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Automation and Unemployment:
Help is on the Way
Hideki Nakamura
Osaka City University∗
Joseph Zeira
Hebrew University of Jerusalem, CEPR and RCEA♣
December 2018♥
Abstract This paper examines the evolution of unemployment caused by automation in a
task-based framework that allows for two types of technical change. One is
automation that turns labor tasks to mechanized ones. The second is addition of
new labor tasks, as in the expanding variety literature. The main result of the paper
is that if the share of labor in output does not converge to zero over time, the rate
of unemployment, which is caused by automation, converges to zero over time.
JEL Specification: J64, O14, O30, O40.
Keywords: Automation, Unemployment, Technical Change, Growth, Labor Income Share.
∗ E-mail: [email protected]. ♣ E-mail: [email protected]. ♥ We are grateful to Daron Acemoglu, Gadi Barlevy, Matthias Doepke, Marty Eichenbaum, Eugene Kandel, Marti Mestieri, and Hernando Zuleta for very helpful comments. Remaining errors are all ours.
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Automation and Unemployment: Help is on the Way
1. Introduction This paper examines a theoretical model of automation, namely of machines that replace
workers in various tasks. The paper focuses on the effect of automation on unemployment.
It is obvious that if automation replaces human labor in various tasks, it increases
unemployment, as the replaced workers lose their jobs for some time, during which they
need to search for another job. This is one of the short run costs of economic growth. But
the paper also shows that under a very plausible assumption, namely that the share of labor
in output does not converge to zero over time, the rate of automation unemployment can
decline over time and converge to zero in the long run.
The paper presents a model of production by tasks, which we also describe as
intermediate goods (and use the two terms interchangeably). Initially all tasks are
performed by labor, but gradually new machines are invented, each specific to a certain
task, which replace the workers performing this task and thus enable automation or
mechanization (also interchangeable terms). In addition to automation, inventors can also
invent new tasks and this is another type of technical change. Hence, the paper combines a
task-based model of automation, as in Zeira (1998), and a variety expansion model similar
to Romer (1990). Below we explain why we combine these two models together.
Using this model the paper analyzes the possible dynamics of automation and their
effect on unemployment. Consider first the possibility that automation has a finite bound.
Then, the rate of automation approaches zero as it gets close to this bound and so does the
rate of unemployment.1 Consider next the more interesting possibility, that automation
continues unbounded. In this case, wages must rise. Otherwise producers might prefer to
hire labor rather than adopt the newly invented more costly machines, and automation
might stop. As the paper shows, wages can rise either if the share of labor declines, or if
the number of labor tasks increases. Hence, if the share of labor is not converging to zero,
automation requires that the number of new tasks increase continuously and faster than
automation. As a result, new automation becomes small relative to the number of existing
1 This holds except for one sub-case, in which the economy reaches an equilibrium where only machines produce, all labor is idle and output per capita is bounded. This is clearly a highly unrealistic case.
2
labor tasks, so that the rate of unemployment becomes low. This is the mechanism, which
drives our main result.
Hence, the paper’s main result is that the dynamics of unemployment depend
crucially on the long run behavior of the share of labor in output. Actually, many studies,
the most famous is Kaldor (1961), have found that the share of labor in output is stable,
around 2/3, both across countries and over time. In recent decades, the share of labor has
experienced a slight decline, but it is still far from going down to zero. Hence, the
assumption that the share of labor does not converge to zero, fits well the stylized facts of
modern economic growth.
The mechanism through which automation unemployment declines is by increasing
the set of labor tasks. This also fits well-known stylized facts. First, the process of
automation and mechanization itself leads to invention of new tasks. The invention of
spinning machines in the early Industrial Revolution enabled a finer division of tasks in the
production of cotton and woolen fabrics. The invention of automobiles replaced horse-
drawn transport, but created many new tasks in the transportation sectors. We have recently
experienced similar trends in IT technologies that lead both to automation and to creation
of new labor tasks. Second, we can also learn on increases in labor tasks, from data on job
titles. According to the National Occupational Classification (NOC) of Canada, there were
approximately 35,000 job titles in 2016. This number increased by 204 new job titles in
the five years between 2011 and 2016.
It is important to stress that the creation of new labor tasks is not an exogenous
assumption, but a requirement of the model. It is important, as some might claim that just
adding expanding variety to a model of automation leads to the main result of the paper, as
creation of new labor tasks increases demand for labor and reduces unemployment.
However, we do not assume a-priori that the number of labor tasks increases. We just
assume that it can increase and show that it does, if automation expands. Hence, the
addition of expanding variety is a result of the ongoing process of automation.
Interestingly, automation and its effect on unemployment has captivated public
debate since the beginning of the Industrial Revolution. Workers feared that the new
machines might drive them out of jobs into unemployment and poverty. This fear first
showed up in the rebellion of the Luddites in 1811-1817. They were artisans, who viewed
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in awe how the textile industry mills threatened their economic existence. They started a
mutiny and the British government had to send a large army to oppress this rebellion.
This issue came up again often. In the 1920s, a period of rapid technical change,
Evan Clark wrote an article to the New York Times on February 26, 1928, titled “March
of the Machine Makes Idle Hands.” He claimed: “It begins to look as if machines had come
into conflict with men – as if the onward march of machines into every corner of our
industrial life had driven men out of the factory and into the ranks of the unemployed.” At
the time, US rate of unemployment was only 4.2 percent. At the same period, Keynes
(1930) coined the term “technological unemployment.”2
Although history taught us that the fears of high technological unemployment did
not materialize, the issue has recently returned to the public debate. The technological
revolution in computers, information technology (IT), and artificial intelligence (AI),
renewed the fear and led many to claim, “This time is different.”3 Recently, the discussion
of technological unemployment appears not only in the public debate, but in economic
research as well. The following survey focuses on this new research.
So far, the best framework to study automation has been task-based models, in
which production of final output is a result of many tasks, where each can be performed
either by labor or by a machine, once it is invented. Champernowne (1961) was the first to
present such a model, in an attempt to build micro-foundations to the aggregate production
function. Zeira (1998) embedded the model in a general equilibrium framework of
economic growth and reached two main results. The first is that adoption of mechanization
requires that wages be sufficiently high, and the second is, that despite lower demand for
labor in mechanized tasks, demand for labor in the remaining tasks increases, since
working with more capital raises marginal productivity of labor. This raises wages and
enables adoption of more automation. Zeira (1998) also has two shortcomings, as it implies
that the Solow Residual is zero and that the share of labor in output converges to zero over
time, in contrast with well-known stylized facts. This paper shows that one way to avoid
these implications is to combine the task-based model with the expanding variety model.
2 Since it is common, we also use this term for the unemployment caused by automation. 3 See Chuah, Loayza and Schmillen (2018).
