Tax reform, unemployment, and fertility

Minoru Watanabe
December 2021
Discussion Paper No. 2128

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANTax reform, unemployment, and fertility
Minoru Watanabe1,2Abstract
This breif article describes the development of a simple overlapping generations
model with unemployment and endogenous fertility to analyze the impact ofincreasing capital income tax. We find that higher capital income tax promotes
employment and fertility.

JEL classification: H20, J13, J60.

Keywords: Capital income tax, Unemployment, Fertility

1. IntroductionAs Fanti and Gori (2010) and Wang (2015) contend, declining fertility and rising
unemployment are serious economic issues in developed countries. Becker and Barro(1988) revealed that an increase in child-rearing costs reduces fertility, while Daveriand Tabellini (2000) revealed that a high cost of labor increases unemployment.
The present study analyzes the role of capital income tax on unemployment andfertility choice. Uhlig and Yanagawa (1996) demonstrated that capital income
taxation can increase economic growth in an overlapping generations model. Theirstudy introduced a revenue-neutral tax reform in which increasing capital income
reduces labor income tax. Kunze and Schuppert (2010) extended Uhlig andYanagawa’s (1996) model by considering involuntary unemployment caused by wagebargaining, showing that higher capital income tax reduces labor income tax and,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

consequently, reduces wages. Therefore, an increase in capital income tax was foundto promote employment. However, Uhlig and Yanagawa (1996) and Kunze andSchuppert (2010) do not consider fertility choice, and Kunze and Schuppert (2010)
assume a constant growth rate of population size.
Fanti and Gori (2010) noted that previous theoretical studies analyzed
unemployment and fertility choice separately. Their study demonstrated that childtax improves both unemployment and fertility in an overlapping generations model.
Wang (2015) extended Fanti and Gori’s (2010) model by including public pensions
and child allowances. However, studies on the effect of capital income tax on
unemployment and fertility in an overlapping generations model are scarce. Thus,
we extended the work of Fanti and Gori (2010) and Wang (2015) by adding capitalincome tax, showing that a tax reform, which increases capital income tax, canpromote both employment and fertility.
We describe our model and the results in Section 2, and conclusions in Section 3.

2. The Model
2-1. Households
Consider a standard one-sector overlapping generations model. Households areidentical and are classified into two periods (young and old). They are endowed with
one unit of time devoted to labor supply and child-rearing during the young period,in accordance with Fanti and Gori (2007) and Hirazawa and Yakita (2009). The time
constraint for young agents is(1)𝑙𝑡 + 𝑧𝑡 = 1,

where 𝑙𝑡 is the labor supply and 𝑧𝑡 is the time cost for child-rearing. According toFanti and Gori (2007), Fanti and Gori (2010), and Wang (2015), a constant minimum
wage increases unemployment, and the unemployment rate is defined as a fractionof time. Note that Hirazawa and Yakita (2009) assumed full employment.
Households derive utility from consumption and number of children. We assume
the following lifetime utility, similar to Fanti and Gori (2010) and Wang (2015).
𝑣𝑡 = log 𝑐𝑡 + 𝛽 log 𝑑𝑡+1 + log 𝑛𝑡 ,2(2)where 𝑐𝑡 and 𝑑𝑡+1 represent the consumption in the young and old periods,
respectively, 𝑛𝑡 is the number of children, and 𝛽 ∈ (0,1) is the discount factor. We
denote 𝑁𝑡 as the population size born at 𝑡 and the evolution of the population isrepresented by 𝑁𝑡+1 = 𝑛𝑡 𝑁𝑡 .
During the young period, households earn wage income, which is spent on
consumption, savings, and childcare. In the old period, they retire and consume theirsavings. The budget constraints for the young and old periods are:𝑐𝑡 + 𝑠𝑡 + 𝑥𝑡 = (1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑙𝑡 𝑤,(3)

