PRI Discussion Paper Series (No.21A-07)

Does Automation Technology increase Wage?

Assistant Professor, College of Economics, Aoyama Gakuin University
Ryosuke Shimizu

Policy Research Institute, Ministry of Finance, Japan
Shohei Momoda

October 2021

The views expressed in this paper are those of theauthors and not those of the Ministry of Finance or
the Policy Research Institute.

Research Department Policy Research Institute, MOF
3-1-1 Kasumigaseki, Chiyoda-ku, Tokyo 100-8940, JapanTEL 03-3581-4111

Does Automation Technology increase Wage?∗
Ryosuke Shimizu†and Shohei Momoda‡

Abstract
This paper examines the relationship between automation technology diﬀusion and the wage. In this model, producers either choose automation or
non-automation technology, whichever is more proﬁtable. Further, when the
producers introduce automation technology, they must pay ﬁxed costs, which
diﬀer between industries. The main results of this paper indicate that the
improving the productivity of automation technology promotes automation
diﬀusion, decreases labor share, and also decreases the wage when the levelof automation technology diﬀusion is suﬃciently high.
Keywords: automation, the wage, labor share decline, technology choice
JEL codes: E24, J23, O3∗ We are deeply grateful to Akihisa Shibata and Kazuhiro Yuki for their helpful comments andadvices. We also thank Takayuki Ogawa, Hideki Nakamura, Masakatsu Nakamura and seminar
participants at 2019 Asian Meeting of Econometric Society; 2020 International Conference
on Trade, Financial Integration and Macroeconomic Dynamics, 2019; Summer Workshop on
Economic Theory; and Uncertainty and Macroeconomics. This work is supported by Japan
Society for the Promotion of Science KAKENHI Grant No.861145 JP.
† Assistant Professor, College of Economics, Aoyama Gakuin University:
R.shimizuecon@gmail.com‡ Policy Research Institute, Ministry of Finance, Japan: shohei.momoda.econ@gmail.com

11 Introduction
In recent years, many studies show that labor income share has been declining overthe last several decades. Technological advancement is considered one of the main
factors inﬂuencing this phenomenon (Karabarbounis and Neiman, 2014; Grossman
et al., 2017; Alvarez-Cuadrado et al., 2018).1 Acemoglu and Restrepo (2018b) andAutor et al. (2018) show that the diﬀusion of labor-saving automation technology, suchas robots and machines, is one of the most important recent technological advances
underlying this decline in labor income share. Labor income share decreases as labor
is replaced by capital in the production process through the diﬀusion of automation.
Bergholt et al. (2021) ﬁnd that automation has become an increasingly importantfactor of the labor share decline since the early 2000s.
To analyze the beneﬁt that individual workers gain from automation diﬀusion, thispaper studies not only the eﬀect on labor income share but also the eﬀect on the wage.
While many empirical studies examine the eﬀect of automation technology on thewage, no consensus exists on whether the automation diﬀusion increases or decreases
the wage. Graetz and Michaels (2018) analyze data on robot adoption and conclude
that, on average, an industry’s adoption of industrial robots positively aﬀects worker
wages. On the other hand, Acemoglu and Restrepo (2020a) focus on local labormarkets data divided by commuting zones between 1990 and 2007 and demonstratethat the introduction of industrial robots negatively aﬀects the wage. Further, Dauthet al. (2019) applies the approach used by Acemoglu and Restrepo (2020a) to data ofGermany and Chiacchio et al. (2018) applies it to data of 6 countries in EU (Finland,
France, Germany, Italy, Spain and Sweden). The research of the former shows that1

Several types of drivers of the decline in labor income share have been identiﬁed in the literature. Elsby et al. (2013) and Boehm et al. (2020) show that globalization, which promotes
oﬀshoring and reduces the ratio of labor-intensive industry in OECD countries, substantiallycontributes to a decline in labor income share. Autor et al. (2017, 2020), and Barkai (2020)
emphasize the rising market concentration of sales and markups; as market sales are concentrated on a smaller fraction of productive ﬁrms, their markup rates increase, and the income
share of proﬁts compresses the income share of workers.

2the average wage does not increase as a result of robot adoption, while the that of
the latter shows the decrease in the average wage but the result is not robust.
A theoretical study on the relationship between automation diﬀusion and the wage
is conducted by Acemoglu and Restrepo (2018b). Using the task-based model presented by Zeira (1998),2 they analyze the impact of advancement in automation technology (i.e. when automation technology becomes available for more tasks) on thewage. The adoption of automation technology by enterprises displaces workers but
simultaneously increases aggregate production and the labor demand. As a result of
this combination of positive and negative eﬀects on the labor demand, Acemoglu andRestrepo’s static analysis shows that the wage might decline. However, the extended
model taking capital accumulation into account concludes that the advancement inautomation technology increases the wage because capital accumulation has an additionally positive eﬀect.
This paper presents a general equilibrium model in which the wage could decrease
even in the presence of capital accumulation. The model shows that an improvementin automation technology decreases the wage if the present level of automation diffusion is suﬃciently high, while it increases the wage otherwise. This result suggeststhat the varying results of the aforementioned empirical researches may be consistently explained by focusing speciﬁcally on the level of automation diﬀusion.
The model is constructed using the task-based model presented by Zeira (1998)
and the over-lapping generations model by Diamond (1965). Households live for two
periods; they inelastically supply labor as workers in the ﬁrst period and manage
production units in industries as managers in the second period. They choose either
automation technology or non-automation technology for the management of the production units to maximize proﬁts. To introduce automation technology, they must
pay ﬁxed costs, which are diﬀerent across industries. The ﬁxed cost is considered as
one of the main factors of technologically choice in previous theoretical studies (e.g.,2

Acemoglu and Autor (2011); Nakamura (2009); Nakamura and Nakamura (2008, 2019); Nakamura and Zeira (2018); Yuki (2016); Hémous and Olsen (2021); Aghion et al. (2019); Martinez
et al. (2018) etc utilize the task-based model.

3

Jovanovic and Lach, 1989; Hall, 2004). Sirkin et al. (2015) specify the ﬁxed costs,
estimate their magnitude, and state that their burden impedes the adoption of robots.
In the model, managers choose the more proﬁtable technology in each industry;
automation technology is chosen in the industries that feature lower ﬁxed costs. As
automation technology improves, more industries with higher ﬁxed costs can adopt
automation technology.
The main result of this paper is that an improvement in automation productivity
causes a decrease (an increase) in the wage when the level of automation diﬀusion is
suﬃciently high (low). Whether the wage decreases or not is associated with the levelof diﬃculty that an industry encounters in introducing automation, that is, how highthe ﬁxed costs are in the industry. The advancement of diﬀusion displaces workers
while it increases aggregate production. The former has a negative eﬀect and the latter a positive eﬀect on the wage. As automation technology diﬀuses, the ﬁxed costs
in industries in which automation technology is newly introduced become higher. To
pay higher ﬁxed costs in these industries, the returns from using automation technology must be larger. Because the returns are proportional to aggregate production,
how much aggregate production increases is associated with how much ﬁxed costs
increase in newly automated industries. This model shows that unless the ﬁxed costsin such industries increase rapidly, the positive eﬀect of the increase in aggregate
production on the wage is smaller than the negative eﬀect of displacement when thelevel of automation diﬀusion is suﬃciently high. In that case, in contrast to Acemoglu
and Restrepo (2018b), the wage decreases when the level of automation diﬀusion is
suﬃciently high, even in the presence of capital accumulation.
In this model, the labor income share decreases with automation diﬀusion. Whilethe income share of managers increases with the diﬀusion, the income share of workers
decreases. Because the impact of the latter is greater than that of the former, thesum of their labor income shares decreases. This result is consistent with the trend
over the last several decades.
This paper also considers two extensions. In the ﬁrst extension, we introduce atechnology frontier. Because of the technology frontier, managers who face high ﬁxed
4

costs cannot utilize automation technology; otherwise, they would use it. Acemoglu
and Restrepo (2018b) adopt this assumption, and the extension of this model highlights the diﬀerence between their model and ours. In the model of Acemoglu andRestrepo (2018b), expanding the technology frontier increases the wage when capital accumulation exists. By contrast, our analysis shows that even in the presenceof capital accumulation, the expansion of the technology frontier decreases the wage
when the level of the frontier is suﬃciently high. The second extension considers apolicy that subsidizes the ﬁxed costs. Because the technology choice depends on the
return net of the ﬁxed cost, such a policy ultimately aﬀects the diﬀusion of automation technology. The beneﬁts of subsidy policy vary across households. Subsidy rates
are shown to exist that which improve the welfare of all households. The analysis
shows that there exists subsidy rates that improve the welfare of all households.
The rest of the paper is organized as follows. Section 2 constructs the model.
Section 3 solves the equilibrium and describes the conditions of the existence of aunique steady state. Section 4 examines the comparative statics and presents anextension of the model considering the creation of new intermediate goods. Section
5 discusses the diﬀerences between this paper and Acemoglu and Restrepo (2018b),
and modiﬁes the model by introducing an automation technology frontier to clearly
demonstrate the diﬀerences. Section 6 introduces a subsidy policy into the model.
Section 7 concludes this paper.

2 The Model
The consumer side of the model is based upon the two-period OLG model presented by Diamond (1965). Each household supplies labor as a worker and chooses atechnology during the ﬁrst period of her life and then manages a production unit producing intermediate goods in an industry during the second period. There are manyindustries over which households are distributed during the ﬁrst period. The producer side consists of managers and ﬁnal goods producers. The ﬁnal goods producers

5use intermediate goods supplied by the managers for production, and the ﬁnal goods
are utilized for consumption and savings. The factor of production for operating aunit in an industry producing intermediate goods is either labor or capital; managers
choose the more proﬁtable input. When capital is chosen, managers can make useof automation technology,3 which enables them to produce without using labor, butinstead they must pay ﬁxed costs that are heterogeneous between industries.

2.1 Households
There is a unit measure of households in each industry j ∈ [0, 1]. Households live
for two periods. During the ﬁrst (young) period, they inelastically supply labor as
workers and allocate their wages to consumption and savings. During the second
period (old), they manage production units in industries of intermediate goods toreceive proﬁts and they consume the returns from their savings and proﬁts. Let
us consider a household born in period t and working in industry j. Her utility
maximization problem is expressed as follows:
Ut (j) = max
log(cyt (j)) + β log(cot+1 (j)), β ∈ (0, 1)
y o

(1)

ct ,ct+1

s.t. cyt (j) + st (j) = wt ,(2)

cot+1 = Rt+1 st (j) + πt+1 (j),where Ut (j), cyt (j), cot+1 (j), st (j), and πt+1 (j) represent lifetime utility, consumption
during the ﬁrst period, consumption during the second period, savings, and the proﬁts
that she receives during the second period, respectively; Rt+1 is the gross interest rate,and β is the discount factor. She is assigned to industry j ∈ [0, 1] before she decides
consumption and savings during the ﬁrst period. Thus, the decision accounts for the
proﬁts she generates during the second period.
From her optimal conditions, savings st (j) are expressed by the following equation:st (j) =β
πt+1 (j)wt −.
1+β(1 + β)Rt+1(3)

3 This structure of the production side is similar to that of Zeira (1998), who is the pioneer inthe literature that examines mechanization by using the task-based model.

