Immigration, Automation and the U.S. Cities:
A Task-based Framework for Welfare Analysis∗
Masahiro Yoshida†
Washington University in St. Louis
Job Market Paper(Frequently updated. Link to the latest version)

December 3, 2019

Abstract
The aging economies facing secular labor shortage are bound to respond by admitting foreign labor or by adopting labor-saving technology. This paper proposes that
inflows of regional foreign labor guides the penetration of automation. I develop a
dynamic spatial framework, in which tasks are optimally allocated across robots, anddomestic or foreign labor. Then, I semi-parametrically recover cross-factor substitution
schedules from a series of commuting zone-level elasticities of economic outcomes withrespect to immigration, which are estimated using a 1940 ethnic settlement pattern.
The model predicts that immigration’s impact on wages during 1980-2015 could bereversed by including eﬀects from immigration-induced adjustments of automation. I
find that low-skilled immigration alone reduces routine occupation native wages, butraises the wages in the long run by retarding the adoption of automation, resulting inenhanced domestic welfare. Finally, I find that a universal basic income policy targetedto U.S. citizens will boost dependence on automation and foreign labor by upshiftingroutine occupation native wages.
JEL Classification: E22, J11, J31, J61, O14
Keywords: immigration, demography, automation, regional labor market∗

I thank my advisors, George-Levi Gayle, Alexander Monge-Naranjo and Stephen Ryan, for their invaluable advices and encouragement. I have also benefited from discussions with David Autor, Michele
Boldrin, Francisco (Paco) Buera, Avi Goldfarb, Barton Hamilton, Xincheng Qiu, Hiroki Saruya, Yongseok
Shin, and Ping Wang. I thank the participants of the NBER AI Young Scholar’s Workshop 2018 and the
Urban Economics Association Conferences in 2018 and 2019 for their helpful discussions. I also thank myformal discussants, Diego Battiston and Wenjian Xu for their insightful comments. I greatly appreciate thefinancial support provided by the Nakajima Foundation, McDonell International Scholar’s Academy and my
dissertation year fellowship.
†
Department of Economics, Washington University in St. Louise-mail: m.yoshida@wustl.edu; Personal Website; Department Website

1

1

Introduction

The rising concern about disruptive automation in the labor market is ubiquitous. (Brynjolfsson and McAfee (2014) and Frey and Osborne (2017)) In contrast to the current conventionalview, which presumes that new technology unambiguously brings net benefits to labor byaugmenting human skills, economists, including Autor and Dorn (2013) and Acemoglu andRestrepo (2017), have re-examined the labor market impacts of skill-replacing technology
and found adverse eﬀects on routine occupation workers. Nearly all the literature, however,frames automation technology as a primitive external force, and underrates this simple observation: automation is implemented by investing in autonomous machines (robots), thus,
automation is a consequence of rational technology choices.1 Overall, in conjunction with thesurging public interest in how evolving automation technologies reshape the labor markets,the literature is dominated by what could be called a technology-push view, and relatively
little is known about the mechanism regarding why, when and how much the production
economy adopts the new automation technology.
To help fill this void, this paper theoretically and empirically investigates the mechanismof automation adoption, and advances the hypothesis that foreign labor inflow, a consistently increasing class of labor across the developed economies, aﬀects the penetration of
automation in the long run. If the adoption of automation is based on the cost advantage
of using robots instead of human labor, then, immigration presumably has the potential toshift the human skill prices, and thus, alter the optimal magnitude of automation. This relatively unexplored link is important because immigration is a policy variable to reconfigure
the national demography, and automation significantly reshapes the labor market landscape,as the recent automation literature demonstrates.2 Therefore, the country could potentiallynavigate the adoption of automation via immigration policy.
To formalize the theory described above and quantitatively assess how immigration has
played a role in shifting automation, I develop a dynamic spatial task-based model of endogenous automation based on Acemoglu and Restrepo (2018a).3 In a spatial economy, acompetitive establishment with heterogeneous productivities produces a final good, com1

A notable exception is a recent work by Acemoglu and Restrepo (2018a), which explores how demographic change nudges the production system to develop automation, in the context of directed technicalchange. This paper complements their work by featuring the distinctive role of foreign labor and by adopting
a more structural, empirically implementable approach.
2

Except Autor and Dorn (2013) and Acemoglu and Restrepo (2017) mentioned above, see for example,
Eden and Gaggl (2018).
3

Ottaviano, Peri and Wright (2013) employ a similar vein of task-based model to jointly examine theinteraction and impacts of immigration and oﬀshoring, in contrast to my focus on automation.

2

bining tasks with diﬀerent routineness (easiness-to-automate). The production factors arerobots, routine skills and non-routine skills, and each skill is a composite of domestic andforeign skills. Each production factor has arbitrary task-specific factor productivities on
the task spectrum, allowing for a richer cross-factor substitution pattern than conventional
iso-elastic technology.
Three non-stationary drivers (immigration inflow, demographic transitions, and technological progress, proxied by quality-adjusted investment price of robots) shapes the equilibrium skill prices. Given the expected future menu of factor prices and task-specific factor
productivities, the central problem for establishments is dynamically optimal task allocation.
Automation is implemented by task reallocation from routine skills to robots; establishmentsof heterogeneous productivities decide when and how much tasks to be automated by the
updated vintage of robots by paying the cost of adjustment friction,4 accompanied by theirinvestments in robots.5
All the implications from the model are tightly disciplined by regional-level cross-factor
substitution. The model yields a key relationship between elasticities of task reallocation
with respect to factor prices and regional cross-factor substitution. Since elasticities of task
allocation are unobservable per se, I analytically derive higher-level estimatable elasticities ofregional economic outcomes (robot deployments, routine skill prices, and native premium ofskill prices) under the steady state with respect to low-skilled (or high-skilled) immigration
as functions of regional-level cross-factor substitutions. Having developed a model, the identification and estimation proceeds in three steps. First, I conduct quasi-experiments using
reduced-form regressions across the CZs and years to estimate the regional-level elasticities
of robots, routine skill price, and native premium of skill prices with respect to low- andhigh- skilled immigration inflow in manufacturing sector, and in information service sector,
respectively, during 1980-1990, regarding this initial decade as the change under the steadystate. To address the routinely invoked issue of non-randomness of regional immigration
inflows, I use the location preferences of immigrants as non-market forces, as immigrants
strongly prefer a neighborhood with larger numbers of people of their own ethnicity (Munshi(2003)). Operationally, I use the historical settlement patterns of 26 ethnic groups in 1940 to4
Adjustment costs for automation could allow for many interpretations: unions, legal labor regulations,
initial learning costs, or inertia.
5