𝑑𝑡+1 = [1 + (1 − 𝜏𝑟,𝑡+1 )𝑟𝑡+1 ]𝑠𝑡 ,(4)

where 𝑠𝑡 is the savings, 𝑥𝑡 is the childcare cost captured by purchased final goods,
𝜏𝑤,𝑡 is the labor income tax, 𝑢𝑡 is the unemployment rate, 𝑤 > 0 is the constant
minimum wage, 𝜏𝑟,𝑡+1 is the capital income tax, and 𝑟𝑡+1 is the interest rate. Inequation (3), (1 − 𝑢𝑡 )𝑙𝑡 denotes the working period. Note that households cannot
provide childcare while they supply labor, including during unemployed periods, asexplained by Fanti and Gori (2007). According to Apps and Rees (2004) andHirazawa and Yakita (2009), child-rearing requires both final goods and time costs.
We thus assume the following fertility function, as put forth by Apps and Rees (2004)
and Hirazawa and Yakita (2009):
𝑛𝑡 = 𝑥𝑡𝜀 𝑧𝑡1−𝜀 ,(5)0 < 𝜀 < 1,Note that if we omit the time cost for child-rearing (i.e., 𝑧𝑡 = 0), and assume equation(5) as 𝑛𝑡 = 𝑥𝑡𝜀 , our results do not change qualitatively.
According to Hirazawa and Yakita (2009), households choose consumption and thenumber of children to maximize equation (2) subject to equations (1), (3), (4), and (5).
We obtain the following optimal allocations:
𝑥𝑡 =𝜀(1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑤,2+𝛽(6)1−𝜀,2+𝛽(7)𝑧𝑡 =𝜀𝜀 𝜀 (1 − 𝜀)1−𝜀 [(1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑤]
𝑛𝑡 =,2+𝛽3(8)1−𝜀𝑠𝑡 𝛽[(1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑤]=𝑛𝑡
𝜀 𝜀 (1 − 𝜀)1−𝜀.

(9)Suppose 𝑝𝑡 is the cost of rearing one child, 𝑝𝑡 ≡ [(1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑤𝑧𝑡 + 𝑥𝑡 ]⁄𝑛𝑡 and(1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑤𝑧𝑡 is the opportunity cost for childcare. We obtain 𝑝𝑡 as follows:1−𝜀[(1 − 𝜏𝑤,𝑡 )(1 − 𝑢𝑡 )𝑤]
𝑝𝑡 =𝜀 𝜀 (1 − 𝜀)1−𝜀.

(10)2-2. Firms
Firms produce final goods with capital and labor inputs in a competitive market. We
thus assume the following production similar to Fanti and Gori (2010) and Wang(2015):
𝑌𝑡 = 𝐾𝑡𝛼 𝐿𝑡 1−𝛼 , 0 < α < 1,(11)where 𝑌𝑡 is the total output of the final goods, and 𝐾𝑡 and 𝐿𝑡 are the aggregate
capital and labor inputs, respectively. The labor input is𝐿𝑡 = (1 − 𝑢𝑡 )𝑙𝑡 𝑁𝑡 .

(12)

As mentioned earlier, a constant minimum wage prevails in this economy. We
assume no capital depreciation. The factor demand is𝑤 = (1 − 𝛼)𝑘𝑡𝛼 [(1 − 𝑢𝑡 )𝑙𝑡 ]−𝛼 ,(13)𝑟𝑡 = 𝛼𝑘𝑡𝛼−1 [(1 − 𝑢𝑡 )𝑙𝑡 ]1−𝛼 ,(14)where 𝑘𝑡 ≡ 𝐾𝑡 ⁄𝑁𝑡 is the per-capita capital. From equation (13), we obtain:11−𝛼 𝛼1 − 𝑢𝑡 = () 𝑘𝑡 𝑙𝑡 −1 .
𝑤

(15)

Substituting equation (15) into equation (14), we obtain the following constantinterest rates:
1−𝛼

1−𝛼 𝛼
𝑟𝑡 = 𝑟 = 𝛼 ().
𝑤(16)2-3. Government
The government imposes a tax on labor and capital income to provide non-productive
expenditure under a balanced budget, in accordance with Uhlig and Yanagawa’s
4

(1996) model. The government’s budget constraint is:𝜏𝑤,𝑡 (1 − 𝑢𝑡 )𝑙𝑡 𝑤𝑁𝑡 + 𝜏𝑟,𝑡 𝑟𝑡 𝑠𝑡−1 𝑁𝑡−1 = 𝐺𝑡 ,(17)where 𝐺𝑡 denotes nonproductive expenditure, which contributes to no marginalutility or marginal productivity. The left-hand side of this equation denotes tax
revenue from labor and capital income, respectively. We introduce a tax reform inaccordance with the models by Uhlig and Yanagawa (1996) and Kunze andSchuppert (2010). Thus, we impose the following assumptions:𝜏𝑟,𝑡 = 𝜏𝑟 , 0 < 𝜏𝑟 < 1,(18)𝐺𝑡 = 𝜇𝑦𝑡 𝑁𝑡 , 0 < 𝜇 < 1.