62.2 Producers
There are two types of producers: ﬁnal goods producers and managers producing
intermediate goods. Final goods producers combine a unit measure of intermediate
goods. Their technology is
Z 1
log Yt =log yt (j)dj,(4)

0

where Y is the amount of ﬁnal goods and y(j) is the amount of the intermediate
good j ∈ [0, 1]. 4 From the proﬁt maximization problem of ﬁnal goods producers, thedemand for each intermediate good y(j) is
yt (j) =Yt,
pt (j)(5)

where pt (j) is the price of intermediate good j.
A manager operating a production unit in industry j produces intermediate goodj. In each industry, a unit measure of managers exists.5 To produce intermediate
goods, managers must choose between non-automation technology and automation
technology. When managers choose non-automation technology, they employ only
labor as their factor of production. On the other hand, when they choose automation
technology, they use only capital as their factor of production. To introduce automation technology, they must pay ﬁxed costs before commencing operations. The choiceof technology is made at the end of the ﬁrst period.
Acemoglu and Restrepo (2018b) assume that labor productivity varies between
tasks. Capital has a comparative advantage when used in tasks with low labor productivity and thus automation technology is adopted in such tasks. In our model, the

4 In the model of Acemoglu and Restrepo (2018b), the range of intermediate goods is from N − 1

and N , where N ≥ 1 and N increases over time. They interpret a change in N as the creationof new tasks. We set N = 1 here, but Section 4.3 develops a model in which the range is fromN − 1 to N and analyzes the eﬀect of an increase in N on the wage.
5 This assumption ensures that the intermediate goods market is competitive, in spite of households’ immovability between industries.

7

ﬁxed costs of introducing automation technology are the source of diﬀerent comparative advantages across industries. Sirkin et al. (2015) specify the components of thecost to introduce a spot welding, an example of automation technology, and estimate
their magnitudes. They state that the price of the robotic hardware and software to
introduce a spot welding robot is only one-quarter of the total cost.6 Some of othercomponents of the cost are ﬁxed costs, such as the cost of programming and integrating a robot into a factory and the cost of safety structures that protect workers androbots themselves. These ﬁxed costs account for two-thirds of the total. The argument that ﬁxed costs inhibit the adoption of new technologies has been the subjectof many theoretical papers (e.g., Jovanovic and Lach, 1989; Hall, 2004).
Because managers gain proﬁts from producing intermediate goods, they choose themore proﬁtable technology as follows:lk
πt+1 (j) = max{πt+1(j), πt+1
(j)},

(6)

l
k
where πt+1(j) and πt+1(j) are the proﬁts from producing intermediate goods by using

non-automation technology and automation technology in industry j, respectively.
When a manager utilizes non-automation technology, she can produce intermediate
goods without incurring any ﬁxed costs. The production function is given by asfollowing:7
ytl (j) = lt (j),(7)

where lt (j) is the amount of labor inputs. The proﬁt maximization problem thenbecomesπtl (j) = max plt (j)ytl (j) − wt lt (j),(8)

lt (j)

6 Spot welding is a type of electric resistance welding used to weld various sheet metal products,

and it achieves a labor-saving in a welding line (Takayama and Takahashi, 2014). It is applied
to production processes, such as the production of automobile and aerospace vehicles (Li andDuarte, 2018).
7 For simplicity, the production function for non-automation technology is linear. The case inwhich the function is non-linear is discussed in section 4.2.

8

where wt represents the wage. Since the production function is linear, the price of
the intermediate goods produced by using non-automation technology is as follows:plt (j) = wt .

(9)Thus, the proﬁt from non-automation technology is equal to zero.
The production function for automation technology is8
ytk (j) = γkt (j)α , α ∈ (0, 1), γ > 0,(10)where kt (j) is the capital used in industry j and γ is the productivity of automation technology. The proﬁt maximization problem for managers choosing automation
technology is
πtk (j) = max pkt (j)ytk (j) − Rt kt (j) − C(j),(11)

kt (j)

where C(j) ≡ C · j, C > 0.

(12)

In the above equation, C(j) represents the ﬁxed cost function required to introduce
automation technology into industry j and is paid for by ﬁnal goods.9 Since ﬁxed costs
are increasing in j, the comparative advantage of labor to capital is also increasing inj.10 The ﬁrst order condition is the following equation:
Rt = pkt (j)γαkt (j)α−1 .

(13)

By substituting this equation into (11), proﬁt becomes as follows:
πtk (j) =

α
1
1−α k
(pt (j)γα) 1−α Rtα−1 − C(j).
α(14)

3 Equilibrium
At equilibrium, the optimal conditions for consumers and producers and market
clearing conditions for each market are satisﬁed. By using (5), (10) and (13), from
8 Rosenfeld et al. (2004) study the productivity of foraging robots. They show that the marginal

productivity is decreasing in the number of robots.
9 Chen and Koebel (2017) estimates ﬁxed costs by industry and show that they are diﬀerent.
10 In the model of Acemoglu and Restrepo (2018b), the productivity of labor depends on theindustry index. Our setting diﬀers, but it is the same in terms of the comparative advantage.

9

the intermediate goods market clearing condition, the price of intermediate good j
produced by using automation technology pkt (j) is given by the following:
Yt1−α Rt α
( ) .
γ
α

pkt (j) =(15)

Substituting this equation into (14), πtk (j) can be rewritten as
πtk (j) = (1 − α)Yt − C(j).

(16)

The ﬁxed costs are increasing in the industry index. This paper focuses on thecase in which a threshold j ∗ ∈ (0, 1) exists and managers in industry j ≤ j ∗ choose
automation technology.
Since the proﬁt from non-automation technology is equal to zero, the return fromautomation technology for industry jt∗ is zero. Thus, by using (16) and (12),jt∗ =(1 − α)Yt
.
C

(17)

αYt
.
Rt

(18)

kt (j)dj,(19)

From (13) and (15),kt (j) =The capital market clearing condition is
Z jt∗
Kt =
0

where Kt represents aggregate capital. Thus, the interest rate is equal toRt =jt∗ αYt
.
Kt

(20)

Yt
.
wt

(21)

From (5), (7), and (9),lt (j) =Since the labor supply is unity, the labor market clearing condition isZ 1
1=
lt (j)dj.

(22)

j∗

Thus, the equation for the wage equals the following:
wt = (1 − jt∗ )Yt .
10

(23)

Combining (18) and (21) with (4), (7) and (10) yield the amount of ﬁnal goods Yt
as follows:  α jt∗ 1−jt∗
Kt
1
Yt = γ.
jt∗
1 − jt∗(24)

By using (17), this equation can be rewritten as  α jt∗ 1−jt∗
Cjt∗
Kt
1
= γ
.
1−α
jt∗
1 − jt∗(25)

Thus, Kt equals

Kt =

Cjt∗
1−α

 αj1∗
t

∗
1−jt

1
∗
(1 − jt∗ ) αjt jt∗ γ − α ≡ Φ(jt∗ ).

(26)

From the ﬁnal goods market clearing condition, savings are used to produce capitalfor the next period. Thus, the equation for capital accumulation is
Z 1
Kt+1 =st (j)dj.

(27)

0

By combining (3), (16), (17), (20), and (23), this equation can be rewritten as follows:
Kt+1 =

2αβC(1 − jt∗ )(jt∗ ) ≡ Ψ(jt∗ ).
(1 − α){2α(1 + β) + (1 − α)}(28)∗
Since Kt+1 = Φ(jt+1) from (26), the equation for capital accumulation is thenconverted to the dynamic equation for j ∗ as follows:∗Φ(jt+1) = Ψ(jt∗ ).

(29)

From this equation, the conditions of the existence and the stability of a unique steadystate with j ∗ ∈ (0, 1) are obtained as the following propositions.11
Proposition 1. When the parameter of the ﬁxed cost function C is suﬃciently low,a unique steady state exists in j ∗ ∈ (0, 1).
11 j ∗ = 1 and j ∗ = 0 do not hold in a steady state. When j ∗ = 1, since workers receive no income,tno capital is left and no manager can choose automation technology in the next period (i.e.,∗
jt+1= 0). When jt∗ = 0, workers make savings Kt+1 > 0. When ﬁxed cost function is (12),k∗π (0) > π l (0) = 0, and thus jt+1
> 0.

11

Proposition 2. Suppose that a unique steady state in j ∗ ∈ (0, 1) exists.12 When automation productivity γ is low enough or C is high enough that the level of automation
diﬀusion j ∗ is less than 1/2, this steady state is globally stable. If j ∗ > 1/2 and C is
suﬃciently low or high, a threshold of j ∗ ∈ (1/2, 1) exists below which the steady stateis locally stable.
In Appendix A, we provide more formal conditions and proofs for Proposition 1
and 2 and illustrate the dynamics graphically. The analysis in this paper focuses only
on the cases in which there exists a unique steady state that is globally and locallystable.

4 Analysis
4.1 Comparative Statics
In this section, we focus on the unique steady state and analyze the long-run eﬀects
on the wage and income shares of an improvement in automation technology productivity, γ, and of the cost-saving technological changes which reduces the parameter ofthe ﬁxed cost function, C. The former eﬀect is summarized in the following proposition. Proofs of the propositions presented in this section are provided in Appendix.B.
Proposition 3. Suppose that Proposition 2 holds and thus a unique and locally stablesteady state exists. If the productivity of automation technology γ is high (low) enoughor if the ﬁxed cost parameter C is low (high) enough that the level of automation
diﬀusion j ∗ is greater (less) than 1/2, an improvement in automation productivity γ
decreases (increases) the wage in the long-run.
This proposition shows that the wage decreases (increases) due to the improvementin the automation productivity when the threshold of a unique steady state j ∗ is

12 Strictly speaking, this proposition requires the additional assumption that C is small enoughthat Φ(j ∗ ) is an increasing function in the steady state as shown in Proposition A.2.

