Capital income taxation and trade unions
in an endogenous fertility model
Minoru Watanabe
February 2023
Discussion Paper No. 2302

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCapital income taxation and trade unions in
an endogenous fertility model
Minoru Watanabe12
Abstract
The current study aimed to develop a standard overlapping generations model
incorporating involuntary unemployment caused by the union wage setting andfertility choice within an endogenous growth framework. Our study assumes that
capital income tax finances in-work benefits, which is the income transfer
conditioned on work. The results indicate that increasing capital income tax
promotes employment, and hence, promotes economic growth. Further, wedemonstrate that a rise in capital income tax improves fertility.
JEL Classification: H55, J13, J51
Keywords: Capital income tax, Involuntary unemployment, Fertility

1

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 University

11. Introduction
Rising unemployment and slow economic growth have been observed in several
developed countries (Daveri and Tabellini, 2000; Kunze and Schuppert, 2010). In
addition, many developed countries are experiencing a decline in fertility (Fanti andGori, 2010). These serious issues can be caused by high labor and child care costs.
Fanti and Gori (2010) demonstrated that child tax improves both employment andfertility in an overlapping generations model, although their study does not include
endogenous growth. Kunze and Schuppert (2010) stated that cutting labor income
tax with a higher capital income tax 3 promotes not only employment but also
economic growth in an overlapping generations model. However, Kunze andSchuppert (2010) did not consider fertility choices in their study.
Rising unemployment, slow economic growth, and lower fertility rates in developed
countries are serious economic issues; however, previous studies that haveincorporated involuntary unemployment and fertility choice in an overlapping
generations model that includes endogenous growth are, to the best of our knowledge,
scarce. To bridge this gap in existing literature, the present study develops astandard overlapping generations model to demonstrate that capital income tax isthe solution to the above economic issues, referring to Fanti and Gori’s (2010) andKunze and Schuppert’s (2010) studies. The results indicate that a higher capitalincome tax promotes not only employment and economic growth but also fertility.
The remainder of this paper is structured as follows: Section 2 describes our model,and Section 3 concludes the study.
2. Model2.1. Households
In a standard overlapping generations economy, households are identical andexperience two periods, young and old. We assume that households derive utility
from the consumption in the two periods and the number of children. Following
Fanti and Gori (2010), the lifetime utility for households is described as follows:
𝑣𝑡 = log 𝑐𝑡 + 𝛽 log 𝑑𝑡+1 + log 𝑛𝑡 ,(1)

where 𝑐𝑡 and 𝑑𝑡+1 are the consumption in the young and old periods, respectively,
𝑛𝑡 is the fertility rate, and 𝛽 < 1 is the discount factor. If we denote 𝑁𝑡 as thepopulation born in 𝑡, the population growth is given by 𝑁𝑡+1 = 𝑛𝑡 𝑁𝑡 .
Households endow one unit of time and supply it to the labor market inelasticity
3Uhlig and Yanagawa’s (1996) notable study theoretically indicates that a higher capital income tax
can lead to faster economic growth in an overlapping generations model.

2during the young period, while, during old periods, households retire. The present
study introduces the issue of unemployment caused by young individuals joining
trade unions, in line with Daveri and Tabellini (2000) and Ono (2010). The
unemployment rate is defined as a fraction of time, following Fanti and Gori (2010)
and Ono (2010). This assumption considers work-sharing as explained in Section 5.2
in Ono’s (2010) study.
Following Kolm and Tonin (2011, 2015), the current study introduces in-work
benefits, which comprise the income transfer conditioned on work. The two studies
have theoretically demonstrated that in-work benefits reduce wage demands, andhence, reduce the equilibrium unemployment rate within a search model. Though
Kolm and Tonin (2011, 2015) considered that wage tax finances in-work benefits, we
assume that in-work benefits are financed by capital income tax in this study.
Households receive wage income and in-work benefits when they are employed, andunemployment benefits, financed by wage tax, when they are unemployed.
Households split their disposable income between consumption, savings, and childrearing costs when they are young. The budget constraint during the young period
is indicated as follows:
𝑐𝑡 + 𝑠𝑡 + 𝑞𝑤𝑡 (1 − 𝑢𝑡 )𝑛𝑡 = [(1 − 𝜏𝑤 )𝑤𝑡 + 𝜃𝑡 ](1 − 𝑢𝑡 ) + 𝑏𝑡 𝑢𝑡 ,(2)

