The Society for Economic Studies
The University of Kitakyushu
Working Paper Series
No.2020-4(accepted in March 29, 2021, revised in November 19, 2021)

Limited asset market participation and ﬁscal
sustainability
Noritaka Maebayashi∗

Jumpei Tanaka†

Abstract
We examine the implications of limited stock market participation (the stylized fact that
only a fraction of households directly own stocks) for economic growth and, in particular,ﬁscal sustainability. Constructing an overlapping generations (OG) model where (1) individuals can choose between two types of savings (i.e., physical capital with high returns but
costly to hold and bank deposits with low returns but no costs), and (2) banks invest partof their total deposits in physical capital and use the rest to underwrite government debt,
we show the following. First, an increase in the average level of an individual’s ﬁnancial
literacy worsens ﬁscal sustainability while promoting economic growth under plausible parameter values. Second, if banks decrease (resp. increase) the number of government bonds
they underwrite, ﬁscal sustainability improves (resp. worsens) when the average ﬁnancial
literacy of individuals is relatively low.

JEL classiﬁcation: E13; G11; H63
Keywords: limited asset market participation, ﬁnancial literacy, ﬁscal sustainability, public debt,
overlapping generations, endogenous growth∗

Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata, Kokura
Minami-ku, Kitakyushu, Fukuoka 802-8577, JAPAN; e-mail: non818mn@kitakyu-u.ac.jp. This research was supported by grants-in-aid from the Ministry of Education, Culture, Sports, Science, and Technology of the Japanese
government, KAKENHI (grant number 20K13455).
†
Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata, Kokura
Minami-ku, Kitakyushu, Fukuoka 802-8577, JAPAN; e-mail: j-tanaka@kitakyu-u.ac.jp1

1

Introduction

Accumulated government debt is one of the most important problems common to all advanced
countries. Figure 1 shows the evolution of the ratio of government debt to GDP for the G7
countries over the last 20 years. The ﬁgure shows that the ratio of government debt to GDP in G7
countries, with the exception of Germany, has been on an upward trend, and that among others,
Japan’s ratio is remarkably high. Motivated by this current situation, a number of theoretical andempirical studies on ﬁscal sustainability have been conducted. This study is one such attempt.
Speciﬁcally, it examines how a stylized fact calledʠ limited asset market participation ʡis related
to ﬁscal sustainability (or the maximum sustainable level of government debt).
[Figure 1]
It is well-known that in developed countries, only a fraction of households own stocks, i.e.,their participation in stock markets is limited (Mankiw and Zeldes (1991), Haliassos and Bertaut(1995), Campbell (2006), and so on).1 Various reasons can be adduced as to why stock market
participation is limited,2 but the relationship between ﬁnancial literacy (the ability to properly
gather and analyze investment information) and stock market participation is one reason that hasreceived special attention recently. Holding risky assets, such as stocks, requires a higher levelof ﬁnancial literacy than holding safe assets, such as bank deposits. Since acquiring ﬁnancial
literacy comes at a cost, it is reasonable to forgo participation in the stock market if its cost is high.
In fact, many empirical studies, such as Kimball and Shumway (2006), van Rooij, Lusardi, andAlessie (2011), Yoong (2011), and Thomas and Spataro (2018) show that low ﬁnancial literacy
signiﬁcantly reduces the rate of stock market participation.3
What impact would the above characteristics of limited stock market participation have onﬁscal sustainability (or the maximum sustainable level of government debt)? Consider a situation where participation in the stock market is stimulated by the increased ﬁnancial literacy of
1Thomas and Spataro (2018) reports that the average stock market participation rate of households for 9 European countries in year 2010 was 16.84%. Fujiki, Hirakata, and Shioji (2012) conﬁrm that in Japan at most 15%
of households held stocks for the period from 2007 to 2010. See also Guiso, Haliassos, and Jappelli (2003) andChristelis, Jappelli and Padula (2010) for other data.
2
Haliassos and Bertaut (1995) point to short-sale constraints, income risk, inertia, and departures from expectedutility maximization as reasons for limited participation. Constantinides et al. (2002) argue for the liquidity constraints young people face. Besides these factors, sociological explanations such as trust and culture (Guiso et al.
2008) and the inﬂuence of neighbors and peers (Hong et al. 2004; Brown et al. 2008) have also been oﬀered.
3
See also Guiso and Jappelli (2005) and Christelis, Jappelli and Padula (2010). The former (resp. the latter)
investigate the relationship between ﬁnancial awareness (resp. cognitive abilities) and stock market participation.

2

individuals. This situation, on the one hand, makes it easier for ﬁrms to raise capital investment funds through the stock market, and this could have a positive impact on economic growth.
Since the ratio of government debt to GDP gradually decreases with increased economic growth,
it improves ﬁscal sustainability. Increases in ﬁnancial literacy and stock market participation, on
the other hand, will reduce the share of individuals who hold safe assets such as bank deposits.
Table 1 shows the proportion of domestic outstanding government bonds held by domestic andforeign banks in G7 and several other countries in 2020. This table suggests that private banks
are non-negligible holders of government bonds, and accordingly that savings in the form of bankdeposits are more likely to be used to underwrite government debt. Hence, increases in ﬁnancial
literacy and stock market participation could worsen ﬁscal sustainability by reducing the underwriters of government bonds and pulling up the government bond yield. Which of these two
conﬂicting eﬀects dominates? The purpose of this study is to examine this point theoretically.
[Table 1]
To investigate the above issues, we construct a simple overlapping generations (OG) modelwith the following characteristics:• There are three types of assets: physical capital (which corresponds to stocks), bank deposits, and government bonds.
• When saving, individuals can choose between physical capital, which is more proﬁtable
but costly to hold, and bank deposits, which are less proﬁtable but not costly to hold.
• In investing in physical capital, individuals must allocate a portion of their endowed timeto acquiring ﬁnancial knowledge. The more time they spend on it, the higher the returnfrom their savings. Learning ability (more precisely, the extent to which time spent on
improving ﬁnancial literacy increases the return on savings) varies across individuals.
• Banks are assumed to invest a given proportion of their total bank deposits in physicalcapital and the rest in government bonds.
• The government manages its ﬁnances in such a way that past debts are repaid only by
issuing new bonds (i.e., the government plays the “Ponzi game”).

3Under the above model setup, we show the following results.
First, there exists one stable balanced growth path (BGP) and one unstable BGP in our model.
In the stable BGP, all individuals are physical capital holders; hence, the ratio of government debt
to physical capital in the economy is zero. Contrariwise, the ratio of government debt to physicalcapital in the unstable BGP corresponds to its maximum sustainable value; if the actual ratioof government debt to physical capital in the initial period is larger than this value, the actual
ratio will diverge and the government will go bankrupt. If the initial actual ratio is smaller than
this value, on the other hand, the actual ratio will decline over time, and the economy eventuallyconverges to the stable BGP.
Second, under plausible numerical settings of parameters, an increase in each individual’s
learning ability (or equivalently, each individual’s ﬁnancial literacy) reduces the maximum sustainable ratio of government debt to physical capital, while increasing the economic growth rates
in both stable and unstable BGPs. This result is very important for countries that have introduced
educational programs to improve ﬁnancial literacy, with the increasing importance of ﬁnancial
literacy due to the increasing complexity of ﬁnancial markets and rising life expectancy.4 Our results imply that while such a policy certainly has the positive eﬀect of raising the potential rate of
economic growth, it also has the negative eﬀect of worsening ﬁscal sustainability and increasing
the likelihood of government bankruptcy.
Third, we examine the impact on ﬁscal sustainability when banks change their asset investment behavior. When banks decrease (resp. increase) the number of government bonds they
underwrite, it would intuitively seem to worsen (resp. improve) ﬁscal sustainability. However,
we show that when the average ﬁnancial literacy of individuals is relatively low, the opposite result holds, that is, ﬁscal sustainability improves (resp. worsens) if banks decrease (resp. increase)
the amount of government bonds they underwrite. This result implies that if banks decrease (resp.
increase) the amount of government bonds they underwrite, the need for ﬁscal consolidation will
conversely decrease (resp. increase) when the average ﬁnancial literacy of individuals is relatively low.

