ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E21-5

Robots and Unemployment
by
Noritaka Kudoh
Hiroaki Miyamoto

June 2021

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Robots and Unemployment
Noritaka Kudoh

Hiroaki Miyamoto†

Nagoya University

Tokyo Metropolitan University
June 10, 2021

Abstract
This paper studies the impact of the robotics revolution on the labor market outcomes through the lens of capital-augmenting technological progress. We develop asearch-matching model with labor market segmentation and multi-factor production
to find the condition under which the new technology harms the labor market. For
technology optimism, the elasticity of substitution between routine labor input androbots must be less than 2.5. The robotics revolution hits the labor market for routinetask intensive jobs harder under a more generous unemployment policy. Automationof abstract tasks may cause a disaster for those who are reallocated to routine-task
intensive occupations.
JEL classification: E32, J20, J64.
Keywords: robots, capital-augmenting technological progress, search-matching frictions, unemployment, routinization.

E-mail: kudoh@soec.nagoya-u.ac.jp† E-mail: hmiyamoto@tmu.ac.jp1

1

Introduction

Recent advances in robots and artificial intelligence (AI) technologies create widespread
concerns that the robotics revolution will eventually make human obsolete in the processof production (Brynjolfsson and McAfee, 2014; Ford, 2015; Harari, 2016; Frey and Osborne, 2017; Baldwin, 2019). In response to these concerns, there is a growing numberof formal economic analyses regarding the impact of the robotics revolution on the labor market (Acemoglu and Restrepo, 2018b; Berg et al., 2018; Graetz and Michaels, 2018;
Caselli and Manning, 2019).
An important common element in the recent theoretical literature is that the firm’s
automation decision is explicitly modeled, making one of the shape parameters in the traditional production function endogenous (Acemoglu and Restrepo, 2018b). While thisapproach helps us organize the likely channels through which the robotics revolution
affects the labor demand, the existing models lack testable predictions or concrete conditions under which the new technology harms the labor market (Caselli and Manning,2019). Further, task-based models with automation choice require new deep parameters
to be estimated, and this imposes a serious restriction on making quantitative predictions.
To provide robust predictions, Berg et al. (2018) choose to study standard dynamic
general equilibrium models with robots as equipment capital (Greenwood et al., 1997;
Krusell et al., 2000; Autor et al., 2003; Autor and Dorn, 2013). While automation oftasks is not explicitly modeled, Berg et al. (2018) consider capital-augmenting technological progress (CATP). Berg et al. (2018) exploit their standard model structure to obtaina set of realistic parameters, and quantitatively show that if advances in robots decrease
the marginal product of labor, then there is a series of short-run equilibria in which an increase in the productivity of robot causes the demand for labor to decline, and the periodof short-run pain may take generations.
An important shortcoming of the existing models of CATP is that the labor market isassumed to be perfectly competitive, ruling out unemployment as a result of factor substitution. In this paper, we study the impact of robotics revolution on the labor market
outcomes through the lens of CATP in the standard search-matching model of the labormarket (Diamond, 1982; Mortensen and Pissarides, 1994; Pissarides, 2000), or the DMP
model. This issue has not been fully explored in the literature and we aim to fill this gap.
As a preliminary analysis, we first study the impact of CATP in the context of the
textbook search-matching model with homogeneous workers to ask whether the robotics
revolution can cause joblessness for all workers. We show that CATP reduces unemploy2

ment and increases the wage rate in the long run because CATP cannot reduce the marginal product of labor when there are two inputs and the production technology exhibits
constant returns to scale. Thus, robots cannot be a threat for all workers. This result is
consistent with Caselli and Manning (2019).
We then present our main model, in which there are two types of workers, skilled
and unskilled, and there is a distinct labor market for each skill level. Labor markets
are segmented in that skilled workers participate in the labor market for abstract tasks
whereas unskilled workers participate in the labor market for routine tasks. We show that,with worker heterogeneity and labor market segmentation, CATP can reduce the demand
for routine labor input.
Further, we find and quantify a condition under which CATP increases unemployment
for the unskilled in the long run. Namely, whether the robotics revolution results in technology optimism or technology unemployment depends on whether the elasticity of substitution between routine labor and capital is below or above some threshold level. Ourquantitative model implies that this threshold is 3.91 for the United States.
In our quantitative analyses, we focus on a hypothetical scenario in which the productivity of robots doubles. In our optimistic case when the elasticity of substitution is2.5, which is below the threshold, job creation and the wage rate increase for all types
of workers and the overall unemployment rate decreases. However, the skill premiumincreases and the labor share decreases. In our pessimistic case when the elasticity of
substitution between routine labor and capital is 5 or 10, routine workers face declines inboth job-finding rate and wage rate, and the overall unemployment rate rises. Even inthese pessimistic cases, the magnitude of job loss is quantitatively very small if robots andtraditional capital are pooled as a single input of production.
We extend our basic model to include traditional capital as a distinct input. While capital heterogeneity itself does not cause technology unemployment, it reduces the threshold
elasticity for technology optimism to arise because an investment surge absorbs part ofthe increase in the robot productivity. In this model, the robotics revolution increases thedemand for routine labor if and only if the elasticity of substitution between routine laborand capital is below 2.52. Further, the magnitudes of the impacts of the robotics revolution on the labor market variables are strikingly high. When the elasticity of substitution
between routine labor and capital is 10, CATP increases the unskilled unemployment ratefrom 8.3% to 43.7%.
A policy implication obtained from our model is similar to those found in Mortensen
and Pissarides (1999) and Hornstein et al. (2007). We find that the robotics revolution
3

hits the labor market for routine-task intensive occupations harder under a more generous unemployment policy. When the elasticity of substitution between routine labor andcapital is 10, the monthly job-finding rate for the unskilled falls from 0.42 to 0.05 under
our baseline value of the unemployment benefit replacement rate of 0.25. However, when
the unemployment benefit replacement rate is set 20% higher than our baseline level to be
0.30, it falls from 0.37 to 0.02.
Because our assumption that abstract tasks cannot be automated is controversial (Brynjolfsson and McAfee, 2014; Ford, 2015), we consider the possibility that some of the abstract occupations are subject to automation. In particular, we adopt Frey and Osborne’s(2017) machine learning outcome that 47% of the occupations currently exist in the U.S.
are in the category of high risk of being computerized to consider a scenario in which 47%of the skilled workers are reallocated to the labor market for routine jobs. We show thatcomputerization of abstract tasks may cause a disaster in the routine labor market by reducing the monthly job-finding rate to 0.01 if the elasticity of substitution between routine
labor and robots is 10.
In many scenarios we consider, we find asymmetry in the two labor markets. In thelabor market for abstract-task intensive jobs, the adjustment is mainly through the wage
rate: CATP significantly increases the wage rate while the job-find rate does not changemuch because the additional supply of skilled workers is limited as they are mostly on
the job. On the other hand, in the labor market for routine jobs, the adjustment is mainly
through job creation: CATP significantly reduces the job-finding rate while the wage rate
does not decline much because workers have outside options in wage bargaining.
Cortes et al. (2017) and more recently Jaimovich et al. (2020) argue that workers withroutine-task-intensive occupations are likely to exit from the labor market in response to
the deterioration of the prospect of such occupations. To assess the importance of the labormarket participation margin, we further extend our model in line with Tripier (2004) andVeracierto (2008). We find that part of the upward pressure on the unemployment rate
goes to the participation margin, and its magnitude is up to one half of the overall change
in the unskilled unemployment rate. However, if the household’s income declines byautomation, then the income effect operates to increase the labor force participation rateeven when the job-finding rate for routine jobs becomes extremely low.
This paper is closely related with the literature on routinization and labor market polarization (Autor et al., 2003; Goos and Manning, 2007; Autor and Dorn, 2013; Cortes etal., 2017; vom Lehn, 2020). In particular, our analysis builds on their conceptual framework such as routine versus abstract tasks in production. While this literature focuses on
4explaining the decline in the middle of wage distribution, our model has only two skill
types. This modeling choice is made for quantifying our model within a standard set ofparameters and calibration targets. However, we note that our experiment on the decline
in the proportion of the skilled can be interpreted as the decline in the middle, because theskilled workers in our model include those in the middle of income distribution.
The analytical framework of paper builds on the recent development in the field of
dynamic general equilibrium with search-matching frictions in the labor market. One
technical challenge in integrating search-matching frictions into our rich production technology is that a firm employing a large number of employees takes into account of its size
when negotiating with its workers over the wage rates. To avoid the potential complication that is not the center of our investigation, we adopt a vertically integrated market
structure commonly employed in the recent business cycle literature (Trigari, 2009; Christiano et al., 2016). This modeling device allows us to study the labor market equilibrium
conditions for a given set of competitive factor prices.
The idea that technological progress may destroy jobs has been extensively studied in
the creative destruction literature (Aghion and Howitt, 1994; Mortensen and Pissarides,
1998; Hornstein et al., 2007; Pissarides and Vallanti, 2007). In this literature, jobs are destroyed when the technology embodied in capital becomes obsolete and unemployment
occurs because workers switch from one job to another with a newer technology. Because
the production unit is a capital-worker pair, technological progress itself is accompaniedwith job creation. In contrast, we are more interested in the scenario in which technological progress permanently changes the composition of labor demand.
This paper is organized as follows. Section 2 presents a preliminary analysis basedon a two-factor production function with search-matching frictions in the labor market
to highlight the importance of worker heterogeneity. Section 3 presents our main modelwith worker heterogeneity and labor market segmentation. Section 4 conducts quantitative analyses, and we apply this quantitative model to obtain some policy implications in
Section 5. Section 6 explores the possibility that abstract tasks are automated. Section 7
extends our main model to consider labor force participation. Section 8 concludes. Proofs
and additional results are found in the Appendix.

52

Homogeneous Workers

This section presents a preliminary analysis to clarify the impact of advances in robotics
on the aggregate labor market in a standard search-matching model. We introduce thefollowing three (minimum) components into the textbook DMP model. First, production
requires both labor and capital. Second, technological progress is capital-augmenting.
Third, capital is supplied by the representative household. Within this simple framework,
we ask whether CATP can cause persistently high unemployment.

2.1

Environment

To bypass the issue of intra-firm bargaining between a firm and a continuum of workers,
we assume a vertical market structure commonly employed in the business cycle literature. There is a single consumption good, produced by the representative final good
producer. Production of the final consumption good requires capital and intermediate
inputs (or tasks) supplied by intermediate-good firms.1 Technological progress is capitalaugmenting: yt = F ( xt , at k t ), where yt is output, xt is the amount of tasks supplied byintermediate-good firms, k t is the stock of capital, and at is the level of capital-augmenting
technology. We assume that the production function satisfies the neoclassical properties:
F1 > 0, F2 > 0, F11 < 0, F22 < 0, and the Inada conditions (limx!0 F1 = ∞, limak!0 F2 = ∞,limx!∞ F1 = 0, and limak!∞ F2 = 0). We also assume that F satisfies constant returns
to scale (CRS). The assumptions of CRS and decreasing marginal products jointly imply
F12 (= F21 ) > 0.2
It is assumed that the final-good firm behaves competitively. Thus, the firm chooses its
demand for xt and k t to maximize F ( xt , at k t ) pt xt rt k t , where pt is the price of a unitof task (ultimately performed by workers) and rt is the rental price of capital. Thus, theinput prices are determined as their marginal products:
pt = F1 ( xt , at k t ) = F1 (1, Kt ) ,(1)

rt = F2 ( xt , at k t ) at = F2 (1, Kt ) at ,(2)

1 Each intermediate-good firm acts as an intermediary and plays two important roles. One is to find aworker. The other is to bargain with its employee over the wage rate.
2 Because F is homogeneous of degree 1, F is homogeneous of degree 0, which implies F ( x , x ) x +1
11 1 2 1
F12 ( x1 , x2 ) x2 = 0, from which F12 ( x1 , x2 ) x2 = F11 ( x1 , x2 ) x1 > 0. By symmetry of partial derivatives,
we also obtain F21 > 0. We thank Kenji Yamamoto for his comments on the properties of homogeneousfunctions.