4
Most task-based models focus on long run technical change and on economic
growth. Nakamura and Nakamura (2008) and Nakamura (2009) use them to provide micro-
foundations to CES production functions. Acemoglu (2010) generalizes these models to
‘labor saving technologies.’ Perretto and Seater (2013) examine a version of the theory
without tasks. Zuletta (2008, 2015) tries to consolidate the decreasing share of labor with
the empirics. Aghion, Jones and Jones (2017) offer a variation of the basic task-based
model. Another paper that combines expanding variety with automation is Hémous and
Olsen (2018), but it focuses mainly on skill, while our paper focuses on unemployment.
A number of papers discuss the short-run effects of automation, mainly on
unemployment. Hrdy (2017) provides a survey of this literature. Cortes, Jaimovich and Siu
(2017) and Graetz and Michaels (2017) examine its relation to jobless recoveries. Other
recent contributions are Autor (2015) and Sachs (2017). Nakamura and Nakamura (2018)
examine how automation affects diversity of Job mismatch possibilities. Most relevant to
this paper is a series of papers by Acemoglu and Restrepo (2017a, 2017b, 2018a, 2018b)
that examine the labor market effects of automation in a task-based model.4 They assume
that new labor tasks arrive continuously, as in this paper, but impose a specific structure
on arrival of new tasks and assume that old tasks disappear as new ones arrive. This paper
does not impose any specific growth of new tasks, but examines the growth patterns that
enable continuing automation. It also shows that destruction of tasks is impossible in our
model. The differences in modeling lead to different results, mainly that in Acemoglu and
Restrepo unemployment is stable, while in this paper it disappears over time.
The structure of the paper is as follows. Section 2 presents the model, while Section
3 derives the equilibrium conditions. Section 4 examines equilibrium in various cases.
Section 5 analyzes economic growth while Section 6 examines the dynamics of the rate of
unemployment. Section 7 examines a balanced growth path. Section 8 studies some
potential extensions of the model, like making better machines and eliminating tasks.
Section 9 introduces different skill levels and analyzes skill-biased technical change and
Section 10 summarizes the paper. The Appendix presents proofs and extensions.
4 Acemoglu and Restrepo (2018b) is the basic model, Acemoglu and Restrepo (2017b) extend it to demography and Acemoglu and Restrepo (2017a) examines it empirically. Acemoglu and Restrepo (2018a) presents a less technical summary of their work in the area.
5
2. The Model Consider an economy that produces a single final good, which is used both for consumption
and for investment. In period t the final good is produced by a continuum of intermediate
goods [0, ]tj T∈ in the following way:
(1)
1
0
( ) .tT
t tY x j djθ
θ
= ∫
The quantity Yt is final output, while ( )tx j is the quantity of the intermediate good j. We
can think about the intermediate goods also as tasks, as discussed in the introduction. The
parameter θ is smaller than 1 as in all CES production functions.
The set of available intermediate goods, [ ]0, tT , is exogenous. We assume that
producers can choose which intermediate goods to use in production. In Section 3 we show
that to enable this choice, the parameter θ must be positive. We, therefore, assume that θ is
positive.5 Hence, the elasticity of substitution between tasks 1/ (1 )θ− is higher than 1.
Each task can be performed either by human labor or by a machine specific to the
task, if it has already been invented. Machines can be robots or automats as well.
Production of intermediate good j by labor requires one unit of labor per one unit of the
good.6 For an automated task j the cost of a machine that produces one unit of j is k(j) units
of capital. Assume that the function k is increasing over j. Hence, machines become more
difficult and harder to build with automation.7 Capital is invested one period ahead assume
for simplicity that it fully depreciates in one period, so the depreciation rate is 1.
We next assume that machines are invented by order of difficulty. Hence, the set of
tasks with available machines in period t is [0, ]tM , where Mt is the automation technology
frontier. Hence, the set of tasks performed by labor is equal to ( ],t tM T . There are two types
of technical change in this model. One is automation, inventing new machines, and the
5 Aghion, Jones and Jones (2017) make an opposite assumption, that θ is negative. This implies that their model does not allow producers to choose intermediate goods. 6 Appendix D deals with rising productivity of labor over tasks. 7 Bloom, Jones, Van Reenen and Webb (2017) provide evidence that innovation becomes more costly along technical change. One reason for it could be innovation of more complex machines.
6
other is inventing new tasks. We therefore assume that the automation frontier Mt grows or
remains unchanged over time, 1 for allt tM M t−≥ , and Tt increases or remains unchanged,
1t tT T −≥ for all t. We do not model the technology creation here for simplification, but
assume that Mt grows as long as producers adopt the new automation technologies.
There is a continuum of size 1 of infinitely lived people. Each one supplies one unit
of labor per period and works in a specific task.8 Automation of task j in period t causes
workers in this sector to lose their jobs. During period t they search for a new task and from
t + 1 they work in another task. Hence, the number of unemployed in period t due to loss
of task j, is the number of people who worked in that task last period, 1( )tl j− . In this simple
case, technological unemployment is the only type of unemployment. Note that as T
increases, workers move from old tasks to new ones without unemployment. We make this
simplifying assumption to focus on automation unemployment alone. Appendix E
examines an additional type of unemployment, of mismatch between workers and tasks.
We further assume that the economy is small and open. The final good is tradable,
while the intermediate goods are non-tradable. The economy is open to capital mobility
and the global interest rate is constant over time and is equal to r. We denote the sum of
the interest rate and the rate of depreciation by 1R r= + . Finally, we assume that markets
are perfectly competitive and expectations are rational.
3. The Equilibrium Conditions
3.1 Choice of New Intermediate Goods
Assume that the final good is the numeraire. Denote the price of intermediate good j in
period t by ( )tp j . Profits of producers of the final good are equal to:
1
0 0
( ) ( ) ( ) .t tT T
t t tprofits x j dj p j x j djθ
θ
= − ∫ ∫
8 The model identifies for simplicity labor tasks and jobs. In the real world, a job might consist of a number of tasks. Then, if one task is automated, the worker does not necessarily lose her job, but performs the other tasks with greater quantities. This actually further reduces unemployment and supports our results.
7
Firms decide first how many intermediate goods to use, or how many tasks to activate. This
decision depends on the marginal productivity of adding a task, namely of increasing T:
(2) 11 ( ) ( ) ( ).t t t tY x T p T x Tθ θ
θ− −
Hence, if 0θ ≤ this marginal productivity is negative, if there is production. This means
that producers will not use any intermediate good, which is not acceptable in such a model.
This justifies our assumption that the coefficient θ is positive.
The second FOC is with respect to the quantity used of each intermediate good,
( )tx j , and it implies that:
(3) 1
1( ) ( ) .t t tx j Y p j θ−−= ⋅
Note that if we substitute this condition in (2) we get that for any value of T the marginal
productivity of adding the last task satisfies:
(4) 11 1( ) ( ) ( ) ( ) 1 ( ) 0.t t t t t tx j p j x j p j Y p jθθ
θ θ
−− − = − ≥
Hence, this marginal productivity can be equal to zero only if the price of the intermediate
good is infinite. In all other cases, it is positive. This means that producers adopt all the
new tasks, unless the wage is infinite.
