Capital income taxation and public debt in an endogenous
fertility model
Minoru Watanabe
April 2022
Discussion Paper No. 2209

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCapital income taxation and public debt in an
endogenous fertility model
Minoru Watanabe1,2Abstract
This study constructs an overlapping generations model with fertility choice andunemployment, caused by a constant minimum wage and incorporating public debt.
It shows that a higher capital income tax reduces the public debt burden, and hence
promotes capital accumulation, which leads to an improvement in unemploymentand fertility rates.
JEL classification: H20, J13, J60.
Keywords: Capital income tax, Unemployment, Fertility, Public debt
1. Introduction
Many developed countries face the combination of high unemployment rate and low
fertility rate (Fanti and Gori, 2010; Wang, 2015). High labor and childcare costs are
considered to cause such economic issues. Fanti and Gori (2010) demonstrated thatchild tax can improve unemployment and fertility rates in an overlapping
generations model. Wang (2015) extended Fanti and Gori (2010)’s study byincorporating pensions and child allowances.
To solve the above mentioned economic issues, this study focuses on capital incometax. The previous studies that have shown that capital income taxation canaccelerate economic growth are Uhlig and Yanagawa (1996) and Caballé (1998).
Uhlig and Yanagawa (1996) introduced tax reform, which shows that higher capitalincome tax cuts labor income tax, and demonstrated that such tax reform promotes
savings and capital accumulation. Based on Fanti and Gori (2010) and Wang (2015),
we consider tax reform of cutting public debt with a higher capital income tax. It
shows that increasing capital income tax promotes employment and 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

The rest of the paper is constructed as follows. Section 2 gives a description of themodel. Section 3 concludes our study.
2. Model2.1. Households
We consider an overlapping generation economy where households are identical andexperience two periods: young and old. They derive utility from the number ofchildren in the young period, and from consumption in the old period. Following
Momota (2000) and De la Croix and Doepke (2004), the utility function is given as:
𝑣𝑡 = log 𝑐𝑡+1 + 𝜑 log 𝑛𝑡 ,(1)where 𝑐𝑡+1 is the consumption in the older period, 𝑛𝑡 is the number of children, and
𝜑 > 0 is the taste for children. Suppose 𝑁𝑡 is the population size of younger agentsat period 𝑡, and the evolution of the population is described as 𝑁𝑡+1 = 𝑛𝑡 𝑁𝑡 .
According to Fanti and Gori (2010) and Wang (2015), a constant minimum wage
raises unemployment, and unemployment rates are defined based on a fraction oftime. Each household is endowed with one unit of time and provides it to the labormarket inelastically in the young period. Households earn wage income, accumulatesavings, and invest in childcare in the young period. In the old period, they consume
savings. The lifetime budget constraints for households are given as follows:
𝑠𝑡 + 𝜀𝑛𝑡 = (1 − 𝑢𝑡 )𝑤,(2)

𝑐𝑡+1 = (1 − 𝜏)(1 + 𝑟𝑡+1 )𝑠𝑡 ,(3)

where 𝑠𝑡 is the savings, 𝜀 > 0 is the cost of childcare, 𝑢𝑡 is the unemployment rate,𝑤 > 0 is the constant minimum wage, 𝜏 ∈ (0,1) is the capital income tax, and 𝑟𝑡+1is the interest rate. Equation (3) shows that capital income tax is imposed on thegross interest rate, consistent with Yakita (2003) and Uchida and Ono (2021). The
optimal allocations are given as follows:
𝑛𝑡 =𝜑(1 − 𝑢𝑡 )𝑤,𝜀(1 + 𝜑)(4)𝑠𝑡𝜀= .
𝑛𝑡 𝜑(5)Equation (4) shows that the fertility rate improves if employment increases.

2.2. Firms
Firms produce final goods using capital and labor inputs in a competitive economy.
The production technology is given as:
𝑌𝑡 = 𝐴𝐾𝑡𝛼 𝐿𝑡 1−𝛼 , 𝐴 > 0; 0 < 𝛼 < 1,2(6)where 𝑌𝑡 is the total output, and 𝐾𝑡 and 𝐿𝑡 are the total capital and labor inputs,
respectively. 𝐿𝑡 is given as:(7)

𝐿𝑡 = (1 − 𝑢𝑡 )𝑁𝑡 .
Assuming full depreciation, the factor demands are given as follows:𝑤 = (1 − 𝛼)𝐴𝑘𝑡𝛼 (1 − 𝑢𝑡 )−𝛼 ,(8)1 + 𝑟𝑡 = 𝛼𝐴𝑘𝑡𝛼−1 (1 − 𝑢𝑡 )1−𝛼 ,(9)where 𝑘𝑡 ≡ 𝐾𝑡 ⁄𝑁𝑡 is the capital per capita. From Equation (8), we obtain1(1 − 𝛼)𝐴 𝛼
1 − 𝑢𝑡 = [] 𝑘𝑡 .
𝑤

(10)

It indicates that unemployment reduces with capital accumulation, also explained
by Fanti and Gori (2010) and Wang (2015). Substituting Equation (10) into Equation(9), we obtain the following constant interest rate:1−𝛼
1 1−𝛼 𝛼1 + 𝑟𝑡 = 𝛼𝐴𝛼 ()
.