6where Kt = at k t /xt is the effective capital-labor ratio. Note that F1 and F2 are homogeneous of degree zero.
Consider an infinite-horizon economy with a continuum of ax-ante homogeneous individuals. Total labor force is normalized to 1. Ex-post, individuals are either employedor unemployed. The unemployment rate is denoted by ut . As in Merz (1995) and Andolfatto (1996), all individuals are members of a single representative household and there isperfect risk sharing among the members. The primary role of the household is to supply
capital. The household’s lifetime utility is given by
∞

∑ βt

t =0

Ct1 σ
,
1 σ

(3)

where β is the household’s subjective discount factor, Ct is the level of consumption, whichis common for all individuals, and σ is the elasticity of intertemporal substitution (whichcorresponds to the relative risk aversion in a stochastic environment). The budget constraint is Ct + τt + k t+1 (1 δ)k t = wt (1 ut ) + b̄ut + rt k t + πt , where πt is the profit
from ownership of all firms, τt is lump sum taxes to the government, k t is the stock of capital, δ is the capital depreciation rate, wt is the wage rate, b̄ is the unemployment insurance
benefits provided by the government, and rt is the rental rate of capital. The householdchooses sequences of Ct and k t+1 to maximize (3) subject to the budget constraint. The
first-order conditions imply Ct σ = Λt and Λt = βΛt+1 (rt+1 + 1 δ), with the transversality condition, limt!∞ Λt βt k t = 0, where Λt is the Lagrange multiplier for this problem.
When making decentralized decisions, each individual discounts future payoffs by
Bt =1
1 + r t +1

δ=β

Λ t +1
=β
Λt

Ct+1
Ct

σ

,

(4)

which is a deterministic version of the stochastic discount factor. Note that B = β holds
in any steady state.

2.2

Labor Market

We assume search-matching frictions in the labor market. Total number of matches madein period t is given by a constant-returns-to-scale matching technology m(ut , vt ), where
ut is the number of job-seekers and vt is the number of vacancies. To ensure the vacancyfilling probability and the job-finding probability to be in between 0 and 1, we adopt the
specification of den Haan et al. (2000) and Hagedorn and Manovskii (2008):uv,(5)m(u, v) = ψ(u + vψ )1/ψ
7

where ψ is a parameter.
The vacancy-filling rate, the probability at which a vacancy is filled in period t, isψ
qt = m(ut , vt )/vt = (1 + θt ) 1/ψ , where θt = vt /ut is labor market tightness. Similarly,
the job-finding rate, the probability that a job-seeker is employed in the next period, isgiven by Qt = m(ut , vt )/ut = θ (1 + θ ψ ) 1/ψ = θt qt . The matching elasticity with respect
to unemployment is given by θ ψ /(1 + θ ψ ) and the matching elasticity with respect tovacancies is given by 1/(1 + θ ψ ).
Separations are exogenous. At the end of each period, a current production unit separates with probability λ. Thus, the equilibrium unemployment rate evolves accordingtou t +1 = u t + λ (1 u t ) Q t u t .
(6)
The values of employment (Wt ), unemployment (Ut ), a filled job (Jt ), and a vacancy (Vt )are given by Wt = wt + λBt Ut+1 + (1 λ) Bt Wt+1 , Ut = b + Qt Bt Wt+1 + (1 Qt ) Bt Ut+1 ,Jt = pt wt + λBt Vt+1 + (1 λ) Bt Jt+1 , and Vt = c + qt Bt Jt+1 + (1 qt ) Bt Vt+1 , where cis the cost of posting a vacancy. The flow value of nonwork b is the sum of unemployment
insurance benefits b̄ and the saved disutility of work in units of consumption (Hall and
Milgrom, 2008).
As in Pissarides (2000), we assume free entry of vacancies, which implies that the equilibrium number of vacancies is determined so that Vt = 0 holds for all t. This yields thefollowing job creation condition:c= Bt pt+1
qt

w t +1 +(1λ) cq t +1.

(7)

The wage rate is determined through bilateral wage bargaining. As in Pissarides (2000),
the wage rate is determined so that η ( Jt Vt ) = (1 η )(Wt Ut ) is satisfied for eachperiod, where η is the worker’s bargaining power. Substitute the value functions and thefree entry condition into the bargaining outcome to yield the wage equation:w t = η p t + (12.3

η )b + ηcθt .

(8)Equilibrium

Because each intermediate firm can hire at most one employee and each employee supplies one unit of labor service, the amount of intermediate goods (or, tasks) supplied
equals the number of employed individuals. Thus, xt = `t = 1 ut .
8

The aggregate profit of all firms is given by πt = yt wt (1 ut ) rt k t cvt . The
government finances unemployment insurance benefits by lump-sum taxes. Thus, τt =ut b̄. Eliminate πt and τt from the household’s budget constraint to obtain the resource
constraint as
Ct + k t+1 (1 δ)k t = yt cvt .
(9)A perfect-foresight equilibrium is determined by (1), (2), (4), (6), (7), (8), (9), and Kt =at k t /`t . Given a long-run level of productivity at = a, we can determine a steady-state ofthis economy as the solution to the following system of equations:r=

1β

1 + δ = F2 (1, K ) a,(10)p = F1 (1, K ),
β 1

1+λ c
= (1
q

(11)

(1

η) p

w = η p + (1

η )b

ηcθ,

η )b + ηcθ,

(12)
(13)

with K = ak/` and u = λ/(λ + Q). Given a level of a, (10) determines the steady-statelevel of K. Given K, (11) pins down the level of p. Thus, from (10) and (11), we canimplicitly define p = P ( a). Given this, (12) and (13) determine θ and w.
Lemma 1 P0 ( a) > 0.
Proof. Totally differentiate (10) and (11) to obtain F2 da + aF22 dK = 0 and dp = F12 dK,from which dp/da = ( F12 F2 /aF22 ) > 0 because F12 > 0.
Proposition 1 In any steady state, CATP decreases u and increases w.
Proof. A steady-state is determined by
β 1

1+λ c
= (1
q

η ) P ( a)w = ηP ( a) + (1
λ
u=
.
λ+Q

(1

η )b

η )b + ηcθ,

ηcθ,

(14)
(15)
(16)

Because P0 ( a) > 0, an increase in a increases both θ and w in the long run.
Thus, if workers are homogeneous and the robotics revolution can be expressed ascapital-augmenting technological progress, then the robotics revolution unambiguously
9creates jobs and increases the wage rate in the long run. To put it differently, robots cannot
be a threat for all workers in the long run.
Our next question is whether there is short-run pain for workers. The mechanism that
potentially decreases job creation in the short run is through the discount factor,
Bt =1
1 + F2 (1, Kt+1 ) at+1

δ

,

(17)

which is the inverse of the real interest factor. If the discount factor decreases significantly
in response to the robotics revolution, then it decreases job creation as it is interpreted as
the reverse of the capitalization effect (Pissarides, 2000).
We argue that this effect should not be large enough to increase the unemployment
rate. To see this, suppose for now that σ = 0 (utility is linear in consumption), in whichcase we obtain Bt = β and rt = β 1 1 + δ for all t. Then, the effect through the discountfactor disappears and the equilibrium conditions become

(1

λ) c

c
= β P ( a t +1 )qt

w t +1 +wt = ηP ( at ) + (1η )b + ηcθt ,(19)

Qt ut .

(20)

u t +1 = u t + λ (1ut )q t +1,(18)

In this case, ut unambiguously decreases even in the short run. This result implies thatfor a short-run pain to arise, the curvature of the utility function should be very large. If
this is the case, then through (the reverse of) the capitalization effect, the increase in thereal interest rate reduces job creation because the opportunity cost of creating a vacancy
is higher when the rate of returns on capital is higher.

3

Heterogeneous Workers

The previous section clarifies that in the textbook DMP model with homogeneous workers, CATP stimulates job creation and increases the wage rate. In this section, we explore
the impact of the robotics revolution on the distribution of jobs and wages. For this purpose, we consider worker heterogeneity and labor market segmentation. The key ingredient is a class of constant elasticity of substitution (CES) production technology consideredby Krusell et al. (2000) and in particular Berg et al. (2018).

10

3.1

Environment

There are two types of workers: type-A and type-R. Type-A workers have skills capableof performing abstract tasks, which are hard to be automated. In contrast, type-R workers
have skills capable of performing routine tasks, which are relatively easy to be automated.
The population share of type-A workers is φ and the remaining proportion 1 φ is typeR. The number of employed workers of type i and the number of job-seekers of type i aredenoted by `it and uit , respectively. Thus, φ = `tA + utA and 1 φ = `tR + utR .
Consider a production function with three factors: F (`tA , `tR , at k t ). We assume that the
production function satisfies the neoclassical properties: F1 > 0, F2 > 0, F3 > 0, F11 < 0,
F22 < 0, F33 < 0, and the Inada conditions. We also assume that F satisfies constantreturns to scale. When there are three factors of production, one of the three cross-partial
derivatives can be negative under our maintained assumptions, as shown in Appendix A.
To obtain testable implications from our model, we follow Krusell et al. (2000) and inparticular Berg et al. (2018) to adopt a nested CES function for the final good producer:
yt = A[µ1 ( xtA )ρ1 + (1ρ1µ 1 ) z t 1 ] ρ1 ,(21)

where xtA is the input of abstract tasks (performed by type-A workers) that the producer
purchases from type-A intermediate-good firms, zt is the input of routine tasks, 1/(1 ρ1 )is the elasticity of substitution between abstract and routine inputs, and 0 < µ1 < 1. The
routine input zt is a composite good, given byzt = [µ2 ( xtR )ρ2 + (11µ2 )( at k t )ρ2 ] ρ2 ,(22)

where xtR is the input of routine tasks (performed by type-R workers) that the producer
purchases from type-R intermediate-good firms, 1/(1 ρ2 ) is the elasticity of substitution
between type-R labor input and capital. We assume ρ1 < ρ2 < 1.
Our specification is a natural extension of Autor et al. (2003), who explore the implication of computerization for wage inequality. If we set ρ1 = 0 (so that (21) becomes
Cobb-Douglas) and ρ2 = 1 (so that xtR and at k t become perfect substitutes), then our production function reduces to that of Autor et al. (2003).3 Type-A workers provide abstract
inputs that are essential for production. On the other hand, because routine tasks can besupplied both by type-R workers and capital, a change in the composition of inputs in(22) toward capital can be interpreted as automation.
3 Autor and Dorn (2013), Jaimovich et al. (2020), and vom Lehn (2020) employ similar specifications tostudy polarization of employment.