3.2 Equilibrium in the Goods Markets
Substituting the FOC (3) in the production function (1), we get the following equilibrium
condition in the market for goods:
(5) 10
( ) 1.tT
tp j djθθ
−− =∫
We next examine the markets for intermediate goods and find the prices of each good in
order to substitute in the equilibrium condition (5).
Consider first a firm that produces an intermediate j with machines. Such a firm
produces a quantity x, sells it at a price ( )tp j and its profits are:
( ) ( ) .tp j x Rk j x−
Due to perfect competition and to free entry, profits must be zero. Hence:
(6) ( ) ( ).tp j Rk j=
8
Consider next a firm that produces an intermediate good by labor. If we denote the
wage at period t by wt and the size of labor in the firm by l, the profit is:
( ) .t tp j l w l−
Due to perfect competition and free entry, the price of the intermediate good is equal to the
cost of labor:
(7) ( ) .t tp j w=
Equations (6) and (7) describe the supply prices of the goods produced by machines and
by labor. We substitute them in the equilibrium condition in the goods market (5) and get:
(8) 1 1 10
( ) 1.t t
t
M T
tMR k j dj w dj
θ θ θθ θ θ
− − −− − −+ =∫ ∫
3.3. Determination of Wages
We next use the equilibrium condition (8) to find the equilibrium wage rate. Note that the
first addend on the left hand side of (8) is a function of Mt. Denote this function by φ:
1 10
( ) ( ) .M
M R k j djθ θθ θϕ
− −− −= ∫
The function φ is increasing and since k is increasing and θ is positive, the function φ is
concave. For equation (8) to hold, the following must apply:
( ) 1.tMϕ ≤
We therefore assume that this holds, and we discuss the conditions for it in Section 4.
Next, we derive from condition (8) the equilibrium wage rate:
(9)
1
.1 ( )
t tt
t
T MwM
θθ
ϕ
−
−= −
Clearly, if ( )tMϕ is strictly lower than 1, the wage is finite. If ( )tMϕ is equal to 1, there
are two cases. In the first case, the number of labor jobs T – M is positive, so the wage rate
is infinite and there is no production by labor due to FOC (3). If T – M is equal to zero there
is no production by labor as well. In both cases, there is zero employment.
3.4 The Share of Labor
From the FOC (3) we get for each intermediate good produced by labor:
9
1
1 ( ) .tt t t
t t
EY w x jT M
θ−− = =
−
The variable Et is the amount of employment in period t. Note that labor is equally allocated
across all labor tasks between M and T, since all sectors face the same wage and the same
price. From this equation, we derive the share of labor in output:
( ) 1 .t tt t t
t
w E T M wY
θθ
−−= −
Substituting the equilibrium wage from equation (9) we get that the equilibrium
share of labor in output is:
(10) 1 ( ).t tt
t
w E MY
ϕ= −
In a similar way it can be shown that the share of capital in output is ( )tMϕ . Equation (10)
implies that as automation M increases, the share of labor in output declines, but by less
and less, due to the concavity of φ.
3.5 A Condition for Technology Adoption
Producers adopt an automation technology only if it enables them to produce at a lower
cost. Hence, they produce by machines only if:
( ) .tRk j w≤
Since the function k is increasing, this condition should hold for all the goods between 0
and the frontier Mt. Hence, the condition for adoption of automation technology is:
(11) ( ).t tw Rk M≥
As long as the inequality in (11) is strict, automation continues. Once condition (11)
becomes binding, automation stops.
We next rewrite condition (11) more explicitly, by substituting in it the wage rate
(9). We get the following condition:
(12) [ ]1 1
1 1
1 ( ) 1 ( )( ) 1 ( ) .( )
( )
t tt t t t
tt
M MT M k M R MM
k M R
θ θθ θ
θ θθ θ
ϕ ϕϕϕ
− −− −− −
− −− ≥ − = =
′
Hence, the condition for automation depends only on the variables Mt and Tt and on the
shape of the function φ. We turn to analyze condition (12) in more detail in the next Section.
10
4. The Dynamics of Technology and of the Share of Labor We begin our analysis of the dynamics of technology by rewriting condition (12) in the
following way:
(13) 1 ( ) ( ) 1 ( ) .( ) ( )
t t t tt t
t t
M M M MT MM M
ϕ ϕ ϕϕ ϕ
′− + −≥ + =
′ ′
Note that the RHS of (13) is increasing in Mt, as can be shown by direct derivation. Hence,
there exists a unique level of M, for which there is equality in (13). We denote this value
by ( )tTψ . Clearly, this is a well-defined function, which is increasing in T. Figure 1
describes the determination of this function, which is actually the inverse of:
( ) 1 ( ) .( )
M M MTM
ϕ ϕϕ
′ + −=
′
Figure 1
We assume that automation is determined elsewhere and is therefore exogenous in
this model, but we also assume that there is invention of automation machines as long as
these machines are adopted. This leads to the following constraint on automation:
(14) ( ).t tM Tψ≤
Hence, if the set of tasks is bounded by T*, automation is bounded as well by ( *)Tψ .
M
1-φ(M)
Mφ’(M)
ψ(T)
φ
T
1
11
We therefore divide the dynamics of the economy to two main cases. One is that
automation creation comes to a stop and is bounded by a finite M*. The second case is that
automation continues without a bound. In this case the number of total tasks T is
unbounded as well. Note that this is possible only as long as the function φ is strictly below
1. Hence, we turn next to examine when φ is above or below 1.
The function φ is increasing, concave and begins from (0) 0ϕ = . We distinguish
between three disjoint cases of this function:
Case (a): The function φ is bounded by A < 1.
Case (b): The function φ is bounded by 1, but converges to 1.
Case (c): The function φ rises above 1, so that ( *) 1Mϕ = at a final level of automation M*.
The three different cases depend on the cost of machines k and on the cost of capital
R, as implied by the definition of φ. In cases (a) and (b), where the function φ is bounded,
the cost of machines k must go to infinity as M goes to infinity. Changes in the cost of
capital R do not affect the boundedness of φ and only shift it up or down. The following
diagram shows the three cases:
Figure 2
The following proposition describes the full dynamics of the model in its various
cases:
M
A
1
Case (c)
Case (b)
Case (a)
12
Proposition 1:
1. In case (a) the set of tasks can be bounded and then automation is bounded as well, or
both M and T can grow without bounds. In both cases the share of labor converges to a
positive number.
2. In case (b) the set of tasks and automation can be bounded and then the share of labor
converges to a positive number. If T and M are unbounded, the share of labor converges
to zero.
3. In case (c), automation always converges to a finite level. This level is below M* if T
is below M*. In this sub-case, the share of labor converges to a positive number. If T is
equal to M* or higher, automation stops at M*, there is zero employment, full
automation and output is finite. Hence, the share of labor converges to zero. In case (c)
output per capita is bounded in the long run.
Proof is in Appendix A.
The next proposition shows that in all the above cases there is something common
to the dynamics of automation if the share of labor converges to a positive number. As
shown below in Sections 5 and 6, this proposition is important for the analysis of the
dynamics of economic growth and of unemployment.