12

greater (less) than 1/2. From (17) and (23), the wage in the steady state is
w = (1 − j ∗ )Y,C ∗j (1 − j ∗ ).
=
1−α

(30)
(31)

Lemma B.1 presented in Appendix.B shows that j ∗ increases as γ rises or C falls.
(31) demonstrates that the level of j ∗ determines the sign of the marginal eﬀect of γon the wage. This proposition shows that when γ is high (low) enough or C is low(high) enough that j ∗ > 1/2 (j ∗ < 1/2), an increase in γ decreases (increases) thewage.
From (30), the wage is the product of the mass of industries employing labor 1 − j ∗multiplied by the amount of ﬁnal goods Y . When j ∗ increases, labor is substituted
for capital in newly automated industries. Thus, this channel decreases labor demand
and the wage. On the other hand, an increase in j ∗ causes an increase in Y from(17), thus pulls up the wage from (30). For automation technology to be introduced
in industries with higher ﬁxed costs, Y needs to be large. When j ∗ > 1/2 (j ∗ < 1/2),
the negative (positive) eﬀect is dominant, thus the wage decreases (increases) withan increase in γ. This result might appear to depend on the assumption that theﬁxed cost function is linear. We discuss how robust the result is when the ﬁxed costfunction is more general.
The following proposition shows the eﬀect of a cost-saving technological change on
the wage.
Proposition 4. Suppose that Proposition 1 holds and thus a unique steady state
exists. If γ is high enough or C is low enough that j ∗ > 1/2, the wage decreases
with a decrease in C. If j ∗ < 1/2, there exists a threshold of C, Cw2 , and the wage
decreases with a decrease in C for any j ∗ when C < Cw2 . When C ≥ Cw2 , the wage
increases with a decrease in C if the level of j ∗ is intermediate in (0, 1/2), while thewage decreases if j ∗ is suﬃcierntly high or suﬃciently low.
This proposition shows the eﬀects of a cost-saving technological change that reduces
the ﬁxed cost of automation technology on the wage. When γ is high enough or C is13low enough that j ∗ > 1/2, a decrease in C reduces the wage. When j ∗ < 1/2, it also
reduces the wage if C < Cw2 . Whereas the indirect eﬀect of a decrease in C throughan increase in j ∗ raises the wage when j ∗ < 1/2, the direct eﬀect reduces the wage
(see (31)). When the level of C is suﬃciently low, the negative direct eﬀect is larger
than the positive indirect eﬀect, and thus the wage decreases with a decrease in C.
By contrast, when j ∗ < 1/2 and C ≥ Cw2 , there exists the region in which the wage
increase by the technological change. The wage increases when j ∗ is intermediatein (0, 1/2), whereas the wage decreases when j ∗ is large enough or small enough in(0, 1/2).
From Proposition 3 and 4, when the technology level is high, that is γ is high enoughor C is low enough that j ∗ > 1/2, the wage decreases with both an improvement of
automation productivity and a cost-saving technological progress. On the other hand,when the technology level is low enough that j ∗ < 1/2, the wage increases with γfrom Proposition 3. By contrast, as for the cost-saving technological progress, except
for the case in which C > Cw2 and j ∗ is intermediate in (0, 1/2), the wage decreases
with a decrease in C from Proposition 4. These propositions suggest that when thetechnology level reaches a high enough state of advancement and thus, the diﬀusionlevel is suﬃciently high, both types of technological progress lower the wage.
While many empirical researches examine the eﬀect of automation diﬀusion on thewage, these studies obtain diﬀerent results. Some researches show that the diﬀusion of
automation increases the wage (e.g., Autor and Salomons, 2017; Graetz and Michaels,
2018). Others, such as Acemoglu and Restrepo (2020a), Dauth et al. (2019) and
Chiacchio et al. (2018), show that the eﬀect of automation on the wage is negativeor ambiguous. Proposition 3 and 4 show that whether the technological progress
promoting automation diﬀusion increases or decreases the wage is determined by thelevel of automation diﬀusion in the economy. Existing researches do not empiricallyexamine the inﬂuence of the diﬀusion level upon the eﬀect of an introduction of new
automation technology on the wage. The results in this paper suggest a new empiricalquestion regarding the relationship between automation diﬀusion and the wage.
The following proposition analyzes the eﬀect of an increase in γ or a decrease in C
14

on the labor income share. In this model, workers and managers exist. The deﬁnitionof the labor income share in SNA includes the income share of managers, so in thenext proposition, the labor income share is the sum of the income shares of both
workers and managers.13
Proposition 5. When Proposition 2 holds and thus a unique and locally stable steadystate exists, the income share of managers increases (decreases), and that of workers
decreases (increases) with γ (C). The total income share of managers and workers
decreases (increases) with γ (C).
The income share of managers increases (decreases), and that of workers decreases(increases) with γ (C). Their total income share decreases (increases) with γ (C). This
result is consistent with many researches, such as Elsby et al. (2013) and Karabarbounis and Neiman (2014) that show a decline in the labor income share in many
countries since the 1980s.14
As Piketty and Saez (2003) and Jones and Kim (2018) suggest, the gap betweenof income share of workers and entrepreneurs has been widening in the US since the1970s.
(Figure 1 around here)
Figure 1 illustrates graphically the eﬀect of a productivity improvement of automation technology on the extent of automation technology diﬀusion, the amount of ﬁnal
goods production, wage, the income share of workers, the income share of managers,
and the total labor income share at the steady state. The parameters are set to be
α = 0.3, β = 0.82, and C = 4.4. The time preference β is a standard value for OLG
model. The parameter value of the production function of automation technology αdoes not change qualitative results. The value of C and the range of γ are chosen to
ensure the unique stable steady state. As shown in Proposition 3, when j ∗ is over 1/2
13 Let C be aggregate ﬁxed cost, and Π be aggregate proﬁt.

The income share of managers andworkers are Π/(Y − C) and w/(Y − C), respectively.
14 The deﬁnition of labor income share in Karabarbounis and Neiman (2014) does not include
the element equivalent to proﬁts in our model.

15

(γ is approximately 5), aggregate capital and the wage decrease with γ. As shown in
Proposition 5, the income share of workers decreases, the income share of managers
increases, and the total income share decreases with γ.

4.2 Why Can Wage Decrease?
In the model of Acemoglu and Restrepo (2018b) and many other task-based models
analyzing mechanization, the wage increases because of technological change when
the technology choice depends upon producers’ decision.1516 By contrast, our modelshows that the wage decreases with an increase in the automation productivity γ
when the level of automation diﬀusion j ∗ is suﬃciently large. This section explains
the mechanisms underlying the decrease in the wage presented in Proposition 3 and4.
We assume that the ﬁxed cost function, which was linear in the previous sectionas (12), is a continuous, diﬀerentiable, and increasing function C(j), C ′ > 0. The
equation determining the technology choice threshold j ∗ is transformed from (17) into
the following:C(j ∗ ) = (1 − α)Y.

(32)

In industry j ∗ , the price of intermediate goods is equal to the average cost because the
proﬁt from automation technology is zero. The price depends on the demand for theintermediate goods, which is proportional to Y from (5). The average cost includes
15 Acemoglu and Restrepo (2018b) suggest the possibility of a decrease in the wage when someproducers cannot utilize automation technology because of a technology frontier. The model
considering the technology frontier will be discussed in Section 5. Berg et al. (2018) also obtaina result similar to that of Acemoglu and Restrepo (2018b) using a diﬀerent model.
16 Acemoglu and Restrepo (2020b) analyze the eﬀect of automation diﬀusion on labor markets,
including the wage by focusing on a skill diﬀerence. Then, they suggest theoretically andempirically that the automation diﬀusion enhances the gap between the wages of low- andhigh-skilled labor and the decline in the wage of low-skilled labor. In their model, low-skilledlabor performs tasks that automation technology is able to replace. Still, high-skilled labor is
not replaced because they are engaged in tasks that automation technology cannot perform.
However, as noted by Autor (2015), the progress of automation tends to encroach upward in
abstract tasks in that high-skilled labor is apt to engage. Thus, the approach of Acemoglu andRestrepo (2020b) has a limitation. This paper proposes a mechanism for the decrease in thewage that does not depend on skill diﬀerences.

16the ﬁxed cost. Thus, as shown in (32), Y is related to the ﬁxed cost in industry j ∗ .
Since the ﬁxed cost function is increasing in j ∗ , an increase in Y enhances automation diﬀusion j ∗ . Intuitively, this is because, as the output of ﬁnal goods increases,
the diﬀerence between the returns from automation and non-automation technology
increases, enabling more industries to pay these ﬁxed costs and thus introduce automation.
The wage in the steady state is
w = (1 − j ∗ )Y,⇔ log w = log(1 − j ∗ ) + log Y.

(33)Thus, the eﬀect of an increase in j ∗ on the logarithm of the wage is1
d log Yd log w
+
=−
.
∗
∗
dj
1−j
dj ∗

(34)

Following Acemoglu and Restrepo (2018b), we call the ﬁrst term the displacementeﬀect, and the second term the productivity eﬀect. The diﬀusion of automation technology captured by an increase in j ∗ displaces labor because the industries in which
automation technology is newly introduced cease employing labor. The displacementeﬀect has a negative eﬀect on the labor demand, and then on the wage. On the other
hand, automation diﬀusion has a positive eﬀect on the wage through the increase in
the output of ﬁnal goods. When the negative eﬀect dominates the positive eﬀect, thewage decreases.
The scale of the productivity eﬀect depends on the rate of change in the ﬁxed costfunction from (32):

From (34) and (35):d log C(j ∗ )d log Y
=
.
dj ∗
dj ∗

(35)

1
d log wd log C(j ∗ )
=
−
.
+
dj ∗
1 − j∗
dj ∗(36)

The rate of change of the ﬁxed cost function determines whether the wage increases
or decreases with the automation diﬀusion. When the rate of change is small enough,the wage decreases since the negative displacement eﬀect dominates the positive productivity eﬀect. In the model presented in Section 2, d log w/dj ∗ < 0 when j ∗ > 1/2
17because of the linear ﬁxed cost function. The condition of linearity can be relaxed.
For example, when the ﬁxed cost function is an exponential function C(j) = exp(a·j),
where a > 1, d log w/dj ∗ < 0 when j ∗ > 1 − 1/a. Even when limj→1 C(j) = ∞, thewage can decrease when j ∗ is high enough. Let us consider the case of the following
function.
C(j) =

exp(a1 j + a3 ), where 0 < 1 − a2 < a1 , a2 > 0.
(1 − j)a2

This function satisﬁes that limj→1 C(j) = ∞ and limj→1 C ′ (j)/C(j) = ∞. In thiscase, d log w/dj ∗ < 0 when j ∗ > (a1 +a2 −1)/a1 . Therefore, the results of Proposition
3 and 4 hold even if the ﬁxed cost function increases with j much more rapidly thanthe linear function. For simplicity, we assume the linear function is speciﬁed as (12)except for this subsection.

