Capital income taxation,
public capital and matching frictionsin an overlapping generations model
Minoru Watanabe

December 2023
㻌Discussion Paper No. 2320

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITY㻌ROKKO, KOBE, JAPANCapital income taxation, public capital andmatching

frictionsinanoverlapping

generations model
Minoru Watanabe1,2Abstract
This study develops a standard overlapping generations model to investigate the roleof capital income tax within an endogenous growth framework. It introduces capitalincome tax to finance public investment as a source of endogenous growth.
Furthermore, we assume that matching frictions give rise to unemployment. It was
found that an increase in capital income tax can contribute to improvements ineconomic growth, employment, and welfare.

JEL classification: H51, H20, and J51.
Keywords: Capital income taxation, Endogenous growth, Unemployment
Hokusei Gakuen University, 2-3-1, Ohyachi-Nishi, Atsubetsu-ku, Sapporo 004-8631, Japan.
E-mail: m-watanabe@hokusei.ac.jp
2 Research Fellow, Graduate School of Economics, Kobe University1

1. Introduction
This study investigated the role of capital income tax on economic growth,
employment, and welfare. Traditionally, capital income taxation has been
controversial. As noted by Judd (1985), Chamley (1986), and Lucas (1990), theoptimal capital income tax is zero in the Ramsey model. If the government imposesa tax on capital income, it directly reduces household interest income, andsimultaneously

reducessavings.

The

declineinsavings

inhibits

capitalaccumulation and economic growth. Conversely, Bertola (1996), Uhlig and Yanagawa(1996), and Caballé (1998) found that increasing capital income tax can enhance
economic growth in an overlapping generations (OLG) model. Households decide tosave during young periods and receive the return from savings during old periods ina standard overlapping generation framework. In this case, even if the government
imposes a tax on interest income, savings, and economic growth may not be inhibited.
For example, Uhlig and Yanagawa (1996) introduced a tax reform that involved
cutting labor income tax with a higher capital income tax. Their study revealed thatsuch tax reforms promote savings and economic growth. These studies testify to theimpact of capital income tax on economic growth within a theoretical framework.
According to Chen et al. (2017), some studies empirically demonstrated that capitalincome tax inhibits economic growth, whereas others show that capital income tax
is neutral or contributes to economic growth. Thus, there is a disagreement
regarding the impact of capital income taxation on economic growth.
Although Bertola (1996), Uhlig and Yanagawa (1996), and Caballé (1998)
demonstrated that capital income taxation contributes to economic growth, they
assume full employment. Kunze and Schuppert (2010) insist that several developed
countries face a combination of sluggish growth and rising unemployment rates.
Thus, Kunze and Schuppert (2010) considered tax reform, as did Uhlig andYanagawa (1996), and involuntary unemployment caused by wage bargaining, as
explained by Layard et al. (1991) and Bräuninger (2005). They found that such a tax
reform could contribute to improvements in economic growth, employment, andwelfare in an OLG model.
Studies examining the impact of capital income tax on economic growth,
employment, and welfare using an endogenous growth model are insufficient. Thisis because several previous studies that incorporated endogenous growth assumed
full employment. Therefore, the current study develops a standard OLG model toinvestigate the impact of capital income tax on macroeconomic performance building
as in Kunze and Schuppert (2010). Compared to Kunze and Schuppert (2010), thereare two major differences in our study. First, following Barro (1990) and Futagami
et al. (1993), we assume that public spending generates endogenous growth 3 :Numerous empirical studies have testified to the impact of government spending on
employment and output. Using US data, Monacelli et al. (2010) showed that public
spending promotes employment. Kato and Miyamoto (2013) revealed that fiscal
stimulus policies reduce unemployment in Japan. Abiad et al. (2016) demonstrated
that public investment increases output and reduces unemployment in advanced
countries. Petrović et al. (2021) found that public investment generates positiveeffects on outputs, employment, and wages in Central and East European countries.
Second, we assume that matching frictions lead to unemployment,4 as in Batyra etal. (2019) and Zhang et al. (2022). If we introduce involuntary unemployment, as
Kunze and Schuppert (2010), unemployment may not improve unless wages decline.
However, Zhang et al. (2022) revealed that rising wages and improved employment
are compatible.
Following conventional wisdom, capital income tax is a controversial system. This
study develops a standard OLG model that includes endogenous growth andmatching frictions. The contributions of this study are summarized as follows: First,
this study reveals that a rise in capital income tax without tax reform promotes not
only economic growth and employment but also welfare, although tax reform is thedefinitive assumption in Kunze and Schuppert (2010). Second, this study
demonstrates that increasing capital income tax can improve welfare for the current,
subsequent, and present old generations. In Kunze and Schuppert (2010), oldgenerations uniquely experience welfare losses with a higher capital income tax.
Third, several empirical studies have found that public spending can contribute tooutput and employment. The results of this study reinforce the findings of previous
empirical studies.
The remainder of this paper is organized as follows. Section 2 describes the
proposed model. Section 3 discusses welfare effects. Section 4 presents numericalexamples. Section 5 concludes the paper.

