Robot tax and endogenous fertility in an Overlapping
Generations Model
Minoru Watanabe
April 2023
Discussion Paper No. 2307

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANRobot tax and endogenous fertility in an
Overlapping Generations Model
Minoru Watanabe12
Abstract
This brief study constructs a simple Overlapping Generations Model incorporating
endogenous fertility and automation capital, which can be used as a replacement forlabor inputs. Furthermore, this study introduces a robot tax on automation capital.
In the long run, robot tax promotes not only fertility, but also per capita income.
JEL classification: E10, H50, J11
Keywords: Automation capital, Robot tax, Endogenous fertility, Income growth1

Hokusei Gakuen University, 2-3-1, Ohyachi-Nishi, Atsubetsu-ku, Sapporo 004-8631, Japan.

2

Research Fellow, Graduate School of Economics, Kobe University

E-mail: m-watanabe@hokusei.ac.jp

11. Introduction
The decline in fertility3 has puzzled many developed countries for a long run. Hence,
governments in developed countries have been facing serious economic issues inpromoting both per capita income growth and population growth. According to Fanti
and Gori (2009), there exists a trade-off between population growth and per capita
income growth in the standard neoclassical growth literature. Their study
demonstrated that child taxes lead raise not only population growth, but also percapita income growth using an overlapping generations model.
Thus, this article focuses on building automation capital based on Fanti and Gori(2009). Prettner (2019) stated that many production steps have already been
replaced by machines in the automobile industry. According to Gasteiger andPrettner (2022), many policy makers have been interested in the potential impactsof automation in recent years. Following Prettner (2019), Gasteiger and Prettner(2022) as well as Zhang, Palivos and Liu (2022), this study considers automation
capital as a perfect substitute for labor inputs. Further, we introduce a robot tax
levied on automation capital similar to those proposed in Gasteiger and Prettner(2022) as well as Zhang, Palivos and Liu (2022). The results show that robot tax
promotes population growth as well as per capita income growth.
The remainder of this article is organized as follows. Section 2 describes our
proposed model. Finally, Section 3 concludes the study.
2. Model2-1. Households
We employ a standard overlapping generations model with fertility choice. Identicalhouseholds experience two periods: young and old. They derive utility fromconsumption during these two periods and the number of children. During the young
period, they endow one unit of labor and supply it inelastically to the labor market.
The assumption is that there is full employment. When households are young, they
divide their wage income into consumption, savings, and child care. Whenhouseholds become old, they retire and consume their savings.
Following van Groezen, Leers and Meijdam (2003) as well as Fanti and Gori (2009,2012), the utility function is as follows:
log 𝑐𝑡 + 𝛽 log 𝑑𝑡+1 + log 𝑛𝑡 .

3

(1)

According to Becker and Barro (1988) as well as Barro and Becker (1989), a higher child care cost in

developed countries causes a decline in fertility.

2

Where 𝑐𝑡 and 𝑑𝑡+1 denote the consumption during young and old periods,
respectively, 𝑛𝑡 is the number of children, and 𝛽 < 1 is the discount factor. We
denote 𝑁𝑡 as the population size born in the period 𝑡. The population growth isgiven by 𝑁𝑡+1 = 𝑛𝑡 𝑁𝑡 . The budget constraints are given as follows:
𝑐𝑡 + 𝑠𝑡 + (𝜀𝑤𝑡 + 𝜃)𝑛𝑡 = 𝑤𝑡 , 0 < 𝜀 < 1, 𝜃 > 0(2)

𝑑𝑡+1 = 𝑅𝑡+1 𝑠𝑡 .

(3)

where 𝑠𝑡 is the savings, 𝑤𝑡 is the wage, 𝜀𝑤𝑡 + 𝜃 is the child care cost, and 𝑅𝑡+1 isthe gross interest rate. Following Fanti and Gori (2012), we assume two types ofchild care costs. In Equation (2), 𝜀𝑤𝑡 is the child care cost depending on the working
income, while 𝜃 is the constant child care cost.
The optimal allocations are given as follows:𝑠𝑡= 𝛽(𝜀𝑤𝑡 + 𝜃),
𝑛𝑡
𝑛𝑡 =

𝑤𝑡
.
(2 + 𝛽)(𝜀𝑤𝑡 + 𝜃)

(4)
(5)

An increase in wage improves fertility from Equation (5).
2-2. Firms
Under a competitive market, identical firms provide final goods. Following Prettner(2019), Gasteiger and Prettner (2022), as well as Zhang, Palivos, and Liu (2022), thefactors of production include traditional capital, automation capital, and labor inputs.
It is important to note that automation capital is a perfect substitute for labor inputs.
The aggregate production function is expressed as:
𝑌𝑡 = 𝐴𝐾𝑡𝛼 (𝑁𝑡 + 𝑃𝑡 )1−𝛼 , 𝐴 > 0, 0 < α < 1.