4.3 Eﬀects of New Industries
Acemoglu and Restrepo (2018b) point out that the progress of automation technology creates new tasks and show theoretically that this task creation, which increases
the tasks performed by non-automation technology, stimulates labor demand and increases the wage. This subsection examines the eﬀect of the creation of new industries
on wages. In Acemoglu and Restrepo (2018b), the displacement of labor by capital
due to the diﬀusion of automation technology is viewed as the factor that reduces thewage, and the task creation is viewed as the countervailing factor that increases thewage. By contrast, in this paper, task creation works as a factor reducing the wage.
In this subsection, to introduce new industries, the range of intermediate goodindustries is modiﬁed to j ∈ [N − 1, N ]. The technology of ﬁnal goods is rewritten as
Z N
log Yt =N −1

log yt (j)dj,

where N indicates the number of industries. An increase in N is interpreted as thecreation of new industries. This setup is the same as in Acemoglu and Restrepo

18

(2018b). (23), (26) and (28) are rewritten as:
w = (N − jt∗ )Yt ,1∗
 ∗  α(j∗ −(NN −jt+1
−1))
Cjt+1
1
t+1
α(j ∗
−(N −1))N ∗∗∗Φ (jt+1 ) =(1 − jt+1 ) t+1(jt+1− (N − 1))γ − α
1−α


∗
− N + 1 −1
1 − α jt+1βC
N ∗ ∗(N − jt∗ )jt∗
Ψ (jt , jt+1 ) = 1 +∗2α(1 + β)
jt+1(1 + β)(1 − α)From these equations, let us numerically demonstrate the eﬀects of new industries
on the steady state. The values of α, β, and C are the same as before, and thevalue of γ is 4.65. The range of N is set to (1, 1.15) and the parameter values ensure
the existence of a stable equilibrium. As N increases, the ﬁxed cost for automation
technology increases.17(Figure 2 around here)
Figure 2 demonstrates how each variable in the steady state depends on N . Thisﬁgure shows that an increase in N leads to a decrease in j ∗ , Y, and w. As in theprevious subsections, the eﬀect on the wage is separated into two channels; the eﬀect
through 1 − j ∗ and the eﬀect through Y . The eﬀect of the former on the wage
is positive because an increase in N and the resulting decrease in j ∗ stimulate labordemand. However, the eﬀect of the latter on the wage is negative because the decreasein j ∗ leads to a decrease in Y , and is stronger than the eﬀect of the former. Thus, inthis model, the creation of new industries typically leads to a decrease in wages. 18

17 When N is suﬃciently large, ﬁxed costs are too high to introduce the automation technology

even for managers in industry N − 1, and thus no one uses this technology. In this case, theequilibrium conditions do not hold.
18 When γ is large, the decrease in Y is relatively gentle, and the negative eﬀect is weak. Thus,
the total eﬀect of N on the wage becomes positive when N is small. However, an increase in
N magniﬁes the decrease in Y and the corresponding negative eﬀect. Therefore, even in thiscase, the wage decreases with N when N is not small.

19

5 Automation Technology Frontier
In the model in the previous section, automation technology is potentially applied
to all industries. However, some industries may not adopt automation because thetechnology to automate them does not exist. Thus, this section considers the case inwhich automated industries are limited to industry j ≤ j < j ∗ , where j is the upperbound for automated industries. Managers in j ∈ (j, j ∗ ] who want to introduce
automation cannot do so because of this technological limitation. Therefore, thissection examines the eﬀects on the wage of the technological changes, in particular, an
improvement in automation productivity γ and an incremental change in automation
frontier j. The assumption regarding j is similar to that in the model of Acemoglu
and Restrepo (2018b). The diﬀerences between their model and the present one arehighlighted in this analysis.
In this section, the range of intermediate goods industries is the same as in Section
2, j ∈ [0, 1]. Managers in j ∈ [0, j] choose automation technology. Thus, the grossinterest rate (20), the wage (23), and the ﬁnal goods (24) are rewritten as follows:
αYt
· j̄,
Kt
wt = (1 − j)Yt ,  α  j 
1−j
Kt
1
Yt = γ
.
j
1−j

Rt =

(37)
(38)
(39)

From (37), (38), savings (3), and the proﬁts from automation technology (16), thecapital accumulation equation (27) becomes


β
Kt+1
Cj
Kt+1 =
j(1 − j)Yt −(1 − α) −.
1+βα(1 + β)
2Yt+1

(40)

At ﬁrst, the following proposition examines the eﬀect of γ on the wage.
Proposition 6. When a technological frontier j < j ∗ exists, the eﬀect of an increasein γ on the wage is positive in the long run.
The proof of this proposition is provided in Appendix B. This proposition shows
20

that an improvement in automation technology γ increases the wage. Since an increasein γ does not inﬂuence the number of industries using automation technology, thedisplacement eﬀect does not emerge, and the productivity eﬀect increases the wage.
Let us consider the eﬀect of an incremental change in the automation frontier on
the wage. This eﬀect is represented as the following equation from (38):dw
dY= −Y + (1 − j)
.
dj
dj

(41)

As in the previous section, we call the ﬁrst term the displacement eﬀect and thesecond term the productivity eﬀect.
To analyze the productivity eﬀect, we diﬀerentiate (39) with respect to j:
∂Y dK
∂Y
dY
=
+
,
∂K dj
dj
∂j
 α


Y dK
K
= αj
+ Y (1 − α) + log γ(1 − j) .
K dj
j

(42)

The ﬁrst term of (42) reﬂects the indirect eﬀect through capital accumulation, andthe second term reﬂects the direct eﬀect. From (40), the eﬀect of an increase in j on
capital accumulation is
dK
=
dj




1−α
Cj
−
1+
α(1 + β) 2(1 + β)αY

−1

CK
β
×
(1 − 2j)Y ++1+β2(1 + β)αY



β
1
CKj
j(1 − j) −1+β
Y 2(1 + β)αY



dY
dj


.
(43)

Equations (42) and (43) determine dK/dj and dY /dj. Because the signs of them are
not analytically clear, we conduct a numerical analysis.
(Figure 3 around here)
Figure 3 illustrates how the wage depends on γ and j.19 The x-axis is γ, the y-axis
is j, and the z-axis is the wage. The parameters are α = 0.3, β = 0.82, and C = 4.4.
The ﬁgure shows that when the level of j is suﬃciently high (low), the eﬀect of an
19 This ﬁgure focuses on the ranges of γ and j assuring the existence of a unique steady state.

21

increase in j is negative (positive) when γ is suﬃciently low or high. In the model of
Acemoglu and Restrepo (2018b), the wage always increases with j. By contrast, thewage decreases in some ranges of γ and j in this model.
(Figure 4 around here)
To understand why the wage decreases with j when j is suﬃciently large, we focus
on the productivity eﬀect. As previously described, the ﬁrst term of (42) corresponds
to the indirect eﬀect through capital accumulation, and the second term reﬂects the
direct eﬀect. Figure 4 shows how the amount of aggregate capital K depends on γ
and j. Since K is increasing in j from this numerical analysis, the indirect eﬀect ispositive, which is identical to the result in Acemoglu and Restrepo (2018a).
By using (37) and (38), the direct eﬀect is rewritten as follows:



1
∂Y
γ
=
log − log α + α log α − (1 − α) log Y + 1 − α Y.
w
R
∂j

(44)

The part depending on eﬀective factor prices is the eﬀect pointed out by Acemoglu
and Restrepo (2018a) and many other models based on their work. The remaining
part of (44) emerges because the aggregate ﬁnal goods production function (39) isdecreasing return to scale (when α = 1, the additional part disappears), and thus thispart decreases with Y .
In Acemoglu and Restrepo (2018b), the negative displacement eﬀect is always
dominated by the positive productivity eﬀect because of the indirect eﬀect throughcapital accumulation. In this model, the wage could decrease with j, as shown in
Figure 3, which means that the sum of the displacement eﬀect and the productivity
eﬀect could be negative. Because the displacement eﬀect is the same as Acemoglu
and Restrepo (2018b), the productivity eﬀect is smaller than their model and evenmay be negative. One of the reasons for this diﬀerence is the decreasing return toscale of the ﬁnal good production function.

6 Subsidy Policy and Taxation
Sirkin et al. (2015) suggest that a high initial cost prevents the introduction of22automation technology. This section introduce a subsidy to ﬁxed costs ﬁnanced by
tax on young households that encourage the diﬀusion of automation technology, andexamines the eﬀects of such a policy. Since this model utilizes OLG setting, overaccumulation of capital might occur. A subsidy on the introduction of automation
technology might possibly increase aggregate output and ultimately lead to an improvement in overall economic welfare. Let us consider a subsidy that compensatesa certain percentage of ﬁxed costs and that is ﬁnanced by a lump-sum tax on young
households.
This policy changes the budget constraint for young households such thatct (j) = wt − st (j) − τt ,(45)

where τ represents the lump-sum tax. The proﬁt from automation technology is
πtk (j) = max pt (j)ytk (j) − Rt kt (j) − (1 − σ)C(j),(46)

kt (j)

where σ ∈ [0, 1] is the subsidy rate. A policymaker’s budget constraint is
Z jt∗
τt =

σC(j)dj.

(47)

0This section numerically examines the eﬀects of this subsidy. The parameters are
the same as in the previous numerical example except for γ (γ = 4.65). The valuesof γ and C ensure that a unique steady state exists.
(Figure 5 around here)
Figure 5 illustrates the eﬀects of the subsidy on the wage, the ﬁnal good production
Y , the extent of automation technology diﬀusion j ∗ , aggregate capital K, interest rateR, and economic welfare at the steady state. Welfare is measured by the aggregationof each household’s lifetime utility. Figure 5 shows that this subsidy encourages the
adoption of automation technology because it reduces the associated ﬁxed costs. Thus,
the automation technology diﬀusion level increases when the subsidy rate increases.
Automation diﬀusion leads to a decline in 1 − j ∗ , but an increase in Y . Therefore,
as in the previous section, this subsidy has both positive and negative eﬀects on the23

wage. Figure 5 shows that the negative eﬀect dominates the positive eﬀect, and thus
the subsidy decreases the wage.20 Aggregate capital K declines because the increasein tax and the decrease in the wage depress savings. R increases with σ because thedecrease in the supply of capital caused by the decrease in savings and the increase in
the demand for capital resulting from the increases in j ∗ and Y have negative eﬀects
on R.
Figure 5 also shows that as the subsidy rate σ increases, welfare increases at ﬁrst;
however, it subsequently decreases. The welfare of each household is determined by
lifetime income and R. The lifetime income is
w t − τt +

i
πt+1
(j)
, f or i = l, k, j ∈ [0, 1],
Rt+1

ℓ
k
where πt+1(j) = 0, πt+1(j) = (1 − α)Yt+1 − (1 − σ)C(j).

