (F )
RR+ (1 − p(B0 ))(1 − β)= β p(B0 )
=β .
(63)
C0
C1 (1)
C1 (0)
C1
This leads to
0

β C0
R
1 − β C1
 α
β
K0
=
R.
1−β K

Φ (F ) =0

Because of Proposition 2 and the convexity of Φ(F ), it is clear that F ≥ 0.

42

Table 1: Calibrated Parameters
Parameters

Values

Sources

Capital share α

0.3

Gourio (2013)

Discount factor β

0.995

Sugo and Ueda (2011)

Utility weight on C ν

0.3

Gourio (2013)

IES utility ψ

1.5

Sugo and Ueda (2008); Hirose and Kurozumi (2012)

Depreciation δ

0.08

Gourio (2013)

Trend growth of TFP µ

0.0182

Mean of ∆logYt

SD of TFP shock σe

0.023

SD of ∆logYt

Gov spending g

0.02

Mean of ∆(bGt /yt )

0.2015

Post-war Japan

0.0181

Post-war Japan

0.0760Post-war Japan

0.0157

Post-war Japanproduction θ00.059

Laeven and Valencia (2018)

production; dependence on bad loans θ1

0.202

Laeven and Valencia (2018)

1 − 0.061

Flow of Funds in 1997 and 1998

at b = 0 d0

0.025

Level of CDS spreads when b is low and ω Gelasticity to b d1

4

Elasticity of CDS spreads and ω G

Tax weights
capital levy ω K
consumption tax ωCgovernment bonds (default) ωGlump-sum tax ω T
Capital-use inefficiency

Ddepreciation λCrisis probability

43

Table 2: Comparison of Untargeted Moments
Data

Benchmark model

0.5205 (1.1501)

0.756

R −1

0.019

K

G

RD − RG

b/y
G

R

−R

Data

Benchmark model

σ(dY)

0.021

0.021

0.044

σ(dI)

0.062

0.040

0.008

0.007

σ(dC)

0.016

0.018

0.001

0.005

cor(dY,B/Y)-0.302-0.239cor(dI,B/Y)

0.002-0.212cor(dY,R − 1)

0.284

0.353

K

G

−R )

-0.149-0.188

D

G

0.292

-0.198cor(dI,R − 1)

0.058

0.165

K

G

−R )

-0.012-0.188cor(dI,RD − RG )0.023

-0.196Gcor(dY,Rcor(dY,R − R )Gcor(dI,R

Notes: The data are from 1975 to 2019 for Japan. The model-based moments exclude the time of acrisis. Data b/y represents the ratio of net government debt-to-output, whereas the value in parenthesesis the ratio of gross government debt-to-output. σ(dlogY) is the only targeted moment.

Table 3: Welfare Comparison
Model

b/y

y

U

Benchmark

1.16

0.000

0.000

All T tax

0.80

0.192

0.071

No K tax

1.00

0.114

0.014

No C tax

1.15

0.004

0.000

No G tax

1.00

0.046

0.013

Always C tax

0.70

0.108

0.058

Notes: The model includes the crisis periods. Output (y) and welfare (U, in the unit of consumption)
are indicated as differences from the values in the benchmark model.

44

Japan

1

3

0.5

2.5

0.4

2

0.3

1.5

0.2

1

0.1

0.5

0

US

2

Y (left)

0.8

B G /Y (net, right)
B G /Y (gross, right)

1.5

0.6
1

0.4
0.5

0.2
0
1970

0.5

1980

1990

2000

2010

Euro 17

0
2020

-0.1
1990

2

0.5

0.4

2000

2010

Italy

2

0.4
1.5

0.3

1.5
0.3

0.2

1

0.2

0.1

1

0.1
0.5

0
-0.1
1990

0
2020

0.5
0

2000

2010

0
2020

-0.1
1990

2000

2010

0
2020

Figure 1: Government Debt and GDP
Note: The beginning year is 1975 for Japan and 1992 for the other countries, representing a 15-year
period before the financial crises in each region (i.e., 1990 and 2007, respectively, shown as the verticaldashed line). The thick solid line represents the logarithm of real GDP per capita (set to zero in the firstyear), shown on the left axis. The thin solid line represents its linear trend. On the right axis, the linewith crosses and the line with circles represent the ratio of gross and net government debt, respectively,
to nominal GDP. The data are taken from the OECD.

45

75

%

Public Opinion Survey on the Life of the People

%

40

70

35

65

30
25

60

20

55

15

50

10

45

5
0

40
1981

1986

1991

1996

2003

2008

2013

Feel worries or anxieties in one’s everyday life (left)
The prospect of future living standard is worsening (right)

Reasons for future anxieties
0

10

20

30

40

%

50

Health60

50.3

Natural disasters

47.9

Public service

42.1

Employment

35.7

Fiscal balance

34.8

Inequality

33.3

Environment

32.2

Sluggish economy

31.3

Crimes28.3

Figure 2: Japanese Sentiment
Source: Cabinet Office “Overview of the Public Opinion Survey on the Life of the People”

12

600

90

CDS spread
10

80

Fiscal failure/crisis (left)
500

70

Fiscal failure & tax increase (right)
400

8

60
6

50
300
40

4

200

30
2

20
100
10
0

0
2000

0
1981

1986

1991

1996

2001

2006

2011

05

10

15

20

2016

Figure 3: Concerns about a Government Debt Crisis
Note: The left-hand figure shows the occurrences of specific words in the morning and evening editions of
the Nihon Keizai Shinbun, Japan’s financial newspaper, for each year. The right-hand figure shows thesovereign CDS spread with a five-year maturity (in percentage points), which is taken from Bloomberg.