2. ModelConsider a standard OLG model with discrete times. Agents in this economy arehouseholds, firms, and the government. The identical households experience two
3

Kunze and Schuppert (2010) considered productive externality as a source of endogenous growth asshown by Romer (1986).
4 Raurich and Sorolla (2003) and Greiner and Flaschel (2010) constructed an endogenous growthmodel incorporating public capital and involuntary unemployment caused by union wage setting.

periods: young and old. Households endow one unit of labor and supply it due to labormarket inelasticity. The population size is constant. Identical firms input capital andlabor to provide final goods in a competitive economy. The government imposes a tax
on capital income to provide for public investment, which increases economic growth,
as shown by Barro (1990) and Futagami et al. (1993).

2.1 Matchings
Following Zhang et al. (2022), all young households are unemployed at the beginningof the period. They search for jobs and experience matching frictions. Young
individuals who are unsuccessful in finding jobs remain unemployed. When thehouseholds become old, they retire. Following Den Haan et al. (2000) and Hashimoto
et al. (2020), the matching technology in period ‫ ݐ‬is given byሺ‫ݒ‬௧ ܰሻܰ‫ܯ‬ሺ‫ݒ‬௧ ܰǡ ܰሻ ൌ(1)ଵǡߪ ൐ Ͳሾሺ‫ݒ‬௧ ܰሻఙ ൅ ܰ ఙ ሿఙ
where ‫ݒ‬௧ is the vacant jobs per workforce, and ܰ is the constant population size.
Note that ܰ represents the workforce and number of unemployed workers at the
beginning of the period. Thus, ‫ݒ‬௧ ܰ denotes the total number of vacant jobs. Suppose
ߠ is the labor market tightness and is defined as follows.
ߠ௧ ‫ݒ ؠ‬௧ ܰΤܰ ൌ ‫ݒ‬௧ Ǥ(2)where ߠ௧ is the ratio of the total number of vacancies to the total number ofapplicants. If vacant jobs increase, the tightness of the labor market increases. We
denote ‫ݍ‬௧ as the probability that firms hire workers and is described as follows:‫ݍ‬௧ ൌ

ିଵ‫ܯ‬ሺ‫ݒ‬௧ ܰǡ ܰ ሻൌ ሺߠ௧ఙ ൅ ͳሻ ఙ Ǥ‫ݒ‬௧ ܰ

(3)

Suppose ݈௧ is the employment rate and the employment rate equals total matching,that is, ݈௧ ܰ ൌ ‫ܯ‬ሺ‫ݒ‬௧ ܰǡ ܰ ሻ. From Equations (1)–(3), we obtainିଵ݈௧ ൌ ሺͳ ൅ ߠ௧ିఙ ሻ ఙ Ǥ(4)In this equation, an increase in labor market tightness increases employment, as in
the standard search theory literature5. We assume that the population size is unity
for the remainder of the article; that is, ܰ ൌ ͳ.

2.2 Households
Young individuals search for jobs and earn a wage income if they find jobs. Employed
5

See Pissarides (2000).

households split it between consumption in the young age and savings. Households
that fail to find jobs remain unemployed. We consider “large household’’ in line withLu (2015) and Zhang et al. (2022). This assumption indicates that households share
their income. When households become old, they retire and consume their savings.
Households derive utility from consumption in both periods. As denoted by Josten(2006) and Kunze (2010), The log utility function is empirically supported. Thus, theutility function can be expressed as follows:ܷ௧ ൌ  ܿ௧ ൅ ߚ  ݀௧ାଵ ǡ Ͳ ൏ ߚ ൏ ͳ(5)In this equation, ܿ௧ and ݀௧ାଵ are the consumption in young periods and old periods
respectively, and ߚ is the discount factor. Kunze and Schuppert (2010) assume the
CES utility function. The budget constraints for households are as follows:ܿ௧ ൅ ‫ݏ‬௧ ൌ ‫ݓ‬௧ ݈௧ ǡ