where 𝑠𝑡 is the savings, 𝑞𝑤𝑡 (1 − 𝑢𝑡 )𝑛𝑡 is the purchase of goods for child care, 𝑤𝑡 isthe wage, 𝑢𝑡 is the unemployment rate, 𝜏𝑤 ∈ (0,1) is the wage tax, 𝜃𝑡 is the inwork benefits, and 𝑏𝑡 is the unemployment benefits. Since the unemployment rateis defined as unemployment time, 𝑤𝑡 (1 − 𝑢𝑡 ) captures the working income. We
assume that 𝑞 ∈ (0,1), and this assumption captures the child-rearing cost as afraction of the working income, similar to Fanti and Gori (2014). Note that the righthand side of equation (2) denotes the average income of young households.
Next, households retire and consume their savings when they are old. Thus, thebudget constraint during the old period is given as follows:
𝑑𝑡+1 = [1 + (1 − 𝜏𝑟 )𝑟𝑡+1 ]𝑠𝑡 ,(3)

where 𝜏𝑟 ∈ (0,1) is the capital income tax and 𝑟𝑡+1 is the interest rate. Fromequations (1), (2), and (3), the optimal allocations are described as follows:𝑠𝑡= 𝛽𝑞𝑤𝑡 (1 − 𝑢𝑡 ),𝑛𝑡
𝑛𝑡 =[(1 − 𝜏𝑤 )𝑤𝑡 + 𝜃𝑡 ](1 − 𝑢𝑡 ) + 𝑏𝑡 𝑢𝑡.
(2 + 𝛽)𝑞𝑤𝑡 (1 − 𝑢𝑡 )Note that the right-hand side of equation (4) indicates the child-rearing cost.

3(4)
(5)

2.2. Firms
Firms produce final goods with the help of capital and labor inputs under acompetitive market. We introduce positive externality to ensure endogenous growth
as in Romer (1986). The aggregate production function is described as follows:(6)

𝑌𝑡 = 𝐴𝐾𝑡𝛼 (𝐸𝑡 𝐿𝑡 )1−𝛼 , 0 < α < 1,

where 𝑌𝑡 is the total outputs, 𝐴 > 0 is the constant technology level, 𝐾𝑡 and 𝐿𝑡 are
the aggregate capital and labor inputs, respectively, and 𝐸𝑡 is the efficiency. Theaggregate labor input is expressed as follows:(7)

𝐿𝑡 = (1 − 𝑢𝑡 )𝑁𝑡 .

We assume no capital depreciation. The factor demands are described as follows:
𝑤𝑡 = (1 − 𝛼)𝐴𝑘𝑡𝛼 𝐸𝑡1−𝛼 (1 − 𝑢𝑡 )−𝛼 ,

(8)

1−𝛼
𝑟𝑡 = 𝛼𝐴𝑘𝑡𝛼−1 (𝐸𝑡 (1 − 𝑢𝑡 )) ,(9)where 𝑘𝑡 ≡ 𝐾𝑡 ⁄𝑁𝑡 is the per capita capital. From equation (8), labor input increases
if wage shrinks. In equilibrium, efficiency is assumed as follows:(10)

𝐸𝑡 = 𝑘𝑡 ,where 𝑦𝑡 ≡ 𝑌𝑡 ⁄𝑁𝑡 is the per capita output, and the per capita output is described as
1−𝛼𝑦𝑡 = 𝐴𝑘𝑡𝛼 (𝐸𝑡 (1 − 𝑢𝑡 ))based on equation (6).

2.3. Trade Unions
Following Daveri and Tabellini (2000) and Ono (2010), trade unions set wages tomaximize the average income of union members, taking policy variables as given.
The objective function of trade unions is written as follows:(11)[(1 − 𝜏𝑤 )𝑤𝑡 + 𝜃𝑡 ](1 − 𝑢𝑡 ) + 𝑏𝑡 𝑢𝑡 .
1𝛼1−𝛼)
𝑤𝑡