(48)
(49)

Increases in R and τ and a decrease in w depress the lifetime income, and an increasein the proﬁts net of the payment of ﬁxed costs driven by increases in Y and σ increases
the lifetime income. The increase (decrease) in the lifetime income pulls up (pushes
down) the welfare. R has not only an indirect eﬀect through lifetime income, butalso a direct eﬀect on the welfare because R increases the value of savings. Whenthe positive eﬀects are larger (smaller) than the negative eﬀects, welfare increases(decreases). Figure 5 shows that average welfare increases when σ is low and decreases
when σ is high. The reason for the increase in welfare for a low σ may be that thepolicy solves over-accumulation of capital by imposing tax, which decreases K and
increases R.
In this model, there are diﬀerent types of households. Some households may notreceive beneﬁts from the subsidy policy, particularly those households that do notchoose automation technology in response to the subsidy policy because of the highﬁxed costs (thereafter called Non-automation type). These households receive no direct gain from the policy. In addition, households that face zero ﬁxed costs (thereafter
called Zero ﬁxed cost type) also receive no direct gains.
20 If we set a diﬀerent parameter set, for example, a larger γ, the positive eﬀect becomes stronger,

and a region in which the wage increases with σ emerges.

24Let us consider the dynamics of their welfare to examine whether there exist subsidy rates that beneﬁt every household of every generation. At ﬁrst, the economy isin a steady state, and the subsidy policy is unexpectedly enforced in Period 3.
(Figure 6 around here)(Figure 7 around here)
Figure 6 illustrates the dynamics of the welfare of households who are old in periodt. The upper left panel illustrates the dynamics of the average welfare of households.
The upper right panel illustrates the welfare of the Non-automation type, and thelower left panel illustrates the welfare of the Zero ﬁxed cost type. In each panel, theblue line (round), green line (square), and red line (diamond) indicate the dynamics
when the subsidy rate, σ, is 0.01, 0.04, and 0.06, respectively. The black (dotted)
line in each panel indicates the welfare level in the steady state without the subsidy
policy. Figure 7 demonstrates the dynamics of the lifetime income of each type ofhousehold examined in Figure 6 and of the interest rate. The blue (round), green(square), and red (diamond) lines indicate the dynamics when σ is 0.01, 0.04, and0.06, respectively. The black (dotted) lines indicate the levels without the policy.
The upper left panel of Figure 6 shows that the subsidy policy improves average
welfare compared with the steady state economy without the subsidy policy at any
time. However, not all households may beneﬁt from the policy. Figure 6 focuses on
two types of households — the Non-automation type and the Zero ﬁxed cost type —that receive no direct gain from the policy.21
The upper right panel of Figure 6 illustrates that the welfare of the Non-automation
type households increases in Period 4, and that their welfare in subsequent periods isgreater than their welfare without the policy.22 They pay the tax during the young
21 Households with positive ﬁxed costs that introduce automation technology obtain greater ben-eﬁts than the two types of households.
22 Welfare of old households of the Non-automation type and the Zero ﬁxed cost type in Period

3, in which the policy is implemented, are the same as in previous periods. Figure 7 showsthat R and their lifetime income in Period 3 are the same as in previous periods. The subsidy
policy does not inﬂuence R in Period 3 since the supply of capital depends on savings in Period

25period and receive no subsidy. From Figure 7, their lifetime income after the policy
is lower in all cases than in the steady state without the policy. However, since R
after the policy is greater than the level without the policy in Figure 7, their income
during the old period, funded by savings, is higher. The positive eﬀect is dominant,and their welfare improves.
The lower left panel of Figure 6 shows that the welfare of the Zero ﬁxed cost type
households after Period 5 is greater than their welfare without the policy, while theirwelfare in Period 4 is lower when σ is 0.01 and greater when σ is 0.04 and 0.06 than
the welfare without the policy.
Since households that introduce automation technology by paying positive ﬁxed
costs receive the subsidy directly, they gain greater beneﬁts from the policy than the
Non-automation type and the Zero ﬁxed cost type households. Thus, when σ are 0.04
and 0.06, all households of all generations beneﬁt from the policy.23 However, when σ
is suﬃciently high, an increase in σ decreases the average welfare; consequently, many
households do not beneﬁt from a marginal increase in the subsidy rate.

7 Conclusion
This paper investigates the eﬀects of ﬁxed costs on automation technology diﬀusionand examines the relationship between the diﬀusion and the wage. In this model,
households choose either automation or non-automation technology for production;
when the households choose the automation technology, they pay ﬁxed costs. Thus,
their choice determines the extent of automation technology diﬀusion and ﬁxed costs
play an essential role in their decision.
2, and the demand for capital depends on the technology choice made at the end of Period 2.
Lifetime income of these households do not change because their proﬁts in Period 3 are notinﬂuenced by the policy, i.e., π = 0 for the Non-automation type and π = (1 − α)Y for the Zeroﬁxed cost type, where Y is determined by K and j ∗ Thus, while the subsidy policy increases
the lifetime income of old households with positive ﬁxed costs in Period 3, who introduce
automation technology, it does not aﬀect the lifetime income of the two types examined in
Figure 7 in Period 3.
23 When σ is from about 0.032 to 0.068, the welfare of every household of every generation
increases by the policy. When σ is over 0.068, some households suﬀer welfare loss.

26

The analysis in this paper yields two notable results. First, the wage can decrease
with an improvement in automation productivity or decreases in ﬁxed costs, both ofwhich accelerate automation diﬀusion. The diﬀusion has both positive and negativeeﬀects on the wage. The positive (negative) eﬀect dominates in the long-run whenthe extent of automation technology diﬀusion is low (high) enough. In Acemoglu andRestrepo (2018b), the wage increases in the long-run because of the positive eﬀect
through capital accumulation. By contrast, in this study, the wage decreases when
the level of diﬀusion is suﬃciently high, even in the presence of capital accumulation.
Second, the labor income share, which is the sum of the income share of workers
and managers, decreases with automation diﬀusion. This result is consistent with therecent empirical studies that point out a declining labor income share over the last
several decades (Karabarbounis and Neiman, 2014; Grossman et al., 2017; AlvarezCuadrado et al., 2018).
Subsequently, this paper examines two extensions of this model. First, we introduce
an automation technology frontier. This setting is the same as that in Acemoglu andRestrepo (2018b). Because of the technology frontier, automation is not available
to some industries. In contrast to Acemoglu and Restrepo (2018b), in which an
expansion of the frontier always increases the wage in the long-run, in this study, it
can decrease the wage in the long-run. Second, this paper explores the long- andshort-run eﬀects of a subsidy policy for ﬁxed costs on the economy. The analysis
demonstrates that such a subsidy policy has both positive and negative eﬀects on
the long-run wage and welfare. This paper shows that there exist subsidy rates that
improve the welfare of all households of all generations.

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30

Appendix A. Existence and Stability of Unique Steady StateIn the main part of this paper, Proposition 1 shows the existence of a unique steadystate, and Proposition 2 examines its stability. In this section, we show the conditionof the existence and the stability in a more formal way. Proposition A.1 corresponds
to Proposition 1, and Proposition A.3 and A.4 corresponds to Proposition 2.

A.1 Existence of Unique Steady StateIn the main part of the paper, the two types of capital dynamics are described as:
Kt = Φ(jt∗ )(A1)

Kt+1 = Ψ(jt∗ )
 1

∗
1−jt
1
Cjt∗ αjt∗∗ αjt∗ ∗ − α∗(1 − jt )where Φ(jt ) =
jt γ
1−α
2αβC
(1 − jt∗ )(jt∗ ).
Ψ(j ∗ ) =(1 − α){2α(1 + β) + (1 − α)}

(A2)
(A3)
(A4)

Let us express these dynamics in logarithmic form.
∗

1−jt+1
∗
1
Cjt+1
1
∗
αj ∗
∗
∗
∗
∗
t+1
ϕ(jt+1 ) ≡ log Φ(jt+1 ) = log()
(1 − jt+1 ) αjt+1 jt+1
γ− α ,1−α∗∗ψ(jt ) ≡ log Ψ(jt ) = log AC(1 − jt∗ )(jt∗ ),2αβ,(1 − α){2α(1 + β) + (1 − α)}


1
C
∗
∗
lim ϕ(jt+1 ) = −∞, lim ϕ(jt+1 ) = log
,
j→0
j→1
α
γ(1 − α)lim ψ(jt∗ ) = −∞, lim ψ(jt∗ ) = −∞.
where

j→0

A≡

j→1

(A5)
(A6)
(A7)
(A8)
(A9)

∗
Taking the limits of ϕ(jt+1) and ψ(jt ) to zero and one; For simplicity, we ignore thetime scripts. The gap between ϕ and ψ isϕ(j ∗ ) − ψ(j ∗ ) =


1
C
α
∗
∗
∗
∗
log(
) − j log(γ(AC) ) + log j + (1 − j (1 + α)) log(1 − j ) . (A10)j∗α1−αThus, limj ∗ →0 ϕ(j ∗ ) − ψ(j ∗ ) = −∞ and limj ∗ →1 ϕ(j ∗ ) − ψ(j ∗ ) = ∞31

Therefore, if the slope of ϕ is higher than ψ in [0, 1], the ﬁxed point is unique. The
gap of the slope between ϕ and ψ is∂ϕ(j ∗ ) ∂ψ(j ∗ )
1
1−α
∗
∗
∗ 2
−
=
) + 1 − j + (j ) α(1 − j ) log( ∗
∂j ∗
∂j ∗
α(j ∗ )2 (1 − j ∗ )Cj (1 − j ∗ )
(A11)
1
≡
Q(j ∗ )∗
2∗α(j ) (1 − j )The limits of Q(j ∗ ) are:lim Q(j ∗ ) = ∞,(A12)

lim Q(j ∗ ) = α.

