Robot tax, unemployment and endogenous fertility in anoverlapping generations model
Minoru Watanabe
August 2023
Discussion Paper No. 2313

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANRobot tax, unemployment and endogenous
fertility in an overlapping generations model
Minoru Watanabe1,2Abstract
Increasing unemployment and declining fertility rates are serious economic issues
in many developed countries. This brief article constructs a simple overlapping
generations model incorporating involuntary unemployment, fertility choice, andautomation capital, with the assumption that automation capital is a perfect
substitute for labor inputs. It is shown that robot tax imposed on automation capital
improves employment and fertility as well as per capita income in the long run.
JEL Classification: H50, J13, and J51
Keywords: Automation capital, Robot tax, unemployment, endogenous 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
According to Fanti and Gori (2010) and Wang (2015), many developed countries face
a combination of declining fertility and higher unemployment rates. However, few
studies have made policy proposals to improve fertility and employment. As notedby Fanti and Gori (2010), unemployment and fertility choices have been discussed
separately in previous studies.
The aim of this study is to provide solutions for the declining fertility, andincreasing unemployment, rates. Fanti and Gori (2010) constructed a standard
overlapping generations model, revealing that imposing a child tax enhances
unemployment and fertility. Wang (2015) extended Fanti and Gori (2010) byincorporating social security measures, such as pension benefits and childallowances. Compared with Fanti and Gori (2010) and Wang (2015), this study
incorporates two additional variables: automation capital and robot tax. First,
automation capital is considered a perfect substitute for labor inputs, in line withPrettner (2019), Gasteiger and Prettner (2022), and Zhang, Palivos, and Liu (2022).
Gasteiger and Prettner (2022) noted that automation and its potential impacts haveattracted the attention of economists in recent years. Frey and Osborne (2017)
estimated that 47% of all jobs in the USA will be replaced with automation in thecoming two decades. Hence, automation is a factor that can potentially increase
future unemployment. Second, we consider robot tax, which is levied on automation
capital in line with Gasteiger and Prettner (2022) and Zhang, Palivos, and Liu (2022).
The combination of low fertility and high unemployment rates is a serious economic
issue in many developed countries. This study develops a standard overlapping
generations model that incorporates unemployment, fertility choices, andautomation. A higher robot tax improves unemployment, fertility, and per capita
income in the long run.
The remainder of this paper is organized as follows. Section 2 describes ourproposed model, and Section 3 concludes the study.
2. Model2.1 Households
The framework used in this study is a standard overlapping generations model. The
study considers identical households, which experience two periods: young and old.
They derive utility from consumption and the number of children. Following Fantiand Gori (2010) and Wang (2015), the utility function is expressed as,log 𝑐𝑡 + 𝛽 log 𝑑𝑡+1 + 𝛾 log 𝑛𝑡2(1)𝑐𝑡 and 𝑑𝑡+1 denote consumption in young and old periods, respectively, 𝑛𝑡 is thefertility rate, 𝛽 < 1 is the discount factor and, 𝛾 > 0 is the desire for children.
Suppose 𝑁𝑡 is the population size in period 𝑡, and the evolution of the population is
𝑁𝑡+1 = 𝑛𝑡 𝑁𝑡 .
Households are endowed with one unit of time, which they supply inelastically tothe labor market when they are young. The labor market is imperfect because of thepresence of a minimum wage. Employed households earn working income and split
it into consumption, savings, and childcare in the young period. The budget
constraints are expressed as,
𝑐𝑡 + 𝑠𝑡 + 𝜀𝑛𝑡 = 𝑤𝑚,𝑡 (1 − 𝑢𝑡 ),(2)

𝑑𝑡+1 = 𝑅𝑡+1 𝑠𝑡 ,(3)

where 𝑠𝑡 is the savings, 𝜀 > 0 is the child rearing cost, 𝑢𝑡 is the unemploymentrate, 𝑤𝑚,𝑡 is the minimum wage, and 𝑅𝑡+1 is the gross interest rate. Following
Fanti and Gori (2010) and Wang (2015), the unemployment rate is defined as theunemployed time. Optimal allocation is expressed as,