11Factor markets are perfectly competitive. From (21) and (22), the firm’s profit-maximization
implies11

ptA = A[µ1 + (1µ1 ) Zt 1 ] ρ1

ptR = A[µ1 + (1

ρ1 ρ1
1

µ1 ) Zt ]1

r t = A [ µ1 + (1µ1 ) Zt 1 ] ρ1

1

1

ρ

ρ

µ1 ,

(23)
ρ2 ρ1
2

1

1

1

(1

µ1 ) Zt 1ρ1[ µ2 + (1µ2 ) K t ](1µ1 ) Zt 1ρ1[ µ2 + (1µ 2 ) K t 2 ] ρ2

ρ

µ2 ,

(1

(24)
ρ

µ2 ) K t 2

1

at , (25)where Zt = zt /xtA is the ratio of routine and abstract inputs and Kt = at k t /xtR is the ratioof machines and routine labor.
Proposition 2 (i) F13 (` A , ` R , ak )0; (ii) F23 (` A , ` R , ak ) < 0 holds if and only if
1
1

1
1
<
,
ρ1 s
1 ρ2

(26)

where s = p A ` A /y 2 (0, 1).
Proof. In Appendix B.
Proposition 2-(ii) states that an increase in capital decreases the marginal product oftype-R labor if and only if the elasticity of substitution between routine labor and capitalis sufficiently high. Condition (26) appears in Berg et al. (2018) as the condition for shortrun pain to arise. Our result clarifies how the nested CES specification restricts the crosspartial derivatives of the general production function F.
As in the preceding section, the representative household’s lifetime utility is given by(3) and the budget constraint is now Ct + τt + k t+1 (1 δ)k t = wtA ltA + wtR ltR + b̄ A utA +b̄ R utR + rt k t + πt . The household chooses the sequences of Ct and k t+1 to maximize (3)subject to the budget constraint. The first-order conditions imply Ct σ = Λt and Λt =βΛt+1 (rt+1 + 1 δ), with the transversality condition, limt!∞ Λt βt k t = 0, where Λt is
the Lagrange multiplier for this problem. Each individual discounts future payoffs by Bt
defined in (4).

3.2

Labor Market

There are two segmented labor markets, one for abstract-task intensive jobs (type-A labormarket) and the other for routine-task intensive jobs (type-R labor market). Type-A workers participate in type-A labor market and type-R workers participate in type-R labor12market. There is no reallocation of workers across the sectors.4 The assumption that the
proportion of the skilled φ is unaffected by the robotics revolution is clearly too strong.
We address this issue in Section 6.
The number of new employees in market i (i = A, R) is determined bym ( ui , vi ) =ui vi,[(ui )ψi + (vi )ψi ]1/ψi(27)

where vi is the number of type-i vacancies and ψi is the shape parameter for market i. Each
type-i vacancy is matched to a type-i job-seeker with probability qit = [1 + (θti )ψi ] 1/ψi ,where θti = vit /uit is labor market tightness in market i. Similarly, the probability that a
type-i job-seeker is matched with a type-i vacancy is given by Qit = θti [1 + (θti )ψi ] 1/ψi =θti qit . The matching elasticity with respect to unemployment is given by θ ψi /(1 + θ ψi ).
Separations are exogenous and take place at the end of each period. The law of motion
for the stock of type-i employment satisfies `it+1 = (1 λi )`it + qit vit , where λi is the separation rate in market i. Similarly, the law of motion for the stock of job seekers in market i
is uit+1 = (1 qit )uit + λi `it .
The values of employment (Wti ), unemployment (Uti ), a filled job (Jti ), and a vacancy(Vti ) are given by Wti = wit + λi Bt Uti+1 + (1 λi ) Bt Wti+1 , Uti = bi + Qit Bt Wti+1 + (1Qit ) Bt Uti+1 , Jti = pit wit + λi Bt Vti+1 + (1 λi ) Bt Jti+1 , Vti = ci + qit Bt Jti+1 + (1 qit ) Bt Vti+1 ,respectively. Free entry of jobs for each market (Vti = 0) yields the following job creation
condition:"#
1 λi cici
ii= Bt pt+1 wt+1 +
.
(28)
qit
qit+1
Finally, Nash-bargained wage rates are determined by η ( Jti
from which we obtain the wage equation:
wit = η pit + (13.3

η )bi + ηci θti .

Vti ) = (1η )(WtiUti ),(29)

Equilibrium

A type-A intermediate-good firm employs a type-A worker to complete one unit of abstract task, and sells it to the final-good firm at the competitive price ptA . Similarly, a
type-R intermediate-good firm employs a type-R worker to complete one unit of routine
4 In this model, type-A workers have no incentive to participate in type-R labor market because theequilibrium job-finding rate and wage rate are both lower than those in type-A labor market. On the other
hand, type-R workers have no ability to participate in type-A labor market.

13

task, and supplies it at price ptR . In any equilibrium, xtA = `tA and xtR = `tR . The price ofthe final good is normalized to one.
The aggregate profit of all firms is given by πt = yt wtA `tA wtR `tR rt k t c A vtA
c R vtR . The government finances the unemployment benefits by lump-sum taxes. Thus,τt = utA b̄ A + utR b̄ R . Substitute these equations into the household budget constraint toobtain the resource constraint as
yt

c A vtAc R vtR = Ct + k t+1(1δ)k t .

(30)

Equilibrium conditions are (4), (21)–(25), Zt = zt /xtA , Kt = at k t /xtR , (28), (29), (30),xtA = `tA = φ utA , xtR = `tR = 1 φ utR , uit+1 = uit + λi `it Qit uit , and θti = vit /uit .
A steady-state equilibrium is determined by a set of variables f Z, K, p A , p R , θ A , θ R , u A , u R , z, k gsatisfying
11µ 2 ) K ρ2 ] ρ2

[ µ2 + (1p A = A [ µ1 + (1(r δ + λi ) ci
= (1[1 + (θ i )ψi ] 1/ψi

1

1
ρ1 ρ1

1

1
ρ1 ρ1

1

µ1 ) Z ]1µ 2 ) K ρ2 ] ρ2η )( pi

bi )

1

(1

(1

µ1 ) Z ]p R = A [ µ1 + (1[ µ2 + (11µ 1 ) Z ρ1 ] ρ1

r = A [ µ1 + (1µ 1 ) Z ρ1 1µ2 )K ρ2 1 a,

µ1 ,

(1

(32)
µ 1 ) Z ρ1 1

µ2 ,

ηci θ i , (i = A, R)

φλ A(1 φ ) λ R
R
,
u
=
,
λ A + θ A [1 + (θ A )ψA ] 1/ψA
λ R + θ R [1 + (θ R )ψR ] 1/ψR
h
i1
ρ
R 2
ρ2 ρ2
z = µ2 1 φ u,+ (1 µ2 )( ak)

uA =

z= φ

u A Z,

ak = 1

φu R K,

(31)

(33)
(34)
(35)
(36)
(37)
(38)

where r = β1 1 + δ is constant in any steady state. A steady-state equilibrium is summarized by two loci on the K-Z plane. One locus, defined implicitly by (31), describes
the capital market equilibrium and is labeled as the CM curve. The other locus, defined
implicitly by (32)–(38), describes the labor market equilibrium and is labeled as the LM
curve. Lemma 2 below establishes the properties of the loci.
Lemma 2 (i) (31) implies dZ/dK < 0 holds for each level of a; (ii) (31) implies dZ/da > 0 foreach level of K. (iii) (32)–(38) are independent of a and jointly imply dZ/dK > 0.
14

Figure 1: Steady-State Equilibrium

Proof. In Appendix C.
As shown in Figure 1, the CM locus is downward sloping and the LM locus is upward
sloping. Because an increase in the level of technology shifts the CM curve up, CATP
increases both Z and K, which in turn increases p A . Thus, as expected, CATP increases
the demand for abstract tasks and hence the demand for type-A workers. The impact ofrobotics revolution on the demand for type-R workers depends on the detail of the model.
Proposition 3 CATP decreases p R if and only if
1
1
where
εA =

1
1 s
1
+ εA
<
,
ρ1 s
s
1 ρ2

(39)

dx A p A
pA xA
µ1
>
0,
s
=
2 (0, 1).
=y
µ 1 + (1 µ 1 ) Z ρ1
dp A x A

Proof. In Appendix D.
Thus, CATP decreases the demand for type-R labor input if and only if the elasticity
of substitution between routine labor input and capital is sufficiently high. In this case,
CATP causes joblessness for type-R in the long run. Note that condition (39) is more strict
15

than (26), implying that F23 (` A , ` R , ak ) < 0 is not enough to generate long-run job losses
for type-R workers.
Acemoglu and Restrepo (2018a) question the usefulness of the CATP approach forinvestigating the robotics revolution by claiming that CATP cannot reduce the labor demand.5 We have two objections to this claim. First, as we discussed in Section 2, the CATP
approach in the model with homogeneous workers should be used to address whether
CATP can reduce labor demand for all workers or not. For this specific question, thisapproach provides a clear answer, which is no.
Second, as Proposition 3 states, the CATP approach with worker heterogeneity andlabor market segmentation can generate a steady-state decline in the demand for type-R
workers. Further, Proposition 3 provides a testable condition under which the new technology harms the labor market for routine-task intensive occupations. Thus, while thetwo-factor model may have a serious limitation, the CATP approach, when applied to a
multi-factor model, offers a rich framework for predicting the impact of the new technology.

3.4

Alternative CES Nesting

Our CES nesting implies that the elasticity of substitution between type-A labor and typeR labor is the same as that between type-A labor and capital. Krusell et al. (2000), however,recommend the use of an alternative nesting,y = A[µ1 (` R )ρ1 + (1z = [µ2 (` A )ρ2 + (11µ 1 ) z ρ1 ] ρ1 ,

(40)

1
ρ2 ρ2

µ2 )( ak ) ] ,which implies that the elasticity of substitution between type-R labor and type-A labor isthe same as that between type-R labor and capital. According to this specification, withhigh complementarity between type-A labor and capital, the composite input z in (40) canbe interpreted as broadly-defined capital in a two-factor production function. While thisformulation is useful for exploring capital-skill complementarity (as in Krusell et al., 2000),
it cannot describe the scenario in which the robotics revolution might cause technology
5 They argue, “if automation were to be conceptualized as capital-augmenting technological change, itwould never reduce labor demand or the equilibrium wage, and it would increase the labor share—two
predictions that are neither intuitively appealing nor always consistent with the evidence” (Acemoglu andRestrepo, 2018a, p.49).

16

unemployment because CATP with a two-factor production function necessarily increases
both ` R and z in (40), as we have shown in Section 2.