Proposition 2: If the share of labor converges to a positive number, the rate of automation
satisfies:
1
1
( ) 0.1 ( )
t tt
t
M MM
ϕϕ
−→∞
−
′ ∆→
−
Proof is in Appendix A.
In the rest of the paper, we focus mainly on case (a) and assume that the cost of
machines k and the cost of capital R are such that the function φ satisfies:
( ) 1 for all .M A Mϕ < <
In this case the share of labor is everywhere higher than 1 – A and hence it is always positive
and it converges in the long run to a positive number as well. Since the function φ is
13
increasing, the share of labor might decline over time, as automation M increases, but it
converges to a constant positive level, so from some time on, it is quite stable. This fits the
well-known stylized facts of economic growth and explains why we focus on this case.
Furthermore, Appendix B shows that if we assume that capital is used also in labor tasks,
by structures for example, then the decline of the share of labor is even more moderate, as
it does not begin at 1, but at a lower level. Hence, the result of a stable share of labor
becomes even more realistic in such an extension of the model.
We next present another implication of case (a). As mentioned above, if T grows
unboundedly, so does automation M. Furthermore, inequality (12) implies that:
( ) .
1 ( )t t t
t t t
M M MT M M
ϕϕ
′≤
− −
Note that the numerator of the right hand side of this inequality is converging to zero, as
shown by Figure 1. Since in case (a) the denominator converges to a positive number, the
RHS ratio converges to zero and so does the ratio between automated tasks M and labor
tasks T – M. Hence, if automation is unbounded, the number of labor tasks must increase
faster than automation.
5. Economic Growth In this section, we calculate output per worker and its rate of growth. We show that the
long run rate of growth reflects mainly the growth of new tasks and not automation. This
section also examines the size of the Solow Residual and shows that it is positive in this
model, and that fits the empirical studies.
5.1 Growth of Output per Worker
From equations (9) and (10) we get that output per worker in this model is equal to:
(15) ( )
[ ]
1
1 .1 ( ) 1 ( )
t tt tt
t tt
T MY wyE M M
θθ
θϕ ϕ
−
−= = =
− −
The following proposition examines when output per worker is growing unboundedly and
when it is bounded.
14
Proposition 3: If T is bounded, so is M and output per worker is bounded as well. If T is
unbounded, so is M and output per worker is unbounded as well.
Proof is in Appendix A.
The level of output per worker (15) enables us to calculate its rate of growth:
(16) ( ) 1
1
( )1 1ln ln .1 ( )
t tt t t
t
M My T MM
ϕθθ θ ϕ
−
−
′ ∆−∆ = ∆ − +
−
Hence, the rate of growth of output per worker, namely of labor productivity, depends on
the two types of technical change, the growth of automation, M, and the growth of labor
tasks, T – M. If the share of labor does not converge to zero, as in case (a), Proposition 2
implies that the second addend on the right hand side of (16) converges to zero and hence
the long run rate of growth of output per worker is equal to:
( )1ln ln .t t ty T Mθθ−
∆ ≅ ∆ −
We can therefore conclude that in this model, that combines the Zeira (1998) model of
automation and the Romer (1990) model of expanding variety, the long-run rate of growth
is determined by the expanding variety and not by automation. Interestingly, the shares of
labor and of capital are determined by the automation part of this combined model, rather
than by its expanding variety part.
5.2 Solow Residuals
We next turn to calculate the Solow Residual and to examine whether it is indeed positive,
as all empirical tests show. Using the first order condition (3), we calculate the amount of
capital in task j:
1 1
1 1 1( ) ( ) ( ) ( ) ( ) ( ) .t t t t tK j x j k j Y p j k j Y R k jθ
θ θ θ− − −− − −= = =
Summing over the automated tasks [0, Mt] we get:
1 11 10 0
( ) ( ) ( ).t tM M
t t t t tK K j dj Y R R k j dj Y R Mθ θθ θ ϕ
− −− −− −= = =∫ ∫
Hence, the share of capital in output is ( )Mϕ , and the capital-labor ratio is:
(17) ( ).t tt
t
K y ME R
ϕ=
15
This enables us to calculate the Solow residual, which is defined by:
ln ( ) ln .tt t t
t
KSR y ME
ϕ
= ∆ − ∆
Using equations (16) and (17) we find that the Solow residual is equal to:
(18) [ ] [ ]11 11 ( ) ln( ) ( ) .t t t t t tSR M T M M Mθ θϕ ϕθ θ −− − ′= − ∆ − + ∆
Since both mechanization and the number of tasks performed by labor increase over
time, the Solow residual is positive, according to equation (18). We next consider the
Solow residual in the long run. If automation M is bounded, ΔM converges to zero. If
automation is unbounded, we are in cases (a) or (b), where the function φ is bounded by 1.
In these cases φ’(M) converges to zero. Hence, in all cases ( )M Mϕ′ ∆ converges to zero.
As a result, the long run Solow Residual is approximately equal to:
[ ]11 ( ) ln( ).t t t tSR M T Mθϕθ−
≅ − ∆ −
Hence, the long run Solow residual converges to the rate of growth of output per worker
times the share of labor. This is similar to what we find in the many empirical studies.
5.3 Production by Labor and Capital
Finally, we present output in the economy in the standard way, as a function of capital and
labor, and not as a function of intermediate goods or tasks. We substitute the quantities of
intermediate goods produced by labor and by capital ( )tx j in the production function (1)
and get:
1
1 10
( ) ( ) .( ) ( )
tMt tt t t
t t t
E K RY T M R k j djT M M
θ θθ θ θ θθ θ
θ θϕ
− −− −
= − + −
∫
Since the integral is equal to ( )tMϕ we get:
(19) 1
1 1( ) ( ) .t t t t t tY T M E M R Kθ θ θ θ θ θϕ− − = − +
The production function (19) is in labor E and capital K. Note that the coefficient of capital
does not really depend on the interest rate R, since:
16
1
1 10
( ) ( ) .tM
tM R k j djθθ
θ θ θϕ−−
− −
= ∫
It therefore depends on the technological parameters only.
The coefficients of labor and capital in the production function (19) grow with
technical change, namely with the growth of M and T. Note that the coefficient of capital 1( )tM Rθ θϕ − is bounded even if automation is unbounded, while the coefficient of labor
1( )t tT M θ−− grows unboundedly. Hence, technical change is mainly labor augmenting in
the long run. Note also that the coefficients of both labor E and of capital K in (19) are
correlated with the wage through equation (9). As a result, the elasticity of substitution
between labor and capital in this model is not constant and not equal to 1/ (1 )θ− . Appendix
C calculates this elasticity and shows that it is close to 1.
6. The Rate of Unemployment We next turn to measure the rate of unemployment in the economy. The group of
unemployed are people who worked in the previous period, but their tasks become
automated in the present, in period t. Once they leave their job due to automation, they
search for another task and remain unemployed for one period.