As robots are durable, and their use is de facto irreversible, the decision to invest in robots inherently depends on the current menu of factor prices (both regional skill prices and national quality-adjusted investment
prices), and the expected future path of menus. The model echoes with classical optimal replacement model(e.g. Rust (1987)) and durable good demand literature. (e.g. Adda and Cooper (2000), Gowrisankaran andRysman (2012))3

construct an instrument of ethnically-driven waves of foreign labor settlements, orthogonal
to regional economic climates6 (Altonji and Card (1991), and Card (2001)). The regional
reactions to immigration inflows exhibit remarkable heterogeneity, especially between urban and rural areas; the elasticities of routine occupation wages and robots with respect toimmigration are more negatively skewed in rural areas than in urban areas.7
Second, I semi-parametrically identify and estimate the entire cross-factor substitution
schedules to consistently match the series of region-specific immigration elasticities in eachsector, as obtained in the first step. To do this, I draw on three analytical mappings,
generated from the benchmark model; the threshold-specific substitutions of routine skills
vs. robots, the substitutions of non-routine vs. routine skills, and the substitutions of foreign
vs. domestic skills. I use these analytical mappings as moment conditions in a Generalized
Method of Moments (GMM) procedure to consistently rationalize a trio of regional-specificelasticities of economic outcomes with respect to immigration. To the best of my knowledge,
this paper is the first to empirically recover the task-specific factor productivities in the lineof literature using a task-based approach.8
Third, after inserting these factor productivities recovered in the previous step into the
dynamic model, I estimate a pair of non-convex and convex adjustment parameters forautomation with a product diﬀerentiation parameter, using a Simulated Method of Moments(SMM) matched with patterns of establishment size evolutions across sectors.
Armed with the model including the estimated technology and adjustment costs, I startby quantifying the impact of immigration on the similarly-skilled native wages, both withand without eﬀects from automation upgrading. A comparison to the counterfactual casewith no immigration growth from 1980 to the baseline reveals that low-skilled immigration
alone reduced the routine native wages by -0.05%, but raised the wages by +0.04% from1980-2015 when automation adjustment is included. In contrast, high-skilled immigration
alone raised the non-routine native wages by +0.03%, and inclusive of adjusted automation,raised the wages by +0.05% from 1980-2015.
The results highlight the importance of taking the dynamic eﬀect of adjusted technology
in the production system into account as well as substitution between immigrants and natives
6
The instrumental variable strategy has been predominantly used in the immigration literature. For an
extensive list, see Jaeger, Ruist and Stuhler (2018).
7

The urban areas are the top 37 CZs accounting for the half of the labor force in 1980, and the rural
areas are the other CZs.
8

Recent related works on the task-based approach includes Peri and Sparber (2009) (immigration), Acemoglu and Restrepo (2018b)(automation), and Ottaviano, Peri and Wright (2013) (oﬀshoring). See Acemoglu
and Autor (2011) for a theoretical background.

4

when examining the economic impacts of immigration and crafting an immigration policy.
Low-skilled immigration weakened robotization in manufacturing, and thus, raised routine
native wages in the long run. The measured impacts are stronger by 4 times when theregions are limited to the top 30 CZs of the largest low-skilled immigration growth rate,showing the geographically heterogeneous impacts of immigration.9 In contrast, increased
computerization driven by high-skilled immigration in the information service sector further
benefited non-routine native workers. The findings are applicable to the welfare implication
when social welfare is defined as a weighted sum of consumption utilities from labor andnon-labor incomes for the U.S. citizens. Under the plausible distributional preference of theplanner, I find that both low-skilled and high-skilled immigration during 1980-2015 improved
the domestic welfare at the time of 2015.10
Having reproduced the rational delay of automation upgrading during the target period,
1980-2015, I extend the analysis by another 35 year future period, 2016-2050, and conduct
policy experiments. First, by keeping the current immigration policy, I extend the simulation,
following the demographic projections of the most recent Census (Census Bureau (2017))
and Weldon Cooper Center (2018). This could be labeled as the future baseline case.
Second, motivated by the burgeoning anti-immigration movements occurring across developed countries, I ask what would happen if future immigration is restricted in the U.S..
To consider an extreme case, I mechanically keep foreign labor in the U.S. at the projectedlevel of 2020, and quantify the diﬀerence from the future baseline. The simulation showsthat the immigration regulation is welfare-harming by -0.5% (low-skilled) and -1.9% (highskilled) compared to the baseline at the time of 2050. The policy simulation mirrors thefindings during the target period: low-skilled immigration weakens robotization, shrinking
wage inequality. High-skilled immigration augments the wages of both occupation classes
despite the expansion of wage inequality from greater use of computerization.
Last, motivated by the rising policy debate about a potential universal basic income (UBI)
in the face of looming technological unemployment due to technological advances11 , I explore
what would happen if a part of natives exit from the labor force by a UBI policy. Typically,a UBI is discussed as unconditional minimum (basic) stipend (800-1200 USD per month)9

These top 30 CZs account for 46% of low-skilled immigration inflow in terms of headcounts.

10

I try out sensitivity analysis of welfare protocols, and the implications are stable. See Section 6.1.

11

Recently, UBI has gained a new vive instead of the canonical progressive social security system.
Economists, including Peter Diamond and Chris Pissarides, have publicly expressed their approval. (Forbes(2017)) In 2016, a national referendum was held in Switzerland to vote on adopting a UBI, and it was
rejected.