(6)where 𝑌𝑡 is the total output, 𝐾𝑡 is the aggregate traditional capital, 𝑁𝑡 is the totallabor input and 𝑃𝑡 is the aggregate automation capital. We assume full depreciation.
The total revenue is calculated as follows:
𝐴𝐾𝑡𝛼 (𝑁𝑡 + 𝑃𝑡 )1−𝛼 − 𝑤𝑡 𝑁𝑡 − 𝑅𝑡𝑘 𝐾𝑡 − (1 + 𝜏)𝑅𝑡𝑝 𝑃𝑡 .
where 𝑅𝑡𝑘 and 𝑅𝑡𝑝(7)

are the gross rental prices of traditional capital and automation

capital, respectively, and 𝜏 is the robot tax as in Gasteiger and Prettner (2022) aswell as Zhang, Palivos, and Liu (2022). The factor demands are given as follows:
𝑤𝑡 = 𝐴(1 − 𝛼 ) (𝑅𝑡𝑘 = 𝐴𝛼 (

𝑘𝑡 𝛼
) ,
1 + 𝑝𝑡1 + 𝑝𝑡 1−𝛼
) ,
𝑘𝑡

3

(8)
(9)

𝑝

𝑅𝑡 =

𝐴(1 − 𝛼 )
𝑘𝑡 𝛼
(
) .
1 + 𝜏 1 + 𝑝𝑡

(10)

where 𝑘𝑡 ≡ 𝐾𝑡 ⁄𝑁𝑡 and 𝑝𝑡 ≡ 𝑃𝑡 ⁄𝑁𝑡 represent per capita traditional capital andautomation capital, respectively. We denote 𝑦𝑡 ≡ 𝑌𝑡 ⁄𝑁𝑡 as per capita output and 𝑦𝑡
is described as 𝑦𝑡 = 𝐴𝑘𝑡𝛼 (1 + 𝑝𝑡 )1−𝛼 . If we omit automation capital, that is 𝑝𝑡 = 0,
the per capita output boils down to the standard form, that is 𝑦𝑡 = 𝐴𝑘𝑡𝛼 .
According to Gasteiger and Prettner (2022) as well as Zhang, Palivos, and Liu(2022), a no-arbitrage condition exists between traditional capital and automation
capital:𝑅𝑡𝑘 = 𝑅𝑡𝑝 .

(11)

Based on Equations (9)-(11), we obtain:𝑘𝑡 =𝛼 (1 + 𝜏)[1 + 𝑝𝑡 ]
.
1−𝛼

(12)

2-3. Governments
The government imposes a robot tax on automation capital in order to financegovernment expenditures under a balanced budget. The government is subject to thefollowing budgetary constraint.
𝜏𝑅𝑡𝑝 𝑃𝑡 = 𝐺𝑡 .

(13)

where 𝐺𝑡 is the government expenditure. Following Uhlig and Yanagawa (1996), we
assume that 𝐺𝑡 does not contribute to productivity and welfare. In other words, 𝐺𝑡
is a waste of government resources.
2-4. Equilibrium
In equilibrium, 𝑅𝑡𝑘 = 𝑅𝑡𝑝 = 𝑅𝑡 holds. When we substitute Equation (12) into equation(9), the equilibrium gross interest rate is given by 𝑅 = 𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼−1 . We
assume large enough 𝐴 to ensure 𝑅 > 1. The dynamics 4 of this economy can be
described as follows:
𝑘𝑡+1 + 𝑝𝑡+1 =

𝑠𝑡
.
𝑛𝑡

(14)

Based on Equations (4), (12), and (14), we obtain the following long-run per capita
automation capital:𝑝=4𝛽(1 − 𝛼 )[𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 + 𝜃] − 𝛼 (1 + 𝜏).
1 + 𝛼𝜏

See Appendix for the derivation of equation (14)

4

(15)

If 𝛽(1 − 𝛼 )[𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 + 𝜃] ≤ 𝛼 (1 + 𝜏) holds, then 𝑝 = 0 holds. We
assume that 𝛽 (1 − 𝛼 )[𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 + 𝜃] > 𝛼 (1 + 𝜏) throughout the rest of
the article to ensure the interior solution regarding automation capital. Fromequation (15), we obtain:
𝑑𝑝 𝛼 (1 − 𝛼 )=×𝑑𝜏 (1 + 𝛼𝜏)2
1 + 𝛼𝜏
{𝛽𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 [− 1] − (1 + 𝛽𝜃)} < 0.
1+𝜏
Since

1+𝛼𝜏
1+𝜏

(16)