(11)

𝑤

2.3. Government
The government imposes a tax on capital income, issues new bonds to finance public
expenditure, and redeems public debt. The government’s budgetary constraints aregiven by:
𝐵𝑡+1 = (1 + 𝑟𝑡 )𝐵𝑡 + 𝐺𝑡 − τ(1 + 𝑟𝑡 )𝑠𝑡−1 𝑁𝑡−1 ,where 𝐵𝑡+1 is the amount of public debt issued at period 𝑡 ,(12)(1 + 𝑟𝑡 )𝐵𝑡 is theexpenditure for redemption of public debt, 𝐺𝑡 is non-productive expenditure that
does not contribute to either welfare or productivity, and τ(1 + 𝑟𝑡 )𝑠𝑡−1 𝑁𝑡−1 is thecapital income tax revenue. Following Uhlig and Yanagawa (1996), we assume
𝐺𝑡 = 𝛾𝑦𝑡 𝑁𝑡 , 0 < 𝛾 < 1,(13)𝛼1−𝛼where 𝑦𝑡 ≡ 𝑌𝑡 ⁄𝑁𝑡 = 𝐴𝑘𝑡 (1 − 𝑢𝑡 )is the per capita output (Equation (6)).

2.4. Equilibrium
We denote 𝑏𝑡 ≡ 𝐵𝑡 ⁄𝑁𝑡 as the per capita stock of public debt, and it is equal to 𝑏𝑡+1 =𝑏𝑡 = 𝑏 in a steady state. In equilibrium, 𝑘𝑡+1 + 𝑏𝑡+1 = 𝑠𝑡 ⁄𝑛𝑡 holds. Using thisequation and Equation (5), we have:𝑘𝑡+1 =

𝜀− 𝑏𝑡+1 .
𝜑(14)This equation indicates that 𝜀 ⁄𝜑 is the upper limit of public debt. Further, we3obtain the following equation3:
𝑏𝑡+1 =𝜀(1 + 𝜑) (1 − 𝜏)𝛼𝑏𝑡+ 𝛾 − 𝛼𝜏][ 𝜀𝜑(1 − 𝛼)−𝑏𝑡𝜑(15)

In a steady state, we have the following quadratic equation from Equation (15):(1 − 𝛼)𝜑(1 − 𝛼) 2𝜀𝑏 + [𝛼 − 𝛾 −] 𝑏 + (𝛾 − 𝛼𝜏) = 0.
𝜀(1 + 𝜑)1+𝜑𝜑(16)

Solving this equation, we obtain𝑏=(1 − 𝛼)− [𝛼 − 𝛾 − 1 + 𝜑 ] ± √𝐷
2𝜑(1 − 𝛼)𝜀(1 + 𝜑)(17)2(1 − 𝛼)4(1 − 𝛼)(𝛾 − 𝛼𝜏)𝐷 ≡ [𝛼 − 𝛾 −] −1+𝜑
1+𝜑(18)Suppose 𝑏𝑠 and 𝑏𝑢 are the two different per capita public debts in a steady state;then, from Equation (17), they are described as follows:𝑏𝑠 =(1 − 𝛼)− [𝛼 − 𝛾 − 1 + 𝜑 ] − √𝐷
2𝜑(1 − 𝛼)𝜀(1 + 𝜑)(1 − 𝛼)+1 + 𝜑 ] √𝐷
2𝜑(1 − 𝛼)𝜀(1 + 𝜑)− [𝛼 − 𝛾 −𝑏𝑢 =(19)(20)

If 𝐷 > 0 and 𝑏𝑠 > 0 hold, there exist two positive equilibria where one is locallystable, and the other is unstable. Figure. 1 captures the dynamics of per capita public
debt with two positive equilibria.
[Figure 1 around here]
Figure. 1 describes that 𝑏𝑠 is locally stable and 𝑏𝑢 is unstable. If 𝛾 − 𝛼𝜏 ≤ 0 holds,
the dynamics of public debt have two equilibria as captured by Figure. 2. In this case,a locally stable steady state exists though it converges to zero, i.e., 𝑏𝑠 = 0.
[ Figure 2 around here]3

See Appendix for the derivation of Equation (15).

