Capital income taxation and economic growth underlimited stock market participation
Noritaka Maebayashi∗

Jumpei Tanaka†

Abstract
We examine the eﬀect of capital income tax on the degree of stock market participation and economic growth in an economy with limited stock market participation (i.e., an
economy in which only a fraction of households own stocks directly). We construct an endogenously growing overlapping generations (OLG) model, in which individuals can choose
between two types of savings (i.e., physical capital with high returns but high holding costs
and bank deposits with low returns but no associated costs), and banks allocate deposits
between physical capital investments (such as lending to ﬁrms) and non-productive lending (such as consumption loans). Our ﬁndings show that when the bank’s ratio of physicalcapital investments to deposits (denoted by e) is given exogenously, higher capital income
taxes reduce the share of individuals who save in physical capital (i.e., reduce the degree ofstock market participation), leading to a lower proportion of physical capital investment inaggregate savings and hampering economic growth. Additionally, when banks choose e endogenously, the relationship between the capital income tax rate and the share of individuals
who save in physical capital can be inverted U-shaped. However, higher capital income taxes
still result in a lower proportion of physical capital investment in aggregate savings, thereby
hindering economic growth.

JEL Classiﬁcation: E13, E62, G51
Keywords: Limited asset market participation, capital taxation, endogenous growth, overlapping
generations.
∗

Faculty of Economics, Nanzan University, 18 Yamasato-cho, Showa-ku, Nagoya 466-8673, Japan; e-mail:maebmn@nanzan-u.ac.jp.
†
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.jp

1

1 Introduction
It is well known that in developed countries, only a fraction of households own stocks; that is, their
participation in stock markets is limited (e.g., Mankiw and Zeldes, 1991; Haliassos and Bertaut,
1995; Campbell, 2006).1 The reason often pointed out for limited stock market participation is
diﬀerences in ﬁnancial literacy (the ability to properly gather and analyze investment information)
among individuals. Holding stocks requires a higher level of ﬁnancial literacy than holding bankdeposits does. Because acquiring ﬁnancial literacy comes at a cost, it is reasonable to abandon
participation in the stock market if its cost is high. Many empirical studies show that low ﬁnancial
literacy reduces the rate of stock market participation signiﬁcantly (e.g., van Rooijet al 2011;
Yoong, 2011; Thomas and Spataro, 2018).2 This study investigates how capital income tax aﬀects
economic growth in an economy with limited stock market participation.
The relationship between capital income taxation and economic growth has been debated for along time. In general, capital income taxation aﬀects households’ savings decisions (and hence,
economic growth) by distorting the rate of return on savings, termed the intertemporal distortionary eﬀect of capital income tax, hereafter. In inﬁnitely-lived agent models, the intertemporal
distortionary eﬀect of capital income tax is widely acknowledged to hinder economic growth if
the tax revenue is used for unproductive spending or transferred to households (e.g., Rebelo,1991;
Jones and Manuelli, 1992; Pecorino, 1993).34
This study examines a diﬀerent source of distortionary eﬀect of capital income tax on growthfrom the aforementioned intertemporal distortionary eﬀects of capital income tax, focusing on
the impact of this tax on households’ participation in the stock market.5 When stock market par1

Thomas and Spataro (2018) report that the average stock market participation rate of households for 9 European
countries in year 2010 is 16.84%. Fujiki et al. (2012) conﬁrm that in Japan at most 15% of households held stocks
for the period from 2007 to 2010. See also Guiso et al. (2003) and Christelis et al. (2010) for other data.
2
See also Guiso and Jappelli (2005) and Christelis et al. (2010). The former (resp. the latter) study investigates
the relationship between ﬁnancial awareness (resp. cognitive abilities) and stock market participation.
3
Besides the positive analysis on the growth eﬀect of capital income tax, a huge body of literature exists on
the normative analysis of capital income taxation (i.e., optimal taxation), originally initiated by Judd (1985) andChamley (1986). They demonstrated that a zero capital income tax is optimal in the long-run steady state.
4
By contrast, in an overlapping generations (OLG) model, the intertemporal distortionary eﬀect of capital income
tax does not necessarily impede economic growth even when the tax revenue is used for unproductive spending. See
Uhlig and Yanagawa (1996) for this point.
5
A recent study by Jaimovich and Rebelo (2017) also assesses another source of distortionary eﬀect of capitalincome tax on growth. They show that the tax reduces incentives to be entrepreneurs and causes them to stop
innovating, thus distorting growth . They also show that the growth eﬀect of a capital income tax is nonlinear whenthe entrepreneurial ability varies among individuals. This is because in a low-tax (resp. high-tax) economy, theability of the marginal entrepreneur is low (resp. high), so increasing the tax rate leads to an exit of low-ability (resp.

2

ticipation is limited, only those with high ﬁnancial literacy can participate in the stock market(i.e., only those types of individuals can have direct access to physical capital), and those with low
ﬁnancial literacy are forced to save in the form of bank deposits. Because the banking sector allocates deposits between physical capital investments (such as lending to ﬁrms) and non-productive
lending (such as loans to consumers and holdings of government bonds), a decrease in the shareof individuals who save in physical capital (in other words, an increase in the share of individuals
who save in bank deposits) could increase non-productive lending in aggregate savings and hinder economic growth. In such an economy, a higher capital income tax would burden physicalcapital holders and decrease the share of physical capital holders, thereby hampering economic
growth. This study aims to present a model in which such a hypothetical scenario of the impact
of capital income tax can hold.
To achieve this objective, we develop an endogenously growing OLG model that incorporates
two types of savings options for individuals: physical capital with high returns but high holding
costs, and bank deposits with low returns but no associated costs. For the sake of clarity, we ﬁrst
consider in Section 2 the case in which the ratio of bank’s physical capital investments to deposits(denoted by e in this paper) is given exogenously, and then in Section 3, the case in which e isendogenous (i.e., the bank optimally determines e). To illustrate a hypothetical scenario of the
impact of capital income tax, we consider the case of Cobb-Douglas (log-linear) utility functionand eliminate the usual “intertemporal distortionary eﬀect of capital income taxes”.6
We then show the following results: First, if e is exogenous, the hypothetical scenario described above can hold. In other words, when capital income taxes rise, the share of individuals
who save in physical capital (i.e., the degree of stock market participation) falls because of therelatively large tax burden on physical capital holders. This reduces the proportion of physicalcapital investment in aggregate savings, thereby impeding economic growth.
Second, when e is endogenous, the above result changes slightly. In this case, higher capitalincome taxes have two opposite eﬀects on the share of physical capital holders. On the one hand,
it decreases the share of physical capital holders for the same reasons as when e is exogenous.
On the other hand, it also increases banks’ non-productive lending (i.e., the supply of consumer
loans) and lowers the loan interest rate. The latter eﬀect occurs when e is endogenous, andhigh-ability) entrepreneurs.
6
In an OLG model, the usual “intertemporal distortionary eﬀect of capital income taxes” is known to disappear
under the Cobb-Douglas (log-linear) utility function. See Uligh and Yanagawa (1996) for this point.

3

such decreases in the loan interest rate raise the share of physical capital holders by lowering
the deposit interest rate. These opposite eﬀects can result in an inverted U-shaped relationshipbetween the capital income tax rate and the share of physical capital holders. Even with thisnon-monotonic eﬀect on the share of physical capital holders, the eﬀect of higher capital income
taxes on economic growth is still negative. This is because even if higher capital income taxes
increase the share of physical capital holders (contrary to the case where e is exogenous), theendogenous decline of e by banks still results in a lower proportion of physical capital investment
in aggregate savings.
Finally, we note that our conclusion (i.e., that (higher) capital income taxes impede economic
growth) is not due to the usual “intertemporal distortionary eﬀect of capital income taxes.” Thisis because we demonstrate this result under the Cobb-Douglas (log-linear) utility function thateliminates the intertemporal distortionary eﬀect of capital income taxes. We show that this conclusion stems from the negative eﬀect of capital income taxes on the proportion of physical capital
investment in aggregate savings.