5

to be provided to every (universal ) working-aged U.S. citizen, excluding immigrants.12 The
simulation shows that a UBI boosts automation intensity (proxied by robot task share) by
0.4-0.6%, and 1.5-2.1% across sectors, by raising routine native wages by 4.1-5.8% and 6.29.1%, respectively, as compared to the future baseline for 2020-2050. Parallel to the surging
deployment of automation, dependency on foreign labor, which has been excluded from the
UBI provision, is predicted to sharply rise by 0.5 and 0.3 p.p. in 2050 as compared to thebaseline across sectors. Although a UBI is often touted as a counter-measure to massively
disruptive automation, which is implicitly founded in the technology-push view, my model
delivers a contrary perspective: a UBI would serve as a policy shock resulting in the highly
automated economy.
The paper is structured as follows. Section 2 shows motivating evidences, delivers aperspective on mechanisms of automation, and positions a related literature within the perspective. Section 3 provides a setup and equilibrium of the framework. The framework
under the steady state delivers analytical formula of the economic impacts of immigration
via automation adjustment. Section 4 describes the data sources, and their salient patterns.
Section 5 delivers an identification strategy, outlines the estimation procedures, and reports
the results obtained. Section 6 quantitatively assesses each channel, accompanied by decomposition exercises by multiple forces. Section 7 reports policy experiments. Section 8
concludes.

2

Understanding the Mechanism of Automation by Linking Immigration and Automation

2.1

Motivating graphics

To motivate the hypothesis, I juxtapose foreign labor inflow and automation penetration
at the U.S. regional level. The U.S. commuting zones (CZs, developed by Tolbert andSizer (1996)) provide 722 geographically consistent units suitable for tracking the relationships between immigration inflow and automation penetration. Immigration inflow is stably
asymmetric across the CZs13 , generating a large cross-region labor supply shocks to the aging
12
Unsurprisingly, the largest obstacle preventing the adoption of a UBI is the massive expenditure necessary. To finance a UBI, I consider a regime of heavier taxation of high-income earners, a sizable part ofwhom belong to the information service sector.
13
From 1980-2015, the mean growth rate of working-age immigration (employed in Figure.1 and 2) ismore than double the median. (low-skilled, 5.0% (mean) vs. 2.4% (median), high-skilled, 3.4% (mean) vs.
1.5% (median), implying a remarkably skewed regional immigration inflow. See Figure.13 for a geographic

6

demography, which proceeds in more uniformly across regions, and is mirrored by declining
geographic mobility for the native population (Notowidigdo (2011)).
It is well-known that immigrants into the U.S. are characterized by their polarity of skill
levels (See Data Appendix). Due to the salient skill dispersion among the immigrants, they
are categorized as low-skilled or high-skilled, according to their education attainments.14
I contrast each class of foreign labor according to their plausibly pertinent mode of automation across two sectors: the robotization of production tasks in manufacturing and thecomputerization of non-production tasks in information service. As automation technology
is unobservable per se, it is proxied by the capital embodying each mode of automation,
namely, industrial robots and information technology (IT) capital.15
As Figure 1 and 2 shows, the magnitude of immigration inflow with diﬀerent skill levels
robustly stands in sharp contrast to their pertinent mode of automation; that is, low-skilled
immigration negatively correlates to robotization, but high-skilled immigration is positively
correlated with computerization.16 This observation guides my hypothesis to bridge thetwo factors, by highlighting the key mediating role of regional routine wages: low-skilled
immigration inflow reduces the routine wages by providing an over-supply of routine skills.
Thus, establishments would use less robots in production tasks.17 On the other hand, highskilled immigration inflow raises the demand for complementary routine tasks, and thus,
elevates the routine wages. Establishments would heighten the computerization of routine
tasks previously provided by routine labor.18
illustration.
14
In Figure 1 and 2, immigrants with high-school diplomas or less are low-skilled, while those with bachelor’s
and higher degrees are high-skilled. For the rest of the analysis, high-skilled immigration is extended toincorporate immigrants with some college education.
15

Eden and Gaggl (2018) emphasize the distinct roles of IT capital as opposed to non-IT capital in theirwelfare calculations.
16

Figure 2 interprets IT capitals as skill-replacing to routine skills. but it could be also interpreted as
skill-augmenting to non-routine skills in the context of skill-capital complementarity. (e.g. Krusell et al.
(2000), Autor, Levy and Murnane (2003), Autor and Dorn (2013)). My model explicitly distinguishes two
forms of IT capitals by robots and equipments.
17

A companion paper, Yoshida (2019) examines whether low-skilled immigration inflow impedes robotization, using a more narrowly defined industrial robot dataset from IFR (2015). The paper examines
manufacturing, construction and agriculture.
18

Intuitively, a larger inflow of managers or engineers require a larger back oﬃce (e.g. secretaries, bookkeepers) to support their expert skill level activities. For example, a secretary in San Jose, CA is better paid
than one in Flint, MI, possibly because of more abundant demand for administrative jobs.

7

Figure 1: Low-skilled immigration inflow and robotization (1980-2015)

Figure 2: High-skilled immigration inflow and computerization (1980-2015)
Notes: Computed from the Population Census (1980), American Community Survey (2015) and National
Income and Production Accounts from the Bureau of Economic Analysis. Individuals with high-school
degrees or less are low-skilled , while those with bachelor’s and higher degrees are high-skilled . Robot adoption
is measured by industrial robots per production worker. IT capital adoption is captured by IT capitals per
non-production employment. Regressions are weighted with CZ-level working-age population in 1980, as
depicted by the size of each bubble. Shaded areas are 99% confidence intervals. For details of the data
source, see Section 4.