4

OECD (2014, p. 28) provides a brief overview of ﬁnancial education eﬀorts in such countries as New Zealand,
Australia, Belgium, Shanghai-China, Czech Republic, Estonia, France, U.S., Colombia, Italy, Israel, and Spain.

4Overview of previous literature(A) Theoretical studies of ﬁscal sustainability
Previous studies that have investigated ﬁscal sustainability theoretically can be divided into
two main categories: those based on the Ramsey model and those based on the overlapping
generations model. The latter is relevant to this study.
Tirole (1985) examined the conditions for the existence of asset price bubbles using Diamond’s (1965) neoclassical growth model and showed that positive bubbles can exist when theeconomy is dynamically ineﬃcient. This result can also be interpreted as the sustainability condition of the government’s Ponzi game. King (1992) and Grossman and Yanagawa (1993) reexamined the argument of Tirole (1985) using an AK-type endogenous growth model and showedthat the government Ponzi games can be sustainable when the economic growth rate exceeds theprivate rate of return on physical capital, even if the economy is dynamically eﬃcient. Chalk(2000) studied the sustainability of ﬁscal management with permanent primary deﬁcit using Diamond’s (1965) neoclassical growth model, and showed that it can be sustainable if the size of thepermanent primary deﬁcit is small. Bräuninger (2005) revisited Chalk’s (2000) argument using
an AK-type endogenous growth model and conﬁrmed that a similar conclusion holds. Using an
endogenous growth model with the public investment of Barro (1990), Arai (2011) showed thatwhen public investment/GDP is small, an increase in public investment spending improves ﬁscal
sustainability. Yakita (2008), using an endogenous growth model with public capital accumulation of Futagami et al. (1993) type, showed that a larger initial value of public capital raises
the maximum sustainable level of government debt. Using a model of human capital accumulation through both private and public education, Motoyama (2019) showed that the maximum
sustainable level of government debt does not depend on the initial value of human capital. Usinga model with involuntary unemployment based on the eﬃcient wage hypothesis, Yakita (2014)
showed that an increase in the primary deﬁcit raises unemployment and the amount of capitalstock in the new steady state, while the impact on the maximum sustainable public debt/GDP isambiguous. While all of the above arguments are based on a closed economy model, Farmer andZotti (2010), based on a two-country, two-goods model, showed that there is a negative correlation between the maximum sustainable levels of public debt in both countries, and that raising
the amount of public debt in one country worsens capital accumulation in both countries.

5Thus, previous studies have considered the issue of ﬁscal sustainability in overlapping generations models under various model settings, but to the best of our knowledge, no study has
investigated the relationship between limited asset market participation and ﬁscal sustainability.
The contribution of this study is to derive new insights on this point.
(B) Theoretical studies of limited asset market participation
We now provide a brief overview of previous theoretical studies on limited asset market
participation. Initially, the fact of limited asset market participation attracted attention on the
grounds that asset price behaviors, which are diﬃcult to understand rationally, can be explained tosome extent by taking this fact into account (e.g., Mankiw and Zeldes,1991; Allen and Gale,1994).
However, many recent studies of limited asset market participation have addressed this issue inrelation to the theory of monetary policy. For example, Christiano and Eichenbaum (1995) andAlvarez et al. (2001) showed that the ”liquidity eﬀect” of monetary policy (i.e., the phenomenon
that monetary easing lowers the nominal interest rate), which cannot be explained in the complete market model, can be explained by assuming limited participation in asset markets. Galiet al. (2004) and Bilbiie (2008) point out that under limited asset market participation, local
determinacy may not hold in the New Keynesian model, even if the Taylor principle5 is assumed.
Thus, most of the theoretical studies of limited asset market participation are related to asset
pricing theory or monetary policy theory, and to the best of our knowledge, no studies haveconsidered this issue in the context of long-run economic growth.
Organization of this paper.
The remainder of this paper is organized as follows. We set up the model in Section 2 andexamine the existence and stability of balanced growth paths (BGPs) in Section 3. Eﬀects ofchanges in ﬁnancial literacy (resp. banks’ asset investment behavior) on economic growth andﬁscal sustainability are investigated in Section 4 (resp. 5). Finally, we conclude the paper insection 6.

5

The Taylor principle means that when the inﬂation rate increases, the monetary authority increases the shortterm nominal interest rate more than the inﬂation rate.

6

2

Setting up the model

In this section, we describe the setup of the model. Our model is an OG model consisting offour economic agents (individuals, ﬁrms, banks, and the government) and three assets (physicalcapital, bank deposits, and government bonds). It is basically an extension of Grossman andYanagawa (1993) on the following three points: (1) Individuals can choose between physicalcapital and bank deposits when saving, (2) in investing in physical capital, individuals need tospend part of their endowed time acquiring ﬁnancial knowledge, and (3) banks invest a given
proportion of their total deposits in physical capital and the rest in government bonds.

2.1

Firms

The production sector is composed of homogenous ﬁrms, the number of which is normalized to
1. Each ﬁrm produces ﬁnal goods using two kinds of production factors (i.e., physical capital andlabor). We assume that the ﬁnal goods market and factor markets are perfectly competitive, andthe price of the ﬁnal goods is normalized to 1 (i.e., the ﬁnal goods are numeraire). The production
function of each ﬁrm is given by
Yt = F [Kt , A(K̄t )Lt ].

(where A(K̄t ) ≡ θK̄t ).

Here, Yt , Kt , and Lt are the outputs of the ﬁnal goods, physical capital input, and labor input,
respectively. We assume that the production function F exhibits constant returns to scale andpositive marginal products with respect to each input. A(K̄t )(≡ θK̄t ) is the labor-augmenting
productivity, and it is an increasing function of K̄t (i.e., average physical capital stock per the
young generation’s population). Following Romer (1986), we suppose that each ﬁrm maximizes
its proﬁt, considering A(K̄t ) as given. The ﬁrst-order conditions of proﬁt maximization are as
follows:
FK [Ktd , θK̄t Ldt ] = rtk + δθK̄t FAL [Ktd , θK̄t Ldt ] = wt

(where FK ≡ ∂F/∂Kt ).

(1a)(where FAL ≡ ∂F/∂(θK̄t Lt )).

(1b)Here, Ktd , Ldt , rtk , δ, and wt are the demand for physical capital, demand for labor, (net) rate of7

return on physical capital, depreciation rate of physical capital, and wage, respectively.

2.2

Individuals

Each individual lives for two periods (young and old). The set of individuals born in period t iscalled “generation t.” The population of each generation is assumed to be 1, so the population of
the economy is constant over time.
There are two types of savings instruments available to individuals: physical capital andbank deposits, and individuals choose one of the two instruments when saving.6 They face thekfollowing trade-oﬀ: The rate of return on physical capital (denoted by rt+1
) is higher than that on
dbank deposits (denoted by rt+1), but the actual rate of return on physical capital that individuals

can receive depends on their levels of ﬁnancial literacy.7 (A more detailed explanation of thispoint is provided in the next paragraph.) On the other hand, the rate of return on bank deposits
is lower than that on physical capital, but no ﬁnancial literacy is required to hold them.
Individuals in generation t are endowed with one unit of time in their young period. Whensaving in the form of physical capital, individual i allocates 0 ≤ φi,t ≤ 1 of time to acquiring
ﬁnancial literacy and the remaining 1 − φi,t of time to labor supply. Therefore, the wage incomeof individual i is (1 − φi,t )wt , and she allocates part of it to young-age consumption cyi,t and therest to savings in the form of physical capital si,t . As mentioned before, the actual return fromsavings depends on the level of ﬁnancial literacy. Speciﬁcally, we assume that individual i cankonly receive the fraction 1−σi /φi,t of the total return (1+rt+1)si,t (i.e., individuals fail to receivekk(σi /φi,t ) × (1 + rt+1
)si,t .8 ) The unrecovered amount (σi /φi,t ) × (1 + rt+1
)si,t is smaller when

φi,t (the time spent on acquiring ﬁnancial literacy by individual i) is larger or σi is smaller. Here,σi is an exogenous parameter for individual i’s learning ability, and a smaller σi corresponds toa higher learning ability. We assume that σi is uniformly distributed in the interval [σ, σ̄] andsatisﬁes 0 < σ ≤ σi ≤ σ̄ < 1. Accordingly, the individual with the highest (resp. lowest)
learning ability has σ (resp. σ̄).