Equation (8) can be rewritten as 1 − 𝑢𝑡 = (11−𝛼𝐴𝛼 𝑘𝑡 𝐸𝑡 𝛼 . Substituting this equation

into (11), trade unions set wages as follows:
𝑤𝑡 =1(𝑏 − 𝜃𝑡 ).
(1 − 𝛼)(1 − 𝜏𝑤 ) 𝑡(12)In this equation, 1⁄(1 − 𝛼)(1 − 𝜏𝑤 ) is the constant markup and 𝑏𝑡 − 𝜃𝑡 is the net
benefit from social security. An increase in in-work benefits reduces net benefits fromsocial security and leads to a wage decline. Note that 𝑏𝑡 > 𝜃𝑡 is obtained inequilibrium. See the Appendix for the proof of 𝑏𝑡 > 𝜃𝑡 .
2.4. Government
The government provides unemployment and in-work benefits financed by wage andcapital income taxes, respectively, under a balanced budget. The budget constraints
4

of the government are written as follows:𝜏𝑤 𝑤𝑡 (1 − 𝑢𝑡 )𝑁𝑡 = 𝑏𝑡 𝑢𝑡 𝑁𝑡 ,(13)𝜏𝑟 𝑟𝑡 𝑠𝑡−1 𝑁𝑡−1 = 𝜃𝑡 (1 − 𝑢𝑡 )𝑁𝑡 .

(14)

The left-hand side of equations (13) and (14) are the revenue from wage tax andcapital income tax, respectively. The right-hand side of equations (13) and (14)indicate the government expenditure for unemployment and in-work benefits,respectively.
2.5. Equilibrium
The dynamics in this economy are denoted as follows:
𝑘𝑡+1 = 𝑠𝑡 ⁄𝑛𝑡 .
(15)
1−𝛼𝛼
We recall that 𝑦𝑡 = 𝐴𝑘𝑡 (𝐸𝑡 (1 − 𝑢𝑡 ))holds; therefore, the following equations arederived, using equations (8) and (9):𝑤𝑡 (1 − 𝑢𝑡 ) = (1 − 𝛼)𝑦𝑡 ,(16)

𝑟𝑡 𝑘𝑡 = 𝛼𝑦𝑡 .

(17)

From equations (13)–(17), we obtain the following equations:𝑏𝑡 = 𝜏𝑤 (1 − 𝛼)𝑦𝑡 ⁄𝑢𝑡 ,(18)

𝜃𝑡 = 𝜏𝑟 𝛼𝑦𝑡 ⁄(1 − 𝑢𝑡 ).

(19)

Substituting equations (16), (18), and (19) into (12), we obtain the following constantunemployment rate:𝑢=𝜏𝑤

𝛼𝜏𝑟 < 1.
1−𝛼By differentiating equation (20) with respect to 𝜏𝑟 , we obtain:
𝑑𝑢=𝑑𝜏𝑟1 − 𝛼 + 𝛼𝜏𝑤 +𝛼𝜏− 1 −𝑤𝛼𝛼𝜏 2(1 − 𝛼 + 𝛼𝜏𝑤 + 1 −𝑟𝛼 )

< 0.

(20)

(21)

A rise in capital income tax increases the amount of in-work benefits and reduces
the net benefits from social security, thus, reducing wages from equation (12). A
decline in wages leads firms to increase labor input. Therefore, increasing capitalincome tax reduces unemployment. This finding is qualitatively consistent with thatof Ono’s (2010) study. However, Ono (2010) considered public pension benefits andnot in-work benefits.
Next, this study examines how capital income tax affects the growth rate. Denote𝑔𝑡 as the per capita growth rate given by 𝑔𝑡 = 𝑦𝑡+1 ⁄𝑦𝑡 = 𝑘𝑡+1 ⁄𝑘𝑡 . From equations (4),(8), (10), (15), and (20), the growth rate is described as follows:𝑔 = 𝛽𝑞(1 − 𝛼)𝐴(1 − 𝑢)1−𝛼5(22)1−𝛼= 𝛽𝑞(1 − 𝛼)𝐴 (1 −𝜏𝑤𝛼𝜏 )1 − 𝛼 + 𝛼𝜏𝑤 + 1 −𝑟𝛼,where the growth rate is constant. Recall that the right-hand side of equation (22)denotes child-rearing cost from equation (4). We assume a large enough 𝐴 to ensuresustained growth. Differentiating equation (22) with respect to 𝜏𝑟 , we derive:
𝑑𝑔 𝛽𝑞𝛼(1 − 𝛼)𝐴𝜏𝑤 (1 − 𝑢)−𝛼=> 0.
𝛼𝜏 2𝑑𝜏𝑟
(1 − 𝛼 + 𝛼𝜏𝑤 + 1 −𝑟𝛼 )(23)