𝑑𝑝

− 1 < 0 uniquely holds in this equation, 𝑑𝜏 < 0 uniquely satisfies. By

substituting Equation (15) into Equation (12), the long-run per capita traditional
capital5 can be calculated by:𝑘=𝛼 (1 + 𝜏)[1 + 𝛽𝜃 + 𝛽𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 ].
1 + 𝛼𝜏

(17)

From this equation, we obtain:
𝑑𝑘 𝛼 (1 − 𝛼 )=×𝑑𝜏 (1 + 𝛼𝜏)2{𝛽𝜀𝛼 𝛼 (1−𝛼)−𝛼𝐴(1 + 𝜏

)𝛼 (

(18)

2

1 + 𝛼 𝜏) + 1 + 𝛽𝜃} > 0.

A higher robot tax leads to a shift from automation capital to traditional capital as
indicated by Gasteiger and Prettner (2022). Thus, an increase in robot tax reduces
long-run per capita automation capital though increases in long-run per capita
traditional capital.
To investigate how robot tax affects fertility in the long run, we derive theequilibrium wage. By substituting Equation (12) into Equation (8), we obtain thefollowing constant wage:
𝑤 = 𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 .

(19)

When this equation is differentiated with respect to 𝜏, we obtain:𝑑𝑤= 𝛼 1+𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼−1 > 0.
𝑑𝜏

(20)

A higher robot tax increases per capita traditional capital, despite a reduction in
automation capital. These two effects raise wages. Next, we focus on fertility choicesin the long run. From Equations (5) and (19), the long-run fertility under genericform of 𝜏 is given by:5

If 𝑝 = 0 holds, the long-run per capita traditional capital is implicitly described as 𝑘 =𝛽[𝜀(1 − 𝛼)𝐴𝑘 𝛼 + 𝜃].

5

(21)

𝑛 = 𝑛[𝑤(𝜏)].
When this equation is differentiated with respect to 𝜏, we obtain:
+

+

⏞ 𝑑𝑤
⏞
𝑑𝑛 𝑑𝑛
=
.
⏟ 𝑑𝜏
𝑑𝜏 𝑑𝑤

(22)

+

A higher robot tax increases wages, which leads to improved fertility in the long run.
When we substitute Equation (19) into Equation (5), we obtain:𝑛=𝑤,(2 + 𝛽)(𝜀𝑤 + 𝜃)

𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼=.
(2 + 𝛽)[𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 + 𝜃]

(23)

From Equation (23), we obtain:𝑑𝑛
𝛼 1+𝛼 (1 − 𝛼 )1−𝛼 𝐴𝜃 (1 + 𝜏)𝛼−1=> 0.
𝑑𝜏 (2 + 𝛽)[𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 + 𝜃]2(24)

Therefore, we derive the following proposition.
Proposition 1
A rise in robot tax improves fertility
Now, we focus on the impact of robot tax on the long-run per capita output. Basedon Equations (6), (12), and (14), the long-run per capita output under generic formof 𝜏 is described as follows:𝑦 = 𝑦[𝑘 (𝜏), 𝑝(𝜏)].

(25)

We recall that a rise in the robot tax promotes per capita capital accumulation, while
per capita automation capital reduces, as shown by Equations (16) and (18). Fromequation (25), we obtain:
+

+

+

−

⏞ 𝜕𝑘
⏞ 𝜕𝑦
⏞ 𝜕𝑝
⏞
𝑑𝑦 𝜕𝑦
=
+
.
⏟ 𝜕𝜏 ⏟
𝑑𝜏 𝜕𝑘
𝜕𝑝 𝜕𝜏
+

(26)

−

A higher robot tax has the following effects on the long-run per capita outputs. First,
an increase in the robot tax promotes traditional capital accumulation, which in turnincreases the long-run per capita outputs. Second, an increase in the robot tax
reduces automation capital, thereby reducing the long-run per capita outputs. The
first and second terms in the right-hand side of Equation (26) represent the first andsecond effects, respectively. If the first effect dominates the second effect, income
growth arises with a higher robot tax. From equations (6), (12), and (15), we obtain
6the following long-run per capita outputs:
𝑦 = 𝐴𝑘 𝛼 (1 + 𝑝)1−𝛼=

𝐴𝛼 𝛼 (1 − 𝛼 )1−𝛼 (1 + 𝜏)𝛼 [𝛽𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 + 1 + 𝛽𝜃].
(1 + 𝛼𝜏)(27)After differentiating Equation (27) with respect to 𝜏, we derive:𝑑𝑦 𝛼 1+𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼=×(1 + 𝛼𝜏)2𝑑𝜏
{𝛽𝜀𝛼 𝛼 (