Given that autonomous machines follow the law of one price in the global capital market,
Figure 1 indicates that automation could be framed as regional phenomena induced byregional heterogeneous wages molded by the geographic and demographic constraints of8human workers.19

2.2

Why does automation occur?: the mechanism of automation

To organize the hypothesis on the interaction of immigration and automation, I craft aconceptual chart, Figure 3, which summarizes the mechanism of automation. I sequentiallydivide the mechanism into three channels, drivers of automation, captured by (1) and (2) andthe consequences of automation, as by (3). (4) represents the aggregate eﬀect of immigrationon the labor market.20

Figure 3: Mechanism of automation
Notes: Robot investment price is quality-adjusted so that it proxies the technological progress of automation.
S denotes skills, q denotes skill prices, and the upper-line and lower-line corresponds to non-routine androutine skills, respectively. Formal definitions are given in Section 3.

First, in the short run and under a fixed level of automation, immigration aﬀects skill
prices in conjunction with demographic change. The eﬀect of immigration inflow on skill
prices without eﬀects from adjusted automation is given by:
"
dq ""
χ1 =
"θ
dS!F "(1)

19

Research shows that routine occupation workers exhibit less mobility (Notowidigdo (2011)), and theirwages are presumably more vulnerable to regional demographic change. Moreover, when adopting automation, establishments will be plausibly more sensitive to routine workers’ wages rather than non-routine
workers’ wages.
20

Theoretically, declining robot investment price augments skill prices via investment-specific technology
changes, independently from adoption of automation. (Greenwood, Hercowitz and Krusell (1997)) This
channel is also included in the quantitative model.

9where q ∈ {q, q} represents skill prices including routine and non-routine occupations, S!F islow- or high-skilled foreign skills, and θ is automation intensity, as defined below in Section
3.
In the long run, technology is the choice variable of producers. Second, the routine skill
price changed from the first channel, and guides the automation intensity in parallel to theevolution of quality-adjusted investment price of robots. The eﬀect of routine skill pricechange on automation intensity is given by:
χ2 =

dθ
dq

(2)

where q denotes routine skill prices. Third, in turn, the induced automation change aﬀects
the skill prices. The eﬀect of automation adjustment to skill price is given by:
χ3 =

dq
dθ

(3)

Summing up (1),(2) and (3), the aggregate eﬀect of immigration on skill prices are formulated as
dq
=
χ1
+
χ2 × χ3.
(4)#$%&# $% &dS!F
direct eﬀect
indirect eﬀect via automation adjustment

The key empirical challenge is to quantitively assess each channel of χ1 , χ2 and χ3 . One
can see that what makes this mechanism intractable is the fact that the equilibrium skill
price changes serve both the drivers and the consequences of automation. To distinguishthese eﬀects, this circularity guides me to develop a unified structural model that regards
regional immigration inflow as a driving force from the labor supply side.

2.3

Literature Review

The paper encompasses the intersection of three related lines of literature, each of which
could be mapped on Figure 3. First, this paper is related to a small stream of papers
bridging labor supply and technology adoption. (e.g. Beaudry, Doms and Lewis (2010),
Manuelli and Seshadri (2014), Akerman, Gaarder and Mogstad (2015)) My hypothesis about
the manufacturing sector most closely echoes those of Lewis (2005) and Lewis (2011), by
exploring the negative eﬀect of low-skilled immigration to technology adoption, using microlevel plant data in metropolitan areas. My hypothesis about the information service sector
complements the work by Basso, Peri and Rahman (2017), who analyze the role of low-skilled
immigration in the non-tradable service sector on labor market polarization by adopting
computerization. This paper features high-skilled immigration into the tradable service
10sector; Beaudry, Doms and Lewis (2010) explores the eﬀect of high-skilled labor on computer
adoption across metropolitan areas. This stream of papers do not explicitly frame theirtechnology as labor-saving automation. Also, they consider the link (1) and (2) in Figure 3,
but does not explicitly split the channel into the two by mediating with routine occupation
wages.
Second, this paper belongs to the immigration literature (e.g. Card and Peri (2016), Ottaviano and Peri (2012), and Borjas (1995)), exploring the aggregate eﬀect of (4), mapped by(4) in Figure 3. The literature is characterized by the intensive focus on immigration impacts
on native wages (Dustmann, Schonberg and Stuhler (2016)), and this paper complements thepapers in the stream by highlighting the dynamic impact from adjustment of automation,indexed by (3) in Figure 3. Moreover, this paper allows for regionally heterogeneous impactsof immigration by heterogenous substitution patterns across tasks. All of the papers in thisstream use the canonical constant elasticity of substitution (CES) protocols as a theoreticalframework, and do not adopt a task-based approach.
Third, this paper complements the automation literature. As stressed in the introduction,
the recent automation literature (e.g. Autor, Levy and Murnane (2003), Autor and Dorn
(2013), Acemoglu and Restrepo (2017), Eden and Gaggl (2018)) is founded by the technologypush view, and focuses on the channel of (3) in Figure 3. In contrast, I highlight themechanics of why automation occurs as well as its feedbacks on the labor market.
Overall, no literature jointly analyze the channel of (1)-(4). My framework elucidates
the processes of (1) and (2) and incorporates the feedback eﬀects of automation adjustment
from (3) to evaluate the aggregate impact of immigration, (4). The quantitative assessmentof each channel is presented in Section 6.

3

Dynamic Spatial Model of Endogenous Automation

To quantify how immigration accounts for regional automation adoption, and how automation aﬀects the regional labor markets, I develop a dynamic spatial task allocation modelwith endogenous automation. The model is characterized by three features. First, under
the rationally expected transition of skill prices and a perfectly-foreseen path of constantly
declining quality-adjusted prices of robots, automation upgrading is run by forward-looking
establishments, accompanied by investments to robots with the latest vintage. Note thatthe model implements automation as task reallocation from routine labor to robots, conceptually distinct from the canonical mode of technological progress proxied by declining
quality-adjusted prices of robots.
Second, as is common in capital investment literature, an establishment incurs sunk costs
11

in both non-convex and convex forms, whenever they adjusts automation technology. As automation is factor-biased technical change, adjustment costs of automation conceptuallyaccommodate both capital adjustment (e.g., robot installation, learning) and labor adjustment costs (e.g., layoﬀ costs, renegotiation costs), suﬃciently proxying the non-economic
frictions of adjusting automation.
Third, an establishment has their factor-neutral unobservable total factor productivities (TFP), which could influence automation upgrading, accompanied by frictional investments.21 As a result of diﬀerential upgrading behaviors due to heterogeneous productivities,
establishments with newer vintages enable lower final good prices, and achieve larger market
shares, substituting the market share with establishments with older vintages.