3.5

Importance of Worker Heterogeneity

Berg et al. (2018) emphasize the role of heterogeneous capital in reducing the (short run)demand for human labor. To assess whether heterogeneous capital is more important thanheterogeneous labor, this section presents a model with two types of capital, traditional
capital (denoted by k Tt ) and robot capital (denoted by k tR ). We assume that CATP takes
place only on robots. The rates of returns on traditional capital and robot capital are rtT
and rtR , respectively. For brevity, their depreciation rates are assumed to be the same. We
also assume that workers are homogeneous as in Section 2. Production technology is givenbyyt = F (k Tt , xt , atR k tR ) = A[µ1 (k Tt )ρ1 + (1z t = [ µ 2 ( x t ) ρ2 + (1ρ1µ 1 ) z t 1 ] ρ1 ,

1µ2 )( atR k tR )ρ2 ] ρ2 ,which is identical to Model 1 of Berg et al. (2018). If we replace k Tt with xtA , then thisproduction function becomes (21).
Proposition 4 In any steady state, CATP decreases u and increases w.
Proof. In Appendix E.
In other words, the demand for human labor increases in the long run as the productivity of robot capital increases. We therefore conclude that worker heterogeneity, not capitalheterogeneity, is essential for studying technology unemployment. What drives the result
is rate-of-return equalization between robot capital and traditional capital. On the other
hand, labor market segmentation prevents wage equalization between the two tasks.

3.6

Heterogeneous CapitalThis section extends our main theoretical model presented in section 3.1 to distinguishrobots and computers from traditional capital. While capital heterogeneity alone cannot
generate a decline in the demand for human labor (as clarified in Section 3.5), drawing
a line between the two types of capital allows us to work with a richer model in ourquantitative studies.
17

The stock of traditional capital is denoted by k Tt and depreciates at rate δ T < 1. The
stock of computer capital such as robots, computers, and software is denoted by k tR anddepreciates at rate δ R < 1. Both types of capital are owned by the representative household. The rates of returns on traditional capital and robot capital are denoted by rtT andrtR , respectively.
To maintain the structure of the model in section 3.1, we use the Cobb-Douglas function as in Krusell et al. (2000) to aggregate traditional capital and (21):yt = A(k Tt )α [µ1 ( xtA )ρ1 + (1zt = [µ2 ( xtR )ρ2 + (11 α

ρµ 1 ) z t 1 ] ρ1 ,
1µ2 )( atR k tR )ρ2 ] ρ2 ,

(41)
(42)

where 0 < α < 1 and ρ1 < ρ2 < 1. The equilibrium conditions are derived in Appendix F.
With (41), the income share of abstract-task intensive labor satisfies
s=

pA xA
= (1
y

α)

µ1
.
µ 1 + (1 µ 1 ) Z ρ1

Thus, Proposition 3 is modified accordingly as follows.
Proposition 5 CATP decreases p R if and only if
1
1

4

1
ρ1

α
s

+ εA

1

α
s

s

<

1
1

ρ2

.

(43)

Quantitative Analysis

This section provides quantitative analyses based on the model with heterogeneous workers presented in Section 3.1. We also study a richer quantitative model to allow for capitalheterogeneity presented in Section 3.6. Following Berg et al. (2018), we consider a hypothetical scenario in which the productivity of robots doubles. While Berg et al. (2018) focus
on the transition to a new steady state to highlight a series of short-run pain as their model
exhibits long-run gains for workers, we focus on steady states to highlight the possibilityof long-run pain.

4.1

Calibration

We calibrate our model so that the initial steady state with a = 1 matches the current
U.S. economy. We set the model period to be a month and choose the subjective discount18

factor β to be 0.997, which corresponds to the annual real interest rate of approximately 4
percent. We set the capital depreciation rate δ to be 0.0083, or 10% annually, the standard
value in the literature. We choose the value of the coefficient of relative risk aversion σ
to be 2, which is within a plausible range in the literature. We set the proportion of typeA workers in labor force φ to be the proportion of individuals with college education or
higher, which is 0.45 (Acemoglu and Autor, 2011).
According to Fallick and Fleischman (2004), the monthly transition rate from employment to unemployment equals 0.0097 for the skilled and 0.0378 for the unskilled. We
therefore adopt their estimates and set λ A = 0.0097 and λ R = 0.0378.
Unfortunately, there is no consensus in the literature regarding the worker’s bargaining power in wage determination. We thus focus on the symmetric case, η = 1/2.6 Another controversial parameter in the literature is the value of nonwork bi . We follow Halland Milgrom (2008) to target the flow value of nonwork to be 71% of the wage rate. Thus,
we set bi to satisfy bi = 0.71wi . Because quantitative results of search-matching models are
known to be highly sensitive to the target flow value of nonwork, we present a sensitivity
analysis in Appendix G.
To pin down the shape parameters in the matching function ψi and vacancy costs ci ,we target the aggregate vacancy-unemployment ratio, job finding rates, and vacancy durations in the data. The mean value of the aggregate vacancy-unemployment ratio in the
U.S. is 0.72 (Pissarides, 2009). Using the Current Population Survey, Hagedorn et al. (2016)
estimate that the average monthly job finding rate is 0.3618 for the skilled and 0.4185 forthe unskilled. They also find that a vacancy for the skilled lasts 2.128 times longer than
one for the unskilled. Thus, 1/q A = 2.128 1/q R . Appendix H provides the results under
the Cobb-Douglas specification.
The input share parameters µ1 and µ2 are chosen so as to match the wage premiumof 1.68 (Acemoglu and Autor, 2011) and the labor share of 0.61 (Berg el al., 2018). The
aggregate productivity parameter A is chosen to normalize the steady-state output y tobe 1. The existing studies find that the elasticity of substitution between the skilled andthe unskilled ranges between 1.4 and 2.0 (Katz and Murphy, 1992; Ciccone and Peri, 2005;
Goldin and Katz, 2008). For our benchmark, we set ρ1 = 0.286 so that the elasticity of
substitution 1/(1 ρ1 ) is 1.4.
As Berg et al. (2018) note, there is no econometric estimate available for ρ2 . We there6 There are mainly two strategies in the literature. One is to set η = 1/2, and the other is to set η to matchthe estimated elasticity of the matching function with respect to unemployment.

19fore consider the following three cases for the elasticity of substitution: Case 1 when it ismoderately low (1/(1 ρ2 ) = 2.5); Case 2 when it is moderately high (1/(1 ρ2 ) = 5);
and Case 3 when it is very high (1/(1 ρ2 ) = 10). We set the initial level of technology
to be a = 1 and set the value of A so that y = 1.0 for each case. Under our calibration
strategy, the initial steady-state values are invariant to the value of ρ2 .
The calibrated parameter values are summarized in Table 17 .
Table 1: Calibrated Parameter Values: Homogeneous Capital
cA

cR

ψA

ψR

bA

bR

µ1

µ2

A

0.205

0.211

0.644

1.169

0.587

0.349

0.467

0.891

0.967

The equilibrium job finding rates and equilibrium input prices match their target values. The model generates the aggregate unemployment rate of 5.7 percent, which is close
to the observed average unemployment rate between 1948 and 2019. The equilibrium
numbers of skilled and unskilled job seekers are 0.012 and 0.046, respectively. These numbers imply that the unemployment rate for the skilled u A /φ is 2.6 percent and that for theunskilled u R /(1 φ) is 8.3 percent. The equilibrium wage premium matches its calibration target, which is 1.68.
The implied matching elasticity with respect to unemployment in the type-A labormarket is 0.52, which is within the plausible range of 0.5-0.7 reported by Petronglo andPissarides (2001). The implied matching elasticity with respect to unemployment in thetype-R labor market is 0.36. While this is off the range suggested by Petronglo and Pissarides (2001), this value is consistent with the estimate by Coles and Smith (1996), which
is 0.4.
The model with heterogeneous capital presented in Section 3.6 has robot capital k R andtraditional capital k T as distinct factors of production. To pin down the value of α in (41),
we follow Berg et al. (2018) to target the income share of robot capital to be 0.04 and thatof traditional capital to be 0.35. We set the depreciation rates, δ R and δ T , to be the samelevel at 0.0083 as in the homogeneous capital model.8 As Table 1 and Table 2 show, thecalibrated parameters regarding the labor market are the same for the two models and for
all cases. Only parameters for the production function are slightly different because the
7 The parameter values of µ , µ , and A presented in the table are those for Case 1. For Case 2, µ = 0.449,
1 2
1µ2 = 0.950, and A = 1.108. For Case 3, µ1 = 0.442, µ2 = 0.966, and A = 1.172.
8 We considered several cases under δ R > δ T and verified that our main results are nearly independentof these parameter values.

20

assumed value of ρ1 differs for each case.9
Table 2: Calibrated Parameter Values: Heterogeneous Capital

4.2

cA

cR

ψA

ψR

bA

bR

α

µ1

µ2

A

0.205

0.211

0.644

1.169

0.587

0.349

0.350

0.580

0.958

0.484

Threshold Elasticity

We can now quantify the terms in (39) of Proposition 3 for the U.S. economy. Under theselected set of parameters, we obtain s = 0.3656, ε A = 0.0447, and 1/(1 ρ1 ) = 1.4. Thisimplies that the left-hand side of (39) is 3.91. Thus, the demand for routine labor increases
in response to the robotics revolution if and only if the elasticity of substitution between
routine labor and robots is less than 3.91. It is also easy to verify that the left-hand sideof (26) in Proposition 2 is 3.83, which implies that robots increase the marginal product ofroutine labor if and only if the elasticity of substitution between routine labor and robots
is less than 3.83.
We can also quantify the terms in Proposition 5, which is based on a richer model for
the U.S. economy. Under the selected set of parameters, s = 0.3656, ε A = 0.0436, and1/(1 ρ1 ) = 1.4. This implies that the left-hand side of (43) is 2.52. Thus, for “technology
optimism” to arise, the elasticity of substitution between routine labor and robot must be
less than 2.52.

4.3

Robots and Labor Market Outcomes

We are interested in the long-run quantitative effects of a significant increase in the levelof robot productivity on joblessness as well as wage inequality. As in Berg et al. (2018), we
consider the scenario in which the level of productivity of robots doubles. In Appendix
G, we present how our results depend on our calibration target regarding the flow valueof nonwork. In Appendix H, we show how the results depend on our choice of matching
function.
An important caveat concerning the model with homogeneous capital is that a unit increase in the productivity of robots does not translate into the same magnitude of increase
9 The parameter values of µ , µ , and A presented in the table are those for Case 1. For Case 2, µ = 0.579,
1 2
1µ2 = 0.969, and A = 0.486. For Case 3, µ1 = 0.579, µ2 = 0.974, and A = 0.487.

21

in the productivity of the aggregate stock of capital as the share of robots is relatively small.
Indeed, Berg et al. (2018) document that the income share of robot capital in the U.S. is
0.04 while that of traditional capital is 0.35.
Suppose (implicitly) that there are two types of capital, k T and k R . The aggregate stockof capital satisfies k = k T + k R and we set k T /k R = 0.35/0.04. The effective capital input
is ke = k T + a R k R , where a R is the level of robot productivity. Suppose that a R increases
from 1 to 2. We then obtain
0.35
e0.04 + 2 e
k1 = 0.35
k0 = 1.10
0.04 + 1

ke0 .