(6)݀௧ାଵ ൌ ሾͳ ൅ ሺͳ െ ߬ሻ‫ݎ‬௧ାଵ ሿ‫ݏ‬௧ ,(7)where ‫ݏ‬௧ is the savings,‫ݓ‬௧ is the wage, ݈௧ is the employment rate, ‫ݎ‬௧ାଵ is theinterest rate and ߬ is the capital income tax to finance public capital. The optimal
allocation is as follows:‫ݓ‬௧ ݈௧
ǡሺͳ ൅ ߚሻ

(8)ߚሾͳ ൅ ሺͳ െ ߬ሻ‫ݎ‬௧ାଵ ሿ‫ݓ‬௧ ݈௧Ǥሺͳ ൅ ߚሻ

(9)ߚ‫ݓ‬௧ ݈௧

ͳ൅ߚ

(10)

ܿ௧ ൌ
݀௧ାଵ ൌ

‫ݏ‬௧ ൌ

For later use, we derive the values for households with additional employment inline with Batyra et al. (2019). Using Equations (5)–(7), we obtain:߲ܷ௧ൌ ܷ௖ǡ௧ ‫ݓ‬௧ Ǥ
߲݈௧

(11)

In this equation, ܷ௖ǡ௧ denotes the marginal utility of consumption during the young
periods. Differing from Batyra et al. (2019), households work only during young
periods and retire in old periods in the present study6.

2.3 Firms
The basic structure of a firm’s behavior is similar to that of Zhang et al. (2022). Firms
are identical and employ both labor and capital to provide final goods in acompetitive market. The production technology is as follows:
Batyra et al. (2019) constructed three periods’ OLG model, namely, young, senior, and old. In theirstudy, households work during young and senior periods.

6




‫ݕ‬௧ ൌ ‫݇ܣ‬௧ఈ ሺ݁௧ ݈௧ ሻଵିఈ ǡ ‫ ܣ‬൐ Ͳǡ ߙ ‫ א‬ሺͲǡͳሻ

(12)

where‫ݕ‬௧ is the total output, ݇௧ and ݈௧ are the total capital stocks and total labor
inputs, respectively, and ݁௧ is labor efficiency. In other words, ݁௧ indicates a
positive externality that gives rise to endogenous growth. Differentiating ‫ݕ‬௧ withrespect to ݇௧ and ݈௧ , we obtain:‫ݕ‬௞ǡ௧ ൌ ߙ‫݇ܣ‬௧ఈିଵ ሺ݁௧ ݈௧ ሻଵିఈ ǡ(13)‫ݕ‬௟ǡ௧ ൌ ሺͳ െ ߙ ሻ‫݇ܣ‬௧ఈ ݁௧ଵିఈ ݈௧ିఈ Ǥ(14)

respectively, where ‫ݕ‬௞ǡ௧ and ‫ݕ‬௟ǡ௧ are the marginal products of capital and labor,
respectively. Firms incur the cost of opening up jobs. Denoting ߣ௧ as the vacancy
costs, such as advertising and hiring costs. Then, the total amount of vacancy cost is
described as ߣ௧ ‫ݒ‬௧ . Assuming no capital depreciation, in line with Kunze andSchuppert (2010), profits are given by‫ݕ‬௧ െ ‫ݓ‬௧ ݈௧ െ ‫ݎ‬௧ ݇௧ െ ߣ௧ ‫ݒ‬௧ Ǥ
Following Lu (2015) and Chen et al. (2016), we assume7
ߣ௧ ൌ ߤ݇௧ ǡ

ߤ൐Ͳ

(15)
the following vacancy costs.
(16)

where ݇௧ is average capital. Equation (16) captures the vacancy cost as a proportionof average capital8. As we assume that the population size is unity, ݇௧ ൌ ݇௧ holds inthe equilibrium. The firms choose ݇௧ and ‫ݒ‬௧ to maximize their profits. Recall that‫ݍ‬௧ is the probability that a firm fills vacancies. The employment rate is expressed asfollows:(17)

݈௧ ൌ ‫ݍ‬௧ ‫ݒ‬௧ Ǥ

Using Equations (13)–(17), the first-order conditions for ݇௧ and ‫ݒ‬௧ are given as
follows:‫ݕ‬௞ǡ௧ ൌ ‫ݎ‬௧ ǡ(18)‫ݕ‬௟ǡ௧ ‫ݍ‬௧ ൌ ‫ݓ‬௧ ‫ݍ‬௧ ൅ ߤ݇௧ ǡ(19)