3.1

Setup

Suppose a country consists of an array of islands (regions) l. Production is operated bya finite group of establishments, indexed by i. Time is an infinite horizon, starting at theinitial year t = 0. 22 Establishment i in region l in year t competitively produces a nationallytraded final good, Yi,t . Each island is inhabited by individuals, indexed by j with citizenshipej ∈ {F, D}, consisting of natives (ej = D) and immigrants (ej = F ). On the consumption'side, consumption for individual j, C
j,t is a CES combination of diﬀerentiated final products
from every establishment i in the country, C'
j,i,t , formulated as'' σ−1 σCj,t = {Σi (Cj,i,t ) σ } σ−1'where σ > 1 is an elasticity of substitution.23 The final aggregate consumption good C
j,t isset to be numeraire. On the production side, an establishment i produces the final aggregate
good byη1−η
Y!
i,t = (Ai,t ) (Yi,t )where η is an intellectual property share, stemming from an unobservable productivity, Ai,t ,and Yi,t is a deterministic baseline production. Ai,t follows a persistent AR(1) evolution such
21

In this context, TFP could be well-interpreted as intangible knowledge assets inside the establishment,or the firm it belongs to, including intellectual patents, management know-hows, and engineering knowledge
accumulated, which plausibly aﬀects the adoption of the frontier automation technology.
22

l, i and t are often suppressed for brevity, except where especially required.

23

When σ > 1, if automation lowers the final good price relative to the sectoral aggregate price, theproduction value is upshifted.

12

that
ln Ai,t+1 = ξt + ρ ln Ai,t + ϵi,twhere ξt is an annual TFP shifter, ρ ∈ (0, 1) is a stationarity parameter, and ϵi,t is identicallyand independently distributed (i.i.d.) shock drawn from a standard normal distribution.
Task allocation A final good is produced from a continuum of tasks ω ∈ [0, 1], a continuous, diﬀerentiable, and closed interval with measure one. Yi,t is produced from unit-elastic
aggregation of output from task ω, y(ω) such thatY = exp(( 1

ln y(ω)dω).

(5)

0

24

Competitive labor market in island l oﬀers either routine skills S from routine occupation workers and non-routine skills S from non-routine occupation workers. y(ω) is specified
as follows:(6)y(ω) = {AR r(ω) + Aa(ω)s(ω) + Aa(ω)s(ω)}α k(ω)1−α
where s(ω) is routine skills, s(ω) is non-routine skills, r(ω) is robots, and k(ω) is capital
equipment employed in task ω. {a(ω), a(ω)} are series of task-specific factor productivities
for routine- and non-routine skills, respectively, normalized as a(0) = a(0) = 1. AR , A, A are
factor-augmenting productivities for robots, routine skills, and non-routine skills, respectively. Without loss of generality, A is normalized to unity. 1 − α is intensity of capital
equipment. Lower bars indicate variables of routine occupations, while upper bars show
those of non-routine occupations.
Analogous to the relationships between final goods and tasks, I construct a nested structure of aggregate skills and sub-tasks (Figure 4). Specifically, an aggregate skill S) ∈ {S, S}′′
is unit-elastic aggregation of output from a sub-task ω ∈ [0, 1], y)(ω ) such thatS) = exp[

( 1
0

′

′

ln y)(ω )dω ].

Production from each sub-task is perfect substitutes of eﬀective foreign and domestic skill
productions such that

24

′
′
′
′
′
!F a!
!
!D a!
!
F (ω )sF (ω ) + A
D (ω )sD (ω )y)(ω ) = A(7)

Acemoglu and Restrepo (2018b) features primarily shifting of the technology frontier (most non-routine
task automatable from the engineering point of view) in directed technical change framework, while thepaper endogenizes technology adoption of the realized automation.

13!
!F , A!D are factorD (ω ′ ), a
F (ω ′ )} are sub-task productivities for ω ′ ∈ [0, 1], and A
where {a!
augmenting productivities of sub-tasks for domestic and foreign skills, respectively. Without!D is set to unity.
loss of generality, A

Figure 4: Nested structure of task allocation
Notes: Note that after tasks are allocated to R, S, S, each factor is combined with equipments, K to produce
task outputs.

Dynamic automation problem Establishments periodically employs labor skills fromthe regional labor market, rents equipments from the global capital market, and owns robots
via investments.25 Shaped by regional immigration inflow and demographic transition inisland l, a menu of routine and non-routine skill prices, q l , q l , respectively, is competitively
given. Assume that establishments rationally expect a skill price trajectory of routine andnon-routine skills with nativity distinction in each sector. In nationwide capital markets,robot R and equipment K are elastically supplied with quality-adjusted investment pricesof robots, pR and rental prices q K , respectively. Assume that establishments have perfectforesight of {pR , q K }.26
L
H !
e
Task allocation rule is characterized by Λt = {ΛRt , Λt , Λt , Λt } (e ∈ {D, F }), where
L
H !
e
ΛR
t , Λt , Λt , Λt ⊂ (0, 1) is a set of tasks allocated to robots, routine skills, non-routine skills,
25

Note that automation shifts with friction, and investment is implemented without friction. Even when
automation is not upgraded, investment to robots occurs to compensate for depreciation to retain the
production. Regarding equipment, one could alternatively model establishments investing to equipment.
However, this is mathematically equivalent to rental price payment equal to compensation for depreciation
under the perpetuate inventory method.
26
This assumption is conventional in the investment literature that capitals are produced with marginal
cost pricing, almost exclusively shaped by technological forces. (e.g. Moore’s law)14

and a set of sub-tasks allocated to foreign or domestic skills at period t, respectively. The
periodic profit at period t is specified as− (q S + q S )Πi,t (X t , Ai,t , Λi,t | Θ) = Pi,t Y!
# $% i,t& # l,t i,t $% l,t i,t&revenue−payment to skills