From this calculation, the magnitude of CATP we consider for the model with homogeneous capital is 1.10.
Table 3 presents the steady-state values of some selected endogenous variables for themodel with homogeneous capital (presented in Section 3.1) and with heterogeneous capital (presented in Section 3.6). The column labeled “Initial” shows the steady-state values
under the calibrated parameters. We verify that both models replicate our calibration
targets. The columns labeled “Basic” show the steady-state values of the homogeneous
capital model under a = 1.1, whereas the columns labeled “Capital” show the steady-statevalues of the heterogeneous capital model under a R = 2.
Table 3: CATP and Labor Market Outcomes
Initial Case 1: 1 1ρ2 = 2.5 Case 2: 1 1ρ2 = 5 Case 3: 1 1ρ2 = 10
Model
Productivityy
QA
QR
u A /φu R / (1 φ )uwAwRw A /w R
Labor share

1.00
0.36
0.42
2.6%
8.3%
5.7%
0.83
0.49
1.68
0.61

Basic
a = 1.1

CapitalaR = 2

Basic
a = 1.1

CapitalaR = 2

Basic
a = 1.1

CapitalaR = 2

1.13
0.41
0.44
2.3%
7.8%
5.4%
0.90
0.50
1.78
0.58

1.14
0.41
0.42
2.3%
8.2%
5.6%
0.91
0.49
1.84
0.57

1.18
0.43
0.41
2.2%
8.5%
5.7%
0.93
0.49
1.91
0.55

1.43
0.49
0.28
2.0%
12.0%
7.5%
1.06
0.43
2.45
0.47

1.22
0.44
0.37
2.2%
9.2%
6.0%
0.95
0.47
2.02
0.53

1.85
0.55
0.05
1.7%
43.7%
24.8%
1.27
0.37
3.45
0.37

22

First, consider Case 1 of the homogeneous capital model, in which the elasticity of substitution 1/(1 ρ2 ) is 2.5. Results in Table 3 confirms Proposition 3. Because the elasticity
of substitution is below the threshold of 3.91, the demand for labor will increase in bothlabor markets. Thus, CATP reduces the overall unemployment rate by increasing job creation in both labor markets. Similarly, the wage rates for both abstract and routine tasks
increase. While this case is largely consistent with technology optimism, wage inequality
expands and the labor share decreases.
Next, consider Case 2, in which the elasticity of substitution 1/(1 ρ2 ) is 5, which
is above the threshold level of 3.91. In this case, CATP increases only the demand for
abstract-task intensive jobs and a larger proportion of routine tasks is performed by robots. Thus, the job-finding rate for routine-task intensive jobs decreases. There is littlechange in the overall unemployment rate because the increase in the number of unskilled
job seekers is offset by a simultaneous decrease in the number of skilled job seekers. Albeit
its small magnitude, the wage rate for routine jobs declines. In contrast, the wage rate forthe skilled increases significantly and wage inequality expands.
Finally, consider Case 3, in which the elasticity of substitution 1/(1 ρ2 ) is 10. In thiscase, the increase in job creation for abstract-task intensive jobs cannot compensate for the
decline in the demand for routine-task intensive jobs. As a result, the overall unemployment rate increases. The wage rate for routine-task intensive occupations declines andwage inequality expands significantly.
Overall, our predictions based on the homogeneous capital model are not a disastereven under Case 3 as the magnitudes of job losses and wage reductions are modest in
response to a large increase in the productivity of robots.
In contrast, our predictions based on the heterogeneous capital model are alarming.
The reason is twofold. First, we now face a severer threshold elasticity for technology
optimism, which is 2.52. Our optimistic scenario of Case 1 is now near the border. Second,
the magnitudes of job losses are now significant. In particular, Case 3 is a disaster for
those with routine-task intensive occupations. For type-R workers, the average durationof unemployment increases from 2.4 months to 20.6 months and the unemployment rate
rises from 8.3% to 43.7%. Because the associated increase in the demand for abstracttask intensive occupations is moderate, the overall unemployment rate rises from 5.7% to24.8%.
The large impact on the labor market in our heterogeneous capital model is partly
explained by a surge in capital investment (Brynjolfsson and Hitt, 2003). In Case 2, thelevel of robot capital increases by a factor of about 8 when its productivity doubles, and23

traditional capital increases by 43%, whereas the labor input of type-A increases only by
0.7%. Similarly, in Case 3, the level of robot capital increases by a factor of about 17 and thelevel of traditional capital increases by 85%, whereas the labor input of type-A increases
by 0.9%.

5

Unemployment Policy

This section conducts a policy analysis based on the model with heterogeneous capital.
Our analysis is similar in spirit to Mortensen and Pissarides (1999) and Hornstein et al.
(2007). We are interested in a scenario in which the level of unemployment insurance
benefits is set higher than its current level.
In our benchmark model, the unemployment benefit replacement ratio b̄i /wi is 0.25.
Given the set of model parameters summarized in Table 2, we consider the scenario in
which the unemployment benefit replacement ratio increases by 20% to 0.30. This implies
that the new flow value of unemployment satisfies bi /wi = 0.76.
Table 4: Institution and Labor Market Outcomes
Initial Case 1: 1 1ρ2 = 2.5 Case 2: 1 1ρ2 = 5 Case 3: 1 1ρ2 = 10aRb̄i /wi

1
0.25

1
0.30

2
0.30

1
0.30

2
0.30

1
0.30

2
0.30

u A /φu R / (1 φ )uwAwRw A /w R
Labor share

2.6%
8.3%
5.7%
0.83
0.49
1.68
0.61

2.9%
9.2%
6.4%
0.82
0.49
1.67
0.61

2.4%
9.2%
6.1%
0.91
0.50
1.83
0.57

2.9%
9.2%
6.4%
0.83
0.49
1.67
0.61

2.0%
14.9%
9.1%
1.06
0.44
2.42
0.47

2.9%
9.3%
6.4%
0.83
0.49
1.67
0.61

1.8%
63.1%
35.5%
1.26
0.39
3.26
0.35

Table 4 summarizes the results. The column labeled “Initial” shows the steady-statevalues under bi /wi = 0.71 and a R = 1. All other columns show the values under bi /wi =0.76. For each case, we present the steady-state values under a R = 1 and those undera R = 2.
First, consider the economy prior to CATP. The results are nearly identical across allcases. As is standard in the search-matching literature, a higher flow value of unemployment implies a higher wage rate and a higher unemployment rate. In all cases we24consider, the increases in the wage rates are negligible and the wage premium declines
only slightly. The unemployment rates for both types increase but the unskilled unemployment rate increases by more than the skilled unemployment rate.
The effects of CATP on the overall unemployment rate and especially joblessness for
type-R workers are much more serious than those under a less generous unemployment
policy. In Case 3, the unemployment rate for type-A workers declines from 2.9% to 1.8%.
However, the unemployment rate for type-R workers increases from 9.3% to 63.1%. The
magnitude amounts to 53.9 percentage points. Under a less generous unemployment
policy (in Table 3), the magnitude is 35.4 percentage points. Under the more generous
unemployment policy, the overall unemployment rate increases from 6.4% to 35.5%, while
it increases from 5.7% to 24.8% under the less generous policy.

6

Computerization of Abstract Tasks

The previous sections have assumed that abstract tasks cannot be automated. This view,
however, is controversial (Brynjolfsson and McAfee, 2014; Ford, 2015). The recent breakthrough in the field of machine learning opens the door to a new economy in whichtasks that are considered as sufficiently complicated, such as driving, recognizing handwritings and verbal expressions, and translating one language into another, can be performed by computers.
This section considers the case in which skills that type-A workers acquired througheducation and training are subject to automation. Because our definition of skilled workers is those with skills that are hard to be automated, we consider the impact of a reductionin φ on the labor market outcomes and interpret it as reallocation of workers across the
skill levels caused by computerization of abstract tasks.
Using a machine learning method, Frey and Osborne (2017) categorize 702 occupations currently exist in the U.S. into those facing the risk of automation and those that are
resilient. They obtain the probability of computerization for each of 702 occupations andfind that 47% of the occupations have computerization probabilities greater than 70%.
Given the set of model parameters summarized in Table 2, we change the proportionof the skilled by -47% to set φ = 0.2385. When the level of capital productivity is at its
original level (a R = 1), computerization neutralizes part of human skills, which results inan output loss. When the level of robot productivity is 2, computerization does two things:
reallocating 47% of skilled workers into unskilled and doubling robot productivity.
25

Table 5: Computerization of Abstract Tasks
Initial

Case 1: 1 1ρ2 = 2.5

Case 2: 1 1ρ2 = 5Case 3: 1 1ρ2 = 10

aR
φ

1
0.45

1
0.238520.238510.238520.238510.238520.2385yQA
QR
u A /φu R / (1 φ )uwAwRw A /w R
Labor share

1.00
0.36
0.42
2.6%
8.3%
5.7%
0.83
0.49
1.68
0.61

0.71
0.47
0.09
2.0%
30.0%
23.3%
1.01
0.38
2.68
0.62

0.79
0.50
0.09
1.9%
29.0%
22.5%
1.09
0.38
2.89
0.58

0.71
0.47
0.10
2.0%
27.9%
21.8%
1.01
0.38
2.66
0.63

0.85
0.52
0.06
1.8%
39.2%
30.3%
1.05
0.37
3.11
0.52

0.71
0.47
0.11
2.0%
26.2%
20.4%
1.01
0.38
2.65
0.64

0.99
0.55
0.01
1.7%
74.1%
56.8%
1.28
0.36
3.54
0.38

Table 5 presents the results. The column labeled “Initial” shows the steady-state values
under φ = 0.45 and a R = 1. All other columns show the values under φ = 0.2385. For
each case, we report the steady-state values under a R = 1 and those under a R = 2.
The results are striking. If automation of abstract tasks takes place when the productivity of robots doubles, then the joblessness for routine workers reaches 74.1% and theoverall unemployment rate becomes 56.8% in Case 3. The overall unemployment rate
becomes 30.3% even in Case 2.
If computerization is directed solely at neutralizing the skills that individuals acquired
through education and training without any productivity gain, then the aggregate output
significantly declines because those who were previously considered as skilled are now
considered as unskilled in production. The output loss associated with this effect cannot
be compensated for by the productivity gains by robots if the magnitude is factor of 2.
An important observation is asymmetry between the two labor markets. In the labormarket for abstract-task intensive jobs, the impact of automation on the unemployment
rate is small while the impact on the wage rate is significant. On the other hand, in thelabor market for routine-task intensive jobs, the impact of automation on the wage rate is
negligible while the unemployment rate deteriorates significantly.