2 The basic model
In this section, we set up an endogenous growth model with limited stock market participation
based on Maebayashi and Tanaka (2022). Our model has the following two main features. First,
individuals consider their ﬁnancial literacy and choose between two types of savings (i.e., physical capital with high returns but high holding costs, and bank deposits with low returns but noassociated costs). Second, banks allocate deposits between physical capital investments (i.e.,
lending to ﬁrms) and non-productive loans (i.e., lending to consumers).

2.1 Firms
The production sector is composed of homogenous ﬁrms and the total number of ﬁrms is normalized to one. Each ﬁrm produces ﬁnal goods using two types of production factors (i.e., physicalcapital and labor). We assume that both the ﬁnal goods and factor markets are competitive, andthe price of the ﬁnal goods is normalized to 1 (i.e., the ﬁnal goods are numeraire). The production

4

function of each ﬁrm is given by:
Yt = AKtα (ht Lt )1−α

0 < α < 1,(1)

where Yt , Kt , Lt , A and ht are the output of the ﬁnal goods, physical capital input, labor input,
total factor productivity, and labor-augmenting productivity, respectively. We assume that capitaldepreciates fullly in one period. Following Romer (1986), the labor-augmenting productivity ht
is speciﬁed as
h t = Kt ,(2)

and each ﬁrm maximizes its proﬁt by taking ht (≡ Kt ) as given. The ﬁrst-order conditions for
proﬁt maximization are:(1 − α)AKtα ht (ht Lt )−α = wt

αAKtα−1 (ht Lt )1−α = 1 + rtk ,(3)Here, rtk and wt are the (net) rate of return on physical capital and the wage rate, respectively.

2.2 Individuals
Each individual lives for two periods (young and old). The set of individuals born in period t iscalledgeneration t . The population of each generation is assumed to be 1.

There are two types of individuals in each generation: those who save (hereafter, savers) andthose who do not (hereafter, spenders). The population of savers is 0 < λ < 1. They work onlyin their young age, save a portion of their wages, and use the returns from saving for old-age
consumption. Savers with high ﬁnancial literacy save in the form of physical capital, whereas
those with low ﬁnancial literacy save in the form of bank deposits. In contrast, the populationof spenders is 1 − λ. They work in both young and old periods, borrow for consumption when
young, and repay debt in their old period.7 The reason for introducing spenders into our model
is to depict a situation where a portion of aggregate savings is allocated (through banks) to non7

Macroeconomic analysis that considers the households of such hand-to-mouth type has been introduced by
Mankiw (2000) in a neoclassical growth model. See also the studies by Gali et al. (2004, 2007) in the literature of
the New Keynesian models.

5

productive lending, such as consumption loans. By considering this situation, we shed lighton the importance of increasing stock market participation (the share of individuals who savein physical capital) for economic growth. Capital income taxes can reduce this incentive andhamper economic growth, which constitutes the main result of this study.
2.2.1 Savers
Savers choose one of two types of savings instruments: physical capital and bank deposits, andthey face the following trade-oﬀ between these two. In the former case, the gross rate of returnkk
on physical capital (denoted by Rt+1(≡ 1 + rt+1)) is higher than that on bank deposits (denotedddby Rt+1(≡ 1 + rt+1)); however, the actual rate of return on physical capital that individuals canreceive depends on the level of ﬁnancial literacy.8 (A more detailed explanation of this pointis provided in the next paragraph.) In the latter case, the gross rate of return on bank deposits
is lower than that on physical capital; however, no ﬁnancial literacy is required to hold bankdeposits.
Savers are endowed with one unit of labor when young, and they supply it inelastically. Whensaver i saves in the form of physical capital, he/she allocates a proportion (ϕi,t ) of his/her aftertax wage income ((1 − τL )wt ) to the expenditure to acquire ﬁnancial literacy and allocates therest to young-age consumption (cyt ) and savings (st ). Therefore, the budget constraint in his/her
young period is cyi,t + si,t = (1 − ϕi,t )(1 − τL )wt . As mentioned, the actual return from savings
depends on the level of ﬁnancial literacy. Speciﬁcally, we assume that individual i can onlykreceive fraction 1 − βσi /ϕi,t of the total return Rt+1
si,t (where β is a positive constant). In otherkkwords, the saver fails to receive (βσi /ϕi,t )Rt+1
si,t .9 The unrecovered amount (βσi /ϕi,t )Rt+1
si,t

is smaller when ϕi,t is larger and σi is smaller. Here, σi is an exogenous parameter that represents
saver i’s learning ability, and a smaller σi corresponds to a higher learning ability. We assume
that σi is uniformly distributed in the interval [σ, σ] and satisﬁes 0 < σ ≤ σi ≤ σ < 1/β.10
Accordingly, the saver with the highest (resp. lowest) learning ability has σ (resp. σ).

8This assumption is often employed when modeling the individual’s investment behavior in ﬁnancial literacy.
See Jappelli and Padula (2013) and Lusardi et al. (2017).
9
The resources that individuals fail to receive are assumed to be disposed of without being used for either consumption or investment.
10
See (5) and Figure 1 regarding the reason for setting the range of σi to 0 < σi < 1/β.

6

We assume that the utility function of individual i in generation t is given by
Ui,t = cyi,tacoi,t+11−a,

0 < a < 1,

where cyi,t and coi,t+1 are the young- and old-age consumption, respectively. Each saver compares
the utility level derived from choosing physical capital with that from choosing bank deposits andthen chooses the one that gives higher utility.11 The utility maximization problem when choosing
physical capital is:kok
max Ui,ts.t. cyi,t + si,t = (1 − ϕi,t )(1 − τL )wt , ci,t+1= (1 − τK )(1 − βσi /ϕi,t )Rt+1
si,t ,

si,t ,ϕi,tkwhere Ui,t, τL , and τK are the utility level when the saver chooses physical capital, labor income

tax, and capital income tax rates, respectively. Throughout this study, both τL and τK are assumed
to be constant over time. Note that capital income tax is imposed on the return on savings thatkindividual i actually receives (i.e., (1 − βσi /ϕi,t )Rt+1
si,t ).

By arranging the ﬁrst-order conditions, we have:

∂Ui,t
=0
si,t = (1 − a)(1 − ϕi,t )(1 − τL )wt ,
(4a)
∂si,t


∂Ui,t
(ϕi,t )2∂σi
∂ 2 σi
βσi
=0
σi =
,
> 0,
> 0, 0 <
< 1.
2
∂ϕi,t
β[(1 − a)(1 − ϕi,t ) + ϕi,t ] ∂ϕi,t∂(ϕi,t )
ϕi,t



(4b)
Note here that the level of individual savings does not depend on the capital income tax τK(see (4a)). The assumption of a Cobb-Douglas utility function eliminates “the intertemporal
distortionary eﬀect of capital income tax,” as explained in the Introduction section. However, as
we demonstrate later, the change in τK aﬀects economic growth not by altering individual saving
itself but by changing the allocation of aggregate savings between physical capital investment
and non-productive lending.