6

We assume that individuals cannot directly hold government bonds. This assumption is not unrealistic. For
example, only 1.2% of outstanding government bonds are held by households in Japan.
7This assumption is often employed when modeling the individual’s investment behavior in ﬁnancial literacy.
See Jappelli and Padula (2013) and Lusardi et al. (2017).
8
The resources that individuals fail to receive are assumed to be disposed of without being used for either consumption or investment.

8

We assume that the utility function of individual i in generation t is given by: α  o 1−αci,t+1
Ui,t = cyi,t,where cyi,t and coi,t+1 are the young-age and old-age consumption, respectively. Each individualcompares the utility level in the case of choosing physical capital with the utility level in the caseof choosing bank deposits, and chooses the one that gives higher utility.9
The utility maximization problem in the case of choosing physical capital iskkmax Ui,ts.t. cyi,t + si,t = (1 − φi,t )wt , coi,t+1 = (1 − σi /φi,t )(1 + rt+1
)si,t .

si,t ,φi,tkkHere, Ui,t, si,t , wt , and rt+1
are the utility level when physical capital is chosen, saving in theform of physical capital, wage, and rate of return on physical capital, respectively.
Arranging the ﬁrst order conditions, we have



∂Ui,t
=0
si,t = (1 − α)(1 − φi,t )wt ,
(2a)
∂si,t


(φi,t )2∂σi
∂ 2 σi
σi
∂Ui,t
=0
σi =
,
> 0,
>
0,
0
<
< 1. (2b)∂φi,t(1 − α)(1 − φi,t ) + φi,t ∂φi,t∂(φi,t )2
φi,tFrom (2b), the function φi,t (σi ) satisﬁes the following properties:
φi,t (σi ) = φ(σi ),φ(0) = 0, φ (σi ) > 0, φ (σi ) < 0.

(3)

Figure 2 depicts the function φi,t = φ(σi ). This ﬁgure shows that individuals with higher learning
ability (i.e., lower σi ) spend less time acquiring their ﬁnancial literacy.
Calculating the indirect utility function in this case, we have

k
k
Ui,t
= αα (1 − α)1−α (1 + rt+1
)1−α [1 − φ(σi )]σi
1−
φ(σi )

1−α
wt .

(4)

[Figure 2].

9Based on essentially the same idea, Spataro and Corsini (2017) formulated the portfolio selection problem ofhouseholds and investigated the factors that determine the holding of risky assets. However, their model is not amacroeconomic model.

9On the other hand, the utility maximization problem in the case of bank deposits isddmax Ui,ts.t. cyi,t + di,t = wt , coi,t+1 = (1 + rt+1
)di,t .

cyi,t ,coi,t+1

Here, di,t is the savings in the form of bank deposits. The saving function and the indirect utility
function in this case are
di,t = (1 − α)wt

(5a)

d
d
Ui,t
= αα (1 − α)1−α (1 + rt+1
)1−α wt(5b)Note that di,t does not depend on an individual’s index i. Individual i chooses physical capital if
k
d
k
d
Ui,t
> Ui,tand bank deposits if Ui,t< Ui,t. Therefore, the threshold of the learning ability (σt∗ )kdat which Ui,t= Ui,tholds, can be calculated as follows:d=
1 + rt+1k
1 + rt+1Φ(σt∗ )

where1
Φ(σt∗ ) ≡ [1 − φ(σt∗ )] 1−α



σt∗
1−
φ(σt∗ )

.

(6)

By calculating Φ (σt∗ )(≡ ∂Φ(σt∗ )/∂σt∗ ), we can obtain the following (see Appendix A).
Φ (σt∗ ) < 0.

(7)

If σi is lower (resp. higher) than the threshold (σt∗ ), individual i chooses physical capital (resp.
bank deposits). Hence, the population (or the share) of individuals who hold physical capital is((σt∗ − σ)/(σ̄ − σ)), while the population (or the share) of individuals who hold bank deposits is((σ̄ − σt∗ )/(σ̄ − σ)).

2.3

Government

The government owes a given amount of debt (denoted by Bt ) at the beginning of period t. The
government repays debt only by issuing new bonds (i.e., the government plays a “Ponzi game”).
We assume that government bonds are the discount bonds at 1 period maturity whose face valueis 1. Thus, the government’s budget constraint in period t is given by:
Bt = qt Bt+1 .
10(8)Here, Bt+1 is the quantity of government bonds issued in period t, and qt is their market price.
The (gross) government bond yield is equal to 1/qt .

2.4

Banks

The banking sector is composed of homogenous banks, the number of which is normalized to 1.
From (5a), the total amount of bank deposits in period t is given by:
dt ≡

 σ̄
σt∗

di,t

1
σ̄ − σt∗dσi = (1 − α)wt.
σ̄ − σσ̄ − σ(9)Each bank invests an exogenous fraction (denoted by 0 < e < 1) of dt in physical capital and therest in government bonds.10
The proﬁt of each bank to be realized in period t + 1 can be expressed as:





1
d
k
d
edt + (1 − e)dt − 1 + rt+1
+ η dt .
πt+1= 1 + rt+1
qt
Here, η is the operating cost per unit of deposit.11d
The banking sector is assumed to be perfectly competitive (πt+1= 0), so the following holds:



1
d
k
1 + rt+1
e + (1 − e) − η.
= 1 + rt+1
qt

(10)

d
k
This shows that 1 + rt+1
(the interest rate on bank deposits) is the weighted average of 1 + rt+1

(the return rate on physical capital) and 1/qt (the government bond yield) minus η (the bank’s
operating costs). Thus, in eﬀect, individuals holding bank deposits hold a fraction (e) of theirsavings in physical capital and the rest in government bonds, paying the cost of η per unit ofsavings.

2.5

Market equilibrium

In this subsection, we derive the market equilibrium conditions. First, the equilibrium conditions
for factor markets in period t are given by Ktd = Kt and Ldt = 1 (in both equations, the left10

Even when we assume that e depends positively on the diﬀerence between the return rate on physical capitalk(1 + rt+1) and the government bond yield (1/qt ), the qualitative results do not change.
11
We assume the operating cost takes the form of expenditure of the ﬁnal goods.

11hand side (LHS) means demand and the right-hand side (RHS) means supply). Because thepopulation of each generation is equal to one, K̄t = Kt holds. Substituting these into (1a) and(1b), the equilibrium factor prices are:rtk = FK − δ,

wt = θFAL Kt .

(11)

Here, both FK and FAL become constant.12 Thus, the return rate of physical capital is constant,and the wage is proportional to physical capital stock.
d
Substituting (11) into (6) and (10), the interest rate on bank deposits 1 + rt+1
, and the gov-

ernment bond yield Qt (≡ 1/qt ) can be calculated as follows:d
1 + rt+1= (1 + FK − δ) Φ (σt∗ )Qt (≡ 1/qt ) =(12a)(1 + FK − δ) [Φ (σt∗ ) − e] + η
.
1−e

(12b)

From (12a) and (12b), we can see that when σt∗ (the threshold value of the individual’s learningdability) increases, both 1 + rt+1
and Qt decline. The reasons for this are as follows. When σt∗increases, more individuals choose physical capital as their saving method. In such a situation,dk
1 + rt+1
must decrease because the return rate of physical capital (1 + rt+1
) is ﬁxed (see (11)).
dkd
In addition, 1 + rt+1
is the weighted average of 1 + rt+1
and Qt from (10), so when rt+1
falls, Qt

also falls.
Second, the equilibrium condition for the ﬁnal goods market (i.e., the equilibrium condition
between savings and investment) can be expressed as follows:
Kt+1 =

 σt∗
si,t
σ

1
dσi + edt .
σ̄ − σ(13)

Substituting (2a), (9) and (11) into (13), we have

Gt

Kt+1
≡
Kt




= A [I (σt∗ , σ) + e (σ̄ − σt∗ )] .