26

7

Labor Force Participation

In many scenarios we consider, the robotics revolution seriously hurts the routine labormarket. However, the hike in the unemployment rate may be (at least partially) masked
by the “discouraged worker effect” as the declines in both job-finding rate and wage rateinduce individuals to exit from the labor market (Cortes et al., 2017; Jaimovich et al., 2020),
thereby creating a downward pressure on the unemployment rate. To assess the importance of this adjustment margin, this section extends the model with traditional capitalpresented in Section 3.6 to further allow for endogenous labor force participation.
There are mainly two avenues to modeling labor force participation in the literature.
One way is to consider an exogenous type distribution for individuals and look for a cutoff individual who is indifferent between participation and nonparticipation (Pissarides,
2000; Garibaldi and Wasmer, 2005; Jaimovich et al., 2020). The other way, which we adopt,is to describe the labor force participation decision through the representative family’stime allocation problem (Greenwood and Hercowitz, 1991; Tripier, 2004; Veracierto, 2008).
The reservation wage rate plays a central role in both approaches, but we adopt the latterbecause our model has a representative decision maker. Further, as we shall see below,
this approach can capture the income effect on labor supply.
Let nit denote the measure of type-i individuals who engage in home activity. As in
Veracierto (2008), we assume a linear home production technology so that nit also represents the units of home goods available for the household. With labor force participation
choice, the representative household’s objective function is given by
#
"
1 σ
∞
A
R
t Ct
(44)
∑ β 1 σ + ζ A ln nt + ζ R ln nt ,t =0
where ζ A > 0 and ζ R > 0. As in the basic model, the representative household pools theresources of all its members. The equilibrium conditions are derived in Appendix I.
In any steady state, the participation decision is summarized by the following:
Qi β wi bi
1 σi
ζi i C = b +n
1 β (1 λ i Q i )(45)

for i = A, R. Note that the right-hand side of (45) is the reservation wage rate for eachworker type, (1 β) U i . The labor force participation rates satisfy l pr A = (` A + u A )/φ =(φ n A )/φ and l pr R = (` R + u R )/(1 φ) = (1 φ n R )/(1 φ). The aggregate labormarket participation rate satisfies l pr = ` A + u A + ` R + u R = 1 n A n R . Similarly, the27unemployment rates are u A /(` A + u A ) and u R /(` R + u R ), and the aggregate unemployment rate is u = (u A + u R )/(` A + u A + ` R + u R ).
We now have two additional endogenous variables (n A and n R ) and two additionalparameters (ζ A and ζ R ). To pin down these two parameter values, we need two additional calibration targets. As in Garibaldi and Wasmer (2005), we target the aggregate
unemployment rate and the labor market participation rate. However, we update the target unemployment rate to be 5.7% (so as to be consistent with our benchmark model) andthe target labor market participation rate to be 0.66 (Krusell et al., 2017). The calibrated
parameter values are summarized in Table 610 .
Table 6: Calibrated Parameter Values: Labor Force Participation
cA

cR

ψA

ψR

bA

bR

ζA

ζR

α

µ1

µ2

A

0.310

0.320

0.645

1.172

0.886

0.527

0.256

0.192

0.350

0.585

0.966

0.631

Table 7: Labor Force Participation
Initial

Case 1: 1 1ρ2 = 2.5

Case 2: 1 1ρ2 = 5Case 3: 1 1ρ2 = 10

aR
φ

1
0.45

2
0.45

2
0.238520.45

20.238520.45

20.2385yC
QA
QR
u A /(` A + u A )u R /(` R + u R )ul pr Al pr Rl pr
wAwR
Labor share

1.00
0.72
0.36
0.42
2.6%
8.3%
5.7%
0.67
0.65
0.66
1.25
0.74
0.61

1.11
0.76
0.41
0.43
2.3%
8.0%
5.4%
0.66
0.62
0.63
1.36
0.75
0.57

0.83
0.58
0.50
0.07
1.9%
36.3%
28.9%
0.69
0.78
0.76
1.67
0.56
0.58

1.35
0.83
0.48
0.34
2.0%
10.1%
6.0%
0.65
0.51
0.57
1.56
0.69
0.46

0.90
0.59
0.52
0.05
1.8%
45.5%
35.9%
0.70
0.78
0.76
1.74
0.56
0.52

1.73
0.92
0.55
0.13
1.7%
22.7%
9.0%
0.64
0.28
0.44
1.89
0.59
0.35

1.06
0.59
0.55
0.01
1.7%
76.5%
59.5%
0.73
0.77
0.76
1.91
0.54
0.38

10 The parameter values of µ , µ , and A presented in the table are those for Case 1. For Case 2, µ = 0.584,
1 2
1µ2 = 0.977, and A = 0.634. For Case 3, µ1 = 0.583, µ2 = 0.982, and A = 0.635.

28

Table 7 presents the steady-state values of some selected variables. The column labeled“Initial” shows the steady-state values under φ = 0.45 and a R = 1. All other columns
show the steady-state values under a R = 2. For each case, we provide the results underφ = 0.45 and φ = 0.2385.
First, consider the benchmark case in which the robotics revolution increases the levelof a R without any influence on φ. In our most pessimistic scenario of Case 3, the long-run
unemployment rate for the unskilled is 22.7%, which is nearly a half of that under the basicmodel. This reduction is driven by the decline in the labor force participation rate of theunskilled, which decreases significantly from 0.65 to 0.28. The skilled unemployment rate
is 1.7% and this is the same as that under the basic model. The labor force participation
rate for the skilled changes only slightly from 0.67 to 0.64. Consistent with Cortes et al.
(2017) and Jaimovich et al. (2020), our results verify that labor force participation can be
an important adjustment margin in response to the robotics revolution.
However, this is not the whole story. We find that the unskilled labor force participation rate may increase through the income effect if the robotics revolution decreases the levelof consumption. Labor force participation decision is made so that (45) is satisfied for each
type. The right-hand side of (45) is the reservation wage rate while the left-hand side is
the marginal utility from home production in units of consumption goods. A decrease in
consumption induces the household to supply more labor because the marginal value ofhome production declines. This is the “added worker effect” (Pissarides, 2000). Our results show the importance of the tension between the discouraged worker effect and theadded worker effect.

8

Conclusion

This paper explored whether the robotics revolution would make human obsolete in theprocess of production. We first demonstrated that the right question is whether the newtechnology harms the labor market prospect for a certain group of workers, not the entire
labor force because we proved that the robotics revolution is beneficial for all workers if
workers are homogeneous. We then addressed whether the new technology would cause
joblessness for unskilled workers using a multi-factor model with segmented labor markets and search-matching frictions. We found and quantified a testable condition underwhich the robotics revolution increases the unskilled unemployment. Namely, the threshold elasticity of substitution between routine labor and capital, above which the new tech29

nology harms the routine labor market in the long run, is 3.91. This threshold is further
reduced to 2.52 if robots and traditional capital are modeled as distinct inputs.
Our results are obtained under two important assumptions, flexible wages and perfect competition in the product market. We conjecture that introduction of wage rigidity
in our framework would strengthen our results because the impact of the robotics revolution on the unemployment rate in our model is quantitatively very strong even under
flexible wage adjustment through Nash bargaining. On the other hand, the impact of therobotics revolution on the labor market outcomes under imperfect product market competition, especially with the rising market power of firms, is an open question. For thisinvestigation, one requires a richer model of market structure.
Another important avenue for future work is to study household heterogeneity. For
tractability, we maintained the assumption that the representative family pools all consumption goods to provide perfect consumption insurance among the family members.
While convenient, this assumption prevents us from making any welfare comparison
across individuals. Further, because all individuals share the same level of consumption,
the income effect on labor market participation decision is the same across all individuals.
Further investigations on (the distribution of) nonmarket payoffs such as home production and wealth are needed to make a better prediction on how peoples’ participation
decisions are affected by the robotics revolution.

30

Appendix
A

Properties of Homogeneous Functions

Because F is homogeneous of degree 1, F1 is homogeneous of degree 0.11 This implies
F11 ( x1 , x2 , x3 ) x1 + F12 ( x1 , x2 , x3 ) x2 + F13 ( x1 , x2 , x3 ) x3 = 0.
Similarly, we obtain
F21 ( x1 , x2 , x3 ) x1 + F22 ( x1 , x2 , x3 ) x2 + F23 ( x1 , x2 , x3 ) x3 = 0,
F31 ( x1 , x2 , x3 ) x1 + F32 ( x1 , x2 , x3 ) x2 + F33 ( x1 , x2 , x3 ) x3 = 0.
Decreasing marginal products (F11 < 0, F22 < 0, F33 < 0) imply
F12 ( x1 , x2 , x3 ) x2 + F13 ( x1 , x2 , x3 ) x3 =F11 ( x1 , x2 , x3 ) x1 > 0,

F21 ( x1 , x2 , x3 ) x1 + F23 ( x1 , x2 , x3 ) x3 =F22 ( x1 , x2 , x3 ) x2 > 0,

F31 ( x1 , x2 , x3 ) x1 + F32 ( x1 , x2 , x3 ) x2 =F33 ( x1 , x2 , x3 ) x3 > 0.

By symmetry of cross-partial derivatives, these expressions imply that one of the threecross partials (F12 = F21 , F23 = F32 , F31 = F13 ) can be negative and that if this is the case,then the other two cross partials must be strictly positive.

B

Proof of Proposition 2(i) Substitute (22) into (21) to consider
F ( xtA , xtR , at k t ) = Afµ1 ( xtA )ρ1 + (1µ1 )[µ2 ( xtR )ρ2 + (1ρ11µ2 )( at k t )ρ2 ] ρ2 g ρ1 .

(46)

It is then straightforward to obtain
F1 ( xtA , xtR , at k t ) = Afµ1 ( xtA )ρ1 + (111 Remember that ifµ1 )[µ2 ( xtR )ρ2 + (1ρ11µ2 )( at k t )ρ2 ] ρ2 g ρ11

µ1 ( xtA )ρ1 1 .

f ( x1 , x2 , x3 ) is homogeneous of degree h, then it satisfies f 1 x1 + f 2 x2 + f 3 x3 =h f ( x1 , x2 , x3 ).

31Thus,
F13 ( xtA , xtR , at k t ) = Afµ1 ( xtA )ρ1 + (1ρ1µ1 )[µ2 ( xtR )ρ2 + (1(1ρ1 )µ1 ( xtA )ρ1 1 (1(1µ2 )( at k t )ρ2 1

1

2

ρ1

1

1

1

µ2 )( at k t )ρ2 ] ρ2 g ρ1µ1 )[µ2 ( xtR )ρ2 + (1µ2 )( at k t )ρ2 ] ρ20.
(ii) From (46), we obtain
F2 ( xtA , xtR , ak t ) = A1fµ1 ( xtA )ρ1 + (1
ρ1

ρ1
(1
ρ2

µ1 )[µ2 ( xtR )ρ2 + (1µ1 )[µ2 ( xtR )ρ2 + (1ρ1µ2 )( at k t )ρ2 ] ρ2

ρ1µ2 )( at k t )ρ2 ] ρ2 g ρ1
1

ρ2 µ2 ( xtR )ρ2 1 .

Thus, we obtain
2
6

1
F23 ( xtA , xtR , ak t ) = 6
4

(1

ρ1
ρ2

µ1 xtA

µ1 )

ρ1

+ (1

h

ρ
µ2 xtR 2 + (1µ2 )( at k t )ρ2hρµ1 ) µ2 xtR 2 + (1i 1µ2 )( at k t )ρ2

i ρ1
ρ2

µ2 )( at k t )ρ2

i

ρ1
ρ2

+

ρ2
1 h
µ2 xtR+ (1ρ2hi ρ1ρ ρ2
ρ
1
2
R 2
ρ2
(1 µ1 ) µ2 x t+ (1 µ2 )( at k t )ρ2() 1 ρ1
ρ1
h
i ρρ1
ρ1
ρ2
µ1 xtA+ (1 µ1 ) µ2 xtR+ (1 µ2 )( at k t )ρ2 2

A

ρ2 µ2 xtR

ρ2 1µ2 )( at k t )ρ2 1 .