Denote the number of unemployed due to automation by AtU , which is also the rate
of unemployment, as the supply of labor is 1. This is the number of people who were
employed in period t – 1 in the jobs that later become automated in period t. The number
of workers in period t – 1 in job j is 1( )tl j− , which is equal to:
11
1 1
( ) .tt
t t
El jT M
−−
− −
=−
Hence, the total number of unemployed due to automation in period t is equal to:
(20) 1
11
1 1 1 1
( ) .t
t
MA t t t
t tt t t tM
E M MU l j djT M T M
−
−−
− − − −
∆ ∆= = ≤
− −∫
Adding inequality (12) to inequality (20) implies that:
1
1 1 1
( ) .1 ( )
A t t tt
t t t
M M MUT M M
ϕϕ
−
− − −
′∆ ∆≤ ≤
− −
17
This upper bound on automation unemployment and Proposition 2 together lead to the next
Theorem:
Theorem 1: If the share of labor in output converges to a positive number, the rate of
technological unemployment, AtU , converges to zero over time: 0A
t tU →∞→
Note that Theorem 1 holds not only for case (a) of the function φ, but in cases (b)
and (c) as well, as long as the share of labor converges to a positive number. Hence,
Theorem 1 establishes a strong connection between the long run behavior of the share of
labor, to the dynamics of technological unemployment. This is a surprising result and we
next try to understand the intuition behind it. Actually, there are two types of intuition here.
First, if automation is bounded, the unemployment it creates will diminish and converge to
zero over time, since there is less and less automation over time.
It is more challenging to understand the intuition behind this result when
automation is unbounded. To ensure adoption of automation wages must rise. According
to equation (9), this happens either by increasing the number of labor tasks, T – M, or by
reducing the share of labor. We next explain the intuition behind these two effect implied
by equation (9). If the number of labor tasks T – M increases, there are less workers in each
labor task and hence their marginal productivity is higher, so wages rise. To explain the
negative correlation between the wage and the share of labor, note that due to FOC (3),
higher wages reduce the quantities produced of labor tasks so much, that the labor income
from each labor task falls as well, being equal to: 1Ywθθ
−− . Hence, total labor income is a
sum of income in all labor tasks and is equal to 1( )T M Ywθθ
−−− . As a result, the share of
labor is 1( )T M wθθ
−−− , which is negatively related to the wage rate. We therefore conclude
that if the share of labor converges to a positive number, it cannot raise wages too much,
so the number of labor tasks T – M must increase to raise wages. If the number of labor
tasks is larger, there are fewer workers in each such task and hence automation creates
smaller unemployment and the rate of unemployment becomes smaller over time. This is
therefore the connection between the share of labor and the rate of unemployment.
18
7. A Balanced Growth Path
One of the main stylized facts of economic growth is that the rate of growth of the countries
at the frontier has been rather stable over a long period. GDP per capita in the US has grown
at an average annual rate of 1.8 percent at least since 1870 and has been stable over most
of the period. This is why economists examine a balanced growth path, as an important
case in their growth models. This section analyzes this specific case as well, by assuming
that technology grows at a constant rate. In order to have a full numerical example, we first
assume that the cost of machinery, k, is equal to:
(21) ( ) (1 ) .k j a j α= +
In this case, the function φ is equal to:
(22) ( ) ( )1
1 11( ) 1 1 .
1M aR M
θ θ αθθ θ
θϕθ αθ
− − −− −
− = − + + −
Note that this function is bounded only if 1 0θ αθ− − < , or if (1 ) /α θ θ> − . We therefore
assume that this inequality holds. In this case the function φ satisfies:
( )11( ) .
1MM aR Aθθ
θϕθ αθ
−−
→∞
−→ =
+ −
Clearly, if a is sufficiently large, the limit A is smaller than 1.
We assume for the balanced growth case, that the rate of growth of automation is
constant over time:
.MM g
M∆
=
We also assume that the rate of growth of new jobs is constant over time as well:
.TT g
T∆
=
The two rates of growth are exogenous, but we show below that they are not arbitrary, but
have to satisfy some constraints.
We next use the condition for adopting automation, inequality (12), to establish a
relationship between the two rates of growth. If we substitute equation (22) and its
derivative into equation (12), we get the following inequality:
19
( ) ( ) ( ) ( )1 11 1 1 .t t t tT M aR M A M Aθ αθθ θ− − ≥ + + − + +
Taking this inequality to the long-run implies that the two growth rates of M and T must
satisfy:
.1T M Mg g gαθ
θ≥ >
−
Hence, the rate of growth of new jobs must be higher than the rate of automation, as implied
in the general case. We next assume for the sake of simplicity that:
.1T Mg gαθ
θ=
−
We can now return to equation (14), which describes the rate of growth of output
per worker, and substitute in it the rates of growth of M and of T. We get that in this
example, the rate of growth of output is equal to:
1
1
1 1 ( )ln ln( ) .(1 )(1 ) (1 )
MM aRy T M g
A M A M
θθ
αθθ
θθ θ
−−
−
−∆ = ∆ − +
− + + +
Over time, as M grows to infinity, the rate of growth of output converges to:
1ln .T MMy g gθ αθ→∞
−∆ → =
Hence, in the case of a balanced growth path of technology, the rate of growth of output
per worker converges to a constant long run rate as well, which is equal to Mgα . This
means that we have a balanced growth path in the long run, where the rate of growth of
output per worker is determined by the rate of growth of the number of jobs, which is
reflecting also the rate of growth of automation.
We next examine the rate of unemployment due to automation in this case of
balanced growth path. Combining inequality (20) with the function φ in (22), leads to the
following inequality:
20
(23) ( ) ( )
( ) ( )
1 11 1
11 1 1
11 1
11 1
( ) ( ) (1 )11 ( ) 1 1 1
1
( ) (1 ) .11 1 1
1
A t t t t tt M
t tt
tM
t
M M M aR M MU gM M aR M
aR MgaR M
θ αθθ θ
θ θ αθθ θ
θ θ αθθ θ
θ θ αθθ θ
ϕθϕ
θ αθ
θθ αθ
− −− −
− −− − −
− − −
− − −− −
− − −− −
′∆ +≤ =
−− − − + + −
+≤
− − − + + −
Note that the denominator is converging to 1 – A as time goes on, while the numerator
converges to zero, since (1 ) /α θ θ> − . Hence, the unemployment due to automation goes
to zero in this case of a balanced growth path as well.
8. Improved Machines and Disappearing Tasks In this section, we deal briefly with two possible critiques of our model. The first potential
critique is that in addition to automation and creating new jobs, technical change can also
enables us to reduce the cost of machines continuously, as in Sachs (2017) and as discussed
in Acemoglu and Restropo (2018a). In that case, the condition that the wage exceeds the
cost of machines in equation (11), becomes less stringent and it might even affect the
validity of Theorem 1. The second worry is that unlike our assumption that all tasks take
part in production, we might experience disappearance of some of the older tasks, as in
Acemoglu and Restropo (2018b).