(1 − α) θFALA≡σ̄ − σ.

(14)Here, Gt denotes the gross growth rate of physical capital. Since our model is an endogenous
FK becomes constant, because FK Ktd , θK̄t Ldt = FK (Kt , θKt ) = FK (1, θ) holds. Here, the second
equality holds because the function FK is homogenous of degree 0. FAL also becomes constant for the same reason.
12

12

growth model of the AK type, it is also the gross economic growth rate. The deﬁnition of thefunction I (σt∗ , σ) in (14) is given by
I (σt∗ , σ) ≡

 σt∗


[1 − φ(σi )]dσi ,

σ


∂I
∂I
∗
= 1 − φ(σt ) > 0,
= −[1 − φ(σ)] > 0 . (15)∂σt∗∂σIn this paper, we assume the following:
∂Gt
>0
∂σt∗

⇔ 1 − φ (σt∗ ) − e > 0.

(16)(16) implies that when the threshold σt∗ increases (i.e., when the share of physical capital holders
increases), the economic growth rate rises. This assumption seems natural because it supposesthat a higher share of direct ﬁnancing leads to higher economic growth.13
Third, the equilibrium condition for the government bond market in period t is given by:
qt B t+1 (= Bt ) = (1 − e) dt .

(17)

The LHS of (17) refers to the supply of government bonds, and RHS refers to the banks’ demand
for them. Substituting (9) and (11) into (17) and arranging, we obtain:

bt

Bt
≡
Kt




= (1 − e) A (σ̄ − σt∗ ) .

(1 − α) θFALA≡σ̄ − σ.

(18)Here, bt is the ratio of government debt to physical capital at the beginning of period t, which is
historically given. The threshold σt∗ in period t is determined to satisfy (18).14
Finally, by substituting (14) and (18) into (8) and rearranging, we derive the following autonomous diﬀerence equation for σt∗ .
∗σt+1= σ̄ −Q (σt∗ ) (σ − σt∗ ).
G (σt∗ )(19)

Here, the deﬁnitions of Q (σt∗ ) and G (σt∗ ) are given by (12b) and (14), respectively.
Because the value of σt∗ in the initial period t is determined by (18), the subsequent path13

Without this assumption, the analysis of the existence and stability of balanced growth paths becomes rather
complicated.
14
Namely, σt∗ is not a jump variable at the initial period t.

13

dis also uniquely determined by (19). Once the path of σt∗ is determined, 1 + rt+1
(the interest

rate on bank deposits), Qt (the government bond yield), and Gt (the economic growth rate) aredetermined from (12a), (12b), and (14).

3

Existence and stability of BGPs

In this section, we investigate the existence and stability of balanced growth paths (BGPs) boththeoretically and numerically. Here, a BGP is a growth path through which σt∗ (the threshold of
the individual’s learning ability) is constant over time.
First, we discuss the existence of the BGPs. From (19), it is easy to see that σt∗ = σ̄ is oneof the BGPs. Since all individuals choose physical capital as a means of saving in this BGP, theratio of government debt to physical capital in this BGP is zero.
In the other BGP, where σt∗ = σ̄ holds, (19) becomesG (σ ∗ ) = Q (σ ∗ ) ,(20)

where the LHS of (20) is the (gross) economic growth rate and the RHS is the (gross) government
bond yield. From (7), (12b), and (16), we have∂G∂Q
(1 + FK − δ)Φ (σ ∗ )
∗
=
A[1
−
φ(σ
)
−
e]
>
0,
=
< 0.
∂σ ∗
∂σ ∗
1−e

(21)

Here, ∂G/∂σ ∗ > 0 holds by assumption (16). In contrast, the reason why ∂Q/∂σ ∗ < 0 holds
is that a rise in σ ∗ lowers the deposit interest rate (see (12a)) and the government bond yield is
linked to the deposit interest rate (see (10)).
In the following we assume
G (σ) < Q (σ) , G (σ̄) > Q (σ̄) .

(22)

Under (21) and (22), (20) can be depicted as in Figure 3. It is clear from this ﬁgure that the BGP
exists uniquely in the range σ < σ ∗ < σ̄. In this BGP, the ratio of government debt to physicalcapital is positive because there are individuals who hold bank deposits. Hence, in our model,
there are two BGPs: one where the ratio of government debt to physical capital is zero and the14other where it is positive.
[Figure 3].
Next, we examine the stability of the BGPs. We deﬁne the RHS of (19) as H (σt∗ ). By (21) and(22), we can easily show that the function H (σt∗ ) has the following properties:
H (σ) < σ,H  (σt∗ ) > 0,

H (σ̄) = σ̄,H  (σ̄) < 1.

(23)

From (23) and the uniqueness of the BGP that satisﬁes σ < σt∗ < σ̄, the phase diagram of (19)can be drawn as the upper half of Figure 4, from which we can see that E s (the BGP with σt∗ = σ̄)is stable, whereas E u (the BGP with σ < σt∗ < σ̄) is unstable. To avoid confusion, we denote thevalue of σt∗ in the stable BGP as σs∗ (= σ̄) and in the unstable BGP as σu∗ .
The lower half of Figure 4 depicts the relationship between σt∗ and bt (the ratio of government
debt to physical capital), as shown by (18). The ratio of government debt to physical capital in
the unstable BGP is given byb∗u = (1 − e)A (σ̄ − σu∗ ) .

(1 − α) θFALA≡σ̄ − σ(24)This value corresponds to the maximum sustainable ratio of government debt to physical capital.
If the initial value of bt (which is historically given) is smaller than b∗u , the initial value of σt∗
becomes larger than σu∗ , and σt∗ converges to its upper limit σ̄. In this case, the economy converges
to the stable BGP, along which the value of bt will decline over time. Conversely, if the initialvalue of bt is larger than b∗u , the initial value of σt∗ becomes smaller than σu∗ , and σt∗ will decline
over time. In this case, the government will go bankrupt within a ﬁnite period.
[Figure 4]
Summarizing the results so far, we have the following proposition.
Proposition 1. Under assumptions (16) and (22), the dynamics of σt∗ (the threshold of the individual’s learning ability) shown in (19) has one stable BGP, where the ratio of government debt
to physical capital is 0 and one unstable BGP where its ratio is positive. The economic growthrate and the ratio of government debt to physical capital in both BGPs are given as follows:15

(unstable BGP) G∗u = A [I (σu∗ , σ) + e (σ̄ − σu∗ )] ,(stable BGP)G∗s = AI (σ̄, σ) ,b∗s = 0.

b∗u = (1 − e) A (σ̄ − σu∗ )

(25a)
(25b)