ρ2 (1Thus, F23 ( xtA , xtR , ak t ) < 0 if and only if
1

ρ1
ρ2

(1

µ1 )

ρ
µ1 xtA 1 + (1Rewrite the condition as

(1

ρ1 )

h

ρ
µ2 xtR 2 + (1

µ1 )

h

ρ
µ2 xtR 2 + (1(1µ1 xtAµ2 )( at k t )ρ2

ρ1

i ρ1
ρ2

µ2 )( at k t )ρ2

i ρ1 +ρ2ρµ1 ) z t 1ρµ1 ) z t 1

+ (1
32

+ ( ρ1ρ2 ) < 0,

ρ1

ρ2
ρ2

< 0.

ρ1

3

ρ2 7
7
ρ2 5

or

ρ

(1 µ1 ) z t 1
ρ1>ρρ 2 (0, 1).
ρ1µ1 xtA 1 + (1 µ1 )zt 1

ρ2
1

(47)

For this inequality to be satisfied, ρ2 > ρ1 is necessary, and ρ2 = 1 is sufficient. Observethat the cost share of type-A input, which is given byµ1 ( xtA )ρ1
ptA xtA=ρ .
ytµ1 ( xtA )ρ1 + (1 µ1 )zt 1
Thus, (47) implies
ρ21

ρ

(1 µ1 ) z t 1
ρ1
>
ρ
ρ =1
ρ1
µ1 xtA 1 + (1 µ1 )zt 1µ1 ( xtA )ρ1
ρ =1µ1 ( xtA )ρ1 + (1 µ1 )zt 1

ptA xtA.
yt

Thus, F23 ( xtA , xtR , ak t ) < 0 holds if and only if
ρ2
1

ρ1
>1
ρ1

ptA xtA
pA xA
1, t t >
yt
yt
1

1

ρ2
1
1 ρ
,
> A A1 .
pt xt
ρ1
1 ρ2
y

Note that this inequality is identical to condition (14) in Berg et al. (2018).

C

Proof of Lemma 2(i) Totally differentiate (31) to obtain
da
= (1
a

ρ1 )

dZ
µ1
+ (1
µ 1 + (1 µ 1 ) Z ρ1 Z

ρ2 )

dK
µ2
,
µ 2 + (1 µ 2 ) K ρ2 K

(48)

where all coefficients are positive because ρi
1. If we impose da = 0 on (48), then we
obtain dZ/dK < 0. This defines the shape of the CM locus.
(ii) If we impose dK = 0 on (48), then we obtain dZ/da > 0. This implies that for eachlevel of K, an increase in a shifts the CM locus upward.
(iii) Solve (34) for pi to obtain
pi =

(1

(r δ + λi ) ci
η i i+c θ + bi
ψ
1/ψ
i
i
1 η
η )[1 + (θ ) i ]Θi θ i ,where Θi0 θ i > 0. (36)-(38) imply
Z=

1

φ
φ

uR
uA

[ µ2 + (1331µ 2 ) K ρ2 ] ρ2 .

(49)

Thus, the system of equations determining the LM locus is given by
1

1

1

1

p A = A [ µ1 + (1µ 1 ) Z ρ1 ] ρ1

p R = A [ µ1 + (1µ 1 ) Z ρ1 ] ρ1

µ1 ,

(1

µ 1 ) Z ρ1 1 [ µ 2 + (1

δ + λi ) ci

η i i
(r
+
c θ + bi
1/ψ
ψ
i
i
i
1 η
(1 η )[1 + (θ ) ]AφλMA θ A ,uA = A
ψ
1/ψ
A
A
A
A
λ + θ [1 + ( θ ) ](1 φ ) λ R
uR = R
MR θ R ,λ + θ R [1 + (θ R )ψR ] 1/ψR
1
1 φ uR
ρ2 ρ2
Z=
µ
+
(
1
µ
)
K
.
[
]
2
2
φ uA
pi =

1

µ 2 ) K ρ2 ] ρ2

1

µ2 ,

Θi θ i ,Evidently, this system is independent of a. Now, take total differentiation of these equations to obtaindp A(1 µ1 ) Z ρ1 dZ,= (1 ρ1 )µ 1 + (1 µ 1 ) Z ρ1 Z
pA
dp R
µ1
dZ
= ( ρ1 1)+ (1ρRµ1 + (1 µ1 ) Z 1 Z
p

(50)
ρ2 )

(1 µ2 )K ρ2 dK,µ 2 + (1 µ 2 ) K ρ2 K

(51)

dpi = Θi0 θ i dθ i ,(52)

dui = Mi0 θ i dθ i ,

(53)

dZ
1
(1 µ2 )K ρ2 dK
A
=
du
+
Z
µ 2 + (1 µ 2 ) K ρ2 K
φ uA

1

1
du R ,
φ uR

(54)

where
Θi0 θ i =M0A θ A =MR0 θ R =(r

δ + λi ) ci
η i
[1 + (θ i )ψi ]1/ψi 1 (θ i )ψi 1 +c > 0,(1 η )1 η

φλ A [1 + (θ A )ψA ] 1/ψA 1< 0,
fλ A + θ A [1 + (θ A )ψA ] 1/ψA g2(1 φ) λ R [1 + (θ R )ψR ] 1/ψR 1< 0.
fλ R + θ R [1 + (θ R )ψR ] 1/ψR g2

From these expressions, we obtain
dZ
Z
= Ω,
dK
K

34

(55)

where0Ω=R(1 ρ2 ) p R M R ( θ )1 φ u R Θ0R (θ R )1

0(1 ρ1 ) p A M A ( θ A ) (1 µ1 ) Z ρ1
ρ
A
0
φ u
Θ A (θ A ) µ1 +(1 µ1 ) Z 11(1 µ2 ) K ρ2
µ2 +(1 µ2 )K ρ2µ1(1 ρ1 ) p R MR0 (θ R )1 φ u R Θ0R (θ R ) µ1 +(1 µ1 ) Z ρ1

> 0.

(56)

Thus, Z and K are positively related along the LM locus.

D

Proof of Proposition 3

We totally differentiate the system of equations determining a steady state. From (55), we
obtain
dZ
dK
=Ω ,
(57)
Z
K
where Ω is defined in (56). Substitute (57) into (48) and (51) to obtaindadK(1 ρ1 ) µ1
(1 ρ2 ) µ2
=
Ω+
,
ρ
ρ
2
1
a
µ1 + (1 µ1 ) Zµ2 + (1 µ2 ) KK
dp R(1 ρ2 ) (1 µ2 )K ρ2 dK( ρ1 1) µ1Ω+.
=µ 1 + (1 µ 1 ) Z ρ1µ 2 + (1 µ 2 ) K ρ2
K
pR

(58)
(59)

These expressions imply1) µ(1 ρ )(1 µ )K ρ2µ1 +(1 µ1 ) Z 1µ2 +(1 µ2 )K 2

(ρ11
Ω + µ +(2 1 µ )2Kρ2 p R
dp Rµ1 +(1 µ1 ) Z ρ12
2=.
(1 ρ1 ) µ1
(1 ρ2 ) µ2
da
a
ρ Ω+
ρ

Because the denominator is positive, we can show that dp R /da < 0 holds if and only if(1 ρ 2 ) (1 µ 2 ) K ρ2
(1 ρ1 ) µ1<Ω,ρµ2 + (1 µ2 ) K 2µ 1 + (1 µ 1 ) Z ρ1
which is equivalent to
1
1

ρ1

<

1
1

p A M0A θ A(1 µ 1 ) Z ρ1µ1+,
ρ 2 µ 1 + (1 µ 1 ) Z ρ1
φ u A Θ0A (θ A ) µ1 + (1 µ1 ) Z ρ1

or

1
1

ρ1

<

1
1

pA xA
ρ2 y

εA 1

pA xA
y

,

where
εA =

=

dx A p A=
dp A x A

M0A θ ApA
Θ0A (θ A ) φ u A

φλ A [1 + (θ A )ψA ] 1/ψA 1 fλ A + θ A [1 + (θ A )ψA ] 1/ψA g 2(r δ + λ A ) c A[1 + (θ A )ψA ]1/ψA 1 (θ A )ψA 1 + 1 η η c A
1 η

35

pA
> 0.
φ uA

(60)

E

Proof of Proposition 4

For both types of capital to exist, rtT = rtR must be satisfied in any equilibrium. Other equilibrium conditions are obtained as in the previous sections. We summarize the equilibrium conditions as follows: rtT = A[µ1 + (1ρ1µ1 ) Zt 1 ] ρ1

1

ρ

1

µ1 ) Zt 1 ] ρ1

µ1 , p t = A [ µ1 + (1

1

(1

ρ 1
ρ 1 1
ρ 1
ρ 1 1
µ1 ) Zt 1 [µ2 + (1 µ2 )Kt 2 ] ρ2 µ2 , rtR = A[µ1 + (1 µ1 ) Zt 1 ] ρ1 (1 µ1 ) Zt 1 [µ2 + (1
ρ 1 1
ρ 1
µ2 )Kt 2 ] ρ2 (1 µ2 )Kt 2 atR , Zt = zt /k Tt and Kt = atR k tR /xt , rt = rtT = rtR , xt = `t = 1 ut ,yt cvt = Ct + k Tt+1 + k tR+1 (1 δ)(k Tt + k tR ), (4), (6), (7), and (8).
In any steady state, r T = r R = r = β 1 1 + δ. Thus, we obtainr = A [ µ1 + (1r = A [ µ1 + (11µ 1 ) Z ρ1 ] ρ11
ρ1 ρ1µ1 ) Z ]

1
1

µ1 ,

(1

(61)
µ 1 ) Z ρ1 1 [ µ 2 + (1

1
ρ2 ρ2

µ2 ) K ]1(1µ2 )K ρ2 1 a R . (62)Note that (61) uniquely determines Z. Given this value, (62) uniquely determines K foreach level of a R . Given the value of Z and K as a function of a R , the price of labor service,p = A [ µ1 + (11µ 1 ) Z ρ1 ] ρ11(1µ 1 ) Z ρ1 1 [ µ 2 + (11µ 2 ) K ρ2 ] ρ2

1

µ2 ,

(63)

is determined separately from the rest of the model (just as in the model of Section 2).
Because Z is constant and independent of a R in any steady state, (62) implies that anincrease in a R increases K. Thus, (63) implies dp/da R > 0.