The left-hand sides of Equations (18) and (19) indicate the marginal revenue withadditional capital and vacancies, respectively. The right sides of Equations (18) and(19) capture the marginal costs of additional capital and vacancies, respectively.
Following Kalaitzidakis and Kalyvitis (2004) and Yakita (2008), labor efficiency isgiven by݁௧ ൌ݇௧ఋ ߩ௧ଵିఋǡ ߜ ‫ א‬ሺͲǡͳሻ
݈௧

(20)

where ߩ௧ is public capital. If ߜ ՜ ͳ, the labor efficiency boils down to ݁௧ ൌ ݇௧ Τ݈௧ asAs quoted by Chen et al. (2016), “This setup is natural the more the economy uses capital, the more
the firms compete for resources and the greater vacancy posting cost will be.”8 Hagedorn and Manovskii (2008) assumed vacancy cost is the liner to capital.
7

assumed by Kunze and Schuppert (2010) and Ono (2010). In this case, government
expenditure does not contribute to productivity or welfare, as assumed by Chamley(1986) and Uhlig and Yanagawa (1996). On the contrary, if ߜ ՜ Ͳ, the labor efficiency
boils down to ݁௧ ൌ ߩ௧ Τ݈௧ as assumed by Shioji (2001) and Maebayashi (2013). Notethat if we assume labor efficiency as ݁௧ ൌ ߩ௧ Τ݈௧ , our analytical results do not change
qualitatively.

2.4 Wage bargaining
When firms and job-seekers meet, they negotiate wages: Following Zhang et al.
(2022), the objective function is expressed asሺͳ െ ߛሻ ൫‫ݕ‬௟ǡ௧ െ ‫ݓ‬௧ ൯ ൅ ߛ ߲ܷ௧ǡͲ ൏ ߛ ൏ ͳǤ
߲݈௧

(21)

ߛ and ͳ െ ߛ are the bargaining power for workers and firms respectively. ‫ݕ‬௟ǡ௧ െ ‫ݓ‬௧is the firm’s marginal profit with additional employment. Firms and workers decide
their wages to maximize Equation (21). The negotiated wage9 is given as follows:(22)‫ݓ‬௧ ൌ ߛ‫ݕ‬௟ǡ௧ ǤThis expression is consistent with the results reported by Zhang et al. (2022). The
negotiated wage is a fraction of the marginal product of labor.

2.5 Government
The government provides public investments financed by capital income taxes undera balanced budget. The government’s budget constraint in period ‫ ݐ‬is as follows:(23)߬‫ݎ‬௧ ‫ݏ‬௧ିଵ ൌ ߩ௧ Ǥ

The left-hand side of this equation indicates tax revenue. The right-hand side of thisequation denotes expenditure on public investment. Kunze and Schuppert (2010)
considered labor and capital income taxes to finance unemployment benefits andpublic pensions under a balanced budget. Their study assumed unemployment
benefits and public pension is a fraction of wages.

2.6 Equilibrium
The clearing condition of the capital market is given as follows.
(24)݇௧ାଵ ൌ ‫ݏ‬௧ Ǥ

See the Appendix for the derivation of this equation. Suppose ݃௧ is the growth rateand defined as ݃௧ ‫ݕ ؠ‬௧ାଵ Τ‫ݕ‬௧ ൌ ݇௧ାଵ Τ݇௧ . Using Equations (11), (14), (15), (19), (23),
Substituting Equation (11) into Equation (21), the first order condition with respect to wage is givenଵିఊఊ
ൌ . Solve this equation with ‫ݓ‬௧ we obtain Equation (22).
by

9௬೗ǡ೟ ି௪೟௪೟

(24), and (25), we obtain the following constant growth rate:
݃ൌ
where

భ

‫ ؠ ܣ‬ሺͳ െ ߙ ሻ‫ܣ‬భషሺభషഃሻሺభషഀሻ

ߚߛ‫ܣ‬ሺߙ߬ሻఝ
ǡ
ͳ൅ߚ

(25)
ሺଵିఋሻሺଵିఈሻand߮ ‫ ؠ‬ଵିሺଵିఋሻሺଵିఈሻ ൐ Ͳ . From this equation,transitional dynamics does not occur. We assume enough large ‫ ܣ‬to ensuresustained growth. By differentiating this equation with respect to ߬, we obtain݀݃ ߚߛ߮‫ ߙܣ‬ఝ ߬ ఝିଵ
ൌ
൐ ͲǤ
ͳ൅ߚ
݀߬