We first turn to the first issue and assume that in addition to automation and new
jobs, technical change can also create new improved machines to perform the automated
jobs. To analyze the effect of such a change, assume that the capital cost of machinery k
declines. Without loss of generality, consider the case of the balanced growth path, which
is discussed in Section 7, and assume that the parameter a from equation (21) is declining.
Note that from equation (23) we get:
( ) ( )
11
11 1
(1 ) .1 1 1
1
A tt M
t
MU gaR M
θ αθθ
θ θ αθθ θ
θθ αθ
− −−
− −− −
+≤
− − − + + −
Hence, reduction of the cost of machines increases the bound on the RHS of this inequality,
so such technical change can potentially weaken our main argument.
21
We next show that this possibility is rather limited, as a cannot decline too much.
As equation (22) shows, a reduction of a increases the share of capital ( )Mϕ and if a
becomes too low, the share of capital might reach 1 and in this case the share of labor
becomes zero. The economy than reaches the case of no employment and production by
machines only, which is highly unrealistic. Hence, the cost of machinery a cannot fall by
too much and that means that the rate of unemployment still converges to zero over time,
so that the result of Theorem 1 still holds.
As mentioned in the introduction, Acemoglu and Restrepo (2018a, 2018b) assume
that in addition to automation and creation of new jobs, some of the older jobs are
disappearing. They assume that the disappearing jobs are exactly the set [ ]0, 1tT − , so that
the set of total tasks remains of size 1. This is impossible in our model, since 0θ > , so the
derivative of profits with respect to old tasks is positive and producers have an incentive to
use them. The second argument against disappearing tasks is more descriptive. We think
of tasks as listing the required needs of the human society at a given state of development.
While these needs grow, the previous ones do not usually disappear. We still grow wheat,
barley, olives and figs, as in ancient times. We still produce transportation, though
differently. We still enjoy poetry, art and theater. Tasks change, but do not disappear.
There are two more differences between our model and the Acemoglu and Restrepo
papers. The first is that they derive the rate of growth of M and T from profit maximization
of an R&D sector, while we impose less structure on it. The second difference is that they
assume that labor has increasing productivity over tasks. We examine this possibility in
Appendix D and find that even in this extension, the main result of the model, namely that
automation unemployment converges to zero, still holds.
9. Skill, Wages and Technical Change Finally, we add to our model of automation the issue of skill. The relationship between
skill and automation has received much attention in recent public debate.9 Many have
claimed that the recent automation, mainly by Artificial Intelligence, has negatively
9 Acemoglu and Restropo (2018a) discuss the issue of skill and its mismatch with automation as well. A recent discussion of skill in a model of automation appears also in Alesina, Battisti and Zeira (2018).
22
affected wages of unskilled and is therefore one of the main reasons for the decline of the
share of labor in recent decades. We build an extension of the model where skill bias
depends on the kind of new tasks created, but is not necessarily correlated with automation.
We show that in this case it does not affect the share of labor.
Assume that there are two skill levels, high and low. Assume that the number of
high skilled in the population is H and the number of low skilled is L, and 1H L+ = . For
simplicity, assume that these shares are constant over time. We also abstract in this section
from the issue of unemployment. Instead, we examine the effect of technical change on the
wages of high and low skilled, as this section focuses on labor income instead.
Labor tasks can be high or low skilled. There is an indicator function of tasks S,
that is equal to 1 if a task requires high skill and to 0 if it requires low skill labor. Denote:
(24) 1( , ) ( ) .
y
x
d x y S j djy x
=− ∫
Hence, d is the density of skilled tasks between x and y. The rest of the economy is as in
the benchmark model, so the prices of the intermediate goods satisfy:
1
0
( ) 1.tT
tp j djθθ
−− =∫
The supply prices of the intermediate goods, or tasks, are:
( ), if 0
( ) , if and ( ) 1.
, if and ( ) 0
tH
t t t tLt t t
Rk j j M
p j w M j T S j
w M j T S j
≤ ≤
= ≤ ≤ = ≤ ≤ =
We further assume that all automations are adopted, so that:
(25) ( ).Lt tw Rk M≥
Clearly, this condition holds also for the wage of high skilled, as it is higher.
We next calculate the skill premium, which is the ratio between wages of high and
low skilled. Substituting the supply prices in the equilibrium condition, we get:
1 11 ( ) ( ) ( ) ( ) [1 ( )] .t t
t t
T TH L
t t tM M
M w S j dj w S j djθ θθ θϕ
− −− −= + + −∫ ∫
Some calculation leads to:
23
(26) ( ) ( ) [ ]1 11 ( ) ( , ) 1 ( , ) .H Ltt t t t t t
t t
M w d M T w d M TT M
θ θθ θϕ − −− −−
= + −−
The first order conditions of tasks imply that skilled non-automated tasks satisfy:
(27) ( ) ( ) ( ) ( )1 1
1 1( ) .,
H Ht t t t
t t t t
HY x j w wT M d M T
θ θ− −= =−
Similarly, each low skilled non-automated tasks satisfy:
(28) ( ) ( ) ( ) ( )1 1
1 1( ) .1 ,
L Lt t t t
t t t t
LY x j w wT M d M T
θ θ− −= = − −
Dividing (27) by (28) we get:
(29) ( )1
1 ,.
1 ( , )
Ht tt
Lt t t
d M Tw Lw d M T H
θ− = −
From this equation we see that the skill premium, H Lt tw w , increases with d. It
means that if the density of high skilled jobs among the new jobs increases, the skill
premium increases as well. If the number of high skilled workers increases, the skill
premium declines. Hence, the rise in the skill premium in the last three decades, despite
the parallel rise in education, can be attributed to a rise in the share of skilled tasks among
new jobs.
To simplify notation we use from here on, ( , )t t td d M T= . From equations (26) and
(29) we can solve the two wage levels, of high and low skilled. The wage of low skilled is:
(30)
11
1 1 .1 ( )
L t t tt t t
t t
T M d Hw d dM d L
θθ θ θθ
ϕ
−− − − = + − −
The wage of high skilled is:
(31)
11
(1 ) .1 ( ) 1
H t t tt t t
t t
T M d Lw d dM d H
θθ θ θθ
ϕ
−− − = + − − −
We can now rewrite condition (25) for technology adoption in the following way and get:
( ) 11 1 .1 ( )
t tt t
t t t t
M d H d dT M M d L
θϕϕ
′ − ≤ + − − −
24
Hence, the dynamic condition imposed by the model on the number of labor jobs remains
similar to that in the benchmark case.
Finally, note that equations (27) and (28) enable us to calculate the share of labor
in output:
( ) 1 1(share of labor) ( ) (1 )( ) .H L
H Lt tt t t t t t t
t
w H w L T M d w d wY
θ θθ θ
− −− −
+= = − + −
Applying (30) and (31) we get:
(32) (share of labor) 1 ( ).t tMϕ= −
This means that the share of labor in output does not depend at all on d, namely on the
density of high skilled tasks in the economy.