Before proceeding to further analysis, we conﬁrm the validity of the above result with a numericalexample. We set the benchmark numerical values of the parameters as shown in Table 2. The
two new parameters (γ, Γ) in the table occur from the speciﬁcation of the production function
as Y = ΓK γ (KL)1−γ .15[Table 2]
Details regarding the numerical settings of the parameters are as follows. For the parameter α
in the utility function, we set α = 0.69 because the discount factor should satisfy (1 − α)/α =0.97330 ≈ 0.44 (see Song et al., 2012). The parameter γ in the production function is set to γ =0.37 by averaging the values of the US (γ = 0.35), the EU (γ = 0.38), and Japan (γ = 0.38).16
For the capital depreciation rate δ, we assume δ = 1 (i.e., the case of full depreciation).17 Under
this parameter setting for the production function, we have 1 + rk = Γγ = Γ × 0.37. Assumingthat the average annual rate of return on capital is 4% (see Trabandt and Uhlig (2011)), we have
1 + rk = 1.0430 ≈ 3.24, which means Γ ≈ 8.76. Therefore, the marginal productivity of labor
can be calculated as FAL = Γ (1 − γ) = 5.52. For the parameter e, we set 1 − e = 0.2, because
the ratio of public debt to total deposits in the private banks in the US, the EU, and Japan is 0.140,
0.212, and 0.192, respectively, and their average value is approximately 0.2.18 The parameter η(i.e., the operating cost of private banks) is set to η = 1.69 to satisfy (10).19 Finally, for the
In this speciﬁcation of the production function, θ = 1 is assumed.
See Trabandt and Uhlig (2011) for the values of the US and the EU, and Hansen and İmrohoroğlu (2016) for
the value of Japan17
It is natural assumption because we regard one period as around 30 years.
18
See International Financial Statistics (IFS). Here, the private banking sector refers to “Other Depository Corporations” (i.e., depository corporations other than the central bank) in the statistics. We also calculate the outstanding
government debt held by the private banking sector as the sum of “Claims on Central Government” and “Claimson State and Local Governments” in the statistics, and outstanding deposits as the sum of “Transferable Deposits
Included in Broad Money” and “Other Deposits Included in Broad Money” (accessed on September 17, 2021).
19
The average annual deposit interest rate in Japan, the EU, Canada, and the US is about 0.9%, so we have
1 + rd = 1.00930 ≈ 1.31. The average annual yield on government bonds in Japan, the EU, Canada, and the US
is about 2.4%, so we have 1/q = 1.02430 ≈ 2.04. We have referred to the International Financial Statistics (IFS)
for these data (accessed on September 17, 2021). On the other hand, for the return rate on capital, we use the value
1 + rk ≈ 3.24 derived earlier. Therefore, the value of η that satisﬁes (10) is η = 1.69.
15
16

16

parameters (σ̄, σ), we set (0.07, 0.04) such that the two assumptions (16) and (22) are satisﬁed,
and Q (σt∗ ) in (12b) becomes positive.
We conﬁrm that the analytical results can be reproduced under the above parameter settings.
Figure 5-(a) represents the value of ∂Gt /∂σt∗ = 1 − φ (σt∗ ) − e, and it checks whether assumption(16) holds. This ﬁgure shows that assumption (16) holds for all possible ranges of σi . Figure 5(c) illustrates the existence of an unstable BGP: σu∗ ∈ (σ, σ̄) ((20)), which is in line with Figure
3. Figures 5-(b) and 5-(d) illustrate respectively the stability of the two BGPs ((19)) and therelationship between σt∗ and bt ((18)), and are in line with the upper half and lower half of Figure
4.
[Figures 5].
From Figure 5-(c), we can conﬁrm that the value of σu∗ in the unstable BGP is 0.0572 underthe benchmark numerical settings. Accordingly, from Figure 5-(d), the maximum sustainableratio of public debt to physical capital is b∗u = 0.1464. On the other hand, the actual average
ratio of public debt to physical capital for the EU, the US, and Japan is approximately 0.11.20
If we use b0 = 0.11 (the initial value of bt ), the economy converges to the stable BGP because
b0 (= 0.11) < b∗u (= 0.1464).
[Figure 6].
Figures 6-(a), 6-(b), and 6-(c) show the evolution of σt∗ , Gt and Qt during the transition
process when b0 = 0.11 with the benchmark parameters in Table 2. From these ﬁgures, we cansee that σt∗ converges to σ̄ within 10 periods, and the economic growth rate increases and thegovernment bond yield decreases during the transition. The changes in these variables are also
consistent with the analytical results (see (21)).

20

According to AMECO database, the average value of Y /K (i.e., the ratio of GDP to physical capital) for Japan,
the EU, the US, and Canada is about 0.4. On the other hand, the average value of B/Y (i.e., the ratio of public debt toGDP) for those four countries is approximately 1.41. Here, we used the OECD database for the values for Japan, theUS, and the EU, and the AMECO database for the value for Canada. (Accessed on September 17, 2021). Therefore,
the average value of B/K (i.e., the ratio of public debt to physical capital) is 0.4 × 1.41 = 0.564. According toArslanalp and Tsuda (2014), the share of government bonds held by domestic private banks is approximately 20%,so the ratio of public debt held by the private banking sector to physical capital is 0.564 × 0.2 ≈ 0.11.

17

4Eﬀect of changes in the learning ability of individuals

In this section, we examine, both analytically and numerically, the eﬀect of a decrease in σi (i.e.,
an increase in the individual’s learning ability) on economic growth and ﬁscal sustainability.
Here, a situation in which σi decreases uniformly can be interpreted as one in which the averageﬁnancial literacy of individuals increases. As mentioned brieﬂy in the introduction, with theincreasing importance of ﬁnancial literacy due to the increasing complexity of ﬁnancial markets
and rising life expectancy, a growing number of countries have introduced educational programs
that improve ﬁnancial literacy, especially for younger generations (OECD, 2014). This section
analyzes the implications of such a policy on economic growth and ﬁscal sustainability.
We consider a situation in which σi changes uniformly by dσ(> 0) (i.e., both the lower andupper bounds of the uniform distribution of σi change by dσ = dσ̄ = dσ). From (20), the eﬀectof such a change on σu∗ (the threshold of the individual’s learning ability in the unstable BGP)
can be calculated as follows:∂G∗uA [1 − φ (σ) − e]dσu∗
= − ∂G∗ ∂σ ∂Q∗ => 0,∗ ) − e] − Q (σ ∗ )
u
u
dσ
A
[1
−
φ
(σ
−
u
u
∗
∗
∂σ
∂σ
u

(26a)

u

∂G∗u
= −A [1 − φ (σ) − e] < 0,
where
∂σ
∂Q∗u
(1 + FK − δ)Φ (σu∗ )
∂G∗u
∗
=
A
[1
−
φ
(σ
)
−
e]
>
0,
=
< 0.
u
∂σu∗
∂σu∗
1−e

(26b)
(26c)

(26a) shows that a uniform decrease in σi lowers σu∗ . Note that because both (σ, σ) and σu∗ movein the same direction, it is ambiguous whether the share of physical capital holders (= (σu∗ −σ)/(σ̄ − σ)) in the new unstable BGP increases or decreases.
Next, we examine the eﬀect of the change in σi on G∗u (the economic growth rate in theunstable BGP) and G∗s (the economic growth rate in the stable BGP). From (25a) and (25b), we
have
indirect eﬀect(+)

direct eﬀect(−)

dG∗u
A [1 − φ (σ) − e] Q (σu∗ )∂G∗u ∂σu∗
∂G∗u
=
+
=
< 0,
dσ
∂σu∗ ∂σ∂σ
A [1 − φ (σu∗ ) − e] − Q (σu∗ )dG∗s= −A [φ (σ) − φ (σ)] < 0.
dσ

(27a)
(27b)

(27a) and (27b) show that a uniform decrease in σi raises both G∗u and G∗s . The reason for this18

result is as follows: (27a) indicates that in the unstable BGP a uniform decrease in σi aﬀects G∗uboth indirectly (i.e., via the impact on σu∗ ) and directly (i.e., not via the impact on σu∗ ). From(26a), (26b), and (26c), the sign of the indirect eﬀect is positive, while that of the direct eﬀect isnegative. Because the negative direct eﬀect exceeds the positive indirect eﬀect in our model, thesign of dG∗u /dσ becomes negative. (27b), on the other hand, indicate that a uniform decrease in
σi in the stable BGP reduces the learning time φ (σi ) and raises both savings and G∗s .
Finally, we examine the eﬀect of the change in σi on b∗u (the ratio of public debt to physicalcapital in the unstable BGP, or the maximum sustainable ratio of public debt to physical capital).
From (25a) and (26a), we can derive the following relationship (see Appendix B for details of
the derivation):

db∗u
0
dσ

⇔

dG∗u,tdQ∗u,t
−
−
.
dσ
dσ

(28)

