A note on capital income taxation withinvoluntary unemployment

Minoru Watanabe
October 2021
Discussion Paper No. 2126

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANA note on capital income taxation withinvoluntary unemployment
Minoru Watanabe12

Abstract
This study develops a standard overlapping generations model with imperfect labormarkets. The results indicate that a higher capital income tax promotes not only
economic growth but also employment if pension benefits exist.

JEL classification: H55, H20, J51

Keywords: Capital income tax, Unemployment, Public pension

1. Introduction
According to the conventional perspective, optimal capital income tax inhibits
economic growth if households live infinitely, as demonstrated by Chamley (1986).
However, Uhlig and Yanagawa (1996) prove that capital income tax can promoteeconomic growth in an overlapping generations model.
This short study analyzes how capital income tax affects not only economic growth
but also unemployment. Corneo and Marquardt (2000) consider the unemployment
issue with public pension in an overlapping generations model, indicating
endogenous growth. Following Corneo and Marquardt (2000), Ono (2010) assumes
that trade unions maximize the lifetime income of union members. Yasuoka (2021)
extends Ono’s (2010) study with a discussion on consumption tax. Kunze and1

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Schuppert (2010) find that the tax reform of cutting labor income tax while imposing
a higher capital income tax reduces wages, which, in turn, reduces unemployment.
However, studies proving the impact of capital income tax on employment in an
overlapping generations model remain scarce. Therefore, this study extends Ono’s(2010) and Yasuoka’s (2021) research with regard to capital income tax. The present
study demonstrates that increasing capital income tax promotes not only economic
growth but also employment if pension benefits exist.
The remainder of this paper is organized as follows. Section 2 describes thedevelopment of the proposed model, and concluding remarks are provided in section
3.

2. Model2.1 Households
We consider a standard overlapping generations model with the population size
constant and equal to unity. Households are identical, experience two periods, andderive utility from consumption during the young and old periods. During young
periods, they have one unit of labor and supply it to the labor market inelastically.
Involuntary unemployment caused by trade unions is also included. Thus, we
assume that households join trade unions.
If young individuals are employed, they earn wages and allocate them between
consumption and savings. Similarly, if unemployed, they receive unemployment
benefits and similarly allocate them. When household members are old, they retire
and receive pension benefits. Older individuals in particular consume both savingsand pension benefits. The utility function is expressed as follows:
⁄

⁄

1−1 𝜃
1−1 𝜃
𝑐𝑖,1𝑡
−1
𝑐𝑖,2𝑡+1−1
𝑣𝑖,𝑡 =+𝛽, 𝑖 = 𝑒, 𝑢.
1 − 1⁄𝜃1 − 1⁄𝜃(1)Here, subscript 𝑖 indicates the status of households. If households are employed,
𝑖 = 𝑒, and if unemployed, 𝑖 = 𝑢. Moreover, 𝑐𝑖,1𝑡 and 𝑐𝑖,2𝑡+1 are the consumption in
the young and old periods, respectively, 𝜃 > 0 is the intertemporal elasticity of
substitution, and 𝛽 < 1 is the discount factor. Note that if 𝜃 → 1, the utility function2

boils down to a log form from equation (1). The budget constraints for the employedindividuals are as follows:𝑐𝑒,1𝑡 + 𝑠𝑒,𝑡 = (1 − 𝜏𝑢 − 𝜏𝑝 )𝑤𝑡 ,(2)𝑐𝑒,2𝑡+1 = (1 + 𝑅𝑡+1 )𝑠𝑒,𝑡 + 𝑝𝑡+1 ,(3)

𝑅𝑡+1 ≡ (1 − 𝜏𝑟 )𝑟𝑡+1 .