11

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

7

From (4b), function ϕi,t (σi ) satisﬁes the following properties:
ϕi,t (σi ) = ϕ(σi ),ϕ(0) = 0,ϕ(1/β) = 1, ϕ′ (σi ) > 0, ϕ′′ (σi ) < 0.

(5)

Figure 1 depicts the function ϕi,t (σi ) = ϕ(σi ). This ﬁgure shows that savers with higher learning
abilities (i.e., lower σi ) spend a smaller proportion of their wage income on acquiring ﬁnancial
literacy.
[Figure 1]
By calculating the indirect utility function in this case, we have
k
= ã
Ui,t





1−a
k
(1 − τK )Rt+1[1 − ϕ(σi )]



βσi
1−
ϕ(σi )

1−a(1 − τL )wt ,(6)where ã ≡ aa (1 − a)1−a .
On the contrary, the utility maximization problem when the saver chooses bank deposits isddmax Ui,ts.t. cyi,t + di,t = (1 − τL )wt , coi,t+1 = Rt+1
di,t .
di,tdHere, di,t is the savings in the form of bank deposits and Rt+1
is the gross rate of return on bankdeposits. The savings and indirect utility in this case are
di,t (= dt ) = (1 − a)(1 − τL )wt
1−a
d
d
Ui,t
(= Utd ) = ã Rt+1
(1 − τL )wt

(7a)
(7b)

d
Note that di,t and Ui,tdo not depend on an individual’s index, i.
kk< Utd . Therefore, the> Utd and bank deposits if Ui,tSaver i chooses physical capital if Ui,tk
= Utd holds can be calculated as follows:
threshold of the learning ability (σt∗ ) at which Ui,t


d
k
Rt+1
= (1 − τK )Rt+1
Φ(σt∗ )

where1
Φ(σt∗ ) ≡ [1 − ϕ(σt∗ )] 1−aβσt∗
1−ϕ(σt∗ )

.

(8)

Regarding the function Φ(σt∗ ) in (8), the following properties hold (see Appendix A):
Φ′ (σt∗ ) < 0,

0 < Φ(σ) < Φ(σt∗ ) < Φ(σ) < 1.
8

(9)

If σi is lower (resp. higher) than the threshold (σt∗ ), saver i chooses physical capital (resp. bankdeposits). Hence, the share of savers holding physical capital is (σt∗ − σ)/(σ − σ) ), whereas theshare of individuals holding bank deposits is (σ − σt∗ )/(σ − σ).
2.2.2 Spenders (hand-to-mouth consumers)
Spenders are endowed with one unit of labor in their young period and 0 < ψ < 1 units of labor in
their old period, and supply labor inelastically in each period. We suppose that they are myopic,
in the sense that they prefer a higher level of young consumption. Speciﬁcally, they borrow frombanks to consume more than their wage income when young and repay the debt when old. Thus,
the budget constraints in the young and old periods are as follows.
cy,H= (1 − τL )wt + xt ,
t

L
co,H
t+1 = (1 − τL )ψwt+1 − Rt+1 xt

(10)

o,H
L
Here, cy,Ht , ct+1 , xt , Rt+1 are the spender’s young-age consumption, spender’s old-age consump-tion, amount of bank loans, and gross interest rate on bank loans, respectively.
In this economy, we assume that, because of friction in the ﬁnancial market, the spender canpledge at most a fraction f (∈ [0, 1]) of the maximum amount that spenders can repay.12 Here,the maximum amount that spenders can repay is the discounted present value of after-tax wage
L. To make debt contracts credible, debt repayments
income in the old period (1 − τL )ψwt+1 /Rt+1

cannot exceed the pledgeable value. Therefore, the borrowing constraint becomesxt ≤f (1 − τL )ψwt+1
.
L
Rt+1

They borrow the maximum available amount of bank loans because they are myopic. Thus, theborrowing amount xt is given byxt =f (1 − τL )ψwt+1
.
L
Rt+1

(11)

When f is lower, large frictions exist in the ﬁnancial market, indicating that a lower amount ofbank loan is available. Thus, f is regarded as the degree of imperfection in the ﬁnancial market.

12

See Hart and Moore (1994) and Tirole (2006) for the foundations of this setting.

92.3 Banks
The banking sector is composed of homogenous banks and the total number of banks is normalized to one. From (7a), the total amount of bank deposits in period t is given byZ σ̄
Dt ≡ λ

σt∗

di,t

1
σ̄ − σt∗dσi = λ(1 − a)(1 − τL )wt.
σ̄ − σσ̄ − σ(12)

In this section, for simplicity, we assume that each bank invests an exogenous fraction (denotedby 0 < e < 1) of Dt in physical capital and uses the rest to lend to spenders. The analysis of thecase in which each bank chooses e optimally is presented in Section 3. Then, the proﬁt of eachbank realized in period t + 1 is expressed as
b
k
L
d
πt+1
= (1 − τK )Rt+1
eDt + Rt+1
(1 − e)Dt − (Rt+1+ η)Dt ,(13)

where η is the operating cost per unit deposit.13 Note that capital income tax is levied on banks
because they invest a portion of their deposits in physical capital.
d= 0), we obtain:
Because we assume that the banking sector is competitive (πt+1

d
k
L
Rt+1
= (1 − τK )Rt+1
e + Rt+1(1 − e) − η(14)d(the gross interest rate on bank deposits) is the weighted average ofThis indicates that Rt+1
Lk(the gross interest rate(the after-tax gross return rate on physical capital) and Rt+1
(1 − τK )Rt+1

on bank loans) minus η (the bank’s operating costs).
k
From (8) and (14), we can rewrite the condition Ui,t
= Utd askk
L
(1 − τK )Rt+1
Φ(σt∗ ) = (1 − τK )Rt+1
e + Rt+1(1 − e) − η.

In the following, we assumeΦ(σt∗ ) − e > 0.

(15)kΦ(σt∗ )The implication of this assumption is as follows. An increase in τK results in a loss of Rt+1

13

We assume the operating cost uses the ﬁnal goods.

10

in the physical capital holders’ savings earnings (where Φ(σt∗ ) represents the fraction of savings
that can be recovered depending on their ﬁnancial literacy levels). By contrast, an increase inkτK results in a loss of eRt+1
in the bank deposit holders’ savings earnings because they invest

the proportion e of their savings in physical capital through banks. Thus, the assumption (15)implies that when τK increases, the loss incurred by the physical capital holders is larger thanthat incurred by the bank deposit holders.

2.4 Government
The government uses tax revenue from both labor and capital income taxes for unproductive
government spending because productive government spending for promoting economic growthis beyond the scope of this study. Denoting labor income tax revenue in period t as TtL , we have
TtL = τL wt Lt .

(16)

Next, capital income tax revenue in period t (denoted by TtK ) can be expressed as follows:
TtK = λZ σt−1
∗
τK
σ

βσi
1−
ϕ(σi )


Rtk si,t−11dσi + τK,t Rtk eDt−1
σ−σ

(17)

Here, the ﬁrst term on the right-hand side of (17) refers to the tax revenue from old individuals
who hold physical capital in period t and the second term refers to the tax revenue from banks.
Substituting (4a) and (12) into (17), we haveλ(1 − a)(1 − τL )wt−1∗∗[H(σt−1) + e(σ − σt−1
)]
σ−σ


Z σt−1∗βσi∗∗where H(σt−1 ) ≡1−[1 − ϕ(σi )]dσi , H ′ (σt−1) > 0.
ϕ(σi )σ
TtK = τK Rtk(18)∗
From (18), we can see that TtK depends on σt−1
. This is because the population of old individuals∗.
who hold physical capital in period t depends on σt−1

Thus, from (16) and (18), the government’s budget constraint in period t is:
Gt = TtL + TtK = τL wt Lt + τK Rtk

λ(1 − a)(1 − τL )wt−1∗∗) + e(σ − σt−1)]
[H(σt−1
σ−σ

11

(19)

2.5 Equilibrium
Because all young individuals (whose population is 1) supply one unit of labor and old individualsof the hand-to-mouth type (whose population is 1 − λ) supply 0 < ψ < 1 units of labor, theequilibrium condition in the labor market in period t is given by
Lt = 1 + ψ(1 − λ).