(A13)j ∗ →0j ∗ →1

We take the diﬀerential with respect to j ∗ ,Cj ∗ (1 − j ∗ )1∂Q(j ∗ )= Q′ (j ∗ ) = log(
) + 2(1 + α)j ∗ − ∗ .
∂j
1−α
j

(A14)

And we diﬀerentiate Q(j ∗ ) with respect to j ∗ one more time,∂ 2 Q(j ∗ )−2(1 + α)(j ∗ )3 + 2α(j ∗ )2 + 1=.
∂(j ∗ )2(j ∗ )2 (1 − j ∗ )(A15)(Figure A.1 around here)
Let Qn (j ∗ ) be the numerator of the above equation. Since the denominator ispositive, the sign of Qn (j ∗ ) determines the sign of Q′′ (j ∗ ). The limits of Qn (j ∗ ) arelimj ∗ →0 Qn (j ∗ ) = 1 and limj ∗ →1 Qn (j ∗ ) = −1. Because Q′n (j ∗ ) = [−6(1 + α)j ∗ +4α]j ∗ , the smaller extreme point of Qn (j ∗ ) is zero and its other extreme point is in[0, 1]. Thus, Qn (j ∗ ) = 0 has one solution in [0, 1] and Qn (j ∗ ) > 0 for smaller j ∗ andQn (j ∗ ) < 0 for larger j ∗ (see Figure A.1 (a)). Therefore, Q′ (j ∗ ) is an inverse U-shaped
function (see Figure A.1 (b1) and (b2)). When the maximized value of Q′ (j ∗ ) in [0, 1]
is negative or when the value of Q(j ∗ ) evaluated at the smaller solution for Q′ (j ∗ ) = 0
is positive, the slope of ϕ is higher than ψ and a unique steady state exist (see ﬁgures
except (a) in Figure A.1).
The above discussion describes the sketch of the following proposition. It shows
the condition of the existence of a unique steady state in a more formal way than
Proposition 1.
32

Proposition A.1.
(1) If the following condition holds, the maximized value of Q′ (j ∗ ) in [0, 1] is negative,Q(j ∗ ) > 0 and a unique steady state exists:
C < C1 ,(A16)[−2(1+α)j s1 + 1 ]j s1(1 − α)ewhere C1 =,j s1 [1 − j s1 ]
!
r
r
√
√
1
1
+
α
1
+
α
α
3
3
ν+
ν+ ν−
ν+
j s1 =
,
1+α
2
2
3

ν=

(1 + α)2
α3
+
.
4
27

(2) If the following condition holds, the maximized value of Q′ (j ∗ ) in [0, 1] is positive
and the value of Q(j ∗ ) evaluated at the smaller solution for Q′ (j ∗ ) = 0 is positive,Q(j ∗ ) > 0 and a unique steady state exists:
C1 < C < C 2 ,

(A17)

1
[−2(1+α)j s2 + j s2]

(1 − α)ewhere C2 =j s2 [1 − j s2 ]

, and

(A18)

j s2 is the unique solution of the following equation in [0, 1]:−(1 + α)(j s2 )3 + 2α(j s2 )2 + 2(j s2 ) − 1 = 0.

(A19)

Proof of Proposition A.1.
(1) This case is illustrated in the upper row of Figure A.1. As shown in the paragraphjust below (A15 ), Qn (j ∗ ) = 0 has one solution j s1 in [0, 1]. By applying Cardano
formula, we obtain the value of j s1 . (A16 ) is derived from Q′ (j s1 ) < 0.
(2) This case is illustrated in the lower row of Figure A.1. When Q′ (j s1 ) > 0 andthus C > C1 , Q′ (j ∗ ) = 0 has two solutions (Figure A.1 (b2)). Let j m be the smaller33

solution. When Q(j m ) > 0, Q(j ∗ ) > 0 for any j ∗ ∈ [0, 1] and a unique steady state
exists in [0, 1]. Since Q′ (j m ) = 0, from (A14 ),log C = log

1−αj m (1 − j m )− 2(1 + α)j m +

1
.
jm

(A20)

By using (A20 ), the condition for the uniqueness of the solution is:


1−α
m
m
m
Q(j ) = (1 − j ) log( m)+1−j+ (j m )2 α > 0
Cj (1 − j m )⇔ −(1 + α)(j m )3 + 2α(j m )2 + 2j m − 1 > 0.

(A21)

Let QL (j) be the left-hand side of (A21 ). The values of QL (j) at endpoints are
QL (0) = −1 and QL (1) = α. Because Q′′L (j) = −6(1 + α)j + 4α, the inﬂection pointj = 4α/6(1 + α) is in (0, 1). Thus, since QL (j) is a cubic function, QL (j) = 0 has
one solution in (0, 1). Let the solution j s2 , and thus Q(j m ) = 0 when j m = j s2 .
Therefore, Q(j m ) > Q(j s2 ) = 0 is the condition of the existence of a unique steadystate from (A21 ).
Then, we show the existence of C2 , which is the level of the ﬁxed cost satisfying
Q(j m ) = 0. When C > C1 , the graphs are illustrated as the lower row in Figure
A.1. From (A11 ), Q(j ∗ ) shifts down as C increases since dQ(j ∗ )/dC < 0, andlimC→∞ Q(j ∗ ) = −∞. Thus, there exists C2 such that Q(j m ) = 0. Therefore, when
C1 < C < C2 , as illustrated by Figure A.1 (c2), Q(j m ) > 0 and there exists a uniquesteady state.24 (A18 ) is derived from (A20 ) when Q(j m ) = 0.

A.2 Stability of Unique Steady StateIn this section, we conﬁrm the stability of the dynamic system of this model. The
steady state level of jt∗ is denoted by j ∗ below. We divide two cases in terms of j ∗ :j ∗ ≤ 1/2 and j ∗ > 1/2. For the former case, the global stability holds as shown below
by Proposition A.3. For the latter case, the local stability holds when j ∗ is small
24 We also examine the condition of existence of a unique steady state when C > C , and shows2that a unique equilibrium or multiple equilibria occur depending on the parameters. This
paper focuses on the case of a unique equilibrium for simplicity.

34

enough. Proposition A.4 shows conditions under which there exists a threshold levelof j ∗ such that the dynamic system is locally stable (unstable) when j ∗ is smaller(larger) than the threshold. In this section, we assume that a unique steady state
exists as shown by Proposition A.1 and that ϕ is a monotonically increasing function.
At ﬁrst, we derive the condition of the monotonicity, then show the stability of the
dynamic system.
A.2.1. The Monotonicity of ϕ
To examine the stability of the dynamic system, ϕ and ψ, we derive the condition
to assure that ϕ is a monotonically increasing function. From (A5 ), if diﬀerentiate∗∗,
ϕ(jt+1) with respect to jt+1
 


∗
)
∂ϕ(jt+1
11−α∗+ 1 − (1 − α)jt+1 .
=∗ )2 log Cj ∗ (1 − j ∗ )∗∂jt+1
α(jt+1
t+1
t+1

(A22)

∗
∗
We deﬁne the bracket of this equation as ϕ̃(jt+1). We diﬀerentiate ϕ̃(jt+1) with∗respect to jt+1,∗
∂ ϕ̃(jt+1
)
1
∗
∗
2
(1
−
α)(j
)
+
(1
+
α)j
−
1
.
=
t+1
t+1
∗
∗ (1 − j ∗ )
∂jt+1
jt+1
t+1

Let j s be the value in [0, 1] when the bracket is equal to zero. Then,p−(1 + α) + (1 + α)2 + 4(1 − α)sj =.
2(1 − α)

(A23)

(A24)

∗
∗
∗
∗
For jt+1< j s , ∂ ϕ̃(jt+1)/∂jt+1< 0 and ϕ̃(jt+1) is decreasing function, and for
∗
∗
∗
∗
jt+1
> j s , ϕ̃(jt+1) is an increasing function. Thus, ϕ̃(jt+1) is minimized at jt+1
= js.
∗
∗
∗
Therefore, when ϕ̃(j s ) > 0, ∂ϕ(jt+1)/∂jt+1> 0 and ϕ(jt+1) increases monotonically.

From (A22 ), ϕ̃(j s ) > 0 corresponds to the following condition;(1 − α)e1−(1−α)jC<j s (1 − j s )

s

(A25)

From the above discussion, the following proposition is obtained:
Proposition A.2.
1−(1−α)j
If C < (1−α)esj (1−j s )ss, where j =−(1+α)+√(1+α)2 +4(1−α), ϕ increases monotonically2(1−α)

in [0, 1].
35

A.2.2. The Global Stability When j ∗ ≤ 1/2
We show the global stability when j ∗ ≤ 1/2. Suppose that Proposition A.1 andA.2 hold. From (A1 ) and (A5 ) and the monotonicity of ϕ,∗
jt+1= ϕ−1 [log Kt+1 ].

(A26)

∗
The dynamic system satisﬁes ϕ(jt+1) = ψ(jt∗ ) from (29). Thus, by substituting(A2 ) into (A26 ),∗jt+1= ϕ−1 [ψ(jt∗ )] ≡ h(jt∗ ).

(A27)

We examine the function h(jt∗ ). The derivative of h(jt∗ ) isdh(jt∗ )dψ(jt∗ )/djt∗=∗ )/dj ∗djt∗
dϕ(jt+1
t+1(A28)

Whether h(jt∗ ) is an increasing function or a decreasing function depends on the∗∗) is an inderivatives of ϕ(jt+1) and ψ(jt∗ ). Because Proposition A.2 holds, ϕ(jt+1

creasing function. ψ(j ∗ ) is increasing for jt∗ ≤ 1/2 and decreasing for jt∗ > 1/2.
Hence, h(j ∗ ) is increasing for jt∗ ≤ 1/2 and decreasing for jt∗ > 1/2 from (A28 ).
Moreover, the following two condition hold:limh(jt∗ ) = 0 and limh′ (jt∗ ) = ∞.
∗
∗

jt →0

jt →0

(A29)

The latter holds because limjt∗ →0 dψ(jt∗ )/djt∗ = ∞ from (A6 ) and ϕ is an increasing
function. The reason of the former is that limjt∗ →0 ψ(jt∗ ) = −∞ from (A9 ), and∗∗limψ(jt∗ )→−∞ ϕ−1 [ψ(jt∗ )] = 0 because limjt+1→0 ϕ(jt+1 ) = −∞ from (A8 ) and ϕ is anincreasing function.
Figure A.2 illustrate h(jt∗ ) and patterns of convergence. The upper two panels in∗∗
Figure A.2 illustrate that jt+1
> jt∗ (jt+1< jt∗ ) when jt∗ < j ∗ ≤ 1/2 (j ∗ < jt∗ ≤ 1/2),

and jt∗ converges to j ∗ . The lower two panels illustrate that when jt∗ > 1/2 and
∗
∗
∗
∗
∗
∗
jt+1
≥ j ∗ , j ∗ < jt+2
< jt+1< 1/2, and when jt∗ > 1/2 and jt+1
< j ∗ , jt+1
< jt+2<j ∗ . Therefore, when j ∗ ≤ 1/2, this dynamic system is globally stable.
From the above discussion, the following proposition is obtained:
Proposition A.3. Suppose that Proposition A.1 and A.2 hold. Then, if j ∗ ≤ 1/2,
the steady state is globally stable.
36

(Figure A.2 around here)

A.2.3. The Local Stability When j ∗ > 1/2
Then, we consider the case of j ∗ > 1/2. Suppsoe that Proposition A.1 and A.2
hold. From (A5 ), (A6 ) and (A28 ),
∗
djt+1
=
djt∗

nh

1

∗
α(jt+1
)2

−

h

∗
Cjt+1

1 − log 1−α1
11−jt∗ − jt∗ii∗ ) − (1 − α)j ∗− log(1 − jt+1
t+1

o.