(19)

In equation (19), 𝑦𝑡 represents the per-capita gross domestic product (GDP), where𝑦𝑡 ≡ 𝑌𝑡 ⁄𝑁𝑡 = 𝑘𝑡𝛼 [(1 − 𝑢𝑡 )𝑙𝑡 ]1−𝛼 . If we assume capital income tax as an exogenous
variable, the labor income tax is determined such that it balances the government’s
budget constraint. Equation (19) indicates that the government’s expenditure is afraction of GDP, as shown by Uhlig and Yanagawa (1996).

2-4. Equilibrium
The dynamics in this economy are
𝑘𝑡+1 = 𝑠𝑡 ⁄𝑛𝑡 .

(20)

Recall that 𝑦𝑡 = 𝑘𝑡𝛼 [(1 − 𝑢𝑡 )𝑙𝑡 ]1−𝛼 ; therefore, using equations (13) and (14), we arriveat(1 − 𝑢𝑡 )𝑙𝑡 𝑤 = (1 − 𝛼)𝑦𝑡 ,(21)𝑟𝑡 𝑘𝑡 = 𝛼𝑦𝑡 .

(22)

From equations (17)–(22), we obtain the constant of the labor income tax rate:𝜏𝑤 =𝜇 − 𝛼𝜏𝑟.
1−𝛼(23)

Following Uhlig and Yanagawa (1996), we assume that 0 < 𝜏𝑤 < 1. Equation (23)indicates that increasing capital income tax reduces labor income tax. Using
equations (1), (7), (9), (15), (20), and (23), we obtain𝑘𝑡+1 = 𝜌𝑘𝑡1−𝜀 .

5(24)

1−𝜀

1−𝛼

𝛽
1−𝛼 𝛼2+𝛽𝜌≡ 𝜀)() [1 − 𝛼(1 − 𝜏𝑟 ) − 𝜇]}{(1−𝜀𝜀 (1 − 𝜀)𝑤1+𝛽+𝜀(25).

Note that 0 < 1 − 𝛼(1 − 𝜏𝑟 ) − 𝜇 < 1 holds in equation (25) because 𝜏𝑤 is given byequation (23), and we assume 0 < 𝜏𝑤 < 1. Thus, we obtain 𝜌 > 0 in equation (25).
Recall that 0 < ε < 1 from equation (5). Equation (24) reveals the existence of astable and unique steady state in this economy. Suppose 𝑘 is the per-capita capitalin the steady state. From (24) and (25), 𝑘 is
1𝜀

1−𝛼𝛼

𝛽1−𝛼𝑘=[ 𝜀)] {(1−𝜀𝜀 (1 − 𝜀)𝑤

2+𝛽() [1 − 𝛼(1 − 𝜏𝑟 ) − 𝜇]}1+𝛽+𝜀1−𝜀𝜀.

(26)Clearly, we obtained 𝑑𝑘⁄𝑑𝜏𝑟 > 0 and 𝑑𝑘⁄𝑑𝑤 < 0 . Equation (23) reveals that ahigher capital income tax reduces labor income tax, which increases disposable
income and savings. Thus, a higher capital income tax promotes capital
accumulation, as revealed by Uhlig and Yanagawa (1996). From equations (7) and(15), we denote the long-run unemployment rate as 𝑢 and obtain11−𝛼 𝛼2+𝛽𝑢 =1−() () 𝑘.
𝑤
1+𝛽+𝜀(27)Equation (27) indicates that long-run unemployment rate declines with capital
accumulation as shown by Fanti and Gori (2010) and Wang (2015). Suppose 𝑘𝑓 isthe long-run per-capita capital under full employment: 𝑢 = 0. Combining equations(1), (7), and (15), and taking 𝑢 = 0, we describe 𝑘𝑓 as
1𝑤 𝛼1+𝛽+𝜀𝑘𝑓 = (.
)
1−𝛼2+𝛽(28)If 𝑘𝑓 > 𝑘 holds, unemployment exists in the long run. Substituting equation (26)
into equation (27), we obtain:
1−𝜀(29)𝑢 = 1 − 𝜎[1 − 𝛼(1 − 𝜏𝑟 ) − 𝜇] 𝜀 ,1−𝛼(1−𝜀)𝛼𝜀1−𝛼𝜎≡()𝑤