4Recall that the present article introduces tax reform that represents cutting public
debt with a higher capital income tax. If public debt does not exist in a steady state,
tax reform will become impossible. Therefore, we analyze the neighborhoods of thepositive and locally stable steady states, i.e., 𝑏𝑠 > 0 , as captured by Figure. 1.
Differentiating 𝑏𝑠 with respect to 𝜏 from Equation (19), we have𝑑𝑏𝑠
𝛼𝜀 −1= − 𝐷 2 < 0.
𝑑𝜏𝜑(21)

Note that 𝐷 is defined by Equation (18) and 𝐷 > 0 holds if 𝑏𝑠 > 0 holds.
Equation (21) indicates that a higher capital income tax reduces the public debt
burden. Substituting Equation (19) into Equation (14), we derive(1 − 𝛼)𝜀𝜀 [𝛼 − 𝛾 − 1 + 𝜑 ] + √𝐷
𝑘𝑠 = − 𝑏𝑠 = +,2𝜑(1 − 𝛼)𝜑𝜑𝜀(1 + 𝜑)(22)

where 𝑘𝑠 is the capital per capita in a stable steady state. Differentiating Equation(22) with respect to 𝜏, we derive𝑑𝑘𝑠
𝑑𝑏𝑠 𝛼𝜀 −1=−=
𝐷 2 > 0.
𝑑𝜏𝑑𝜏
𝜑(23)

A higher capital income tax decreases public debt stock and hence, promotes capital
accumulation. According to Caballé (1998), public debt hinders acquiring capital,
which leads to reduction of economic growth. In other words, capital accumulation
can increase due to a decline in public debt as shown Equation (23).
We now investigate how capital income taxes affect unemployment. Suppose 𝑢𝑠 is
the long-run unemployment rate in a stable steady state. By using Equations (10),(14), (19), and (22), we obtain the following unemployment rate as a generic functionof capital income tax (𝜏):𝑢𝑠 = 𝑢{𝑘[𝑏(𝜏)]}.

(24)

From Equations (10), (21), (23) and (24), we have−𝑑𝑢𝑠 ⏞𝑑𝑢𝑠 𝑑𝑘𝑠 𝑑𝑏𝑠=.
𝑑𝜏𝑑𝑘𝑑𝜏
⏟𝑠 𝑑𝑏⏟𝑠 ⏟<0<0(25)<0In this equation, increasing capital income tax reduces public debt, which in turnpromotes capital accumulation, and this reduces unemployment. Substituting
Equation (22) into Equation (10), we derive51(1 − 𝛼)𝐴 𝛼
𝑢𝑠 = 1 − [] 𝑘𝑠 ,𝑤
1(1 − 𝛼)𝐴 𝛼(1 − 𝛼)[𝛼 − 𝛾 − 1 + 𝜑 ] + √𝐷𝑤

2𝜑(1 − 𝛼)𝜀(1 + 𝜑)=1−[𝜀] { +𝜑(26)}.

From this equation, we derive1(1 − 𝛼)𝐴 𝛼 𝛼𝜀 −1𝑑𝑢𝑠= −[]𝐷 2 < 0.
𝑑𝜏𝑤
𝜑(27)Using Equation (27), we obtain the following proposition:
Proposition 1A higher capital income tax promotes employment.
Next, we investigate how capital income tax affects fertility rates. 𝑛𝑠 denotes thelong run fertility rate in a locally stable steady state. By using Equations (4) and(24), the generic form of 𝑛𝑠 is written as:(28)

𝑛𝑠 = 𝑛{𝑢[𝑘(𝑏(𝜏))]}.
From Equations (4), (10), (21), (23), and (28), we obtain+𝑑𝑛𝑠 ⏞
𝑑𝑛𝑠 𝑑𝑢𝑠 𝑑𝑘𝑠 𝑑𝑏𝑠=.
𝑑𝜏𝑑𝑢𝑑𝜏
⏟𝑠 𝑑𝑘
⏟𝑠 𝑑𝑏
⏟𝑠 ⏟
<0<0(29)<0<0As denoted above, increasing capital income tax promotes employment, and hence,it also improves the long run fertility rate from Equation (4). Substituting Equation(26) into Equation (4), the long-run fertility rate can be written as:
𝑛𝑠 ==𝑤111−𝛼[(1 − 𝛼)𝐴]𝛼(1 + 𝜑)𝜑(1 − 𝑢𝑠 )𝑤,𝜀(1 + 𝜑)(1 − 𝛼)+ 𝐷
1+𝜑 ] √}.
2(1 − 𝛼)(1 + 𝜑)[𝛼 − 𝛾 −{1 +(30)Differentiating Equation (30) with respect to 𝜏, we have
𝑑𝑛𝑠 𝑤
=𝑑𝜏

1−

11𝛼 𝛼[(1 − 𝛼)𝐴]𝛼(1 + 𝜑)Thus, we obtain the following proposition:
6−1𝐷 2 > 0.