Here, dG∗u,t /dσ and dQ∗u,t /dσ means that when σi changes in period t in the unstable BGP, howmuch the economic growth rate and government bond yield will change in the same period t (notin the new unstable BGP). As shown in Appendix B, if σi declines in period t, both the economic
growth rate and the government bond yield in the same period t increase (i.e., dG∗u,t /dσ < 0 anddQ∗u,t /dσ < 0). (28) indicates that the sign of db∗u /dσ is ambiguous in general, and it depends
on whether −dG∗u,t /dσ or −dQ∗u,t /dσ is larger.
Consider the case where −dG∗u,t /dσ < −dQ∗u,t /dσ holds, because we will show numericallybelow that this case indeed is realized. Then, (28) shows that a decline in σi reduces b∗u . The
reasons for this are as follows. When σi declines in period t in the unstable BGP, the following
two cases can occur: In the ﬁrst case, the economy may converge to a stable BGP. In the second
case, it may proceed to a divergent path and eventually go bankrupt. The latter case is more
likely to occur because −dG∗u,t /dσ < −dQ∗u,t /dσ shows that an increase in the government bondyield becomes larger than that in the economic growth rate, inducing the ratio of government
debt to physical capital to explode. Thus, a decline in σi reduces the maximum sustainable ratioof government debt to physical capital b∗u .
[Figure 7].
We check the sign of db∗u /dσ under plausible parameter values. Figure 7 shows that a uniform
decrease in σi lowers b∗u (i.e., db∗u /dσ > 0 holds) under the benchmark numerical settings of19

the parameters in Table 2.21 This numerical result indicates that as individuals become moreﬁnancially literate, the maximum sustainable ratio of public debt to physical capital declines (i.e.,ﬁscal sustainability worsens). Therefore, if the actual ratio of public debt to physical capital in theinitial period is close to its maximum sustainable value, a uniform decrease in σi will cause σt∗ todecline (and accordingly, cause bt to rise) over time, and the government will go bankrupt within
a ﬁnite period of time. On the other hand, if the actual ratio of public debt to physical capitalin the initial period is suﬃciently smaller than its maximum sustainable value, the economy still
converges to the stable BGP even when σi decreases.
[Figure 8].
Figures 8-(a), 8-(b), and 8-(c) show the dynamic paths of the economy when the ratio of public
debt to physical capital at the initial period (i.e., period t = 0) is given by 0.11 and the valuesof (σ̄, σ) are respectively (i) (0.07, 0.04), (ii) (0.08, 0.05), and (iii) (0.06, 0.03). Case (i) is thebenchmark case, and the dynamic path for this case has already been explained in Section 3. Case(ii) is the case where the individual’s learning ability is low compared to the benchmark case, andcase (iii) is the case where the individual’s learning ability is high compared to the benchmark
case. When the individual’s learning ability is relatively high (case (iii)), σt∗ declines sharply(accordingly, bt rises sharply) and reaches its lower bound (σ) at period t = 3. Along this pathto bankruptcy, the government bond yield grows rapidly and the economic growth rate declines
deeply. By contrast, when the individual’s learning ability is relatively low (case (ii)), σt∗ rises(accordingly, bt declines), and σt∗ reaches the stable BGP at period t = 3. Along this stable path,the government bond yield declines and the economic growth rate rises steadily.
Figure 9 illustrates the welfare level of each generation in the benchmark case (i.e., (σ̄, σ) =(0.07, 0.04)) and the case when the individual’s learning ability is relatively low (i.e., (σ̄, σ) =(0.08, 0.05)).22 Here, we deﬁne the welfare of generation t as1
Wt =σ̄ − σ

 σt∗
σ

k
Ui,t
dσi +

21

 σ̄
σt∗


d
Ui,t
dσi .

This result is robust even when the value of e (i.e., the ratio of physical capital investment to total deposits inthe banking sector) is 0.75 or 0.85.
22
In the case where the learning ability is relatively high (i.e., (σ̄, σ) = (0.06, 0.03)), the government goesbankrupt within a ﬁnite period and the behavior of the economy after that is not modeled. The welfare analysis
in this case is thus omitted.

20

Figure 9 shows that when the learning ability declines compared to the benchmark case, thewelfare of each generation decreases, and that the size of decline in welfare becomes larger overtime. This is mainly because a decline in the learning ability lowers the economic growth rate
through more limited participation in the capital market, which lowers wage levels of future
generations.
[Figure 9]
Summarizing the results in this section, we have the following proposition.
Proposition 2. Suppose that σi decreases uniformly (i.e., the learning ability or ﬁnancial literacyof individuals increases). Then, the following holds:.
(1) Such a change raises the economic growth rates of both unstable and stable BGPs.
(2) The eﬀect of such a change on b∗u (the maximum sustainable ratio of public debt to physicalcapital) is ambiguous in general, but under the parameter settings in Table 2 such a change
lowers b∗u .
(3) Under the parameter settings in Table 2, such a change improves the welfare of each generation and the degree of welfare improvement increases as time passes, if the economy
converges to the stable BGP by such a change.
This proposition shows that an improvement in the average level of an individual’s ﬁnancial
literacy certainly increases the potential economic growth, whereas it worsens ﬁscal sustainability and increases the likelihood of the government’s bankruptcy. This result implies that in acountry with a high ratio of public debt to GDP, such as Japan, public policy to improve individuals’ ﬁnancial literacy should be combined with ﬁscal consolidation to avoid the government’s
bankruptcy.

5Eﬀect of changes in banks’ asset investment behavior

In this section, we examine, both analytically and numerically, how the change in banks’ asset
investment behavior (speciﬁcally, the change in the parameter e, i.e., the ratio of physical capital
investment to total deposits) aﬀects economic growth and ﬁscal sustainability. It seems likely that21

(i) an increase in e will improve economic growth because it increases physical capital investment
by banks, while (ii) an increase in e worsens ﬁscal sustainability because it reduces the amountof government debt underwritten by banks. The purpose of this section is to examine whether
such a prediction is correct.
First, we examine the eﬀect of an increase in e on σu∗ (the threshold of an individual’s learning
ability in the unstable BGP).
By totally diﬀerentiating (20), we have
∗

∗

∂Gu
u
− ∂Q
dσu∗
∂e
∂e
,
= − ∂G
∗
∂Q∗u
u
de
−
∗
∗
∂σ
∂σ

where

∂G∗u

u

(29a)

u

∂Q∗u
− (1 + FK − δ) [1 − Φ (σu∗ )] + η
,
=
∂e
∂e
(1 − e)2∂Q∗u(1 + FK − δ) Φ (σu∗ )
∂G∗u
∗
=
A
[1
−
φ
(σ
)
−
e]
>
0,
=
< 0.
u
∂σu∗
∂σu∗
1−e
= A (σ − σu∗ ) > 0,

(29b)
(29c)

Because the sign of ∂Q∗u /∂e in (29b) is ambiguous, so is the sign of dσu∗ /de in general. However,the sign of dσu∗ /de is found to be negative if we assume ∂Q∗u /∂e < 0 as follows:
dσu∗
∂Q∗u
< 0 if< 0 ⇔ η < (1 + FK − δ)[1 − Φ(σu∗ )]
de
∂e

(30)

Figure 10 shows that the assumption (i.e., η < (1 + FK − δ)[1 − Φ(σu∗ )]) in (30) is satisﬁed
under the benchmark parameter settings in Table 2.23 Thus, in the following analyses, we willassume (30) (i.e., an increase in e reduces the share of individuals who choose physical capital
as a means of saving in the unstable BGP).
[Figure 10].
Next, we examine the eﬀect of an increase in e on G∗u (the economic growth rate of theunstable BGP). Before addressing this, note that G∗s (the economic growth rate of the stable
BGP) is independent of e (see (25b)) because in the stable BGP, all individuals hold their savingsin the form of physical capital (i.e., there are no bank deposits in the economy). Thus, we focus

More precisely, Figure 10 shows that the assumption in (30) holds not only for (σ̄, σ) = (0.07, 0.04) (i.e.,the benchmark case), but also for (σ̄, σ) = (0.08, 0.05) and (σ̄, σ) = (0.06, 0.03). We can also conﬁrm throughnumerical calculations that this assumption holds even in the case of e = 0.75 or 0.85.
2322

on G∗u . From (25a), the eﬀect of an increase in e on G∗u can be calculated as follows:

dG∗u
de

indirect eﬀect(−)

direct eﬀect(+)∂G∗u ∂σu∗∂σu∗ ∂e

∂G∗u
∂e

=

+

∂σu∗
∗
= A [1 − φ (σu ) − e]
+ A (σ̄ − σu∗ ) .
∂e

(31)