11𝜀2+𝛽 𝜀𝛽() [ 𝜀> 0.
]
1+𝛽+𝜀𝜀 (1 − 𝜀)1−𝜀(30)Note that 𝑑𝑢𝑡 ⁄𝑑𝑤 > 0 holds for any period 𝑡. Differentiating equation (29) with6

respect to 𝜏𝑟 , we derive
1−𝜀𝑑𝑢𝛼𝜎(1 − 𝜀)[1 − 𝛼(1 − 𝜏𝑟 ) − 𝜇] 𝜀 −1 < 0.
=−𝑑𝜏𝑟𝜀(31)Daveri and Tabellini (2000) found that the negative correlation between
unemployment and labor income tax holds. In this article, increasing capital income
tax was found to reduce unemployment because a higher capital income tax cuts thelabor income tax, which increases savings and per-capita capital. As described byequation (27), capital accumulation promotes employment. Therefore, we obtain
Proposition 1.

Proposition 1
An increase in capital income tax promotes capital accumulation, thereby reducing
unemployment in the long run.

Suppose 𝑦 is the long-run per-capita output, where 𝑦 = 𝑘 𝛼 [(1 − 𝑢)𝑙]1−𝛼 . Note that 𝑙
is constant in equilibrium because 𝑙 = 1 − 𝑧 =1+𝛽+𝜀2+𝛽from equations (1) and (7).

Recall that a rise in capital income tax increases not only per-capita capital but also
employment. Thus, increasing capital income tax increases long-run per-capita
output.
Next, we examine the impact of capital income tax on fertility. Assuming that 𝑛 is
the long-run fertility rate, from equations (8), (23), (26), and (27), under a genericform, it is𝑛 = 𝑛{𝜏𝑤 (𝜏𝑟 ), 𝑢[𝑘(𝜏𝑟 )]}.

(32)

The total derivative of equation (32) with respect to 𝜏𝑟 yields++𝑑𝑛 ⏞𝜕𝑛 𝜕𝜏𝑤 ⏞𝜕𝑛 𝜕𝑢 𝜕𝑘=+.
⏟ 𝜕𝑘⏟ 𝜕𝜏𝑑𝜏𝑟 𝜕𝜏⏟𝑟 𝜕𝑢⏟𝑟
⏟𝑤 𝜕𝜏−−−−(33)+Equation (33) demonstrates that the capital income tax has two effects. First,increasing it reduces the labor income tax, which leads to an increase in disposable
income, thereby improving fertility. The first term on the right-hand side of equation
7(33) describes this effect. Second, capital accumulation with a higher capital income
tax promotes both employment and fertility. The second term of the right-hand sideof equation (33) denotes this effect. Substituting equations (23), (29), and (30) into(8), we obtain:1−𝛼𝛽1−𝛼 𝛼[1 − 𝛼(1 − 𝜏𝑟 ) − 𝜇].
𝑛=()1+𝛽+𝜀𝑤(34)

By differentiating equation (34) with respect to 𝜏𝑟 , we derive:1−𝛼𝑑𝑛𝛼𝛽
1−𝛼 𝛼=()> 0.
𝑑𝜏𝑟 1 + 𝛽 + 𝜀 𝑤(35)Thus, we arrive at the following proposition.

Proposition 2
Higher capital income tax improves fertility in the long run.

Our results are not affected by considering the labor income tax reductions. However,many developed countries have come up against a deterioration of finances in recentyears, as indicated by Ueshina (2018), and tax reduction will worsen their fiscalsituation. Compared to the labor income tax reduction, the tax reform in the current
study is revenue-neutral.

3. Conclusion
Many developed countries are plagued by not only a decline in fertility but also ahigh unemployment rate. In this study, we built an overlapping generations modeland demonstrated that capital income taxation can resolve these economic issues.

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