(20)

From (1), (2), (3), and (20), the factor prices and output in equilibrium can be calculated as:
Rk = αÂ,wt = (1 − α)ÂKt [1 + ψ(1 − λ)]−1 ,Yt = ÂKt ,(21)

where Â ≡ A[1 + ψ(1 − λ)]1−α , Because our model is an endogenous growth model of AK-type,
Rk is constant over time and wt is proportional to Kt .
The equilibrium condition for the ﬁnal goods market (or equivalently, the equilibrium condition between savings and investment) is given by:
Z σt∗
Kt+1 = λsi,tσ1dσi + eDt .
σ−σ

(22)

Substituting (4a), (12) and (21) into (22), we have
Kt+1(≡ 1 + κt ) = B1 (1 − τL ) [I(σt∗ ) + e(σ − σt∗ )] ,Kt
Z σt∗λ(1 − a)(1 − α)Â∗, I(σt ) ≡B1 ≡[1 − ϕ(σi )]dσi , I ′ (σt∗ ) = 1 − ϕ(σt∗ ) > 0. (23)
[1 + ψ(1 − λ)](σ − σ)σHere, 1 + κt (≡ Kt+1 /Kt ) is the gross economic growth rate.
In this study, we assume the following:∂ [I(σt∗ ) + e(σ − σt∗ )]= 1 − ϕ(σt∗ ) − e > 0.
∂σt∗

(24)

The implication of this assumption is as follows. Suppose individuals whose learning abilities
σi equal σt∗ . If they choose to save in deposits, they save 1 − a proportion of one unit of income,and the bank allocates the proportion e of it to physical capital investment, leading to e(1 − a)being contributed to capital accumulation. By contrast, if they save in physical capital, they12

save (1 − a)[1 − ϕ(σt∗ )] proportion of one unit of income, all of which contributes to capital
accumulation. Thus, under assumption (24), an increase in σt∗ (i.e., an increase in the share ofphysical capital holders) promotes economic growth.
L )wt−1
Because (22) is rewritten into λ(1−a)(1−τ= I(σ∗σ−σKt∗t−1 )+e(σ−σt−1 )by (4a) and (12), substitut-ing it into (19) and using (21), we can derive the government budget constraint as follows.
∗
Gt = τL wt Lt + τK Rtk Kt Θ(σt−1),∗Θ(σt−1)≡∗∗H(σt−1) + e(σ − σt−1).
∗∗
I(σt−1 ) + e(σ − σt−1 )(25)∗Here, the function Θ(σt−1) satisﬁes the following properties (see Appendix B for the proof):∗∗Θ′ (σt−1) < 0, 0 < Θ(σ) < Θ(σt−1) < Θ(σ) = 1.

(26)

∗
∗
) < 0 holds is as follows. An increase in σt−1
aﬀects capital income tax revThe reason Θ′ (σt−1enue positively because it increases the share of physical capital holders in period t. Conversely,∗
an increase in σt−1
aﬀects capital income tax revenue negatively because such a change reduces

Kt−1 (for a given Kt ) and lowers wages (and thereby savings) in period t − 1. Because the latter∗) < 0 holds.
negative eﬀect exceeds the former positive eﬀect, Θ′ (σt−1

From (21) and (25), the government budget constraint can be rewritten as∗
gt (≡ Gt /Yt ) = τL (1 − α) + τK αΘ(σt−1
).

(27)

The equilibrium condition for the bank loan market is given by(1 − λ)xt = (1 − e)Dt ,(28)

where the left-hand side (LHS, hereafter) and right-hand side (RHS, hereafter) represent thedemand and supply for bank loans, respectively. By substituting (11) and (12) into (28), andarranging by (21) and (23), we can derive the equilibrium interest rate on bank loans as follows:(1 − τL )B2
L
=
Rt+1
1−e




I(σt∗ )+ e ≡ RL (σt∗ ),∗σ − σtf (1 − λ)ψ Â(1 − α)B2 ≡,1 + ψ(1 − λ)∂RL (σt∗ )
> 0.
∂σt∗
(29)

13

L
From (29), we see that Rt+1
increases as threshold σt∗ increases. This is because an increase inσt∗ reduces the share of bank deposit holders in period t + 1, thus lowering the supply of bankloans.

2.6 Equilibrium analysis
In this subsection, we characterize the equilibrium dynamics and examine the growth eﬀect of
an increase in capital income tax.
R σ∗
1∗
Deﬁnition 1. Given the initial state (K0 , D−1 , σ−1dσi +, si,−1 , di,−1 , x−1 R0L ) where K0 = λ σ −1 si,−1 σ−σRσ1
eD−1 by (22), D−1 ≡ λ σ∗ di,−1 σ−σdσi by (12), and x−1 satisﬁes (1 − λ)x−1 = (1 − e)D−1−1

by (28), a competitive equilibrium in the economy where τL is exogenous is a sequence ofo,H∗∞{cyi,t , coi,t , cy,Ht , ct , si,t , di,t , ϕi,t , σt , xt , Dt , Yt , Gt , Kt+1 }t=0Land prices {Rtk , Rtd , Rt+1
, wt }∞
t=0 such that (a) taking prices, tax rates (τL , τK ), and the distri-bution of σi ∈ [σ, σ] as given, ﬁrms and households (savers) optimize their solutions ((3), (4a),(4b), and (7a)), consumption and borrowing by households (spenders) follows (10) and (11); (b)zero proﬁt condition of the banking sector is satisﬁed ((14)); (c) government’s budget is balanced((25)) under the given tax rates (τL , τK ); (d) the threshold value σt∗ satisﬁes (8) ; (e) markets clear
with (20), (22), and (28).14
Substituting (14), (21), and (29) into (8), we can obtain the following equation that determines
the cutoﬀ value of σt∗ :(1 − τK )αÂ [Φ(σt∗ ) − e] = (1 − e)RL (σt∗ ) − η.

(30)(30) indicates that σt∗ becomes constant over time (i.e., σt∗ = σ ∗ ) because σt∗ is the only variable that depends on time t in (30). We denote the LHS and RHS of (30) as l(σt∗ ) and r(σt∗ ),respectively. Then, we obtain the following proposition.

Note that R0L in period 0 is given exogenously. In our model, the market clearing condition for bank loans inLperiod t determines Rt+1
(see (28) and (29)); therefore, there is no condition that determines R0L in period 0. EvenL
if R0 is given arbitrarily, it only aﬀects the share of income between the old deposit holders and the old spenders inperiod 0 but not the equilibrium dynamics.
14

14

Proposition 1. A unique balanced growth path (BGP) exists if the following condition holds:l(σ)(= (1 − τK )αÂ [Φ(σ) − e]) > r(σ) (= B2 (1 − τL )e − η) .

(31)

The growth rate (1 + κ) of the BGP is given by
1 + κ = B1 (1 − τL ) [I(σ ∗ ) + e(σ − σ ∗ )] .