(4)

where 𝑠𝑒,𝑡 is the savings of employed individuals; 𝜏𝑢 ∈ (0,1) and 𝜏𝑝 ∈ (0,1) are thelabor income tax to finance unemployed benefits and pension benefits, respectively,
𝑤𝑡 is the wage, 𝜏𝑟 ∈ (0,1) is the capital income tax to finance non-productive
expenditure, 𝑟𝑡+1 is the interest rate, 𝑝𝑡+1 is the pension benefits, and 𝑅𝑡+1 is theinterest factor. Combining equations (2) and (3), the lifetime budget constraints forthe employed individuals are as follows:𝑐𝑒,1𝑡 +

𝑐𝑒,2𝑡+1
𝑝𝑡+1= (1 − 𝜏𝑢 − 𝜏𝑝 )𝑤𝑡 +.
1 + 𝑅𝑡+1
1 + 𝑅𝑡+1(5)

Then, the budget constraints for unemployed individuals are as follows:
𝑐𝑢,1𝑡 + 𝑠𝑢,𝑡 = 𝑏𝑡 ,(6)

𝑐𝑢,2𝑡+1 = (1 + 𝑅𝑡+1 )𝑠𝑢,𝑡 + 𝜂𝑝𝑡+1 , 0 < 𝜂 < 1.

(7)

where 𝑠𝑢,𝑡 is the savings of unemployed agents and 𝑏𝑡 is the unemployment
benefits. Following Yasuoka (2021), we assume 𝜂 ∈ (0,1) . This indicates thatunemployed individuals benefit from pensions without contribution, although asmaller amount compared with employed individuals. This assumption reflects theJapanese pension system. Note that 𝜂 < 1 is the definitive assumption in this study.
Ono (2010) assumes 𝜂 as a binary variable, that is, 𝜂 = 0 or 𝜂 = 1.
Combining equations (6) and (7), the lifetime budget constraint for unemployedindividuals is expressed as follows:
𝑐𝑢,1𝑡 +

𝑐𝑢,2𝑡+1
𝜂𝑝𝑡+1= 𝑏𝑡 +.
1 + 𝑅𝑡+1
1 + 𝑅𝑡+1(8)

The right-hand side (RHS) of equations (5) and (8) is the lifetime incomes of theemployed and unemployed individuals, respectively. The optimal savings for bothindividuals are as follows:3

𝑠𝑒,𝑡 =

1𝛽 𝜃 + (1 + 𝑅𝑡+1 )1−𝜃𝑠𝑢,𝑡 =[𝛽 𝜃 (1 − 𝜏𝑢 − 𝜏𝑝 )𝑤𝑡 −1
𝛽 𝜃 + (1 + 𝑅𝑡+1 )1−𝜃

[𝛽 𝜃 𝑏𝑡 −𝑝𝑡+1],(1 + 𝑅𝑡+1 )𝜃

𝜂𝑝𝑡+1
].
(1 + 𝑅𝑡+1 )𝜃

(9)

(10)

2.2 Firms
Firms produce final goods with capital and labor inputs in competitive markets.
Based on Kunze and Schuppert (2010), Ono (2010), and Yasuoka (2021), we assume
the following production technology:
𝑌𝑡 = 𝐴𝐾𝑡𝛼 (𝐸𝑡 𝐿𝑡 )1−𝛼 , 0 < 𝛼 < 1.

(11)where 𝑌𝑡 is the total output, 𝐴 > 0 is the constant technology level, 𝐾𝑡 is theaggregate capital input, 𝐸𝑡 is the labor efficiency, 𝐿𝑡 is the aggregate labor input,
and 𝛼 is a constant parameter. When the population size is unity, 𝐿𝑡 describes
employment rate. By assuming zero depreciation, the factor demand is expressed asfollows:
𝑤𝑡 = (1 − 𝛼)𝐴𝐾𝑡𝛼 𝐸𝑡1−𝛼 𝐿−𝛼
𝑡 ,

(12)

𝑟𝑡 = 𝛼𝐴𝐾𝑡𝛼−1 (𝐸𝑡 𝐿𝑡 )1−𝛼 .

(13)The labor efficiency is expressed as follows:(14)

𝐸𝑡 = 𝐾𝑡 ⁄𝐿𝑡 .
The labor efficiency generates endogenous growth similar to Romer (1986).