RpR
i Ii,t# $% &

investments to robots" " " R "R"
" "!
− I("ΛRi,t > Λi,t−1 )c(X t , Λi,t , ζ, τ )Pi,t−1 Yi,t−1#$%&
adjustment costs payment

where X t is a set of all observable state variables (a history of task allocation {Λi,t } andK
factor prices {q l,t , q l,t , pRt , qt }), Θ = {ζ, τ, σ, ξt , ρ} as a set of parameters to be estimated, Pi,tR
is a final good price, Ii,tis i’s investment to robots in period t, I(·) is an indicator function
and c(X, θ, ζ, τ ) is adjustment cost function per production value at the previous period," "t − 1. "ΛR " captures an size of task share allocated to robots. To simplify the analysis,
automation upgrading is irreversible and the intensity level could only be allowed to berising monotonically over time.27
R
Investment into robots Ii,tis specified by task allocation Λi,t along with the following
the law of motion, since task allocation, especially automation choice, determines the robot
demand.
RIi,t= Ri,t (Λi,t ) − (1 − δ R )Ri,t−1 (Λi,t−1 )(8)Note that investment to robots occurs both from task reallocation and from depreciation
δ R . Following the line of adjustment cost literature (e.g., Cooper and Haltiwanger (2006)),
adjustment cost function is parameterized by non-convex and convex adjustment parameters,ζ and τ such thatc(X, θ, ζ, τ ) =" " " "" ′"+ ( "Λ R " > "Λ R ") 2 τ .
$%&#non-convex costsζ#$%&convex costs

τ is interacted by a square of expansion of robot task share.
Since technology is constant returns to scale, optimization problem for every establishment is isomorphic with respect to the production value. Introduce πi,t (X t , Ai,t , Λi,t | Θ) as
a normalized profit function, denominated by their standard production value, Pi,t−1 Y!
i,t−1
27

This is not so restrictive because under the environment that quality-adjusted prices is constantly declining and skill prices are relatively stable, establishments’ behaviors are only incentivized to upgrading.
Alternatively, allowing for frictional downgrading does not severely alter the result.

15

such that
πi,t (X t , Ai,t , Λi,t | Θ) ≡Πi (X t , Ai,t , Λi,t | Θ).
Pi,t−1 Y!
i,t−1

Using Bellman’s principle of optimality, a forward-looking establishment’s problem is recursively defined as dynamic task allocation problem; a value function is recursively defined as
a
(
′
′
′
′
V (X, A) = max π(X, A, Λ | Θ) + β V (X , A )dFΛ (X , A | X, A)Λ′′

where β is a common discount factor for establishments and FΛ (X , A | X, A) is a choiceΛ-dependent transition distribution. In a lower level, an establishment statically minimizes)
the cost to satisfy the demand of aggregate skills S:!F + q!
!D
FS
DS
min q!

!
"
F ,Λ
D
Λ

s.t. exp{

( 1
0

′
′
) where y)(ω ′ ) is given by (7).
ln y)(ω )dω } ≥ S,

!e =
S

( 1
0

′

s)e (ω )dω′(e ∈ {D, F })In the higher level, an establishment optimally allocates production factors over a spectrum of tasks, anticipating a trajectory of skill prices and quality-adjusted capital prices.
In the lower level, occupation manager minimizes the skill expenditure to satisfy the aggregate skill demand S) from the establishment manager to employ the best mix of foreign anddomestic skills.

3.2

Equilibrium

To highlight the non-degenerate equilibrium, I impose a set of regularity conditions. These
weak regularities are routinely imposed in the task-based literature. (Grossman and RossiHansberg (2008), Acemoglu and Autor (2011), and Ottaviano, Peri and Wright (2013),
among others)
R0 [Task productivities of skills]
Task (or sub-task) productivities for human skills (foreign skills) are strictly increasing (ordecreasing) and continuously diﬀerentiable in task non-routineness. Formerly,′′′
F (ω )/dω < 0 ∀ω, ω ∈ [0, 1]
da(ω)/dω > 0 da(ω)/dω > 0 da!

16

holds.
Brackets are for sub-tasks. Recall that the task-specific productivities of robots andsub-task productivities for domestic skills are set to unity across all (sub-) tasks. The
former part regularity implies that task-specific productivities of human skills are strictlyincreasing when tasks become more non-routine, compared to robots. This naturally comesfrom the definition of task routineness (easiness-to-automate). The latter part states that
sub-task productivities for foreign skills are strictly decreasing when sub-tasks become more
non-routine. This simply serves to capture diﬀerentiation between foreign and domestic
skills across sub-tasks.28 Three additional regularities, R1-R329 are imposed to regulate thebalance of relative cost eﬃciency of three pairs of production factors to ensure every factor
is employed in some non-empty tasks. R0, together with these corner conditions R1-R3,
establishments follow the cutoﬀ strategies to split the task spectrum across factors.
Proposition 1. [Optimal Task Allocation over Production Factors]
Suppose the regularities R0-R3. Then, there exists an automation intensity θ ∈ (0, 1)
and an occupation threshold ψ ∈ (θ, 1) such that ΛR = [0, θ], ΛL = (θ, ψ], and ΛH = (ψ, 1]!F = [0, φ]holds.30 Moreover, there exists an immigration threshold φ ∈ (0, 1) such that Λ!D ∈ (φ, 1] holds.31 Under no friction of adjusting automation, θ,ψ and φ are implicitly
and Λ
defined to satisfy
!F a!
F
F (φ)
q A
q
q!
a(θ)
a(ψ)
=
.
(9)
=
=
,
,
D (φ)
D
AR
q R Aa(ψ)
qa!
q!
where robot rental price q R is computed as q R = δ R pR using a depreciation δ R .32
28