(32)

Proof: l(σt∗ ) and r(σt∗ ), together with (9) and (29), yieldl′ (σt∗ ) = (1 − τK )αÂΦ′ (σt∗ ) < 0, r′ (σt∗ ) = (1 − e)

∂RL (σt∗ )
> 0.
∂σt∗

(33)

We can also conﬁrm that l(σ), that is, the value of l(σt∗ ) evaluated at σt∗ = σ, takes a ﬁnite value
because 0 < Φ(σ) < 1 holds (see (9)), whereas r(σ) diverges to inﬁnity (see (29)). Thus, aunique σ ∗ exists under (31). Once σ ∗ is determined, 1 + κ is determined by (23), Rd by (8), with
Rk = αÂ ((21)), and RL by (29). ■Under assumption (15), the following holds.
l(σ) = (1 − τK )αÂ [Φ(σ) − e] > 0.
Accordingly, we can depict (30) as shown in Figure 2.
[Figure 2]
Before proceeding to a comparative static analysis, we check the validity of the above result
using a numerical example. We set the baseline numerical values of the parameters, as listed inTable 1.
[Insert Table 1 here]
The details regarding the numerical settings of the parameters are as follows: For parametera in the utility function, we set a = 0.69 because the discount factor should satisfy (1 − a)/a =0.97330 ≈ 0.44 (see Song et al. (2012)). According to Trabandt and Uhlig (2011), we use(τL , τK ) = (0.28, 0.36) in the US economy. Parameter α in the production function is set to15α = 0.38 (see Trabandt and Uhlig (2011)). The scale parameter A is set to 12.5579, whichyields an after-tax gross return from physical capital of 3.24 (an annual rate of approximately
4% as in Trabandt and Uhlig (2011)). We set ψ = 0.5, under which the old spender works until
approximately 65 years old, because we regard one period as 30 years. λ = 0.8 is based on
Jappelli (1990) and Jappelli et al. (1998), both of which ﬁnd that the ratio of individuals whobind liquidity condition is approximately 0.2. (σ, σ) = (10−5 , 0.03), η = 0.8, and β = 0.05 areselected to satisfy (15), (24), and (31). We set (e, f ) = (0.8, 0.7) as a benchmark case because
it yields a positive long-run growth rate of 1.2437 and the ratio of government spending to GDPdof gBGP = 0.3067. Furthermore, we obtain steady-state gross interest rates of RBGP
= 2.8828
L
and RBGP= 5.4625.

Under these parameter settings, we can conﬁrm that the analytical results obtained thus far
can be reproduced. Figure 3-(a) shows that condition (24) holds, and Figure 3-(b) shows thatcondition (15) is satisﬁed because of l(σ) > 0.
[Figure 3 here]
In the following, we examine how a change in τK aﬀects the economy. Diﬀerentiating σ ∗ in(30) with respect to τK and considering (15) and (33), we have∂σ ∗αÂ[Φ(σ ∗ ) − e]< 0.
=− ′ ∗∂τKr (σ ) − l′ (σ ∗ )(34)

From (34), we see that an increase in capital income tax τK lowers the threshold σ ∗ (i.e., decreases
the share of physical capital holders). This is because an increase in τK lowers the after-tax ratekof return on physical capital, (1 − τK )Rt+1, which reduces (resp. raises) the share of individuals

who choose physical capital (resp. bank deposits) when saving (see (8)).
Furthermore, diﬀerentiating 1 + κ (the gross rate of economic growth) in (32) with respect
to τK and considering (24) and (34), we obtain∂(1 + κ)∂σ ∗= B1 (1 − τL )[1 − ϕ(σ ∗ ) − e] < 0.
∂τK
∂τK

(35)

From (35), we can see that an increase in τK impedes economic growth. Because this study
assumes a situation in which savers live for two periods, earn wages only in their youth, and have16

a Cobb-Douglas utility function15 , this result is not driven by the intertemporal distortionaryeﬀect of the capital income tax (i.e., changes in the relative prices between present and future
consumption). The negative growth eﬀect of the capital income tax in (35) occurs because thetax decreases (resp. increases) the share of physical capital holders (resp. bank deposit holders).
In this sense, the result shown in (35) arises from the limited stock market participation.
Summarizing the discussion in this subsection, we obtain the following proposition.
Proposition 2. An increase in capital income tax lowers the threshold σ ∗ (i.e., the share of physical capital holders) under assumption (15) and impedes economic growth under assumption(24).

3 Extension of the model: endogenous decisions of e by banks
In the basic model in the previous section, the parameter e (i.e., the bank’s ratio of physical capital
investment to deposits) was assumed to be given exogenously. In this section, we assume a more
realistic situation in which banks optimally choose e, implying that changes in τK aﬀect banks’lending behaviors. The objective is to examine whether this extension alters the main conclusionof the previous section. Because the behaviors of ﬁrms and individuals are the same as those in
the previous section, we focus on the banking sector.

3.1 Endogenous decisions of e by banks
Banks use deposits for physical capital investment (i.e., lending to ﬁrms) and consumption loans
to spenders, as in the previous sections.
We introduce the following operation cost function into the bank sector and replace the constant η in the previous section with it.
η(et ) =η1 ω η2et + (1 − et )ω + η3 .
ωω(ω > 1, η1 > 0, η2 > 0, η3 > 0)(36)

( η ′ (et ) = η1 etω−1 − η2 (1 − et )ω−1 , η ′ (ẽt ) = 0, where ẽt ∈ (0, 1) )Figure 4-(a) depicts (36) and implies that as more management resources are invested in lending
15

In this sense, our model in this subsection corresponds to Section 3 in Uhlig and Yanagawa (1996) in which theutility function is log-linear and there is no intergenerational income transfer through changes in tax rates .

17

to ﬁrms (resp. hand-to-mouth consumers); namely, et approaches 1 (resp. 0), the marginal costof lending to ﬁrms (resp, hand-to-mouth consumers) increases.
[Figure 4 here]Thus, the bank’s proﬁt maximization problem is given by:

 d
L
k
b
+ η(et ) Dt
(1 − et )Dt − Rt+1
et Dt + Rt+1
= (1 − τK )Rt+1
max πt+1
et(37)

The ﬁrst-order condition of this problem isΛt+1 = η ′ (et ),kLΛt+1 ≡ (1 − τK )Rt+1− Rt+1(38)

where Λt+1 is the gap between returns on physical capital and bank loans. From (36) and (38),
we haveet = e(Λt+1 ),e′ (Λt+1 ) =1η ′′ (et )

> 0,

(39)

where e(Λt+1 ) > ẽt (resp. 0 < e(Λt+1 ) < ẽt ) for 0 < Λt+1 < η1 (resp. −η2 < Λt+1 < 0) as
represented in Figure 4-(b). (39) implies that banks raise et as the return gap Λt+1 increases.
From (37) and (39), the bank’s proﬁt is expressed as


b
L
d
πt+1
= Λt+1 e(Λt+1 ) + Rt+1
− Rt+1− η (e(Λt+1 )) Dtdbis= 0 holds in the long run under perfect competition, the deposit interest rate Rt+1
As πt+1

given bydLRt+1
= Λt+1 e(Λt+1 ) + Rt+1
− η (e(Λt+1 )) .

(40)

Finally, the behavior of the government is the same as that in the previous section, but e in(17) (i.e., capital income tax revenue Ttk ) changes to e(Λt ).16

16

Note that e in (17) is not e(Λt+1 ) but e(Λt ), because it is the ratio of physical capital investment to deposits
Dt−1 in period t − 1.