(A30)

To assure local stability, the following condition is necessary:∗
djt+1
djt∗ j ∗ =j ∗tt+1 =j∗< 1 ⇔ −1 < −−dψ(j ∗ )/dj ∗< 1.
dϕ(j ∗ )/dj ∗(A31)

When Proposition A.2 holds, dϕ/dj ∗ > 0. Thus, (A31 ) is equivalent to:dϕ(j ∗ ) dψ(j ∗ )+> 0 anddj ∗dj ∗dϕ(j ∗ ) dψ(j ∗ )
−
>0
dj ∗
dj ∗

(A32)

Evaluating the conditions above at a steady state, dϕ(j ∗ )/dj ∗ −dψ(j ∗ )/dj ∗ is equivalent to (A11 ). Thus, the latter condition is satisﬁed when Q(j) > 0 for any j ∈ [0, 1].
Because Proposition A.1 holds, Q > 0, and thus the latter condition is satisﬁed.
The former condition of (A32 ) is:dϕ(j ∗ ) dψ(jt∗ )1+=
Z(j ∗ ) > 0,
(A33)
∗
∗
∗
2
∗
dj
dj
α(j ) (1 − j )


1−α
∗
∗
∗
) + 1 − j + 2αj ∗ (1 − j ∗ ) − α(j ∗ )2 .
where Z(j ) ≡ (1 − j ) log( ∗Cj (1 − j ∗ )(A34)
When j ∗ = 1/2, Q(j ∗ ) = Z(j ∗ ) from(A11 ). This relationship implies thatdϕ(j ∗ ) dψ(j ∗ )dϕ(j ∗ ) dψ(j ∗ )
+
=
−
.
dj ∗
dj ∗
dj ∗
dj ∗
j ∗ =1/2
j ∗ =1/2
∗

(A35)

∗

)
dψ(j )
Because Q(j ∗ ) > 0 for any j ∗ when Proposition A.1 hold, dϕ(j> 0 from
dj ∗ − dj ∗∗∗)dψ(j )(A11 ). Hence, dϕ(j> 0 around j ∗ = 1/2, and thus the dynamic system
dj ∗ + dj ∗is locally stable. Because limj ∗ →1 Z(j ∗ ) = −α, the steady state is unstable aroundj ∗ = 1. This means that both stable areas and unstable areas exist in j ∗ ∈ (1/2, 1].
37

We derive the suﬃcient conditions that there exists a unique threshold, which divides
the stable area with the unstable area. Before presenting a proposition, we show thefollowing lemma.
Lemma A.1. Suppose that Proposition A.1 and A.2 hold. Then, in j ∗ ∈ (1/2, 1],
Z ′′ (j ∗ ) = 0 has only one solution j z1 , and Z ′′ is positive for j ∗ < j z1 and negative
for j ∗ > j z1 .
Proof of Lemma A.1. From (A34 ), the second derivative of Z is:2(3α − 1)(j ∗ )3 − 6α(j ∗ )2 + 1∂ 2 Z(j ∗ )=.
∂(j ∗ )2(j ∗ )2 (1 − j ∗ )(A36)

The numerator determines the sign of Z ′′ . When j ∗ = 0 and j ∗ = 2α/(3α − 1), it
achieves extreme values. Taking the limits to 1/2 and 1, the numerator of Z ′′ are:lim [2(3α − 1)(j ∗ )3 − 6α(j ∗ )2 + 1] =∗j →1/23(1 − α) > 0
4lim [2(3α − 1)(j ∗ )3 − 6α(j ∗ )2 + 1] = −1.

j ∗ →1Taking it into account that the numerator is the cubic function and the above twovalues, the shape of Z ′′ is one of the two forms depicted in Figure A.3. In both cases,
Z ′′ (j ∗ ) = 0 has only one solution j z1 in j ∗ ∈ (1/2, 1], and Z ′′ is positive for j ∗ < j z1
and negative for j ∗ > j z1 .
(Figure A.3 around here)
Next, we consider when there is a unique threshold dividing the area featuring
stable steady state with the area featuring unstable steady state. As we described,
this is satisﬁed when Z(j ∗ ) = 0 has a unique solution. To examine the shape of Z(j ∗ ),we focus on the relationship between Z(j ∗ ) and Z ′ (j ∗ ), which is illustrated in Figure
A.4.
Lemma A.1 implies that Z ′ (j ∗ ) has a single peak. Z ′ (j ∗ ) equals

 ∗


Cj (1 − j ∗ )
1 ∗
′ ∗
∗ 2
∗
+ 2(1 − 3α)(j ) + 2αj − 1 .
Z (j ) = ∗ j log
j
1−α

38

(A37)

Taking the limits to 1/2 and 1, the values of Z ′ (j ∗ ) are:


C
′ ∗
lim Z (j ) = log− (1 + α),4(1 − α)j ∗ →1/2lim
Z ′ (j ∗ ) = −∞.
∗

(A38)
(A39)

j →1

If (A38 ) is non-negative, that is C ≥ 4(1 − α)e1+α ≡ C z1 , since Z ′ (j ∗ ) has asingle peak, Z ′ (j ∗ ) = 0 has a unique solution in j ∗ ∈ [1/2, 1], and thus Z(j ∗ ) has asingle peak in [1/2, 1]. Z(j ∗ ) is positive around j ∗ = 1/2 since Z(1/2) = Q(1/2), andnegative around j ∗ = 1 since limj ∗ →1 Z(j ∗ ) = −α. Thus, Z(j ∗ ) = 0 has also a uniquesolution (see Figure A.4 (a)).
If (A38 ) is negative, Z ′ (j z1 ) ≤ 0 or Z ′ (j z1 ) > 0. Since Z ′ (j z1 ) is extreme value ofZ ′ (j ∗ ), Z ′ (j ∗ ) ≤ 0 in j ∗ ∈ [1/2, 1] when Z ′ (j z1 ) ≤ 0. Thus, Z(j ∗ ) is weakly decreasingin j ∗ , and Z(j ∗ ) = 0 has a unique solution (see Figure A.4 (b1)). When Z ′ (j z1 ) > 0,
Z ′ (j ∗ ) = 0 has two solutions. Let the solutions be j z2 and j z3 , and assume thatj z2 < j z3 . The extreme values, Z(j z2 ) and Z(j z3 ), determine the shape of Z(j ∗ ).
When Z(j z2 ) > 0, Z(j ∗ ) = 0 has one solution between j z3 and 1 (see Figure A.4(b21)). When Z(j z2 ) < 0, if Z(j z3 ) < 0, Z(j ∗ ) = 0 has one solution between 1/2 andj z2 (see Figure A.4 (b23)).
Following proposition is obtained from the above discussion:
Proposition A.4. Suppose that Proposition A.1 and Proposition A.2 hold, and thatj ∗ > 1/2. Unless the following condition holds, there is a unique threshold of j ∗ , andthe dynamic system is locally stable (unstable) for smaller (larger) j ∗ :C z2 ≤ C ≤ C z3 ,−α(j z2 )2(A40)+(2α−1)j z2 +1(1 − α)e 1−jz2
where C z2 =j z2 (1 − j z2 )−α(j z3 )2(A41)+(2α−1)j z3 +1(1 − α)e 1−jz3
C z3 =
j z3 (1 − j z3 )and j z2 and j z3 are solutions of Z ′ (j) = 0 and j z2 < j z3 .

39

,

(A42)

Proof of Proposition A.4. Above this proposition, we explained all cases in which
there exists a unique threshold of j ∗ such that the dynamic system is locally stablefor j ∗ smaller than the threshold. When Z(j ∗ ) = 0 has multiple solutions, there does
not exist such unique threshold. Since dZ(j ∗ )/dC < 0 from (A34 ), an increase in C
shifts down Z(j ∗ ). Let C z2 (C z3 ) be C satisfying Z(j z2 ) = 0 (Z(j z3 ) = 0), where
j z2 < j z3 . When C z2 ≤ C ≤ C z3 , Z(j ∗ ) = 0 has multiple solutions (see Figure A.4(b22)). Because Z(j ∗ ) has a unique solution when C < C z2 and C > C z3 (see Figure
A.4 (b21) and (b23)), there exists the unique threshold of j ∗ unless C z2 ≤ C ≤ C z3 .
(A41 ) and (A42 ) are derived from (A34 ) evaluated at j z2 and j z3 .
(Figure A.4 around here)
Following ﬁgures show graphically the cases of dynamics. In each ﬁgure, the solid∗∗line and the dot line represents ϕ(jt+1) and ψ(jt∗ ), respectively. Since jt∗ = jt+1

in the steady states, the intersections correspond to the steady states. Figure A.5
demonstrates the case of the unique steady state with monotonic convergence, andFigure A.6 does the case of the unique steady state with cyclical convergence.
(Figure A.5 around here)(Figure A.6 around here)

Appendix B. Proofs of Propositions
B.1. The Proofs of Proposition 3 and 4
At ﬁrst, we propose a lemma that shows the relationship between γ and j ∗ .
Lemma B.1.
Suppose that Proposition A.1 holds. Then:
dj ∗
>0
dγ
dj ∗
<0
dC
40

Proof of Lemma B.1. When Proposition A.1 holds, ϕ(j ∗ ) − ψ(j ∗ ) in the uniquesteady state is increasing in j ∗ since Q(j ∗ ) > 0, j ∗ ∈ [0, 1]. From (A11 ), in thesteady state,dΩ(j ∗ )1=
Q(j ∗ ) > 0,
∗
∗
2
∗
dj
α(j ) (1 − j )∗where Ω(j ) ≡ ϕ(j ∗ ) − ψ(j ∗ ).

(B1)
(B2)

Since ϕ(j ∗ )−ψ(j ∗ ) = 0 in the steady state, by applying the Implicit Function Theoremto (A10 ),−1dΩ(j ∗ )
> 0.
dj ∗

−1
1 − αj ∗ dΩ(j ∗ )
dj ∗
=−
< 0.
dC
Cαj ∗
dj ∗
dj ∗
1
=
dγ
γα



(B3)
(B4)

By using this lemma, Propositions 3 is proved.
Proof of Proposition 3. From (17) and (23) in the main script, the wage in a
steady state are characterized by j ∗ ,w = j ∗ (1 − j ∗ )

C
.
1−α

(B5)

Thus, the ﬁrst order diﬀerentiation with respect to γ isdw
dw dj ∗
C(1 − 2j ∗ ) dj ∗
= ∗
=
.
dγ
dj dγ
1−α
dγ

(B6)

Since dj ∗ /dγ > 0 from Lemma B.1, (B6 ) implies that if the threshold at equilibrium j ∗ is lower than 1/2, the productivity improvement of automation technology
pushes the wage up. On the other hand, if j ∗ > 1/2, it pulls the wage down.
Proof of Proposition 4. Suppose that the condition of Proposition A.1 holds. From

41

(B1 ), (B4 ) and (B5 ),dw∂w∂w dj ∗
+ ∗
=
dC
∂C
∂j dC


∗
j (1 − j ∗ )(1 − αj ∗ )(2j ∗ − 1)=1+
1−αQ(j ∗ )j ∗ (1 − j ∗ )=[Q(j ∗ ) + (1 − αj ∗ )(2j ∗ − 1)]∗(1 − α)Q(j )j ∗ (1 − j ∗ )=Wc (j ∗ ),(1 − α)Q(j ∗ )where Wc (j ∗ ) ≡ Q(j ∗ ) + (1 − αj ∗ )(2j ∗ − 1).