(26)

A rise in capital income tax increases public capital and, hence, increases working
income and savings. Therefore, higher capital income tax can lead to faster economic
growth. If the public investment is financed by the labor income tax, a growthmaximizing tax rate exists, as explained by Barro (1990). Higher labor income tax
for financing public investment has two opposite effects on savings. First, wages
increase with higher labor income tax. It promotes savings. Second, an increase in
the labor income tax directly reduces disposable income. It reduces savings. Whether
higher labor income tax promotes economic growth depends on the level of labor
income tax.
Next, we analyze how capital income taxation affects unemployment. By
substituting Equation (22) into Equation (19), the first-order condition with respectto ‫ݒ‬௧ can be rewritten as(27)‫ݕ‬௟ǡ௧ ‫ݍ‬௧ ൌ ߛ‫ݕ‬௟ǡ௧ ‫ݍ‬௧ ൅ ߤ݇௧ Ǥ

The left-hand side of the equation indicates the marginal revenue with additional
vacancies, while the right-hand side of this equation denotes the marginal cost withadditional vacancies. Using Equations (2), (4), (14), (17), (25), and (27), we obtain thefollowing tightness of the labor market and employment rate:ߠൌ݈ ൌ ቊͳ ൅ ቈሺͳ െ ߛሻ‫ܣ‬ሺߙ߬ሻఝ
ǡ
ߤ

(28)
ఙ

ߤ
ሺͳ െ ߛሻ‫ܣ‬ሺߙ߬ሻఝ

቉ ቋ

ିଵ
ఙ

ǡ

(29)

From Equations (28) and (29), we deriveିሺఙାଵሻ ݈݈߮݀݀ ݀ߠൌൌ ሺߠ ఙ ൅ ͳሻ ఙߠ ൐ ͲǤ݀߬ ݀ߠ ݀߬
߬(30)

An increase in capital income tax promotes public investment. It improves themarginal products of labor, that is, ‫ݕ‬௟ǡ௧ , and hence improves marginal revenue, asexplained by the left-hand side of Equation (27). Conversely, a higher capital income
tax raises the marginal cost of opening a new job, as explained by the right-hand sideof Equation (27). As the former effect uniquely dominates the latter, firms create
vacant jobs. Labor market tightness increases with an increase in vacant jobs, which
improves employment, as shown in Equation (4). Consequently, an increase incapital income tax promotes employment.
Public investment increased wage levels in this study. The wage level in period ‫ݐ‬is given by‫ݓ‬௧ ൌ ߛ‫ݕ‬௟ǡ௧ ǡൌ ߛ‫ߙܣ‬ఝ ቊ߬ ఝఙ

൅ቈ

ଵ
ఙ ఙ

ߤ
ሺͳ െ ߛሻ‫ ߙܣ‬ఝ(31)

቉ ቋ ݇௧ Ǥ

As ߪ ൐ Ͳ and ߮ ൐ Ͳ hold, we have ߲‫ݓ‬௧ Τ߲߬ ൐ Ͳ from this equation. Therefore, ahigher capital income tax increases both employment and wages. These results are
similar to those reported by Zhang et al. (2022). Zhang et al. (2022) introduced
automated capital and robotic taxes10. Their study demonstrated that robot taxes
increase both employment and wages. Therefore, we obtain the following proposition.
Proposition 1
A rise in capital income tax promotes not only economic growth but also employment.
Kunze and Schuppert (2010) found that cutting labor income tax with a higher
capital income tax promotes savings. Further, such a tax reform moderates
bargained wages and, hence, reduces unemployment. Therefore, an increase incapital income tax promotes economic growth and employment, although wages
decline with a higher capital income tax. In contrast, this study demonstrates that arise in capital income tax promotes economic growth and employment, while wages
rise with higher capital income tax.

3. Effects on the welfare
Following Kunze and Schuppert (2010), this section analyzes how capital income
tax affects welfare for the current, subsequent, and present old generations. First,
we verified the welfare effects for the current generation. Using Equations (8), (9),
Zhang et al. (2022) introduced two types of capital, i.e. traditional physical capital and automation
capital. Automation capital is the perfect substitute to labor inputs and robot tax is imposed on theuse of automation capital.