𝑛𝑡 =𝑠𝑡 𝛽𝜀= ,𝑛𝑡𝛾(4)𝛾𝑤𝑚,𝑡 (1 − 𝑢𝑡 )(1 + 𝛽 + 𝛾)𝜀(5)2.2 Firms
Identical firms use capital and labor to produce final goods in a competitive market.
Following Prettner (2019), Gasteiger and Prettner (2022), and Zhang, Palivos, andLiu (2022), we assume two types of capital: traditional and automation capital.
Automation capital and labor are perfect substitutes. The production technology isexpressed as,
𝑌𝑡 = 𝐴𝐾𝑡𝛼 (𝐿𝑡 + 𝑃𝑡 )1−𝛼 , 𝐴 > 0, 0 < α < 1(6)where 𝑌𝑡 is the total output, 𝐿𝑡 is the total labor input, 𝐾𝑡 is the total traditional
physical capital, and 𝑃𝑡 is the total automation capital (e.g., robots and artificialintelligence). The total labor input is as follows,
𝐿𝑡 = (1 − 𝑢𝑡 )𝑁𝑡(7)Assuming full depreciation, the total profit is expressed as,
𝐴𝐾𝑡𝛼 (𝐿𝑡 + 𝑃𝑡 )1−𝛼 − 𝑤𝑚,𝑡 𝐿𝑡 − 𝑅𝑡𝑘 𝐾𝑡 − (1 + 𝜏)𝑅𝑡𝑝 𝑃𝑡
where 𝑅𝑡𝑘and 𝑅𝑡𝑝(8)

are the gross rental prices for traditional and automated capital,

respectively, and 𝜏 is the robot tax, as indicated by Gasteiger and Prettner (2022)
and Zhang, Palivos, and Liu (2022). The factor demands are expressed as,3𝑤𝑚,𝑡 = 𝐴(1 − 𝛼 ) (𝑅𝑡𝑘 = 𝐴𝛼 (𝛼𝑘𝑡) ,1 − 𝑢𝑡 + 𝑝𝑡

1 − 𝑢𝑡 + 𝑝𝑡 1−𝛼) ,𝑘𝑡(1 + 𝜏)𝑅𝑡𝑝 = 𝐴(1 − 𝛼 ) (𝛼𝑘𝑡)
1 − 𝑢𝑡 + 𝑝𝑡

(9)
(10)
(11)

Here, 𝑘𝑡 ≡ 𝐾𝑡 ⁄𝑃𝑡 and 𝑝𝑡 ≡ 𝑃𝑡 ⁄𝑁𝑡 are the per capita traditional and automation
capital, respectively. Suppose 𝑦𝑡 ≡ 𝑌𝑡 ⁄𝑁𝑡 , as the per capita output, and 𝑦𝑡 𝑎𝑠 𝑦𝑡 =𝐴𝑘𝑡𝛼 (1 − 𝑢𝑡 + 𝑝𝑡 )1−𝛼 from Equation (6). If 𝑝𝑡 = 0 , 𝑦𝑡 simplifies to 𝑦𝑡 = 𝐴𝑘𝑡𝛼 (1 −𝑢𝑡 )1−𝛼 , and this per capita production technology is consistent with that of Fanti andGori (2010) and Wang (2015). As noted above, the economy has a minimum wage.
We denote 𝑤𝑐,𝑡 as the competitive wage that satisfies 𝑢𝑡 = 0. Following Irmen andWigger (2006) and Fanti and Gori (2011), we assume that minimum wage is
proportional to competitive wage. Thus, the minimum wage is expressed as,(12)𝑤𝑚,𝑡 = 𝜇𝑤𝑐,𝑡where 𝜇 > 1 is a constant markup. Substituting 𝑢𝑡 = 0 into Equation (9), thecompetitive wage is expressed as𝑤𝑐,𝑡 = 𝐴(1 − 𝛼 ) (𝑘𝑡 𝛼)
1 + 𝑝𝑡(13)

From Equations (9) and (13), the relative wage between the minimum andcompetitive wage, under given 𝑘𝑡 and 𝑝𝑡 , is expressed as𝛼𝑤𝑚,𝑡
1 + 𝑝𝑡=()𝑤𝑐,𝑡
1 − 𝑢𝑡 + 𝑝𝑡

(14)

By combining Equations (12) and (14), we obtain the following unemployment rate:−1𝑢𝑡 = (1 + 𝑝𝑡 ) (1 − 𝜇 𝛼 )(15)