Hence, this extension of the model shows that if the skill bias is in the new jobs
created, but is not related to automation, the skill bias has no effect on the share of labor,
even if it reduces the wage of low skilled workers. Hence, SBTC cannot explain the decline
of the share of labor in recent decades. Understanding this decline requires other
explanations, like a rise in monopoly power, as suggested by Barkai (2017) and by Autor,
Dorn, Katz, Patterson and Van Reenen (2017), or a rise in monopsony power in labor
markets as suggested by Benmelech, Bergman and Kim (2018). Recently, Autor and
Salomons (2018) discuss how much of the decline in labor share can be attributed to
automation and how much to other factors.
The main result of this Section depends on our assumption that demand for skill is
not correlated with automation. We leave examination of this assumption to future
research, but just note that it is a reasonable assumption. It is possible that the determination
whether a new task is high or low skilled, might not always reflect technological
considerations, but social considerations as well. For example, if the workers in a certain
area prefer to interact with highly educated people, it might affect the definition of the job.
10. Summary This paper presents a simple model that enables us to analyze the relationship between the
long-run process of automation and the short-run phenomenon of unemployment. Our
model implies that technological unemployment, which is created by automation, is
actually declining over time and converges to zero in the long run. This is a surprising
25
result and its intuition depends on the need of wages to be sufficiently high in order to
enable continuous adoption of automation technologies. The wage depends on having
sufficient number of new tasks created. As the number of new tasks rises sufficiently fast,
sooner the share of jobs that are automated each period will be relatively small and as a
result, the rate of unemployment declines as well.
The model supplies a strong result and that is a good reason to treat it with some
caution. In general, economic models should help us to explain processes and mechanisms
and dwell less on predicting future developments. We wish to treat this model in a similar
way. Its main message is that although automation causes unemployment, by turning labor
tasks into mechanized tasks, it might also ignite a mechanism that reduces this
unemployment. We show that automation also contributes to creation of new labor tasks,
to produce and to maintain the new machines. If the number of the new tasks grows
sufficiently fast, the labor jobs will absorb automation better and better, since automation
will affect a smaller and smaller share of this growing set of tasks. This is a point to bear
in mind when we consider the effect of automation on unemployment and on the labor
market in general.
The prediction that automation unemployment is converging to zero should be dealt
with caution for an additional reason, as it relies strongly on the assumption that the share
of labor in output does not converge to zero over time. This sounds like a reasonable
assumption, as until now shares of labor have been quite stable over time and across
countries, at an average level of two thirds. However, we should be aware that the
experience of the last two centuries might not continue in the far future, as no one knows
for sure how the future will look like. If indeed the share of labor will decline and even go
all the way to zero, it might as well happen that the unemployment due to automation will
not converge to zero. We shall wait and see… We actually view this paper as pointing at
the connection between the two seemingly separate processes.
26
Appendix
A. Proofs
Proof of Proposition 1:
The proof for cases (a) and (b) follows immediately, as a result of the monotonicity of the
function ψ. The proof for case (c) follows from Figure A.1:
Figure A.1
From this Figure we learn that if the bound on tasks T is below M*, then automation stops
at a level below T, and there is therefore production of many tasks by labor. In this case,
growth stops at a final level of output and output per worker. The long run share of labor,
denoted by sL in the figure, is positive in this case as well. If T keeps growing and passes
M*, then automation reaches M* and stops there. In this case, the share of labor is zero,
wages are infinite and there is no production by labor, namely no production of
intermediate goods beyond M*. Output in this equilibrium is finite, and so is output per
capita, but output per worker is infinite, as no one works.
QED.
Proof of Proposition 2:
The share of labor converges to a positive number:
M M* ψ(T)
φ
T
1
sL
27
1 ( ) 0.t tM Bϕ →∞− → >
Hence, we get:
[ ]ln 1 ( ) ln .t tM Bϕ →∞− → > −∞
If a sequence converges to a finite number, its differences must converge to zero. Hence:
[ ] 1
1
( )ln 1 ( ) 0.1 ( )
t tt t
t
M MMM
ϕϕϕ
−→∞
−
′− ∆∆ − = →
−
QED.
Proof of Proposition 3:
The first part of the proposition, on bounded T, follows immediately from Proposition 1.
To prove the second part, note that inequality (12) implies that:
1 .1 ( ) ( )
t t
t t
T MM Mϕ ϕ
−≥
′−
Hence, using equation (9) we get:
1
1 .( )t
t
wM
θθ
ϕ
−
≥ ′
Using equation (14) we get:
(A.1) [ ][ ]
11 .
1 ( ) 1 ( ) ( )
tt
tt t
wyM M M
θθϕ ϕ ϕ−= ≥
− ′−
If T is unbounded, φ is bounded by 1, due to Proposition 1. Hence, its derivative converges
to zero as M goes to infinity. Inequality (A.1) implies that output per worker is unbound.
QED.
B. Adding Capital to Production by Labor
We construct an extension of the benchmark model, by assuming that production by labor
of intermediate good ( , ]t tj M T∈ requires not only one worker, but also a structure of size
s. Assuming that this capital also fully deteriorates within one period, the price of such an
intermediate good becomes:
( ) .t tp j w Rs= +
28
Substituting the price in the equilibrium condition (5), we get that the wage is equal to:
(A.2)
1
.1 ( )
t tt
t
T Mw RsM
θθ
ϕ
−
−= − −
We next calculate the share of labor in output in this economy in a similar way to the
calculation of equation (10) and get:
(A.3) [ ]11
1 ( ) 1 ( ) ( ) .t tt t t t
t
w E M Rs M T MY
θθθϕ ϕ−
= − − − −
This equation implies that even at the start of automation (in the beginning of the
industrial revolution), when 0tM = and ( ) 0tMϕ = as well, the share of labor is lower
than 1. This is of course the result of using capital for structures, even before automation.
We next show that as automation proceeds and is unbounded, the share of labor in output
still converges to 1 – A, as in the benchmark model. To see this consider the condition for
adopting automation:
( ).t tw Rs Rk M+ ≥
Applying (A.2) to condition, we get:
(A.4) [ ]1 1
( ) ( ) 1 ( ) .t t t tT M Rk M Mθ θθ θϕ− −
− ≥ −
Substituting inequality (A.4) into equation (A.3) leads to:
(A.5) [ ]11 1 ( )1 ( ) 1 ( ) ( ) .
( )t t t
t t t tt t
w E MM Rs M T M sY k M
θθθ
ϕϕ ϕ− −
− − = − − ≤
The RHS of (A.5) is monotonically decreasing and it converges to zero as M goes to
infinity. Hence, the share of labor converges to 1 – A if automation is unbounded. Hence,
the share of labor fluctuates between a number lower than 1 and 1 – A. Hence it fluctuates
less than in the benchmark case.