F

Heterogeneous Capital: Equilibrium Conditions

The representative household’s lifetime utility is given by (3) and the budget constraint is
now Ct + τt + k tR+1 + k Tt+1 (1 δR )k tR (1 δT )k Tt = wtA `tA + wtR `tR + b A utA + b R utR +rtR k tR + rtT k Tt + πt . The household chooses the sequences of Ct , k Tt+1 , and k tR+1 to maximize(3) subject to the budget constraint. The first-order conditions imply Ct σ = Λt and Λt =βΛt+1 (rtT+1 + 1 δT ) = βΛt+1 (rtR+1 + 1 δR ), from which we obtain rtT δT = rtR δRjfor all t. The transversality condition is limt!∞ Λt βt k t = 0, (j = T, R) where Λt is the
Lagrange multiplier for this problem. Each individual discounts future payoffs by Bt =(rtT+1 + 1 δT ) 1 = (rtR+1 + 1 δR ) 1 = β(Ct+1 /Ct ) σ , from which r T δT = r R δR =β 1 1 in any steady state. Thus, δT < δR implies r T < r R .

36

The production function for the final good producer is now
yt = A(k Tt )α [µ1 ( xtA )ρ1 + (11 α

ρµ 1 ) z t 1 ] ρ1 ,

(64)

1

zt = [µ2 ( xtR )ρ2 + (1µ2 )( atR k tR )ρ2 ] ρ2 ,(65)

where 0 < α < 1 and ρ1 < ρ2 < 1. Profit maximization implies ptA = (1(1

ρ1 1ρ α1µ1 ) Zt ]1µ1 , ptR = (1α) A(KtT )α [µ1 + (1

ρ1 1ρ α
1

1

(1

1

ρ

µ1 ) Zt 1[ µ2 + (1
1 α
1
ρ
1
ρ
µ2 )(KtR ) ]µ2 , rtR = (1 α) A(KtT )α [µ1 + (1 µ1 ) Zt 1 ] ρ1 (1 µ1 ) Zt 1 [µ2 + (1
1
1
ρ 1 α
µ2 )(KtR )ρ2 ] ρ2 (1 µ2 )(KtR )ρ2 1 atR , and rtT = αA(KtT )α 1 [µ1 + (1 µ1 ) Zt 1 ] ρ1 , where Zt =zt /xtA , KtT = k Tt /xtA , and KtR = atR k tR /xtR .
1 αBecause r T = β 1 1 + δT in any steady state, r T = αA(K T )α 1 [µ1 + (1 µ1 ) Z ρ1 ] ρ1
determines K T as a function of Z. We then substitute it into the other expressions above1ρ2 ρ2µ1 ) Zt ]α) A(KtT )α [µ1 +1to obtain the following:1

1

r R = Ã[µ1 + (1µ 1 ) Z ρ1 ] ρ1

[ µ2 + (1µ2 )(K R )ρ2 ] ρ2

1

p A = Ã[µ1 + (1p R = Ã[µ1 + (11µ 1 ) Z ρ1 ] ρ11
ρ1 ρ1µ1 ) Z ]whereÃ = (1

α) A

αA
rT

1
1

α
1 α

(1

µ 1 ) Z ρ1 1

1µ2 )(K R )ρ2 1 a R ,

(1

(66)

µ1 ,

(1

(67)
µ 1 ) Z ρ1 1 [ µ 2 + (1

= (1

1

α) A 1 α

β 1

1
ρ2 ρ2

µ2 )(K R ) ]α
1 + δT

1

µ2 ,

(68)

α
1 α

.

(69)

A steady-state equilibrium under heterogeneous capital is determined by a set f Z, K T , K R ,p A , p R , θ A , θ R , u A , u R , z, k g that satisfies (66)-(68), (34)-(38), and k T = x A K T = (φ u A )K T .

GFlow Value of Nonwork

Our calibration strategy in Section 4 follows that of Hall and Milgrom (2008) so that theflow value of nonwork satisfies bi = 0.71wi . This value is the sum of an estimate of the unemployment benefit replacement rate, which is 0.25, and other components including the
saved disutility of work measured in units of consumption. However, it is widely recognized in the business cycle literature that unemployment tends to be more responsive to achange in the demand for labor when the target flow value of nonwork, bi in our model, is37

closer to the flow value of work wi (Hagedorn and Manovskii, 2008). It is therefore useful
to conduct a sensitivity analysis.
Suppose now that the flow value of nonwork bi is calibrated to satisfy bi = 0.4wi . This
corresponds to Shimer (2005), who assumes that the unemployment benefit replacement
rate is 0.40 and that job seekers do not receive any utility value such as the saved disutilityof work.
Table 8: Flow Value of Nonwork
Case 1: 1 1ρ2 = 2.5 Case 2: 1 1ρ2 = 5 Case 3: 1 1ρ2 = 10
bi /wiA∆u A p∆p A u A
R∆u R pR∆p u RA∆w A p∆p A w A
R∆w R pR∆p w R

0.71

0.40

0.71

0.40

0.71

0.40

1.25

0.73

0.88

0.60

0.62

0.47

2.10

1.05

3.70

1.29

17.00

2.00

0.99

1.00

0.99

1.00

0.99

1.00

0.99

1.02

0.99

1.02

0.99

1.02

The results, summarized in Table 8, are in line with the business cycle literature. Whenthe (target) flow value of nonwork satisfies bi = 0.4wi rather than bi = 0.71wi , the unemployment elasticity in response to a change in the demand for labor is lower. Interestingly,
the wage elasticity is about 1.0 for all models and for all cases.

H

Cobb-Douglas Matching Function

Our specification of the matching function, which we adopt from den Haan et al. (2000)
and Hagedorn and Manovskii (2008), is motivated by the restriction that the job-finding
probabilities Qi (as well as the vacancy-filling probabilities qi ) must be restricted to be
in between 0 and 1 even under a large persistent shock such as the robotics revolution.
An important drawback of this choice is that this specification is not widely used in theliterature.
Because the most widely accepted specification in the literature is a Cobb-Douglas
form, it is useful to study this case as a robustness analysis. In particular, we are interested
in whether the large magnitudes of the effects of the robotics revolution on the labor market variables depend on our specification of the matching function. With Cobb-Douglas,
we can no longer guarantee 0 < Qi < 1 and 0 < qi < 1, but we can interpret the results38

as those for a continuous-time model because the steady-state equations are essentially
identical to those for a continuous-time counterpart of our model.
Suppose now that the number of new employees in market i (i = A, R) is determinedby mi (uit )ξ (vit )1 ξ . We assume that each market has a distinct matching constant mi > 0
and that both markets share the same matching elasticity, 0 < ξ < 1. Each type-i vacancy
is matched to a type-i job-seeker with probability qit = mi (uit )ξ (vit )1 ξ /vit = mi (θti ) ξ ,where θti = vit /uit is labor market tightness in market i. Similarly, the probability that a
type-i job-seeker is matched with a type-i vacancy is given by Qit = mi (uit )ξ (vit )1 ξ /uit =mi (θti )1 ξ = θti qit . The calibrated parameters for Case 2 is given in Table 9.
Table 9: Calibrated Parameter Values: Cobb-Douglas Matching Function
cA

cR

mA

mR

bA

bR

α

µ1

µ2

A

0.205

0.211

0.340

0.534

0.587

0.349

0.350

0.579

0.969

0.486

Table 10 shows that the two different matching functions generate nearly identicalresults, except for the unemployment rate for the unskilled.
Table 10: Comparison of Matching Functionsyu A /φu R / (1 φ )uwAwRw A /w R
Labor share

I

Initial

Case 1: 1 1ρ2 = 2.5

Case 2: 1 1ρ2 = 5Case 3: 1 1ρ2 = 10

aR = 1

CD

HRW

CD

HRW

CD

HRW

1.00
2.6%
8.3%
5.7%
0.83
0.49
1.68
0.61

1.14
2.3%
8.2%
5.6%
0.91
0.49
1.84
0.57

1.14
2.3%
8.2%
5.6%
0.91
0.49
1.84
0.57

1.43
1.9%
10.7%
6.7%
1.06
0.43
2.46
0.48

1.43
2.0%
12.0%
7.5%
1.06
0.43
2.45
0.47

1.87
1.5%
25.2%
14.6%
1.28
0.36
3.55
0.38

1.85
1.7%
43.7%
24.8%
1.27
0.37
3.45
0.37

Labor Market Participation: Equilibrium Conditions

The household chooses a set of sequences fCt , k tR+1 , k Tt+1 , `tA+1 , `tR+1 , utA , utR g∞
t=0 to maximize
"
#
1 σ
∞
t Ct
A
R
∑ β 1 σ + ζ A ln nt + ζ R ln nt
t =039

subject to Ct + τt + k tR+1 + k Tt+1 (1 δR )k tR (1 δT )k Tt = wtA `tA + wtR `tR + b̄ A utA +b̄ R utR + rtR k tR + rtT k Tt + πt , φ = `tA + utA + ntA , 1 φ = `tR + utR + ntR , `tA+1 = 1 λ A `tA +QtA utA , and `tR+1 = 1 λ R `tR + QtR utR . The Lagrangian for this problem is
(
∞
Ct1 σ
L = ∑ βt+ ζ A ln ntA + ζ R ln ntR
1 σ
t =0
"
#
wtA `tA + wtR `tR + b A φ `tA ntA + b R 1 φ `tR ntR + rtR k tR + rtT k Tt + πt
+ Λt
Ct τt k tR+1 k Tt+1 + (1 δR )k tR + (1 δT )k Tt
i
h
A
A
A
A
A
A
A
` t +1nt
1 λ `t + Qt φ `t
+ Λt
io
h
`tR+1 .
+ΛtR 1 λ R `tR + QtR 1 φ `tR ntR

The first-order conditions are:Ct σ

Λt = 0,1
Λt b A ΛtA QtA = 0,
ntA1
ζ R R Λt b R ΛtR QtR = 0,
n
h
i
i t
QtA+1 + βΛt+1 wtA+1 b A = 0,ihi
QtR+1 + βΛt+1 wtR+1 b R = 0,hiRΛt + βΛt+1 rt + (1 δR ) = 0,hiΛt + βΛt+1 rtT + (1 δT ) = 0.
ζAhΛtA + βΛtA+1 1hΛtR + βΛtR+1 1

λA
λR

The transversality condition is limt!∞ Λt βt k tR = limt!∞ Λt βt k Tt = 0.
The first-order conditions with respect to Ct , k Tt+1 , and k tR+1 imply Ct σ = Λt and Λt =βΛt+1 (rtT+1 + 1 δT ) = βΛt+1 (rtR+1 + 1 δR ), from which we obtain rtT+1 δT = rtR+1 δR .
In any steady state,1= Λb A + Λ A Q A
nA
1
ζ R R = Λb R + Λ R Q R ,
n

ζA

βΛ

1

βΛ

1

wA bA
= ΛA,
A
A
β (1 λQ )R
Rwb= ΛR ,β (1 λ R Q R )40from which we obtain
QA β wA bA
1 σ
AζA A C = b +,n
1 β (1 λ A Q A )QR β wR bR
1.
ζ R R Cσ = bR +n
1 β (1 λ R Q R )

(70)
(71)

Accordingly, the steady-state conditions (36)-(38) must be replaced with z = [µ2 (1 φu R n R )ρ2 + (1 µ2 )( ak )ρ2 ]1/ρ2 , z = (φ u A n A ) Z, ak = (1 φ u R n R )K, respectively. Other steady-state conditions are identical to those presented in Appendix F.

41

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