In this equation, an increase in automation capital increases unemployment.
Furthermore, unemployment increases with the markup. If we omit automation
capital, that is, 𝑝𝑡 = 0, equilibrium unemployment rate is expressed by 𝑢𝑡 = 1 −𝜇 −1⁄𝛼 from Equation (15). This rate is consistent with that reported by Fanti andGori (2011). Fanti and Gori (2010) and Wang (2015) assumed that a constant
minimum wage leads to unemployment. In contrast, Zhang, Palivos, and Liu (2022)
assume that matching frictions cause unemployment.
According to Gasteiger and Prettner (2022) and Zhang, Palivos, and Liu (2022), theno-arbitrage condition between traditional and automation capital is expressed asfollows:
4𝑅𝑡𝑘 = 𝑅𝑡𝑝(16)

From Equations (10), (11), and (16), we derive,
𝑘𝑡 =𝛼 (1 + 𝜏)[1 − 𝑢𝑡 + 𝑝𝑡 ]1−𝛼(17)Note that 𝑅𝑡𝑘 = 𝑅𝑡𝑝 = 𝑅𝑡 holds in equilibrium.
2.3 Government
The government levies taxes on automation capital to finance consumption with abalanced budget. Thus, we obtain:
𝜏𝑅𝑡𝑝 𝑃𝑡 = 𝐺𝑡 ,(18)where 𝐺𝑡 is the government consumption. We assume that 𝐺𝑡 does not contributeto welfare or productivity. The present study does not incorporate social security,
such as unemployment benefits. If all young households are unemployed, savingsand capital accumulation would be impossible. Thus, the economy is unsustainable
if 𝑢𝑡 = 1 holds.
2.4 Equilibrium
The dynamic in this economy is expressed as:
𝑘𝑡+1 =

𝑠𝑡− 𝑝𝑡+1𝑛𝑡(19)

From Equations (4), (9), (10), (11), (15), (16), (17) and (19), we obtain the following
long run per capita automation capital:−1𝜀𝛽(1 − 𝛼 ) − 𝛼 (1 + 𝜏)𝜇 𝛼𝛾

𝑖𝑓 (1 − 𝛼 )−1𝑝=1 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼{−1𝜀𝛽> 𝛼 (1 + 𝜏)𝜇 𝛼 ,𝛾(20)−1𝜀𝛽𝑖𝑓 (1 − 𝛼 )≤ 𝛼 (1 + 𝜏)𝜇 𝛼𝛾0𝜀𝛽−1We assume (1 − 𝛼 ) 𝛾 > 𝛼(1 + 𝜏)𝜇 𝛼 to ensure the interior solutions of automation
capital in the rest of paper. Differentiating Equation (20) with respect to 𝜏, we have:−1𝜀𝛽𝛼 (1 +()−𝛼1−𝛼𝜇𝑑𝑝𝛾)=<0(21)−1 2𝑑𝜏
[1 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼 ]A higher robot tax reduces the long-run per capita automation capital. This isbecause robot taxation causes firms to switch from automation to traditional
physical capital, as noted by Gasteiger and Prettner (2022). From Equations (15) and5

(20), the equilibrium unemployment rate under the generic form of robot tax isexpressed as,(22)𝑢 = 𝑢{𝑝(𝜏)}Differentiating this equation with respect to 𝜏, we obtain:+−⏞ 𝑑𝑝⏞𝑑𝑢 𝑑𝑢=.
𝑑𝜏 𝑑𝑝
⏟ 𝑑𝜏

(23)

−

If automation capital exists in the long run, a higher robot tax reduces
unemployment, which can be explained as follows: A higher robot tax reduces
automation capital, which raises the relative wage between minimum andcompetitive wages, as shown in Equation (14). Recall that we assume a constantrelative wage in Equation (12). Unemployment declines with a higher robot tax, inorder to maintain a constant relative wage.
By substituting Equation (20) into Equation (15), we obtain the following
unemployment rate:𝑢=−1𝜀𝛽(1 − 𝛼 ) (1 + ) (1 − 𝜇 𝛼 )𝛾−11 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼>0(24)As previously noted, the economy is unsustainable if 𝑢 = 1. To ensure 𝑢 < 1, thefollowing conditions are imposed3:(1 − 𝛼 ) (1 +−1−1𝜀𝛽) (1 − 𝜇 𝛼 ) < 1 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼𝛾

Differentiating Equation (24) with respect to 𝜏, we obtain:−1−1𝜀𝛽𝛼 (1 − 𝛼 ) (1 + ) (1 − 𝜇 𝛼 ) 𝜇 𝛼𝑑𝑢𝛾=−<0−1 2𝑑𝜏[1 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼 ]

(25)

(26)

Next, we investigate how robot taxation affects minimum wage levels. FromEquations (4), (15), (19), and (20), we obtain the long-run per-capita traditional
capital as𝑘=−1𝜀𝛽𝛼 (1 + 𝛾 ) (1 + 𝜏)𝜇 𝛼−11 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼(27)From Equation (27), we have:

3Kunze and Schuppert (2010) introduced parameter restriction to ensure positive unemploymentrate.

6𝑑𝑘=𝑑𝜏𝛼 (1 − 𝛼 ) (1 +−1𝜀𝛽) (1 + 𝜏)𝜇 𝛼
𝛾−1 2[1 − 𝛼 + 𝛼(1 + 𝜏)𝜇 𝛼 ](28)>0As previously explained, a higher robot tax leads to a shift from automation totraditional capital. Thus, an increase in robot tax promotes long-term per capita
traditional capital. Substituting Equation (17) into Equation (9), we obtain thefollowing long-run minimum wage4:
𝑤𝑚 = 𝐴(1 − 𝛼 ) (𝛼𝑘) = 𝐴𝛼 𝛼 (1 − 𝛼 )1−𝛼 (1 + 𝜏)𝛼1−𝑢+𝑝(29)Differentiating this equation with respect to 𝜏, we obtain:
𝑑𝑤𝑚= 𝐴𝛼 1+𝛼 (1 − 𝛼 )1−𝛼 (1 + 𝜏)𝛼−1 > 0𝑑𝜏(30)

An increase in robot taxes reduces automation capital and, hence, increases
traditional physical capital, which increases wages. By contrast, an increase in therobot tax improves employment, as shown in Equation (26). This effect moderates
the wage levels. The first effect dominates the second. Thus, a higher robot tax
increases wages compared with a lower one. Note that Zhang, Palivos, and Liu (2022)
demonstrated that robot taxation increases both, wages and employment.
We now analyze the impact of a robot tax on long-term fertility. Based on Equations(5), (24), and (29), long-run fertility under the generic form of the robot tax is
expressed as,(31)𝑛 = 𝑛{𝑢(𝜏), 𝑤𝑚 (𝜏)}From Equations (5), (26), and (30), we derive:−−++⏞ 𝜕𝑢⏞ ⏞⏞𝑑𝑛 𝜕𝑛𝜕𝑛 𝜕𝑤𝑚=+⏟𝑑𝜏 𝜕𝑢 𝜕𝜏 ⏟𝜕𝑤𝑚 𝜕𝜏+(32)+We demonstrate that an increase in robot taxation increases employment and wages.
Therefore, a higher robot tax rate improves fertility. Substituting Equations (24) and(29) into Equation (5), long-run fertility is expressed as,
𝑛=4𝛾𝑤𝑚 (1 − 𝑢),
(1 + 𝛽 + 𝛾)𝜀Note that the long-run competitive wage, that is 𝑤𝑐 , is denoted by 𝑤𝑐 =Equations (12) and (27).

7(33)

𝐴𝛼 𝛼 (1−𝛼)1−𝛼 (1+𝜏)𝛼𝜇from

−1𝜀𝛽(1 − 𝛼 ) (1 + ) (1 − 𝜇 𝛼 )𝛾𝐴𝛼 𝛼 (1 − 𝛼 )1−𝛼 (1 + 𝜏)𝛼𝛾
{1 −}.
=−1(1 + 𝛽 + 𝛾)𝜀𝛼()
1−𝛼+𝛼 1+𝜏 𝜇

Recall that we assume 𝑢 < 1 in Equation (25). Therefore, the sign in parentheses inEquation (33) is positive. By differentiating this equation with respect to 𝜏, weobtain,−1𝜀𝛽(1 − 𝛼 ) (1 + ) (1 − 𝜇 𝛼 )𝑑𝑛 𝛾𝐴𝛼 1+𝛼 (1 − 𝛼 )1−𝛼 (1 + 𝜏)𝛼−1𝛾{1 −}=−1(1 + 𝛽 + 𝛾)𝜀𝑑𝜏𝛼()1−𝛼+𝛼 1+𝜏 𝜇+−1−1𝜀𝛽𝛾𝐴𝛼 1+𝛼 (1 − 𝛼 )2−𝛼 (1 + 𝜏)𝛼 (1 + 𝛾 ) (1 − 𝜇 𝛼 ) 𝜇 𝛼−1 2(1 + 𝛽 + 𝛾)𝜀 [1 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼 ](34)>0Overall, we obtain the following proposition:
Proposition 1A rise in robot tax improves not only employment, but also fertility.
Finally, we investigate how robot taxes affect per capita output. From Equation (6),the long-run per capita output under the generic form of the robot tax is expressed
as,
(35)