18

3.2 Equilibrium analysis
In this subsection, we derive the equilibrium dynamics and examine the growth eﬀect of an increase in τK .
The equilibrium factor prices are given by (21), as before, inducing the return gap Λt+1 intoL
Λt+1 = (1 − τK )αÂ − Rt+1,

∂Λt+1
= −αÂ < 0,
∂τK

∂Λt+1
= −1(< 0).
L
∂Rt+1

(41)

Regarding the equilibrium condition between saving and investment, e in (22) changes to e(Λt+1 ).17
Thus, the economic growth rate (23) can be rewritten as:
Kt+1(≡ 1 + κt ) = B1 (1 − τL )[I(σt∗ ) + e(Λt+1 )(σ − σt∗ )].
Kt

(42)

∗
) changes
The government’s budget constraint is given by (19) (or (27)), as before, but e in Θ(σt−1to e(Λt ). Furthermore, e in (28) (the equilibrium condition for the bank loan market) changes toe(Λt+1 ): Thus, (29) can be rewritten as:L
Rt+1= B2 (1 − τL )[1 − e(Λt+1 )]−1I(σt∗ )+ e(Λt+1 ) .
σ − σt∗(43)

Appendix C shows that under a certain condition, 0 < e(σt∗ , τK ) < 1 (or, equivalently, σ < σt∗ <σt∗ < σ) exists uniquely such that (43) is satisﬁed, which ensures the uniqueness of RL (σt∗ , τK )satisfying (43). Appendix C also establishes the relationship e(σt∗ , τK ) = e(Λ(σt∗ , τK )). Appendix D illustrates that RL (σt∗ , τK ) satisfying (43) has the following properties.
∂RL (σt∗ , τK )
> 0,
∂σt∗

RL (σt∗ , τK )
< 0.
∂τK

(44)

L
The main diﬀerence from (29) in Section 2.5 (where e is exogenous) is that here Rt+1
depends

on τK . When e is endogenous (i.e., e(Λ(σt∗ , τK ))), an increase in τK lowers the return gap Λt+1
L
by lowering e.
(see (41)), which, in turn, reduces Rt+1

17

Note that e in (22) is not e(Λt ) but e(Λt+1 ), because it is the ratio of physical capital investment to deposits Dt
in period t.

19

Accordingly, Λ(σt∗ , τK )(= (1−τK )αÂ−RL (σt∗ , τK )) and e(σt∗ , τK )(= e(Λ(σt∗ , τK ))) satisfy∂RL (σt∗ , τK )∂Λ(σt∗ , τK )∂Λ(σt∗ , τK )∂RL (σt∗ , τK )
=
−
<
0,
=
−α
Â
−
: ?,
∂σt∗
∂σt∗
∂τK
∂τK
∂e(σt∗ , τK )∂Λ(σt∗ , τK )∂e(σt∗ , τK )∂Λ(σt∗ , τK )
′
′
=
e
(Λ)
<
0,
=
e
(Λ)
: ?.
∂σt∗
∂σt∗
∂τK
∂τK

(45)
(46)

∗
Deﬁnition 2. Given the initial state (K0 , D−1 , σ−1, si,−1 , di,−1 , x−1 , e−1 , R0L ) where K0 =
Rσ
R σ∗
1
1
dσi + e−1 D−1 by (22), D−1 ≡ λ σ∗ di,−1 σ−σdσi by (12), and x−1 satisﬁes
λ σ −1 si,−1 σ−σ−1(1 − λ)x−1 = (1 − e−1 )D−1 by (28), a competitive equilibrium in the economy where τL is
exogenous is a sequence ofo,H∗∞{cyi,t , coi,t , cy,Ht , ct , si,t , di,t , ϕi,t , σt , et , xt , Dt , Yt , Gt , Kt+1 }t=0Land prices {Rtk , Rtd , Rt+1
, w t }∞
t=0 such that (a) taking prices, tax rates (τL , τK ), and the distribu-

tion of σi ∈ [σ, σ] as given, ﬁrms, households (savers) and banks optimize their solutions ((3),(4a), (4b), (7a), and (38)), consumption and borrowing by households (spenders) follows (10)and (11); (b) the zero proﬁt condition of the banking sector is satisﬁed ((40)); (c) government’s
budget is balanced ((25)) under the given tax rates (τL , τK ); (d) the threshold value σt∗ satisﬁes(8); and (e) markets clear with (20), (22), and (28).
k
From (8), (40), and Rt+1
= αÂ in (21), we can derive the equation that determines σt∗ as:(1 − τK )αÂΦ(σt∗ ) = Λ(σt∗ , τK )e(σt∗ , τK ) + RL (σt∗ , τK ) − η(e(σt∗ , τK ))(47)

We denote the LHS (resp. RHS) of (47) as ˜l(σt∗ ) (resp. r̃(σt∗ )). Then, the following proposition
holds:
Proposition 3. A unique BGP exists under the following conditions.
r̃ σt∗ > ˜l σt∗r̃(σ) < ˜l(σ),
The growth rate (1 + κ) of the BGP is given by

1 + κ = B1 (1 − τL )[I(σ ∗ ) + e(σ ∗ , τK )(σ − σ ∗ )].
20

(48)

Proof: From (47) with (38), (44), (45), and (46), we have
∂RL (σt∗ , τK )
> 0.
∂σt∗

(49)

0 < ˜l(σ) < ˜l(σ) < (1 − τK )αÂ.

(50)r̃′ (σt∗ ) = [1 − e(σt∗ , τK )]From (9), we can obtain
˜l′ (σ ∗ ) = (1 − τK )αÂΦ′ (σ ∗ ) < 0,ttHence, (47) can be drawn as shown in Figure 5 under condition (48), indicating that the BGP
exists uniquely. ■[Figure 5 here]
Before proceeding to a comparative static analysis, we check the validity of the above result using
a numerical example. We select the values of the newly appearing parameters as follows:(η1 , η2 , η3 , ω) = (1.0, 5.0, 0.1, 2.0),(51)

whereas the other parameter values remain unchanged. We select a relatively high value of η2 thatsatisﬁes η2 > η1 (under the baseline η1 = 1) to obtain a large value of e(σ ∗ , τK ) (i.e., banks lend
more funds to ﬁrms than households (spenders)).18 In this benchmark case, (47) can be depicted
as in Figure 6-(d), which shows that (48) is satisﬁed and that a unique BGP exists. In this BGP,Ld
= 3.9335, and= 2.8795, RBGP
we obtain e∗ = 0.7177, σ ∗ = 0.0208, 1 + κ = 1.2128, RBGPgBGP = 0.3065. Furthermore, σt∗ deﬁned by (C.2), is 0.0287 (Figure 6-(c)).
[Figure 6 here]
We now examine the growth eﬀect of an increase in capital income tax. Diﬀerentiating σ ∗ in(47) with respect to τK , we obtain:∗∂RL

αÂ[Φ(σ ) − e] + (1 − e) ∂τK
∂σ ∗
=−
: (?)
∂τK
r̃′ (σ ∗ ) − ˜l′ (σ ∗ )(52)When e is endogenous, an increase in τK aﬀects σ ∗ through two distinct eﬀects. First, an increase
in τK reduces the share of physical capital holders because it imposes a relatively large tax burden
18

See Figure 4-(b) to comﬁrm that high value of η2 is associated with large values of e(σ ∗ , τK ) and ẽt .