2.3 Trade unions
According to Daveri and Tabellini (2000), Ono (2010), and Yasuoka (2021), tradeunions increase unemployment. In line with Ono (2010) and Yasuoka (2021), tradeunions determine wages to maximize the lifetime income of union members withgiven interest rates and policy variables. We obtain the following objective function
for trade unions:
𝑉𝑡 = [(1 − 𝜏𝑢 − 𝜏𝑝 )𝑤𝑡 +

𝑝𝑡+1
𝜂𝑝𝑡+1
] 𝐿𝑡 + [𝑏𝑡 +] (1 − 𝐿𝑡 ).
1 + 𝑅𝑡+1
1 + 𝑅𝑡+1(15)

Substituting equation (12) with equation (15), the first-order condition is as follows:
4

𝑤𝑡 =

1
(1 − 𝛼)(1 − 𝜏𝑢 − 𝜏𝑝 )[𝑏𝑡 −(1 − 𝜂)𝑝𝑡+1].
1 + 𝑅𝑡+1(16)

where 𝑏𝑡 − (1 − 𝜂)𝑝𝑡+1 ⁄1 + 𝑅𝑡+1 is the net benefit from social security. Specifically,
increased pension benefits reduce this net benefit and, thus, reduce wages.
Therefore, firms have an incentive to increase their labor input as indicated by Ono(2010) and Yasuoka (2021).

2.4 Government
The government imposes a tax on labor income to finance social security and a tax
on capital income to provide non-productive expenditure under balanced budgets.
Note that we assume a pay-as-you-go pension system. The revenue constraints for
unemployment and pension benefits are as follows:𝜏𝑢 𝑤𝑡 𝐿𝑡 = 𝑏𝑡 (1 − 𝐿𝑡 ),(17)

𝜏𝑝 𝑤𝑡 𝐿𝑡 = 𝑝𝑡 𝐿𝑡−1 + 𝜂𝑝𝑡 (1 − 𝐿𝑡−1 ).

(18)

The left-hand side (LHS) of equations (17) and (18) is the tax revenue from labor
income tax to finance unemployment and pension benefits, respectively. The RHS ofequations (17) and (18) is the expenditure on unemployment and pension benefits,
respectively. The revenue constraint for non-productive expenditure is denoted as
𝜏𝑟 𝑟𝑡 𝑠𝑡−1 = 𝐺𝑡 ,(19)

where 𝑠𝑡−1 is the aggregate savings in the previous periods and 𝐺𝑡 is the nonproductive expenditure. This is in agreement with the findings of Chamley (1986).
The LHS of equation (19) denotes the tax revenue for the capital income tax. The
aggregate savings in the present period are described as follows:
𝑠𝑡 = 𝑠𝑒,𝑡 𝐿𝑡 + 𝑠𝑢,𝑡 (1 − 𝐿𝑡 ).

(20)

2.5 Equilibrium
The clearing condition of the capital market is expressed as follows:(21)

𝐾𝑡+1 = 𝑠𝑡 .

We denote 𝑔𝑡 ≡ 𝑌𝑡+1 ⁄𝑌𝑡 = 𝐾𝑡+1 ⁄𝐾𝑡 as the growth rate. Using equations (9), (10), (12),(13), (14), (16), (17), (18), and (20), we obtain the following constant growth rate:5𝑔=𝛽 𝜃 (1 − 𝜏𝑝 )(1 − 𝛼)𝐴,𝛽 𝜃 + (1 + 𝑅)1−𝜃 + (1 + 𝑅)−𝜃 𝜏𝑝 (1 − 𝛼)𝐴

(22)
(23)

𝑅 ≡ (1 − 𝜏𝑟 )𝛼𝐴.
Differentiating equation (22) with respect to 𝜏𝑟 , we derive𝜏𝑝 (1 − 𝛼)𝐴𝛽 𝜃 (1 − 𝜏𝑝 )𝛼(1 − 𝛼)𝐴2 {1 − 𝜃 [1 + 1 + 𝑅 ]}𝑑𝑔=.
𝑑𝜏𝑟 (1 + 𝑅)𝜃 [𝛽𝜃 + (1 + 𝑅)1−𝜃 + (1 + 𝑅)−𝜃 𝜏 (1 − 𝛼)𝐴]2

(24)