𝑦 = 𝑦{𝑘(𝜏), 𝑢 (𝜏), 𝑝(𝜏)}Differentiating this equation with 𝜏, we derive:++−−+−⏞ 𝜕𝑘⏞ 𝜕𝑦⏞ 𝜕𝑢⏞ 𝜕𝑦⏞ 𝜕𝑝⏞𝑑𝑦 𝜕𝑦=++⏟ 𝜕𝜏 𝜕𝑢⏟ 𝜕𝜏 ⏟𝑑𝜏 𝜕𝑘𝜕𝑝 𝜕𝜏
++(36)−An increase in the robot tax has opposite effects on the long-run per capita output.
First, a higher robot tax promotes both, traditional capital accumulation andemployment, contributing to an increase in per capita output. Second, a higher robot
tax reduces automation capital, thereby reducing long-run per capita output. If theformer effect dominates the latter, a higher robot tax increases long-run per capita
output. Using Equations. (17) and (27), 𝑦 can be expressed as,
𝑦 = 𝐴𝑘 𝛼 (1 − 𝑢 + 𝑝)1−𝛼 ,𝐴𝛼 𝛼 (1 − 𝛼 )1−𝛼 (1 +=−1𝜀𝛽) (1 + 𝜏)𝛼 𝜇 𝛼𝛾−11 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼

Differentiating Equation (35) with respect to 𝜏, we obtain:8(37)𝑑𝑦=𝑑𝜏𝐴𝛼 1+𝛼 (1 − 𝛼 )2−𝛼 (1 +−1𝜀𝛽 −1) 𝜇 𝛼 (1 + 𝜏)𝛼−1 [1 − (1 + 𝜏)𝜇 𝛼 ]𝛾−1 2[1 − 𝛼 + 𝛼 (1 + 𝜏)𝜇 𝛼 ](38)−1An increase in the robot tax increases per capita output if 1 > (1 + 𝜏)𝜇 𝛼 holds.
Therefore, we obtain the following proposition.
Proposition 2−1

If 1 > (1 + 𝜏)𝜇 𝛼 holds, the long run per capita output increases with a higher robottax.
Figures 1-4 present numerical examples5. We investigate the impact of τ on percapita automation capital, unemployment, fertility and long run per capita outputs,respectively. The parameters are set as follows: 𝛼 = 0.33, 𝛽 = 0.35, 𝛾 = 0.1, 𝜀 =0.2, 𝜇 = 1.03, and 𝐴 = 16.
.
[Figure 1 here][Figure 2 here][Figure 3 here][Figure 4 here]
These figures indicate that a higher robot tax promotes not only employment andfertility, but also per capita output in the long run, if the robot tax is sufficientlysmall.
3. Conclusion
A combination of a higher unemployment rate and a decline in the fertility rate has
been reported in many developed countries. This study constructs a simple
overlapping generations model that incorporates automation and robot taxes. Our
results demonstrate that robot taxation is effective in improving unemployment,fertility, and per-capita income.

51−𝛼1−𝛼Note that the equilibrium gross interest rate is denoted by 𝑅 = 𝐴𝛼 [𝛼(1+𝜏)]large enough to ensure that 𝑅 > 1.

9. We assume 𝐴 to beReferences
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Gasteiger, E., and Prettner, K. (2022). Automation, stagnation, and the implicationsof a robot tax. Macroeconomic Dynamics, 26(1), 218-249.
Irmen, A., and Wigger, B. U. (2006). National minimum wages, capital mobility, andglobal economic growth. Economics Letters, 90(2), 285-289.
Kunze, L., and Schuppert, C. (2010). Financing social security by taxing capitalincome: A bad idea? FinanzArchiv/Public Finance Analysis, 66 (3), 243-262.
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