21

on them under assumption (15). The ﬁrst term in the numerator of (52) shows this eﬀect. Thiseﬀect is common to the case where e is exogenous (see (34)). Second, when τK rises, banks
increase the supply of non-productive loans (i.e., loans to spenders), which lowers the loan interest
rate RL . This, in turn, lowers the deposit interest rate Rd and reduces the share of bank deposit
holders (in other words, increases the share of physical capital holders). The second term in the
numerator of (52) shows this eﬀect and is speciﬁc to the case where e is endogenous.
These two opposing eﬀects on σ ∗ make the sign of (52) ambiguous in general. Figure 7-(a)shows the numerically calculated eﬀects of ∂σ ∗ /∂τK under the parameter settings in Table 1 and(51), leading to the following result: Under relatively low (resp. high) values of τK , the sign of∂σ ∗ /∂τK is positive (resp. negative). In other words, the relationship between the capital income
tax rate and the share of physical capital holders is inverted U-shaped.
Next, we examine the eﬀect of an increase in τK on the growth (1 + κ). From Proposition 3,
along with (24), (46), and (52), we obtain
 ∗



∂σ
∂e
∂e ∂σ ∗∂(1 + κ)∗∗= B1 (1 − τL )[1 − ϕ(σ ) − e] ++(σ − σ ) : (?).
∂τK
∂τK
∂τK ∂σ ∗ ∂τK|{z}|{z} | {z }
(#1)

(#2)

(53)

(#3)

When e is endogenous, an increase in τK aﬀects 1 + κ through two distinct eﬀects. First, an
increase in τK aﬀects economic growth by changing the share of physical capital holders. The
term (#1) in (53) illustrates this eﬀect. The sign of ∂σ ∗ /∂τK is ambiguous from (52), which
renders the sign of this eﬀect ambiguous even under assumption (24). Second, an increase in τK
aﬀects economic growth through an endogenous change in e. Banks allocate deposits between
physical capital investments and non-productive loans. An increase in τK not only reduces theafter-tax rate of return on physical capital (1 − τK )αÂ, but also has direct and indirect impacts
on the loan interest rate RL . Banks set e considering changes in these return rates, which, in turn,aﬀects economic growth. The terms (#2) and (#3) in (53) illustrate these eﬀects. The signs ofthese eﬀects are also generally ambiguous.
However, Figure 7-(b) shows that under the parameter settings in Table 1 and (51), an increase in τK lowers 1 + κ, which is common to the case when e is exogenous (see (35)). Numerical studies also demonstrate that other ambiguous properties of variables in (45) and (46) are∂Λ(σ ∗ , τK )/τK < 0 (Figure 7-(c)) and ∂e(σ ∗ , τK )/τK < 0 (Figure 7-(d)).

22

These numerical results indicate the following. On the one hand, when ∂σ ∗ /∂τK < 0 in
Figure 7-(a) (i.e., when a higher τK reduces the share of physical capital holders), the negative
growth eﬀect (#1) in (53) becomes relatively stronger. On the other hand, when ∂σ ∗ /∂τK > 0
in Figure 7-(a) (i.e., when a higher τK raises the share of physical capital holders), the negative
growth eﬀect of (#2) and (#3) in (53) becomes relatively stronger. Consequently, irrespectiveof whether ∂σ ∗ /∂τK is positive or negative, a higher τK hinders economic growth.
[Figure 7 here]
Summarizing the discussion in this subsection, we obtain the following proposition:
Proposition 4. Suppose the bank optimally chooses e (i.e., the ratio of physical capital investments to bank deposits). The eﬀects of an increase in τK (capital income tax) on both the thresholdσ ∗ and the economic growth rate 1 + κ are generally ambiguous. Under the parameter settings
listed in Table 1 and (51), the relationship between the capital income tax rate and the share ofphysical capital holders becomes inverted U-shaped. Despite this, higher capital income taxes
reduce economic growth.

4 Conclusion
We developed an endogenously growing OLG model in which (1) individuals can choose between two types of savings (i.e., physical capital with high returns but high holding costs andbank deposits with low returns but no associated costs) and (2) banks allocate deposits between
physical capital investments (i.e., lending to ﬁrms) and non-productive loans (i.e., lending tospenders). Our investigation focused on the eﬀects of capital income tax on the degree of stock
market participation and economic growth.
If the bank’s ratio of physical capital investments to deposits (denoted by e) is exogenous,
higher capital income taxes can reduce the share of physical capital holders (i.e., reduce thedegree of stock market participation). Such a change lowers the proportion of physical capital
investment in aggregate savings, which can impede economic growth.
By contrast, if banks determine e endogenously, higher capital income taxes have both negative and positive eﬀects on the share of physical capital holders. The negative eﬀect is common

23to the case where e is exogenous. However, the positive eﬀect arises because higher capital income taxes increase the supply of non-productive consumption loans by banks and then lowers
the loan interest rate, which, in turn, lowers the interest rate on deposits. Consequently, the relationship between the capital income tax rate and the share of physical capital holders becomesinverted U-shaped. Even with this non-monotonic eﬀect on the share of physical capital holders, the growth eﬀect of the capital income tax is still negative. This is because even if higher
capital income taxes increase the share of physical capital holders (contrary to the case where eis exogenous), the endogenous reduction of e by banks results in a lower proportion of physicalcapital investment in aggregate savings.

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26

Appendix
A Proof of (9)By diﬀerentiating Φ(σt∗ ) in (8) with respect to σt∗ , we haveαΦ (σt∗ ) = −(1 − ϕ) 1−α

′

1
ϕ
1−α
′



βσ ∗
1− t
ϕ




ϕ − σt∗ ϕ′+ (1 − ϕ)β
.
ϕ2

(A.1)

From (4b) and (5), the sign of the ﬁrst term in the square brackets is positive. ϕ − σt∗ ϕ′ in thesecond term in square brackets is also positive because function ϕ is strictly concave (see (5)).
Hence, Φ′ (σt∗ ) < 0 holds true. Clearly, 0 < Φ(σ) < Φ(σt∗ ) < Φ(σ) < 1 holds, because both0 < 1 − ϕ(σt∗ ) < 1 and 0 < 1 − βσt∗ /ϕ(σt∗ ) < 1 are satisﬁed (see (4b) and (5)).

B Proof of (26)∗∗Diﬀerentiating Θ(σt−1) in (25) with respect to σt−1
, we obtain∗∗)] − [H + e(σ − σt−1)](I ′ − e)
(H ′ − e)[I + e(σ − σt−1.
∗
[I + e(σ − σt−1)]2∗
) + e(H − I)
(H ′ · I − I ′ · H) + (H ′ − I ′ )e(σ − σt−1=∗
[I + e(σ − σt−1)]2∗
Θ′ (σt−1)=(B.1)

Here, we calculate H − I, H ′ − I ′ , and H ′ · I − I ′ · H as follows:
H −I =

Z σt−1∗σβσi
1−ϕ(σi )

[1 − ϕ(σi )]dσi −Z σt−1∗[1 − ϕ(σi )]dσiσ

Z σt−1∗βσi=−[1 − ϕ(σi )]dσi < 0(B.2)
ϕ(σi )
σ


∗
βσt−1
′
′
∗
∗
H −I = 1−[1 − ϕ(σt−1)] − [1 − ϕ(σt−1)]∗
ϕ(σt−1 )∗βσt−1∗[1 − ϕ(σt−1)] < 0(B.3)=−∗
ϕ(σt−1 )Z σt−1
∗
∗
βσt−1
′
′
∗
H ·I −I ·H = 1−
[1 − ϕ(σt−1 )][1 − ϕ(σi )]dσi∗)ϕ(σt−1σZ σt−1∗βσi∗[1 − ϕ(σi )]dσi− [1 − ϕ(σt−1 )]1−ϕ(σi )σ 
Z σt−1
∗
∗
βσt−1
βσi
∗
=[1 − ϕ(σt−1 )]1−− 1−[1 − ϕ(σi )]dσi∗)ϕ(σt−1ϕ(σi )
σ
(B.4)