1−𝛼

)1−𝛼

𝐴(1 + 𝜏)𝛼 [2(1 + 𝛼𝜏)1 + 𝛼𝜏
− 1] + (1 + 𝛽𝜃) [
− 1]}
1+𝜏
1+𝜏

(28)

Whether long-run per capita outputs increase with a higher robot tax depends onthese parameters. If the sign of the large brackets of Equation (28) is positive, anincrease in robot tax is likely to promote long-run income growth. Finally, we havethe following proposition.
Proposition 2
If 𝛽𝜀𝛼 𝛼 (1 − 𝛼 )1−𝛼 𝐴(1 + 𝜏)𝛼 [2(1+𝛼𝜏)1+𝜏

1+𝛼𝜏

− 1] + (1 + 𝛽𝜃) [ 1+𝜏 − 1] > 0 holds, a higher

robot tax promotes per capita income growth.
Table 1 presents a numerical example. The present study sets the following
parameters: 𝛼 = 0.3, 𝛽 = 0.5, 𝜀 = 0.1 𝜃 = 1 and 𝐴 = 14.
[Table 1]

3. Conclusion
Low fertility has been observed in many developed countries. Several developed
countries face the serious economic issue of promoting population growth and per
capita income growth. The current study constructs a simple overlapping
generations model incorporating endogenous fertility and automation capital. Our
results demonstrate that a robot tax is effective in promoting not only population
growth but also income growth.

7Appendix
The clearing condition for the goods market is given as follows:
𝑌𝑡 = 𝐶𝑡 + 𝐾𝑡+1 + 𝑃𝑡+1 + 𝐺𝑡 .

(A.1)

where 𝐶𝑡 is the aggregate consumption at period 𝑡. From Equations (8), (9), and (10),
we have:
𝑌𝑡 = 𝑤𝑡 𝑁𝑡 + 𝑅𝑡 𝐾𝑡 + (1 + 𝜏)𝑅𝑡 𝑃𝑡
As indicated above, 𝑅𝑡𝑘 = 𝑅𝑡𝑝 = 𝑅𝑡

(A.2)

satisfies the equilibrium. Using Equations (2), (3),

(A.1), and (A.2), we obtain:
𝑤𝑡 𝑁𝑡 + 𝑅𝑡 𝐾𝑡 + (1 + 𝜏)𝑅𝑡 𝑃𝑡= (𝑤𝑡 − 𝑠𝑡 )𝑁𝑡 + 𝑅𝑡 𝑠𝑡−1 𝑁𝑡−1 + 𝐾𝑡+1 + 𝑃𝑡+1 + 𝐺𝑡

(A.3)

From Equations (12) and (A.3), we obtain:
𝑅𝑡 (𝐾𝑡 + 𝑃𝑡 − 𝑠𝑡−1 𝑁𝑡−1 ) = 𝐾𝑡+1 + 𝑃𝑡+1 − 𝑠𝑡 𝑁𝑡
To satisfy equation (A.4) for a period 𝑡, Equation (14) should hold.

8

(A.4)

References
Barro, R. J., and Becker, G. S. (1989). Fertility choice in a model of economic growth.
Econometrica, 57(2), 481-501.
Becker, G. S., and Barro, R. J. (1988). A reformulation of the economic theory offertility. The Quarterly Journal of Economics, 103(1), 1-25.
Fanti, L., and Gori, L. (2009). Population and neoclassical economic growth: A new
child policy perspective. Economics Letters, 104(1), 27-30.
Fanti, L., and Gori, L. (2012). A note on endogenous fertility, child allowances andpoverty traps. Economics Letters, 117(3), 722-726.
Gasteiger, E., and Prettner, K. (2022). Automation, stagnation, and the implicationsof a robot tax. Macroeconomic Dynamics, 26(1), 218-249.
Prettner, K. (2019). A note on the implications of automation for economic growth
and the labor share. Macroeconomic Dynamics, 23(3), 1294-1301.
Uhlig, H., and Yanagawa, N. (1996). Increasing the capital income tax may lead tofaster growth. European Economic Review, 40(8), 1521-1540.
Van Groezen, B., Leers, T., and Meijdam, L. (2003). Social security and endogenous
fertility: pensions and child allowances as Siamese twins. Journal of Public
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Zhang, X., Palivos, T., and Liu, X. (2022). Aging and automation in economies withsearch frictions. Journal of Population Economics, 35(2), 621-642.

9

Table 1
𝜏

0

0.1

0.2

0.3

0.4

0.5

𝑝

0.316

0.285

0.256

0.227

0.200

0.174

𝑛

1.727

1.755

1.781

1.805

1.827

1.848

𝑦

14.28914.35914.40014.41714.416

14.399𝛼 = 0.3, 𝛽 = 0.5, 𝜀 = 0.1 𝜃 = 1 and 𝐴 = 14.

10