10




(13), (14), (18), (20), (22), (23), and (24), we have the following consumption levels
during the young and old periods:ߛ‫ܣ‬ሺߙ߬ሻఝ ݇௧
ǡ
ͳ൅ߚ

(32)

ߚሾͳ ൅ ሺͳ െ ߬ሻ߰߬ ఝ ሿߛ‫ܣ‬ሺߙ߬ሻఝ ݇௧
ǡ
ͳ൅ߚ

(33)

ܿ௧ ൌ
݀௧ାଵ ൌ

భ

respectively, where ߰ ‫ ؠ‬ሺߙ‫ܣ‬ሻభషሺభషഃሻሺభషഀሻ . Note that the interest rate is constant, thatis ‫ݎ‬௧ାଵ ൌ ‫ ݎ‬ൌ ߰߬ ఝin this study although the equilibrium interest rate is෡௧ as theindependent of the tax rate in Kunze and Schuppert (2010). Denoting ܷ
indirect utility for the current generations, we obtain the following indirect utility
for the current generations using Equations (5), (32), and (33):෡௧ ൌ  ܿ௧ ൅ ߚ  ݀௧ାଵ ǡܷൌ ሺͳ ൅ ߚሻ߮  ߬ ൅ ߚ ሾͳ ൅ ሺͳ െ ߬ሻ߰߬ ఝ ሿ ൅ ‫ܪ‬௧ ,where ‫ܪ‬௧ ‫ ؠ‬ሺͳ ൅ ߚሻ 

ఊ஺ఈ ക
ଵାఉ

(34)

൅ ߚ ߚ ൅ ሺͳ ൅ ߚሻ  ݇௧ and ‫ܪ‬௧ is independent of the

tax rate. As previously mentioned, we assume that there are large households. Thus,
Equation (34) is an economy-wide indirect utility. Kunze and Schuppert (2010)delivered the indirect utility for employed and unemployed agents separately.
Differentiating Equation (34) with respect to ߬, we have෡௧ ሺͳ ൅ ߚሻ߮ሺͳ െ ߬ሻ߮
߲ܷ
ߚ߰߬ ఝ
൦
ൌ
൅
െ ͳ ൪Ǥఝᇣᇧᇧᇤᇧᇧᇤᇧ߬ ᇧᇥ ͳ ൅ ሺͳ െ ߬ሻ߰߬ ᇣᇧ߲߬߬ ᇧᇥ ሺ‫כ‬ଷሻሺ‫כ‬ଵሻ

(35)ሺ‫כ‬ଶሻ෡௧ Τ߲߬ ൐ Ͳ holds, that a higher capital income tax enhances the welfare ofIf ߲ܷ
current generations and this intuition is explained as follows. A higher capitalincome tax has three effects on indirect utility. First, a rise in capital income tax
raises wages11 under given capital stocks and hence promotes consumption during
the young and old periods. This effect improves welfare. Second, as increasing capitalincome tax raises the interest rate, it increases consumption during old periods andthereby improves welfare. Third, interest income is directly reduced by a higher
capital income tax. This reduces the old period’s consumption and hence worsens
welfare. The terms ሺ‫ͳ כ‬ሻ-ሺ‫͵ כ‬ሻ in Equation (35) capture the first, second, and third
effects, respectively. If the first and second effects dominate the third, capital income
taxation improves the welfare of the current generation.
෡௧ା௝ as the indirect utility
Next, we focus on the subsequent generations. Denoting ܷ
11

From Equation (31), wage rises with a higher capital income tax

for subsequent generations12, the indirect utility for subsequent generations is givenby෡௧ା௝ ൌ  ܿ௧ା௝ ൅ ߚ  ݀௧ା௝ ǡܷൌ ሺͳ ൅ ݆ ሻሺͳ ൅ ߚሻ߮  ߬ ൅ ߚ ሾͳ ൅ ሺͳ െ ߬ሻ߰߬ ఝ ሿ ൅ ‫ܪ‬௧ା௝ ǡ

(36)

ఊ஺ఈ ക

where ‫ܪ‬௧ା௝ ‫ ؠ‬ሺͳ ൅ ݆ ሻሺͳ ൅ ߚሻ  ଵାఉ ൅ ሾ݆ ൅ ߚሺͳ ൅ ݆ ሻሿ  ߚ ൅ ሺͳ ൅ ߚሻ  ݇௧ and ‫ܪ‬௧ା௝ isalso independent of the tax rate. It is noteworthy that ݇௧ା௝ ൌ ݃ ௝ ݇௧ holds. From (36),the following equation is obtained:ሺ‫כ‬ସሻ