From (31), we can see that an increase in e has the following two eﬀects: the indirect eﬀect onG∗u via the impact on σu∗ (i.e., the share of physical capital holder) and the direct eﬀect on G∗u ,not via the impact on σu∗ . The sign of the direct eﬀect on G∗u is positive (by (29b)), while that of
the indirect eﬀect on G∗u is negative (by (29c) and (30)). This is because an increase in e raises
the physical capital investment by banks directly, whereas it reduces the share of individuals whochoose physical capital as a means of saving. Therefore, the total eﬀect is generally ambiguous.
In the following, we examine the sign of dG∗u /de using numerical calculations.
Figure 11-(b) shows how G∗u reacts to the changes in e under the benchmark parameter settingsin Table 2. We can ﬁnd that an increase in e raises G∗u in the range of e from 0.75 to 0.85. This
result implies that the positive direct eﬀect exceeds the negative indirect eﬀect in (31). This result
still holds when the value of (σ̄, σ) takes from (0.07, 0.04) to (0.08, 0.05). However, if the valueof (σ̄, σ) is set to (0.06, 0.03), the above result changes. Figure 11-(a) illustrates this case andshows that there is a value of e that minimizes G∗u . If e is larger (resp. smaller) than that value,
an increase in e raises (resp. lowers) G∗u . Therefore, the eﬀect of the change in e on G∗u varies
depending on the value of (σ̄, σ).
[Figure 11]
Finally, we examine the eﬀect of the change in e on b∗u (i.e., the maximum sustainable ratioof public debt to physical capital). From (25a), we can derive the following:

db∗u
de

=

indirect eﬀect(+)

direct eﬀect(−)∂b∗u ∂σu∗∂σu∗ ∂e

∂b∗u

+

∂e

(−) by (30)= −A(1 − e) ×

∂σu∗
∂e

−A(σ̄ − σu∗ ).

(32)

As before, the eﬀect of an increase in e on b∗u can be divided into indirect and direct eﬀects. From(30), the indirect eﬀect on b∗u is positive, while the direct eﬀect on b∗u is negative, so the total

23

eﬀect is generally ambiguous.24
Figure 12-(b) illustrates how b∗u reacts to the changes in e under the benchmark parametersettings in Table 2. From this ﬁgure, we ﬁnd that a value of e that maximizes b∗u exists in the rangeof e from 0.75 to 0.85.2526 This result is consistent with aforementioned result that a value of e thatminimizes G∗u exists, because a larger outstanding public debt induces a smaller economic growthrate due to its crowding-out eﬀect on physical capital. Figures 12-(c) and 12-(a) show the graphs
when the values of (σ̄, σ) are re-set from the benchmark case of (0.07, 0.04) to (0.08, 0.05) and(0.06, 0.03), respectively. When (σ, σ) = (0.08, 0.05) (i.e., when the average level of learning
ability (or ﬁnancial literacy) is relatively low), an increase in e lowers b∗u in the range of e from0.75 to 0.85. This is because the direct eﬀect exceeds the indirect eﬀect (see (32)) in this case.
On the other hand, when (σ, σ) = (0.06, 0.03) (i.e., when the average level of learning ability(or ﬁnancial literacy) is relatively high), an increase in e raises b∗u in the range of e from 0.75 to
0.85. This is because the indirect eﬀect exceeds the direct eﬀect (see (32)) in this case.27[Figure 12].
Summarizing the results of this section, we have the following proposition.
Proposition 3. Suppose that banks raise e (the ratio of physical capital investment to total deposits). The following then holds:(1) Such a change reduces the share of physical capital holders in the unstable BGP under(30). (30) is satisﬁed under the parameter settings in Table 2.
(2) The eﬀect of such a change on G∗u (the economic growth rate in the unstable BGP) is
ambiguous in general, but under the parameter settings in Table 2, such a change raises
24

For the eﬀect of an increase in e on b∗u , we can derive a similar condition as (28):dG∗u,tdQ∗u,t
db∗u
0⇔

.
de
de
de

Here, LHS (resp. RHS) of the second inequality means that when e rises in period t in the unstable BGP, how muchthe economic growth rate (resp. government bond yield) changes in the same period t. This relationship shows that
if the change in the economic growth rate is larger (resp. smaller) than the change in government bond yield, anincrease in e increases (resp. lowers) b∗u .
25
From (31) and (32), we can see that the value of e that minimizes G∗u (i.e., the value of e that satisﬁes dG∗u /de =0) is not equal to the value of e that maximizes b∗u (i.e., the value of e that satisﬁes db∗u /de = 0).
26This result will still hold when the value of (σ̄, σ) is varied slightly (e.g. (σ̄, σ) = (0.065, 0.035) or(0.075, 0.045)).
27
These results do not necessarily mean that a value of e that maximizes b∗u does not exist when (σ̄, σ) is(0.08, 0.05) or (0.06, 0.03). Even under these values of (σ̄, σ), a value of e that maximizes b∗u will exist if therange of e is set to be wider.

24

G∗u . Furthermore, when the value of (σ̄, σ) (upper and lower bounds of σi ) is low comparedto the benchmark case (i.e., when the individual’s learning ability is relatively high), there
exists a value of e that minimizes G∗u .
(3) The eﬀect of such a change on b∗u (the maximum sustainable ratio of public debt to physicalcapital) is ambiguous in general. The following results hold based on numerical calculations.
(a) When the value of (σ̄, σ) is the benchmark value shown by Table 2, there exists a value ofe that maximizes b∗u . (b) When the value of (σ̄, σ) is high compared to the benchmark case(i.e., the individual’s learning ability is relatively low), such a change lowers b∗u . (c) Whenthe value of (σ̄, σ) is low compared to the benchmark case (i.e., the individual’s learning
ability is relatively high), such a change raises b∗u .
As mentioned at the beginning of this section, it seems likely that an increase in e promotes
economic growth, while it worsens ﬁscal sustainability. This is because an increase in e raises thephysical capital investment by banks, while it reduces the amount of government bonds underwritten by banks (i.e., the direct eﬀect). However, Proposition 3 shows that the result can be more
complicated than such a prediction. This is because an increase in e also has the indirect eﬀectof reducing the share of physical capital holders in the new unstable BGP, which has the opposite
impact to the direct eﬀect. Depending on which eﬀect dominates, the eﬀects of the change in e
on economic growth and ﬁscal sustainability vary.
The following result is particularly noteworthy: When the average ﬁnancial literacy of individuals is relatively low, an increase in e raises b∗u (i.e., when banks decrease the amount ofgovernment bonds they underwrite, ﬁscal sustainability conversely improves). This result implies that in countries where the average ﬁnancial literacy of individuals is relatively low, theneed for ﬁscal consolidation will conversely decrease (resp. increase) if banks decrease (resp.
increase) the number of government bonds they underwrite.

6

Conclusion

We examined the implications of limited stock market participation (the stylized fact that only afraction of households directly own stocks) for economic growth and, in particular, ﬁscal sustain25

ability (or the maximum sustainable level of government debt). Constructing an OG model where(1) individuals can choose between two types of savings (i.e., physical capital with high returns
but costly to hold versus bank deposits with low returns but no costs), and (2) banks invest part of
the total deposits in physical capital and use the rest to underwrite government debt, we mainly
showed the following: First, an increase in the average level of an individual’s ﬁnancial literacy
worsens ﬁscal sustainability while promoting economic growth under the benchmark numericalsettings of parameters. With the increasing importance of ﬁnancial literacy due to the increasing
complexity of ﬁnancial markets and rising life expectancy, a growing number of countries haveintroduced educational programs that improve ﬁnancial literacy. The above result implies thatwhile such a policy certainly has the positive eﬀect of raising the potential economic growth rate,
it also has the negative eﬀect of worsening ﬁscal sustainability and increasing the likelihood of
the government’s bankruptcy. Second, the eﬀect of the change in banks’ asset investment behavior on ﬁscal sustainability varies depending on the average level of ﬁnancial literacy. Speciﬁcally,
if banks decrease (resp. increase) the number of government bonds they underwrite, ﬁscal sustainability conversely improves (resp. worsens) in the case where the average ﬁnancial literacyof individuals is relatively low. This result implies that in countries where the average ﬁnancial
literacy of individuals is relatively low, the need for ﬁscal consolidation will conversely decrease(resp. increase) if banks decrease (resp. increase) the amount of government bonds they underwrite.
In this study, we adopted an ad hoc formulation of the banks’ asset investment behavior tosimplify the analysis, but it would be more desirable to make the optimization behavior of banks
explicit.28 This is an issue for future research.