The normalization in sub-tasks loses no generality. For example, one could run the analogous argument′′
F (ω )/dω > 0 or normalize sub-task productivities of foreign skills as unity instead of domestic
even if da!
skills. This would require a variant of R3, but the subsequent argument would be unaﬀected.
29

Three regularities specifies the range of routine skills relative to robots, of routine skills relative of nonψ − θ AR
S
routine skills, of foreign skills relative to domestic skills in eﬀective terms as<< (1 − ψ)AR
θ a(1)RS
φφ(ψ − θ)Aa(1)(ψ − θ)Aa(θ)S!F<(R2), and
<
<
(R1),
<
(R3). In!F
!F a!
(1 − ψ)a(θ)(1 − ψ)a(1)
D
F (1)
S
S!
(1 − φ)A(1 − φ)A
Model Appendix, I show that R1-R3 is rewritten as corner conditions, graphically assuring single-crossingof marginal cost curves, used in Proposition 1.
30

Each factor is combined with equipments by intensity 1 − α.

31Without loss of generality, a tie-break condition is imposed at the crossing point on the atomless task
(or sub-task) spectrum.
32

This is a special case of a perpetuate inventory method with no rental markets, nor financial markets.
(See, for example, Barkai (2016))

17

Figure 5: Equilibrium task allocation (left: upper level tasks, right: lower level sub-tasks)
Notes: q R is imputed rental price of robots such that q R ≡ δ R pR .

Proposition 1 states that given the task productivities and factor-augmenting productivities (intensive-margin of technology) and a menu of regional skill prices, national rental
prices of equipments, and national investment price of robots, the optimal task allocationis characterized by a trio of task thresholds (extensive-margin of technology), automation
intensity θ ∈ (0, 1), occupation threshold ψ, and immigration threshold φ.
Graphically, the regularities R0-R3 ensure the specialization of factors over the task
spectrum in Figure 5. Since most aﬀordable factors are allocated to each task, robots performthe most routine range, routine skills cover a middle range, and non-routine skills cover themost non-routine range. In the low-level sub-tasks, foreign skills cover the routine range, anddomestic skills cover the non-routine range. Given Proposition 1, the marginal cost functionof each task when tasks are exclusively allocated to each factor is explicitly formulated as afunction of factor price and (sub-) task productivity. Denote the marginal costs of routine(non-routine) task outputs by routine- (or non-routine) skills as p(ω), p(ω), respectively.
Then, some algebra yields
p(ω) = (q
q K 1−αqq K 1−α) , p(ω) = (
)α (
) .
)α (
a(ω)α 1 − αAa(ω)α 1 − α(10)Analogously, let the marginal cost of sub-task outputs implemented by ethnic skills as18′p)e (ω ). Then,′p)e (ω ) =q)e(e ∈ F, D)!e a)e (ω ′ )A(11)holds. Given the regularities and the optimal task allocation policy, the dynamic equilibriumis defined as follows. The formulation of marginal costs will be used as an analytical device
to examine the sensitivity of task reallocation in response to factor price changes in Section
3.5.
Assume that as in the benchmark model, the establishment vintage is the same as theRfrontier level in the baseline year t0 = 1980 such that pR(∀i). As shown in Figure
i,t0 = pt0′
R14, quality-adjusted robot price is monotonically decreasing such that for t < t , pRt > p t′ .
Once θ′ > θ is chosen at period t, the vintage of automation technology is updated from s
to t where t > s is the current vintage of automation, indexed by i’s last year of upgrading.
pR
i,t =

⎧
⎨ pR
t

⎩ pR
s

(i upgrades)(12)(i does not upgrade)

Whenever i adjusts automation intensity θ, the vintage is taken to the frontier level. Thus,
automation adjustment involves not only expansion of automation intensity θ′ −θ but capital
deepening by vintage upgrading (from s to t). The higher vintage of automation is reflectedby lower imputed rental price of robots, serving to the aﬀordable final good price in theRpartially diﬀerentiated good market. i’s imputed robot rental price at period t is qi,t= δ R pi,t .
Definition 1. [Dynamic Task Allocation Equilibrium]
Impose the regularities of R0-R3. Then, given the skill prices q)le , q)l in region l, equipment
rental price q K and investment price of robots pR , the equilibrium requires (a)-(d):′(a) Optimal task allocation: Given task and sub-task productivity schedule {a(ω), a(ω), a)e (ω )},an establishment i in island l, maximizes the profit by optimally allocating the tasks by specifying automation intensity θ, occupation threshold ψ, and immigration threshold φ. ψ andφ are characterized by (9).
(b) Regional skill market clearing:
(i)

Σi∈l
#

( 1

s)i (ω)dω = Σj∈l s)j = S)j
#
$%
&
0
$%
&
aggregate skill supply

aggregate skill demand

(ii)

Σi∈l
#

( 1

′

′

s)ei (ω )dω0$%&domestic/foreign skill demand=Σj∈l s)ei Llj = S)le#$%&domestic/foreign skill supply

19(e ∈ {D, F })(c) Nationwide capital market clearing:(i) Σi∈l#( θ0ri (ω)dω = Rl ,$%&(ii) Σi∈l#

robot demand0ki (ω)dω = Kl$%&equipment demand(d) Nationwide goods market clearing:Σj C'# $%j,i,t&( 1=

final goods demand

Y!
i,t
#$%&

∀i

final goods supply

In dynamic equilibrium under frictional adjustment of automation, automation intensity,θ not necessarily follows a frictionless benchmark shown in Proposition 1. Observe that in
the spatial economy, the skill market of the aggregate skills and foreign (domestic) skills
regionally clears (a la Autor and Dorn (2013), and Beaudry, Doms and Lewis (2010), among
others) and the capital market clears nationwide. Recall that final goods are tradable nationwide with imperfect product diﬀerentiation. Based on the equilibrium definition, nationwide
prices, allocations and productions are formulated as follows.
Proposition 2. [Equilibrium Outcomes]
Suppose the regularities R0 − R3 hold. Then, a dynamic task allocation equilibrium
uniquely exists. A deterministic production Yi,t is given by
S i,t
Si,t 1−ψl,t αlRi,t θi,t′!
F) ()ψl,t −θi,t ()} Ki,t 1−αlYi,t = B(θi,t , ψl,t , αl | AR
l , Al , {a(ω), a(ω), a (ω )}){(
θi,t
ψl,t − θi,t
1 − ψl,tR
and a final good price Pi,t depends on i’s vintage of robots proxied by qi,t
= δpR
i,s .
ψ