(B7)
(B8)

From Proposition A.1, Q(j ∗ ) > 0 for j ∗ ∈ (0, 1). If j ∗ ≥ 1/2, since (1−αj ∗ )(2j ∗ −1) ≥0, Wc > 0 and thus dw/dC > 0. If j ∗ < 1/2, there exists the region of j ∗ in which
dw/dC < 0 depending on C. We show below that there exists a threshold Cw2 , anddw/dC > 0 for all j ∗ when C < Cw2 .
The second derivative of Wc (j ∗ ) is:
Wc′′ (j ∗ ) = Q′′ (j ∗ ) − 4α.

(B9)Because Q′′ (j ∗ ) is a decreasing function for j ∗ ∈ (0, 1/2) and Q′′ (1/2) = 2α + 6 > 4αfrom (A15 ), Wc′′ (j ∗ ) > 0 for j ∗ ∈ (0, 1/2). Thus, Wc′ (j ∗ ) = Q′ (j ∗ ) − 4αj ∗ + α + 2 isan increasing function for j ∗ ∈ (0, 1/2). Because limj ∗ →0 Wc′ (j ∗ ) = −∞, Wc (j ∗ ) forj ∗ ∈ (0, 1/2) is a decreasing function when Wc′ (1/2) ≤ 0, and is an U-shaped function
when Wc′ (1/2) > 0. Figure B.1 illustrates the two cases: the upper row shows theformer case and the lower row shows the latter case.
For the former case, since Proposition A.1 assures Q(1/2) > 0, Wc (1/2) > 0 from(B8 ). Therefore, if the following condition holds, Wc (j ∗ ) > 0 and thus dw/dC > 0
for j ∗ ∈ (0, 1/2):
Wc′ (1/2) ≤ 0 ⇔ C ≤ Cw1 ≡ exp{log[4(1 − α)] − 1}.

(B10)

When Wc (j ∗ ) is an U-shaped function, if the value of Wc (j ∗ ) at the extreme pointj c is positive, that is Wc (j c ) > 0, dw/dC > 0 for j ∗ ∈ (0, 1/2). Since an increase in
C lowers Q from (A11 ), it also lowers Wc from (B8 ). Let Cw2 be the level of C
42and j cc be the level of j c such that Wc (j c ) = 0 Then, when following condition holds,
dw/dC > 0 for any j ∗ :C < Cw2 ≡(1 − α) exph(1−α)(j cc )2 +αj cc
1−j ccj cc (1 − j cc )i,where − (1 − α)(j cc )3 − 2α(j cc )2 + (3 + α)j cc − 1 = 0.

(B11)
(B12)

Conversely, when C ≥ Cw2 , there exists the region of j ∗ in which dw/dC ≤ 0. The
rightmost two ﬁgures at the lower row in Figure B.1 illustrate the cases of C < Cw2
and C ≥ Cw2 . The ﬁgure of the case of C ≥ Cw2 shows that Wc (j ∗ ) > 0 and thus
dw/dC > 0 when j ∗ is suﬃciently small and suﬃciently large, and Wc (j ∗ ) < 0 andthus dw/dC < 0 when j ∗ is intermediate.
We numerically examine the relationship between Cw1 , Cw2 and C2 because therelationship aﬀects the sign of the marginal impact on the wage, but the relationshipcannot be derived analytically. From (A18 ), (A19 ), (B10 ), (B11 ) and (B12 ), Cw1 ,
Cw2 , and C2 depend only on α. Proposition A.1 shows that C < C2 is the conditionof the existence of a unique steady state. Thus, if Cw2 < C2 for any α, the region inwhich dw/dC ≤ 0 could exist when C ≥ Cw2 . Figure B.2 illustrates the relationshipbetween Cw1 (red circle), Cw2 (black diamond), C2 (blue square) and α. The ﬁgureshows Cw1 < Cw2 < C2 for all α. Because Cw2 < C2 , when Cw2 ≤ C < C2 , there
exists the region in which dw/dC ≤ 0. From the graph of Wc (j ∗ ) in the bottom rightﬁgure of Figure B.1, Wc > 0 ⇔ dw/dC > 0 when j ∗ ∈ (0, 1/2) is low enough andhigh enough, and Wc ≤ 0 ⇔ dw/dC ≤ 0 when j ∗ is intermediate.

B.2. The Proof of Proposition 5
From (17) and (11) in the main script, aggregate proﬁt of managers at the steadystate is:
Z j∗
Π≡

Z 1
k

π l (j)dj =π (j)dj +
0

j∗

43

1
1
C(j ∗ )2 + 0 = C(j ∗ )2 .
2
2

(B13)

The aggregate ﬁxed cost is:Z j∗
C≡C · jdj =0

1
C(j ∗ )2 .
2

(B14)

Thus, the income share of managers isΠ(1 − α)j ∗=.
Y −C
2 − (1 − α)j ∗(B15)

The marginal eﬀect of of an improvement of automation technology on the labor shareof income is:dΠ/(Y − C)
2(1 − α)
dj ∗
=
·
>0
dγ
[2 − (1 − α)j ∗ ]2 dγ(B16)When Proposition A.1 holds, Lemma B.1 assures the inequality above.
On the other hand, the income share of labor is given by (B5 ):2(1 − j ∗ )w=
Y −C
2 − (1 − α)j ∗(B17)Thus, since the marginal eﬀect isdw/(Y − C)
2(1 + α)
dj ∗
=−
< 0,
dγ
[2 − (1 − α)j ∗ ]2 dγ(B18)

The wage decreases as automation technology improves. As for total income share:


w
−4α
d
Π
dj ∗
+
=
< 0.
(B19)
dγ Y − C
Y −C[2 − (1 − α)j ∗ ]2 dγThus, the total labor income share decreases with γ. The eﬀect of a change in C onlabor income share can be proved in the same manner.

B.3. The Proof of Proposition 6
In the steady state, (40) is:


Cj
β
K
H≡
j(1 − j)Y −1−α−− K = 0.
1+βα(1 + β)
2Y

44

(B20)

By totally diﬀerentiating the above equation, the eﬀect of an increase in γ on K is:
dK
∂H
=−
dγ
∂Y
where



∂H
∂K

−1

dY
,
dγ

(B21)

1 K · Cj∂Hβj(1 − j) −
=
,
∂Y
1+β
Y 2(1 + β)αY

(B22)

∂H
1−α
Cj
=−
+
− 1.
∂K
α(1 + β) 2(1 + β)αY

(B23)

By using (B20 ), (B23 ) is:
∂H
β
Y
=−
j̄(1 − j̄) .
∂K
1+β
K

(B24)

From (39), the eﬀect of an increase in γ on Y is:
Y
Y dK
dY
= j + αj
.
dγ
γ
K dγ

(B25)

By substituting (B21 ) into (B25 ):
"

dY
Y
=j
dγ
γ

Y
1 + αj̄
K



∂H
∂K

−1

∂H
∂Y

#−1
.

(B26)

From (B22 ) and (B24 ),


∂H
∂K

−1



∂H
K
2αβ(1 − j̄)Y 2
1−
=−
.
∂Y
Y
CK

(B27)

Substituting (B27 ) into (B26 ):

−1
dY
Y
CK
=j
(1 − αj) + αj
> 0.
dγ
γ
2αβ(1 − j)Y 2(B28)

Thus, eﬀect of an increase in γ on the wage as the following equation is positive:dw
dY= (1 − j)
> 0.
dγ
dγ

45

(B29)

Figure 1

Eﬀects of an improvement of automation productivity in the steady state.

46

Figure 2

Eﬀects of a creation of new industries in the steady state.

47

Figure 3 The wage when there is a technological limitation j. This ﬁgure
illustrates the case of j < j ∗ .

Figure 4 Aggregate capital when there is a technological limitation j. Thisﬁgure illustrates the case of j < j ∗ .

48

Figure 5

Eﬀects of a subsidy policy in the steady state.

49

Figure 6 Dynamics of welfare: aggregate welfare of all households, welfare ofhouseholds who choose non-automation technology and welfare of households
who face zero ﬁxed costs.
Note: In each panel, the blue line (round), green line (square), and red line (diamond) indicate
the dynamics in cases in which the subsidy rate, σ, is 0.01, 0.04, and 0.06, respectively. The
black (dotted) line in each panel indicates the level of welfare in the previous steady state.

50

Figure 7 Dynamics of the interest rate and lifetime income of each type ofhousehold presented in Figure 6.
Note: In each panel, the blue line (round), green line (square), and red line (diamond) indicate
the dynamics in cases in which the subsidy rate, σ, is 0.01, 0.04, and 0.06, respectively. The
black (dotted) line in each panel indicates the level of welfare in the previous steady state.

51

𝑄(𝑗 ∗ )𝑄′(𝑗 ∗ )
0

𝑗 𝑠1

1

𝜙−𝜓

𝑗∗
1

0
𝑄𝑛

𝑗∗

𝛼

0

(The numerator of 𝑄′′)

1

𝑗∗

(c1)

(b1)

𝑗∗

(d1)

1

1

0

𝑗 𝑠1
∗
−1 𝑗

(a)

𝑄(𝑗 ∗ )

𝑄′(𝑗 ∗ )0𝑗 𝑚 𝑗 𝑠1

𝜙−𝜓

1 𝑗∗
0 𝑗𝑚

(b2)

Figure A.1

1

(c2)

Cases of Proposition A.1.

52

1 𝑗∗

0
𝛼 ∗

𝑗

(d2)

Figure A.2

Patterns of convergence when j ∗ ≤ 1/2.

53

Figure A.3

Shape of Z ′′ (j ∗ ) in Lemma A.1.

54

Figure A.4

Shape of Z(j ∗ ) in Proposition A.4.

55

Figure A.5

Case of the unique steady state with monotonic convergence.

56

Figure A.6

Case of the unique steady state with cyclical convergence.

57

Figure B.1

The patterns of the shape of Wc (j ∗ ).

58

Figure B.2 Thresholds of ﬁxed cost C: Cw1 (red circle) , Cw2 (black diamond),
and C2 (blue square) with respect to α.
Note: When C < Cw1 , Wc′ (1/2) < 0. When C < Cw2 , there exists an area of j ∗ in which
dw/dC < 0. When C < C2 , a unique steady state exists.

59