Acknowledgments
The authors thank an anonymous referee for their valuable comments. The ﬁrst author was supported by grants-in-aid from the Ministry of Education, Culture, Sports, Science, and Technologyof the Japanese government KAKENHI (grant number 20K13455).

28

Ono (2019) is an example of such an analysis.

26

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AppendixA

Proof of Φ (σt∗ ) < 0

Calculating Φ (σt∗ ) in (6), we haveαΦ (σt∗ ) = −(1 − φ) 1−α

1
φ
1−α





σt∗
1−
φ




φ − σt∗ φ.
+ (1 − φ)
φ2

(A.1)

∗ 
From (2b) and (3), the sign of the ﬁrst term in square brackets
 is positive.
 φ − σt φ in the seconddφterm in square brackets can be rewritten as φ − σt∗ φ = σt∗ σφ∗ − dσ
∗ . From Figure 2, we can
t
t
∗ 

∗
conﬁrm that the sign of φ − σt φ is also positive. Hence, Φ (σt ) < 0 holds.

B

Derivation of (28)

From (25a) the following holds.


dσu∗
db∗u
= (1 − e) A 1 −
,
dσ
dσ

db∗u
dσ ∗
 0 ⇔ u  1.
dσ
dσ

(B.1)

Substituting (26a) into (B.1) and arranging, we have
db∗u
0
dσ

⇐⇒

A [φ (σu∗ ) − φ (σ)]  −Q (σu∗ ) .

29

(B.2)

On the other hand, when σi changes uniformly in period t in the unstable BGP, its eﬀect on σt∗(i.e., the threshold value in period t) can be calculated from (18) as follows:
∂σt∗
=1
∂σ

(B.3)

Therefore, from (14) and (B.3), the eﬀects of the uniform change in σi in period t on the economic
growth rate and the government bond yield in the same period t are as follows:dG∗u,t
∂G 
∂σt∗
∂G 
+
=
×
dσ
∂σ
∂σt∗ σ∗ =σu∗∂σ σ∗ =σu∗tt

= A [1 − φ (σu∗ ) − e] − A [1 − φ (σ) − e]
= −A [φ (σu∗ ) − φ (σ)] (< 0),
dQ∗u,t∂σt∗∂Q = Q (σu∗ ) (< 0).
=
×
dσ
∂σ
∂σt∗ σ∗ =σu∗t

Substituting (B.4) and (B.5) into (B.2), we obtain (28).

30

(B.4)

(B.5)

Australia Canada

FranceGermany

Italy

Japan

Spain

Sweden

UK

US

26.1%

22.2%

25.4%

33.1%

23.6%

31.3%

34.2%

7.7%

19.0%

22.2%

Table 1: Percentage of outstanding domestic government bonds held by private banks in 2020
(Source: ʠ Sovereign Debt Investor Base for Advanced Economies ʡbuilt based on the methodology of Arslanalp and Tsuda (2014))

Parameter

α

γ

Γ

δ

e

η

σ̄

σ

Value

0.69

0.37

8.76

1

0.8

1.69

0.07

0.04

Table 2: The benchmark numerical settings of parameters

31Figure 1: Government debt/GDP (%) in G7 countries (Source: IMF database)

32

( )

1
dφ/dσ

φ/σ

0

1
Figure 2: The function φ(σi )G( ), Q( )
G( )

Q( )

G( )

Q( )
Q( )

G( )
0

Figure 3: Existence of the BGP with σ < σ ∗ < σ̄

33

*
σt+1

45 DEGREE LINEEs

σ = H(¾)
H(¾t)

Eu
σ
H(¾)
0

σ

σu*

σs*(= σ ) σ *tbu*b

PUBLIC DEBT GOES BANKRUPTbt

b t EXPLODES

Figure 4: The function H (σt∗ ) (the upper half) and bt = (1 − e) A (σ̄ − σt∗ ) (the lower half)

34

(a)

(b)

0.08

0.06

0.07
0.06

0.04
0.05
0.04

0.02

0.03
0.02
0.04

0.05

0.06

0

0.07

(c)

0

0.02

0.04

0.06

(d)
0.4

3
2.5

0.3

2
0.2
1.5
0.1

1
0.5
0.04

0.05

0.06

0
0.04

0.07

0.05

0.06

0.07

Figure 5: The relationship between σt∗ and 1 − φ (σt∗ ) − e, G (σt∗ ), Q (σt∗ ), H (σt∗ ), and bt

35

0.07

(a)

0.065

0.06

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

1.45

(b)

1.445
1.44
1.435

1.2

(c)

1
0.8
0.6

Figure 6: Time paths of σt∗ , G (σt∗ ), and Q (σt∗ ): benchmark case36

e = 0.80 (Benchmark Case)
e=0.80

0.3
0.25
0.2
0.15
0.1
0.05
0
0.06

0.065

e = 0.75

0.07

0.08

e = 0.85

e=0.750.3

0.075

e=0.85

0.25

0.25

0.2

0.2
0.15

0.15
0.1

0.1

0.05
0
0.06

0.065

0.07

0.075

0.05
0.06

0.08

0.065

Figure 7: The relationship between σ̄ and b∗u

37

0.07

0.075

0.08

(a)
0.08
0.07
0.06
0.05

Economy is bankrupt

0.04

Public debt goes bankrupt

0.03
0

2

4

6

8

10

12

14

16

18

20

(b)
1.46
1.44
1.42
1.4

Economy is bankrupt

1.38

Public debt goes bankrupt
1.36
0

2

4

6

8

10

12

14

16

18

20

(c)
4

Public debt goes bankrupt
3

Economy is bankrupt

2
1
0
0

2

4

6

8

10

Case (ii)Case (i)

12

14

16

18

20

Case (iii)

Figure 8: The relationship between σ̄ and time paths of σt∗ , G (σt∗ ), and Q (σt∗ )

38

4

{
{ {
{

3.5

3

2.5

2

1.5

1

0.5

0
0

1

2

3

4

5

6

7

Figure 9: Welfare of each generation

39

8

9

10

e = 0.80 (Benchmark Case)
e=0.80

0.37

0.365

0.36

0.355

0.35
0.06

0.065

e = 0.75

0.07

0.08

e = 0.85

e=0.75

0.475

0.075

e=0.85

0.268

0.47

0.267

0.465
0.46

0.266

0.455

0.265

0.45
0.264

0.445
0.44
0.06

0.065

0.07

0.075

0.263
0.06

0.08

0.065

0.07

0.075

0.08

Figure 10: The relationship between σ̄ and (1 + FK − δ) [1 − Φ (σu∗ )] − η (δ = 1 in all cases)

40

(a)

=0.06

1.484
1.482
1.48
1.478
1.476
1.474
1.472
1.47
0.75

0.8

(b)

0.85

=0.07

1.47
1.46
1.45
1.44
1.43
1.42
1.41
1.4
0.75

0.8

0.85

(c)
=0.08

1.46
1.44
1.42
1.4
1.38
1.36
1.34
0.75

0.8

0.85

Figure 11: The relationship between e and G (σu∗ )

41

(a)
=0.06
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.75

0.8

0.85

(b)
=0.07

0.148
0.146
0.144
0.142
0.14
0.138
0.136
0.134
0.75

0.8

0.85

(c)
=0.08

0.27
0.26
0.25
0.24
0.23
0.22
0.21
0.75

0.8

Figure 12: The relationship between e and b∗u42

0.85