27

∗
Since the function ϕ(σt−1) is strictly concave by (5), the following relationship holds within the∗range of σi ∈ [σ, σt−1 ] (see Figure 8):∗∗)ϕ(σt−1βσt−1ϕ(σi )
βσi
>
>
β
⇐⇒
0
<
1
−
<1−
<1
∗
∗
σi
σt−1
ϕ(σt−1 )ϕ(σi )(B.5)

This together with (B.4) indicates H ′ · I − I ′ · H < 0. Thus, H − I < 0, H ′ − I ′ < 0, and∗H ′ · I − I ′ · H < 0 imply Θ′ (σt−1) < 0.
Furthermore, from the deﬁnitions of functions H(σt∗ ) and I(σt∗ ) given in (18) and (23), we
can easily conﬁrm that H(σ) = I(σ) = 0. Thus, the following holds:
Θ(σ) =e(σ − σ)=1e(σ − σ)(B.6)∗
Next, noting that 0 < 1 − βσi /ϕ(σi ) < 1 is satisﬁed for any σi ∈ [σ, σt−1
] (see (B.5)), we obtainRσ
0 < Θ(σ) =σβσi
1 − ϕ(σ
i)

Rσ
σ



[1 − ϕ(σi )]dσi[1 − ϕ(σi )]dσi

< 1.

(B.7)

∗
) < Θ(σ) = 1 holds true.
Thus, 0 < Θ(σ) < Θ(σt−1[Figure 8 here]

C Uniqueness of e(σt∗ , τK ) and RL (σt∗ , τK )From (36), (38), (39), (41), and (43), we haveB2 (1 − τL )[1 − et ]−1I(σt∗ )
L+ et (= Rt+1
) = (1 − τK )αÂ − η1 eω−1+ η2 (1 − et )ω−1 .
tσ − σt∗(C.1)(C.1) is represented in (the upper right-hand side of) Figure 9. A unique value of et (≡ e(σt∗ , τK )) ∈(0, 1) that satisﬁes (C.1) exists under the following condition (i.e., the LHS of (C.1) is smaller
than the RHS for et = 0):
B2 (1 − τL )I(σt∗ )< (1 − τK )αÂ + η2σ − σt∗I(σ ∗ )

where B2 (1 − τL ) σ−σt ∗ = (1 − τK )αÂ + η2 .
t[Figure 9]28⇔

σt∗ < σt∗ (< σ),

(C.2)

L
Figure 9 also shows the uniqueness of Rt+1(≡ RL (σt∗ , τK )) by (C.1) (and (43)) and Λt+1 (≡Λ(σt∗ , τK ) = (1 − τK )αÂ − RL (σt∗ , τK )) by (41). From (39), e(σt∗ , τK ) = e(Λ(σt∗ , τK )) holds
between e(σt∗ , τK ) and Λ(σt∗ , τK ).

D Derivation of (44)
By totally diﬀerentiating (41) and (43), we obtainL(D.1)
dΛt+1 = −αÂdτK − dRt+1B2 (1 − τL ) ∂B2 (1 − τL ) ′I(σt∗ )I(σt∗ )
L
∗
=
dRt+1
+
dσ
e
(Λ
)
+ 1 dΛt+1 .
t+1t
1 − e(Λt+1 ) ∂σt∗ σ − σt∗[1 − e(Λt+1 )]2σ − σt∗(D.2)
From (D.1), (D.2), and ∂σ∂ ∗thI(σt∗ )σ−σt∗i
> 0 using (23), we obtain

L
∂Rt+1
=
∂σt∗
1+
L
∂Rt+1

∂τK

=−

B2 (1−τL ) ∂1−e(Λt+1 ) ∂σt∗h

I(σt∗ )σ−σt∗

B2 (1−τL )e′ (Λt+1 )[1−e(Λt+1 )]2ih

I(σt∗ )+1σ−σt∗

B2 (1−τL )′αÂ [1−e(Λ2 e (Λt+1 )t+1 )]
B2 (1−τL )′1 + [1−e(Λ2 e (Λt+1 )t+1 )]

Therefore, (44) is obtained.

29

dσt∗

h

i > 0,I(σt∗ )
+1
σ−σt∗

h

(D.3)

i

I(σt∗ )
+1σ−σt∗i < 0.

(D.4)

Parameter

a

α

A

ψ

β

λ

e

Value

0.69

0.38

12.5579

0.5

0.05

0.8

0.8

Parameter

f

η

τL

τK

σ

σ

Value

0.7

0.8

0.28

0.36

0.03

10−5

Table 1: The benchmark numerical settings of parametersφ( σi )
1
φ( —
σ)

φ( σ
—)

σ
—

—

σ

Figure 1: The function ϕ(σi )

30

1/β

σi

r( —
σ)

r( σt*)l( σ—)l( σt*)
—
l( σ
)

—

σ*

σ
—

σt*

σ

r( σ
—)
Figure 2: Uniqueness of σ ∗

(a)

(b)
0.2

1

0.195

0.8

0.19

0.6

0.185

0.4

0.18

0.2

0.175

0
0

0.01

0.02

0.03

0

0.01

0.02

0.03

Figure 3: Numerical demonstrations of (a) Eq. (24) and (b) Eq. (30)

31

(a)

η(et)

(b)
η'(et)
η'(et)
η10 < Λ t+1 < η1

0

~
et

1

et
-η 2 < Λ t+1 < 0e(Λt+1 )

-η 2
e~t

0

1

et

Figure 4: The cost function η(et )~r ( σt*)
~

~

l( σ
—)

l( σt*)~r( σt*)

~

r( σ
)
—

~

l( σt*)
σ
—

σ*

σt*

Figure 5: Uniqueness of σ ∗

32

—

σ

σt*

(a)

(b)
1

3

0.8

2.5

0.6

2

0.4

1.5

0.2

1

0

0.5
0

0.01

0.02

0.03

(c)

0

0.5

1

(d)
4

1
0.8

3.5

0.6
3
0.4
2.5

0.2
0

2
0

0.01

0.02

0.03

0

0.01

0.02

0.03

Figure 6: Numerical demonstrations of (24), η(et ), and the relationship between σt∗ and et , r̃(σt∗ ),and ˜l(σt∗ )

33

(a)

(b)

0.021

1.4

1.38
0.02
1.36
0.019
1.34

0.018

1.32

1.3
0.017
1.28
0.016
1.26

0.015

1.24
0

0.5

1

0

(c)

0.5

1

(d)
0.85

0

0.8
-0.5

0.75
0.7

-1

0.65
0.6

-1.5

0.55
0

0.5

1

0

0.5

1

Figure 7: The relationship between τK and σ ∗ , 1 + κ, Λ(σ ∗ , τK ), and e(σ ∗ , τK )34

φ( σi )1

φ( σt*–1)/σt*–1
β
—

σt*–1

σ
—

σi

1/β

σ

Figure 8: Representation of (B.5)

RLt+1

RLt+1

LHS of (C.1)(41):^ − RLΛt+1 = (1− τΚ )αΑt+1

R (σ t∗, τΚ)L

RHS of (C.1)Λ(σt∗, τΚ)Λ t+1 0045°e (σ*t , τΚ)1ete(Λ(σ t∗, τΚ))(38):

Λt+1 = η′(et )

(

e~t

)

(39):
e t = e(Λt+1 )etetFigure 9: Uniqueness of e(σt∗ , τK ), RL (σt∗ , τK ), and Λ(σt∗ , τK )35