𝑝

The sign of 𝑑𝑔⁄𝑑𝜏𝑟 is ambiguous, and we obtain the following condition:
𝑑𝑔⁄𝑑𝜏𝑟 > 0, 𝑖𝑓 𝜃 < 𝜃 ≡1< 1.
𝜏𝑝 (1 − 𝛼)𝐴
1+
1+𝑅

(25)

Increased capital income tax reduces the interest factor, 𝑅 ≡ (1 − 𝜏𝑟 )𝛼𝐴, which has
the following opposite effects on aggregate savings. First, if the income effectdominates substitution effects, a decline in the interest factor increases aggregate
savings, as in Uhlig and Yanagawa (1996). Second, a decline in the interest factor
increases the discounted pension benefits, thereby reducing aggregate savings.
Therefore, if the first effect is relatively large compared to the second effect, a higher
capital income tax promotes aggregate savings and, hence, economic growth. Notethat if 𝜃 → 1, equation (24) can be rewritten as follows:−𝛽𝜏𝑝 (1 − 𝜏𝑝 )𝛼(1 − 𝛼)2 𝐴3
𝑑𝑔
=
< 0.
𝑑𝜏𝑟 [(1 + 𝑅)(1 + 𝛽) + 𝜏 (1 − 𝛼)𝐴]2

(26)

𝑝

From equation (26), 𝑑𝑔⁄𝑑𝜏𝑟 < 0 holds. If the utility function is log form, the income
and substitution effects are canceled. In this case, increasing the capital income tax
increases the discounted pension benefits, thereby reducing savings. Thus, a higher
capital income tax inhibits economic growth.
We now investigate the impact of capital income tax on the employment rate.
Substituting equations (12), (14), (17), and (18) into (16), we obtain the following
equilibrium employment rate:(1 − 𝛼)(1 − 𝜏𝑢 − 𝜏𝑝 )(1 − 𝜂)𝜏𝑝 𝑔+= 𝜏𝑢 .
𝐿
𝐿(1 + 𝑅) [
+
𝜂]
1−𝐿
1−𝐿
6

(27)

Figure 1 illustrates how the equilibrium employment rate is derived.

[Figure 1 here]

The equilibrium employment rate is constant and has a unique solution, as shownin Figure 1. Recall that 𝑔 and 𝑅 are described in equations (22) and (23),respectively. From the total differentiation of equation (27) with respect to 𝐿 and 𝜏𝑟 ,we derive(1 − 𝜂)𝜏𝑝 (1 − 𝐿)𝑑𝐿=𝑑𝜏𝑟 (1 − 𝛼)(1 − 𝜏𝑢 − 𝜏𝑝 )(1 + 𝑅)[𝐿 + 𝜂(1 − 𝐿)](1 − 𝜂)𝜏𝑝 𝑔+
2
𝐿+ 𝜂(1 − 𝐿)
𝐿
(∗1)

(28)

(∗2)

⏞
𝛼𝐴𝑔 ⏞
𝑑𝑔
×
+
.
1 + 𝑅 𝑑𝜏𝑟[]
A higher capital income tax reduces the interest factor, 𝑅 ≡ (1 − 𝜏𝑟 )𝛼𝐴, which has
the following effects on the employment rate. First, a decline in the interest factor
directly increases discounted pension benefits. According to Ono (2010) and Yasuoka(2021), increased pension benefits increase employment. In equation (28), the term(∗ 1) describes the first effect. Second, a higher capital income tax reduces theinterest factor, thereby affecting the growth rate. Recall that the sign of 𝑑𝑔⁄𝑑𝜏𝑟 is
ambiguous from equation (24). If a higher capital income tax promotes economic
growth, then the discounted pension benefits increase, thus promoting employment.
In equation (28), the term (∗ 2) describes the second effect. Substituting equations(22) and (24) into equation (28), we derive:(1 − 𝜂)𝜏𝑝 (1 − 𝐿)𝑑𝐿=𝑑𝜏𝑟 (1 − 𝛼)(1 − 𝜏𝑢 − 𝜏𝑝 )(1 + 𝑅)[𝐿 + 𝜂(1 − 𝐿)](1 − 𝜂)𝜏𝑝 𝑔+
2
𝐿 + 𝜂(1 − 𝐿)𝐿×𝛽 𝜃 𝛼(1 − 𝛼)(1 − 𝜏𝑝 )[(1 + 𝑅)𝜃 𝛽 𝜃 + (1 + 𝑅)(2 − 𝜃) + (1 − 𝜃)𝜏𝑝 (1 − 𝛼)𝐴]𝐴2(1 + 𝑅)1+𝜃 [𝛽 𝜃 + (1 + 𝑅)1−𝜃 + (1 + 𝑅)−𝜃 𝜏𝑝 (1 − 𝛼)𝐴]