(31)

Proposition 2A higher capital income tax improves fertility.
A numerical example is presented in Table. 1. The parameters are set as follows: 𝛼 =0.25, 𝜑 = 0.1, 𝜀 = 0.1, 𝛾 = 0.105, 𝑤 = 3.2, and 𝐴 = 4.2.
[Table 1 around here]
3. Conclusion
Many developed countries face high unemployment rates and a decline in fertility
rates. The current study developed an overlapping generations model with
unemployment and fertility choice. The results show that capital income taxation iseffective in improving employment and fertility rates.
Appendix
Dividing Equation (12) with 𝑁𝑡 and considering Equation (13), we have:𝑏𝑡+1 𝑛𝑡 = (1 + 𝑟𝑡 )𝑏𝑡 + 𝛾𝑦𝑡 − τ(1 + 𝑟𝑡 )𝑠𝑡−1.
𝑛𝑡−1(A.1)

Recall that the per capita output is 𝑦𝑡 ≡ 𝑌𝑡 ⁄𝑁𝑡 = 𝐴𝑘𝑡𝛼 (1 − 𝑢𝑡 )1−𝛼 . From Equations (8)and (9), we derive:(1 − 𝑢𝑡 )𝑤 = (1 − 𝛼)𝑦𝑡 ,(A.2)(1 + 𝑟𝑡 )𝑘𝑡 = 𝛼𝑦𝑡 ,(A.3)

In equilibrium, 𝑘𝑡+1 + 𝑏𝑡+1 = 𝑠𝑡 ⁄𝑛𝑡 holds. Substituting Equations (4), (A.2), (A.3),
and 𝑘𝑡 + 𝑏𝑡 = 𝑠𝑡−1 ⁄𝑛𝑡−1 into Equation (A.1), we derive:𝑏𝑡+1𝜑(1 − 𝛼)𝑦𝑡 𝛼𝑦𝑡 𝑏𝑡
τ𝛼𝑦𝑡 𝑏𝑡=+ 𝛾𝑦𝑡 − τ𝛼𝑦𝑡 −𝜀(1 + 𝜑)𝑘𝑡
𝑘𝑡(A.4)

It can be rewritten as follows:
𝑏𝑡+1 =𝜀(1 + 𝜑) (1 − 𝜏)𝛼𝑏𝑡[+ 𝛾 − 𝛼𝜏]𝜑(1 − 𝛼)𝑘𝑡

(A.5)

Adding one period to Equation (A.5), we have:𝑏𝑡+2 =𝜀(1 + 𝜑) (1 − 𝜏)𝛼𝑏𝑡+1[+ 𝛾 − 𝛼𝜏]𝜑(1 − 𝛼)𝑘𝑡+17(A.6)

By substituting Equation (14) into Equation (A.6) and going back by one period, weobtain Equation (15).

8References
Caballé, J. (1998). Growth effects of taxation under altruism and low elasticity ofintertemporal substitution. The Economic Journal, 108(446), 92–104.
De la Croix, D., & Doepke, M. (2004). Public versus private education whendifferential fertility matters. Journal of Development Economics, 73(2), 607–629.
Fanti, L., & Gori, L. (2010). Child policy solutions for the unemployment problem.

Economics Letters, 109(3), 147–149.
Momota, M. (2000). The gender gap, fertility, subsidies and growth. Economics

Letters, 69(3), 401–405.
Uchida, Y., & Ono, T. (2021). Generational conflict and education politics:Implications for growth and welfare. Journal of Macroeconomics, 69, 103315.
Uhlig, H., & 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.
Yakita, A. (2003). Taxation and growth with overlapping generations. Journal ofPublic Economics, 87(3-4), 467–487.

9Figure 1
𝑏𝑡+145° line𝜀(1 + 𝜑)(𝛾 − 𝛼𝜏)𝜑(1 − 𝛼)𝑏𝑠𝑏𝑢𝜀 ⁄𝜑𝑏𝑡

Figure 2
𝑏𝑡+145° line𝑏𝑢10𝜀 ⁄𝜑𝑏𝑡

Table 1𝜏00.1

0.2

0.3

0.4

𝑢𝑠0.400.250.17

0.12

0.07

𝑛𝑠

1.74

2.19

2.40

2.57

2.71

𝛼 = 0.25, 𝜑 = 0.1, 𝜀 = 0.1, 𝛾 = 0.105, 𝑤 = 3.2, 𝐴 = 4.211