෡௧ା௝
߲ܷ
ൌ
߲߬

൭ͳ ൅ ฎ݆ ൱ ሺͳ ൅ ߚሻ߮߬൅ሺͳ െ ߬ሻ߮
ߚ߰߬ ఝ
ቈ
െ ͳ቉Ǥ
ఝ
ͳ ൅ ሺͳ െ ߬ሻ߰߬
߬

(37)

This expression is similar to Equation (35). From equations (35) and (37), if෡௧ା௝ Τ߲߬ ൐ Ͳ holds simultaneously. Thus, increasing capital෡௧ Τ߲߬ ൐ Ͳ holds, then ߲ܷ߲ܷincome tax can be enhanced for subsequent generations. This intuition isqualitatively consistent with the case of the current generation, except for the growtheffect. An increase in capital income tax directly raises wage levels and thus
promotes capital accumulation. This effect can enhance the welfare of subsequent
generations. Term ሺ‫ כ‬Ͷሻ in Equation (37) represents the growth effect.
෡௢ǡ௧Finally, we investigate the welfare effects for the old generations. We denote ܷ෡௢ǡ௧ is given as
as the indirect utility for the current old generation in period ‫ ݐ‬and ܷ
follows:෡௢ǡ௧ ൌ ߚ  ݀௧ ൌ ߚ ሾͳ ൅ ሺͳ െ ߬ሻ߰߬ ఝ ሿ ‫ݏ‬௧ିଵܷ(38)

Note that ‫ݏ‬௧ିଵ is decided for period ‫ ݐ‬െ ͳ. Differentiating this equation with respectto ߬, it makes෡௢ǡ௧ሺͳ െ ߬ሻ߮
߲ܷ
ߚ߰߬ ఝ
ቈ
ൌ
െ ͳ቉Ǥ
ఝ
߲߬
ͳ ൅ ሺͳ െ ߬ሻ߰߬
߬

(39)

A higher capital income tax raises the interest rate in period ‫ݐ‬, which not only
increases interest income but also consumption for old generations. It enhances
welfare. Conversely, a higher capital income tax reduces interest income and, hence,
reduces consumption for the current old generations. This effect worsens welfare.
෡௢ǡ௧ Τ߲߬ ൐ Ͳ holds, then ߲ܷ෡௧ Τ߲߬ ൐ Ͳ holds. Thus,
From Equations (35) and (39), if ߲ܷ
we arrive at the following proposition.
Proposition 2
12

We assume ݆ as ݆ ൌ ͳǡ ʹǡ ͵ǡ ‫ڮ‬.




If

ሺଵିఛሻఝఛെ ͳ ൐ Ͳ holds a higher capital income tax improves welfare for the current,

subsequent, and presently old generations.
According to Kunze and Schuppert (2010), a higher capital income tax can improvewelfare for the current and subsequent generations, while old generations
experience welfare loss. A higher capital income tax directly reduces interest income
for old generations; hence, it worsens welfare in their study. In contrast, the present
study demonstrates that increasing capital income tax improves welfare for thecurrent, subsequent, and old generations.

4. Numerical example
In this section, we provide a numerical example to investigate the impacts of capitalincome tax on the growth rate, labor market tightness, employment rate, and welfare
for the current and old generations. The present model is composed by eightparameters, that is ߙǡ ߚǡ ߛǡ ߜǡ ߤǡ ߪǡ ‫߬ܣ‬.We assume tax rate from 0.1 to 0.5. The
remaining parameters 13 are set as follows: ߙ ൌ ͲǤ͵͸ǡ ߚ ൌ ͲǤͶͷǡ ߛ ൌ ͲǤͷǡ ߜ ൌ ͲǤͷǡ ߤ ൌʹǤ͹ǡ ߪ ൌ Ͷ‫ ܣ‬ൌ ͳͶǤ The results are summarized in Table 1.
[Table 1: Impacts of capital income tax]
In this table, an increase in capital income tax promotes not only economic growth
but also employment. Further, welfare for both current and present old generations
improves with a higher capital income tax if the capital income tax is not high.