2

(29)
.

Recall that a higher capital income tax accelerates economic growth if 𝜃 < 𝜃 holds
and 𝜃 is less than unity from equation (25). Further, equation (29) indicates that𝑑𝐿⁄𝑑𝜏𝑟 > 0 holds if 𝜃 < 1. Thus, we obtain the following proposition.
7

Proposition 1.
If pension benefits exist and the intertemporal elasticity of substitution is
sufficiently small, that is, 𝜃 < 𝜃, a higher capital income tax promotes not only
economic growth but also employment.

Wang (2015) demonstrates that child allowance improves employment if pension
benefits exist. Similarly, the present study shows that capital income tax promotes
employment if pension benefits exist.
Next, we analyze how the capital income tax affects employment under the log
utility function. Taking 𝜃 → 1, equation (29) can be rewritten as follows:(1 − 𝜂)𝜏𝑝 (1 − 𝐿)𝑑𝐿=𝑑𝜏𝑟 (1 − 𝛼)(1 − 𝜏𝑢 − 𝜏𝑝 )(1 + 𝑅)[𝐿 + 𝜂(1 − 𝐿)](1 − 𝜂)𝜏𝑝 𝑔̂+2
𝐿 + 𝜂(1 − 𝐿)𝐿×𝛽𝛼(1 − 𝛼)(1 − 𝜏𝑝 )(1 + 𝑅)(1 + 𝛽)𝐴2[(1 + 𝑅)(1 + 𝛽) + 𝜏𝑝 (1 − 𝛼)𝐴]
𝑔̂ ≡

2

(30)

> 0,

𝛽(1 − 𝜏𝑝 )(1 − 𝛼)𝐴.
1 + 𝛽 + (1 + 𝑅)−1 𝜏𝑝 (1 − 𝛼)𝐴(31)

Therefore, the following proposition holds.

Proposition 2.
If pension benefits exist and 𝜃 → 1 , a higher capital income tax improves
employment, although the growth rate declines.

Recall that 𝑑𝑔⁄𝑑𝜏𝑟 < 0 holds if 𝜃 → 1 from equation (26). Further, the term (∗ 1)
in equation (28) is larger than the term (∗ 2) in equation (28) if 𝜃 → 1. Therefore,
increased capital income tax improves employment.

3. Conclusion
The present study constructs a standard overlapping generations model toinvestigate the impact of capital income tax on employment. It reveals that if a8higher capital income tax promotes economic growth, it also promotes employment.

9References
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Kunze, L., and Schuppert, C. (2010). Financing social security by taxing capitalincome: A bad idea? FinanzArchiv: Public Finance Analysis, 243-262.
Ono, T. (2010). Growth and unemployment in an OLG economy with public pensions.

Journal of Population Economics, 23(2), 737-767.
Romer, P. M. (1986). Increasing returns and long-run growth. Journal of Political

Economy, 94(5), 1002-1037.
Uhlig, H., and Yanagawa, N. (1996). Increasing the capital income tax may lead tofaster growth. European Economic Review, 40(8), 1521-1540.
Wang, L. (2015). Fertility and unemployment in a social security system. Economics

Letters, 133, 19-23.
Yasuoka, M. (2021). How should a government finance pension benefits? Australian

Economic Papers, 60(1), 138-152.

10

Figure 1 Equilibrium employment rate

LHS of (27)

RHS of (27)

𝐿

1

11

𝐿𝑡