46

10

12

0.2

Credit spread
Capital cost spread
I/K (right)

Bond yield

9

10

8

7
8

0.15

6

5
6
4

3
4

0.1

2

1

2

0

-1
1975

80

90

00

10

0
1975

20

80

90

00

10

0.05
15

Figure 4: Bond Yield, Credit Spread, and Investment
Note: The government bond yield in real terms is defined as that with a five-year maturity minus the annual CPI inflation rate in the following year. The credit spread is defined as the bankloan rate with one-year maturity or longer minus the government bond yield with five-year maturity. The data on investment, capital, and capital cost are obtained from the JIP database(https://www.rieti.go.jp/en/database/jip.html). The capital cost spread is defined as the rental costminus the five-year government bond yield.

Output loss

0.3

Ireland (2008)

0.25

y =0.059 +0.202log(1+x)

0.2
Finland (1991)

Greece (2008)
Spain (1977)

0.15

Japan (1997)

Luxembourg (2008)

Turkey (2000)

0.1

Iceland (2008)

0.05

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ratio of non performing loans

Figure 5: Relation Between Output Loss and Banks’ Nonperforming Loans
Source: Laeven and Valencia (2018).

47

bG /y

y

i

3

0.2

0.4

2

0.1

0.2

1

0

0

0
1990

2000

2010

2020

2030

-0.1
1990

2000

R G -1

2010

2020

2030

2000

R K -RG

0.1

2010

2020

2030

2020

2030

2020

2030

R D -RG

0.05

0.05

0

0.05

0
-0.05

0
-0.05
1990

-0.2
1990

2000

2010

2020

2030

-0.1
1990

CDS

2000

2010

2020

2030

-0.05
1990

2000

2010

Y

Crisis prob
0.78

1
0.4

0.77
0.5

0.2
0
1990

0.76

2000

2010

2020

2030

0
1990

2000

2010

2020

2030

0.75
1990

All G tax
Data
2000

2010

Figure 6: Simulation Results when Tax is Imposed Only on Government Bonds

48

bG /y

y

i

2

0.2

0.5

1

0

0

0
1990 2000 2010 2020 2030 2040

-0.21990 2000 2010 2020 2030 2040

R G -1R K -RG

0.06-0.5
1990 2000 2010 2020 2030 2040

R D -RG
0.04

0.04
0.04

0.02

0.02

0.02

0

0

0
-0.02
1990 2000 2010 2020 2030 2040

-0.02
1990 2000 2010 2020 2030 2040

-0.02
1990 2000 2010 2020 2030 2040

K

CDS
0.15

Y
1

0.2

0.1

0.9
0.1

0.05

0.8

0
1990 2000 2010 2020 2030 2040

01990 2000 2010 2020 2030 2040

0.71990 2000 2010 2020 2030 2040

Figure 7: Simulation Results of the Benchmark Model

49

Benchmark model
All T tax
DatabG /y
2

1.5

y

i

0.1

0.4

0.05

0.2

0

0

-0.05

-0.2

-0.1

-0.4

-0.15
1990 2000 2010 2020 2030 2040

-0.61990 2000 2010 2020 2030 2040

1

0.5

0
1990 2000 2010 2020 2030 2040

R G -1Y

R K -RG

0.08

0.06

0.78

0.06

0.04

0.77

0.04

0.02

0.76

0.02

0

0.75

0
1990 2000 2010 2020 2030 2040

-0.02
1990 2000 2010 2020 2030 2040

0.74
1990 2000 2010 2020 2030 2040

Figure 8: Simulation Results in Different Tax Scenarios at a Crisis

50

BenchmarkNo K tax
No G tax
No C taxbG /y
2

1.5

y

i

0.1

0.4

0.05

0.2

0

0

-0.05

-0.2

-0.1

-0.4

-0.15
1990 2000 2010 2020 2030 2040

-0.61990 2000 2010 2020 2030 2040

1

0.5

0
1990 2000 2010 2020 2030 2040

R G -1
0.07

Y

R K -RG
0.06

0.82

0.04

0.8

0.02

0.78

0

0.76

0.06
0.05
0.04

Benchmark
Zero elast of output loss
(theta1=0)
No depreciation(lambdaD=1)

0.03
0.02
0.01
1990 2000 2010 2020 2030 2040

-0.02
1990 2000 2010 2020 2030 2040

0.74
1990 2000 2010 2020 2030 2040

Figure 9: Simulation Results under Low Capital-Use Inefficiency

51bG /y

y

2

0.1

1.5

0.05

1

0

i
0.4
0.2
0
-0.2

-0.05

0.5

0
1990

2000

2010

2020

2030

-0.1
1990

-0.4

2000

G

2010

K

R -1

2020

2030

-0.6
1990

2000

G

R -R

2010

2020

2030

2020

2030

CDS

0.07

0.05

0.05

0.06

0.04

0.04

0.05

0.03

0.04

0.02

0.03

0.01

0.02

0

BenchmarkConstant
crisis prob

0.03
0.02

0.01
1990

2000

2010

2020

2030

-0.01
1990

0.01

2000

2010

2020

2030

0
1990

2000

2010

Figure 10: Simulation Results under a Constant Crisis Probability

54Outward
Inward

3

2

1

0
1990

1995

2000

2005

2010

2015

-1

Figure 11: Foreign Direct Investment
Note: The data show the foreign direct investment financial flows (outward and inward in the solid anddashed lines, respectively) as a share of GDP (%). The data are taken from the OECD.

52