5. Conclusions
According to conventional wisdom, the optimal capital income tax is zero. This isbecause the capital income tax inhibits savings and economic growth in the Ramseymodel. Conversely, capital income tax can accelerate economic growth within an
OLG model.
This study constructs an OLG model that incorporates endogenous growth andinvoluntary unemployment to investigate how capital income tax affects economic
growth, employment, and welfare. This study assumes that public investment arises
from endogenous growth and that matching frictions cause unemployment.
The results of this study are summarized as follows. First, a rise in capital income
13

Den Haan et al. (2000) and Hashimoto et al. (2020) set ߙ ൌ ͲǤ͵͸. Hashimoto et al. (2020) set ߪ ൌ Ͷ.

tax increases public capital and, hence, contributes to improvements in economic
growth, employment, and welfare. Furthermore, wages rise with higher capitalincome taxes. Compared to Kunze and Schuppert (2010), tax reform is not essential
for obtaining the above results. Second, a higher capital income tax improves welfare
for the current, subsequent, and present old generations, in this study, whereas
increasing capital income tax leads to welfare loss for the present old generations inKunze and Schuppert (2010). This study complements Kunze and Schuppert (2010).
Furthermore, our results reinforce the empirical study that testifies to the impact ofpublic spending. It shows that financing government spending through capitalincome taxes is a good idea.

Appendix
We derive Equation (24) as follows. Using Equations (17)–(19), we obtain:‫ݕ‬௞ǡ௧ ݇௧ ൌ ‫ݎ‬௧ ݇௧ ǡ(A.1)

‫ݕ‬௟ǡ௧ ݈௧ ൌ ‫ݓ‬௧ ݈௧ ൅ ߤ݇௧ ‫ݒ‬௧ Ǥ(A.2)

From Equations (12)-(14), we have‫ݕ‬௧ ൌ ‫ݕ‬௞ǡ௧ ݇௧ ൅ ‫ݕ‬௟ǡ௧ ݈௧ Ǥ

(A.3)

Substituting Equations (A.1) and (A.2) into Equation (A.3), we obtain‫ݕ‬௧ ൌ ‫ݓ‬௧ ݈௧ ൅ ‫ݎ‬௧ ݇௧ ൅ ߤ݇௧ ‫ݒ‬௧ Ǥ(A.4)

Denoting ܿǁ௧ and ݅௧ as the total consumption and investment in period ‫ ݐ‬,
respectively, we have ܿǁ௧ ൌ ܿ௧ ൅ ݀௧ and ݇௧ାଵ ൌ ݅௧ ൅ ݇௧ Ǥ Recall that we assume thepopulation size to be unity with zero depreciation. In this economy, the marketclearing condition of the goods market is‫ݕ‬௧ ൌ ܿǁ௧ ൅ ݅௧ ൅ ߤ݇௧ ‫ݒ‬௧ ൅ ߩ௧ ǡൌ ܿ௧ ൅ ݀௧ ൅ ݇௧ାଵ െ ݇௧ ൅ ߤ݇௧ ‫ݒ‬௧ ൅ ߩ௧ ǡ(A.5)

ൌ ‫ݓ‬௧ ݈௧ െ ‫ݏ‬௧ ൅ ሾͳ ൅ ሺͳ െ ߬ሻ‫ݎ‬௧ ሿ‫ݏ‬௧ିଵ ൅ ݇௧ାଵ െ ݇௧ ൅ ߤ݇௧ ‫ݒ‬௧ ൅ ߩ௧ Ǥ
From Equations (23), (A.4), and (A.5), we have:ሺͳ ൅ ‫ݎ‬௧ ሻሺ݇௧ െ ‫ݏ‬௧ିଵ ሻ ൌ ݇௧ାଵ െ ‫ݏ‬௧ Ǥ(A.6)

To satisfy Equation (A.6) for any period ‫ݐ‬, Equation (24) should hold. Even if we
assume full depreciation, we can obtain Equation (A.6).



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Table 1: Impacts of capital income tax
߬

0.1

0.2

0.3

0.4

0.5

݃

1.007

1.396

1.689

1.934

2.148

ߠ

1.202

1.665

2.016

2.308

2.563

݈

0.907

0.970

0.985

0.991

0.994

෡௧ Τ߲߬
߲ܷ

8.064

3.810

2.326

1.528

0.986

෡௢ǡ௧ Τ߲߬
߲ܷ

1.240

0.398

0.051

-0.178-0.379ߙ ൌ ͲǤ͵͸ǡ ߚ ൌ ͲǤͶͷǡ ߛ ൌ ͲǤͷǡ ߜ ൌ ͲǤͷǡ ߤ ൌ ʹǤ͹ǡ ߪ ൌ Ͷ‫ ܣ‬ൌ ͳͶ