A higher capital income tax increases both employment and working income
described as 𝑤𝑡 (1 − 𝑢𝑡 ), which simultaneously increases the child-rearing cost. Sincea rise in child-rearing costs temporally reduces the fertility rate, per capita capital
accumulation can be promoted (Fanti and Gori, 2010)4. Therefore, increasing capitalincome tax promotes economic growth. Though tax reform is an essential assumptionin Kunze and Schuppert’s (2010) study to indicate that increasing capital income tax
promotes employment and economic growth, the present study does not utilize tax
reforms to obtain the results.
This study investigates the impact of capital income tax on the fertility rate.
Substituting equations (18) and (19) into (5), the equilibrium fertility rate is written
as follows:𝑛=1 − 𝛼 + 𝛼𝜏𝑟.
(2 + 𝛽)𝑞(1 − 𝛼)(24)

From equation (24), the fertility rate is constant. Differentiating equation (24) withrespect to 𝜏𝑟 , we obtain the following equation:
𝑑𝑛𝛼
=> 0.
𝑑𝜏𝑟 (2 + 𝛽)𝑞(1 − 𝛼)(25)Long-term fertility rate improves with a higher capital income tax. Increasing
capital income tax enhances the amount of in-work benefits, which improves fertility.
Furthermore, a rise in capital income tax increases working income, which leads toopposite effects on fertility. First, an increase in working income directly improves
fertility. Second, an increase in working income also increases child-rearing costs,
which reduces fertility. These opposite effects are canceled in equilibrium, and we
obtain the following proposition:

4In Fanti and Gori (2010), the fertility rate temporarily declines with a higher child tax, which
promotes per capita capital accumulation.

6

Proposition 1
A rise in capital income tax promotes not only employment and economic growth but
also fertility.
Note that Fanti and Gori (2010) introduce wage subsidy, which is the income transfer
conditioned on work, similar to in-work benefits. In their study, wage subsidy has
positive impacts on the fertility rate. Similarly, in-work benefits contribute to animprovement in fertility as shown in the present study.
Table 1 presents a numerical example. The parameters are set as follows: 𝜏𝑤 =0.1, 𝛼 = 0.33, 𝛽 = 0.55, 𝑞 = 0.2, and 𝐴 = 16.
[Table 1]
3. Conclusion
Low fertility rates, rising unemployment, and slow economic growth are reported inseveral developed countries. This study employed an overlapping generations modelwith unemployment and endogenous fertility and found that capital income taxation
is effective in promoting employment, economic growth, and fertility rates.

7Appendix:In this appendix, we demonstrate that 𝑏𝑡 > 𝜃𝑡 holds. Using equations (18)–(20), weobtain:
𝑏𝑡 𝜏𝑤 (1 − 𝛼) 1 − 𝑢=.
𝜃𝑡𝜏𝑟 𝛼𝑢(A.1)

Substituting equation (20) into equation (A.1), we obtain:
𝑏𝑡 𝜏𝑤 (1 − 𝛼) 1 − 𝑢=,𝜃𝑡𝜏𝑟 𝛼𝑢(1 − 𝜏𝑤 )(1 − 𝛼)2=+ 1.
𝜏𝑟 𝛼𝑏In this equation, 𝜃𝑡 > 1 holds. Therefore, 𝑏𝑡 > 𝜃𝑡 holds in equilibrium.
𝑡8(A.2)

References
Daveri, F., and Tabellini, G. (2000). Unemployment, growth and taxation inindustrial countries. Economic Policy, 15(30), 48-104.
Fanti, L., and Gori, L. (2010). Child policy solutions for the unemployment
problem. Economics Letters, 109(3), 147-149.
Fanti, L., and Gori, L. (2014). Endogenous fertility, endogenous lifetime andeconomic growth: the role of child policies. Journal of Population Economics, 27(2),529-564.
Kolm,

A.

S.,andTonin,M.

(2011).

In-work

benefits

andunemployment. International Tax and Public Finance, 18(1), 74-92.
Kolm, A. S., and Tonin, M. (2015). Benefits conditional on work and the Nordic model.

Journal of Public Economics, 127, 115-126.
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 publicpensions. Journal of Population Economics, 23(2), 737-767.
Romer, P. M. (1986). Increasing returns and long-run growth. Journal of PoliticalEconomy, 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.

9Table 1𝜏𝑟

0

0.1

0.2

0.3

0.4

0.5

𝑢

0.142

0.133

0.125

0.118

0.111

0.105

𝑔

1.064

1.072

1.078

1.084

1.090

1.094

𝑛

1.961

2.057

2.154

2.251

2.347

2.444

𝜏𝑤 = 0.1, 𝛼 = 0.33, 𝛽 = 0.55, 𝑞 = 0.2, and 𝐴 = 16.

10