C. Elasticity of Substitution between Labor and Capital
From combining equations (15) and (17), we get that the capital labor ratio in the economy
is equal to:
(A.6) ( ) .1 ( )
t t t
t t
K w ME R M
ϕϕ
=−
29
Hence, the logarithm of the capital labor ratio is:
( )ln ln ln .1 ( )
t t t
t t
K w ME R M
ϕϕ
= + −
Hence, the elasticity of substitution between capital and labor is equal to:
(A.7) ( )( ) [ ] ( ) [ ] ( )2 2
ln ( ) ( )1 1 .ln ln ln1 ( ) 1 ( )
t t t t t t
t t tt t
K E M M M Mw R w R w RM M
ϕ ϕϕ ϕ
∂ ′ ′∂ ∂= + = +
∂ ∂ ∂− −
Note that the second addend in the RHS of (A.7) is equal to zero if the rate of automation
is fully exogenous and does not react to the wage at all. Even if the rate of automation
reacts to wages, equation (A.7) implies that in the long run the elasticity converges to 1.
D. Growing Productivity of Labor
In this appendix we assume that the productivity of labor is not constant, but it increases
with tasks, namely the amount of workers required to produce one unit of intermediate
good j is equal to ( )n j , where n is a decreasing function. As a result, the price of a labor
task is:
(A.8) ( ) ( ).t tp j w n j=
Substituting these prices into the equilibrium condition (5) in the benchmark model, which
holds here as well, leads to the following equilibrium wage rate:
(A.9)
1
( , ) ,1 ( )
t tt
t
N M TwM
θθ
ϕ
−
= −
where we use the following notation:
1( , ) ( ) .t
t
T
t t MN M T n j dj
θθ
−−= ∫
We next turn to calculate the share of labor in this extension of the model. Note that
from the first order condition of the final good (3) we get in this case:
1 1
1 1( ) ( ) .t t tx j Y w n jθ θ− −− −=
Hence, the amount of labor producing good j is equal to:
1
1 1( ) ( ) ( ) ( ) .t t t tl j x j n j Y w n jθ
θ θ− −− −= =
30
Summing up the amount of labor for all labor tasks, we get that aggregate employment is
equal to:
1
1( ) ( , ).t
t
T
t t t t t tME l j dj Y w N M Tθ
−−= =∫
We therefore get, using also equation (A.8), that the share of labor in output is:
(A.10) 1 ( , ) 1 ( ).t tt t t t
t
w E w N M T MY
θθ ϕ
−−= = −
Hence, the share of labor in this extension is the same as in the benchmark model.
We next turn to the condition for adoption of automation, which is equal in this
extension to:
(A.11) ( ) ( ).t t tw n M Rk M≥
Raising by the power of / (1 )θ θ− we get from (A.11):
(A.12) 1 1 ( )( , ) ( ) .( )
tt t t
t
MN M T n MM
θθ ϕ
ϕ− −≥
′
Note that the upper bound of automation unemployment is:
1
11 1
1 1 1( ) ( ) .t
t
MAt t t t t tM
U l j dj M n M Y wθθ θ
−
− −− −
− − −= ≤ ∆∫
Using (A.9) we get:
1
11
1 1 1 1
( )( ) .( , ) ( , )
A t tt t t t
t t t t
E n MU M n M MN M T N M T
θθ θθ
−− −
−−
− − − −
≤ ∆ ≤ ∆
Applying inequality (A.12) we get:
(A.13) 1
1 1
1
( ) ( ) .1 ( ) ( )
A t t tt
t t
M M n MUM n M
θθϕ
ϕ
−− −
−
′∆≤ −
We next claim that the RHS of (A.13) converges to zero. If not, 1( ) / ( )t tn M n M− must rise
to infinity, as it multiplies a sequence that converges to zero. However, 1( ) / ( )t tn M n M−
cannot go to infinity, since if we multiply the elements of this sequence over a long period
from t to t + m we get 1( ) / ( )t t mn M n M− + that is bounded. Hence, the rate of technological
unemployment converges to zero as well.
31
E. Unemployment Due to Mismatch and Automation
In this Appendix we extend the model to include unemployment due to mismatch. Assume
that agents belong to overlapping generations. Each one lives L periods, supplying one unit
of labor in each period. Each generation is a continuum of size 1/L, so the overall
population is 1. In the analysis below we assume for simplicity that the number of cohorts
is L = 2. Workers try to work in each period but might find themselves unemployed during
that period for two reasons, one is mismatch between the worker and the job and the other
is due to losing their job to automation.
We assume that workers can fit a specific task or not. If they fit, it becomes their
job, while if they do not fit, they lose this job, remain unemployed for one period and then
search in the next period for another job. This is mismatch unemployment. We assume that
the probability of not fitting a job is q, and it is equal across workers and across tasks.
Workers can also be unemployed if their task becomes automated in the second period.
Assume that the information that a task is automated arrives at the beginning of the period,
so that workers, who search for a new job, do not try this job. Only those who worked in
the job in the past need to leave it and search for a new job, so that q of them become
unemployed. Hence, unemployment due to automation hits older workers only.
There are three groups of unemployed in the population. The first group are young,
who enter the job market for the first time and some of them find themselves mismatched
to the job they try. Their number is: / 2q . The young do not suffer from unemployment
due to automation. The second group of unemployed are older workers, who have not
found a job yet, and are still looking for it. They sample new jobs in their second period of
life and q of them suffer from mismatch. Their total number is 2 / 2q .
The third group of unemployed are people who were matched well and worked in
the previous period, but the tasks they found become automated in the present, in period t.
Once they leave their job due to automation, they search for another job and q of them do
not find one. The number of unemployed of this third group, of older workers, is denoted A
tU . This is the rate of unemployment due to automation.
The number of the third group of unemployed due to automation, is the number of
young people who were employed in period t – 1 in the jobs that later become automated
in period t, when the workers are already old. Notice that young workers are not distributed
32
equally across jobs, since the set of jobs is increasing and every period new jobs are added.
The new jobs don’t have older workers from the past, so they employ more young workers.
The number of old people in each job affects the number of young in the job and that
correlation has a result that the distribution of old and young in a job is not equal across
jobs. This makes the calculation of the size of unemployment due to automation quite
complicated. As an alternative, we can try to find an upper bound to this unemployment.
The unemployed due to automation in period t are people who worked as young in
these jobs in period t – 1. The number of young workers in period t – 1 in job j, which we
denote by 1( )ytl j− , is bounded from above by the total number of workers in the job. This
number is equal across jobs and hence we get:
(A.14) 11
1 1
( ) .y tt
t t
El jT M
−−
− −
≤−
Hence, the total number of unemployed due to automation in period t satisfies:
(A.15) 1
11
1 1 1 1
( ) .t
t
MA y t t t
t tt t t tM
E M MU q l j dj q qT M T M
−
−−
− − − −
∆ ∆= ≤ ≤
− −∫
Note that due to restriction 3, this inequality can be written in the following way:
(A.16) 1
1 1 1
( ) .1 ( )
A t t tt
t t t
M M MU q qT M M
ϕϕ
−
− − −
′∆ ∆≤ ≤
− −
This upper bound on unemployment caused by automation shows that the main result of
the paper, in Theorem 1, holds in this extension as well.
Note that in this case the total rate of unemployment, Ut, satisfies:
2
.2 2t tq qU →∞→ +
This means that the long run rate of unemployment is due to mismatch only and not to
automation.
33
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