−θ

1−ψ

R θi,t{(qi,s) q l,tl,t i,t q l,t l,t }α (qtK )1−αl
Pi,t =αlαl (1 − αl )1−αl B(θi,t , ψl,t , αl )

-ψ
-1
θi,t
where B(θi,t , ψl,t , αl ) = {(AR(Ai )1−ψl,t }αl exp[αl { θi,tl,t ln a(ω)dω + ψl,t ln a(ω)dω}]i )holds. Aggregate skill allocation:
S i,t =

(ψl,t − θi,t )αl Pi,t Yi,t
(1 − ψl,t )αl Pi,t Yi,t
S i,t =
q l,t
q l,t

20

(13)

Skill prices:
F φl,t !
D 1−φl,t
(q!
(ql,t
)
(1 − φl,t ))ql,t S)i,t
φl,t q)l,t S)i,t !
l,t )
D
F
,
q
=
q)l,t =
, q!
=
′
l,t
l,t
F
D
B (φl,t )
S!
S!
i,t

where

!F )φl,t exp[
B (φl,t ) = (Al′holds. Equipment allocation:

Ki,t =( φl,t0i,tF (ω )dω ]
ln a!
′

′

(1 − αl )Pi,t Yi,tqtK

Aggregation of individual skills:!
Σj∈l s!
j,t = Sl,t ,e!eΣj∈l s!
j,t = Sl,tA proof is relegated to Model Appendix. Note that establishments with wider θi,t andRlower qi,twith a newer vintage enables lower final good price Pi,t , yielding larger market shares
under the imperfectly diﬀerentiated market with σ < 1. Skill incomes of each occupation
class are shaped by task thresholds, θi,t , ψl,t and φl,t , which characterizes adjustments oftechnology. As a combination of these skill incomes constitute the basis of the social welfare(discussed in Section 6), the welfare implication is directly tied to task reallocation.

3.3

Solving the model

The key trade-oﬀ of automation upgrading is production benefit from automation versus
fixed lumpy adjustment cost. A policy function is characterized by a threshold rule asfollows. In the baseline year of 1980, automation is presumed to be adopted without friction
to the regionally optimal level by every establishment. Automation intensities and vintages
depart to diverge across establishments within and across regions.
Proposition 3. [Automation Upgrading Policy]
Optimal automation upgrading policy is characterized by a threshold rule such that⎧⎨U pgrade (θ < θ(X,)
A, ∆Qi,t−1 ))⎩N ot)(θ ≥ θ(X,
A, ∆Q
))
i,t−1

)
where a cutoﬀ automation intensity θ(X,A, ∆Qi,t−1 )) is function of state variablesX, A and21

a trend of automation-independent price aggregate price ∆Qi,t−1 , where Qi,t is an automationindependent price bundle such that(q t )αψl,t (q t /A)α(1−ψl,t ) (q K )1−α.
Qi,t ≡-ψ-1
Pt αα (1 − α)1−α exp{α( 0 l,t ln a(ω)dω + ψl,t ln a(ω)dω)}1

1−σ 1−σwhere Pt ≡ (ΣPi,t)is the aggregate good price in the final good market. Specifically, thepolicy is suﬃciently governed by unobservable state A and other reduced observable states
X t ofRqi,t−1}.
Xt = {
θi,t−1,# $% &
q t−1
automation intensity# $% &
vintage value relative to routine skill price

A proof is in the Model Appendix. Proposition 3 states that conditional on dynamic trends
in economic cost environments (e.g. non-routine skill prices, equipment rental prices), automation upgrading is characterized by the current automation intensity and the vintage
value relative to the routine skill price.

3.4

Adjustment costs and size distribution

This sub-section summarizes the role of three dynamic parameters: adjustment parameters
and a product diﬀerentiation parameter on the evolution of establishment size distributions.
In Figure 6, the horizontal line illustrates hump-shaped employment share along employment
size on the horizontal line. Productive establishments are initially large, and frequently conduct upgrading automation by paying the non-convex cost. Simulation under the plausible
range of parameters show that higher intensity of automation shrinks the employment sizeof establishments including routine- and non-routine occupations.
Diminishing non-convex cost expands more automation upgraders, facilitating a balanced′
downward shift. Diminishing convex cost extends upgrading intensity, θ − θ, skewing thesize distribution to the left. Finally, intensified competition accelerates the market share
substitution from automation, spreading the size distribution. This is summarized as follows:
1. Smaller non-convex costs ζ expands the range of automation upgraders. It decreases
mean and skewness (upward-biased) but increases dispersion and concentration due
to inequality in upgrading cycles.
2. Smaller convex adjustment cost τ suppresses the automation intensity θ among the
upgraders. It decreases the mean, dispersion and concentration but increases skewness.

22

Figure 6: The roles of key parameters on the size distribution (conceptual chart) (left: smaller
ζ, middle: smaller τ , right: larger σ)

3. Larger price competition σ governs the profitability of automation by substituting
market shares. It increases dispersion and concentration but decreases skewness. (indeterminate on mean) Carefully keeping directions of parameter changes in mind, thissimulation is summarized as the following table.
Table 1: The roles of key parameters on the size distribution moments (table)

3.5

Comparative Advantage Schedule and Technology Adjustments

In response to factor price changes, an establishment adjusts extensive-margin technologies
by optimally reallocating tasks over production factors, characterized by task thresholds.
This sub-section formalizes the technology adjustments based on the task-specific comparative advantage defined below.
Definition 2. [Comparative Advantage]
Comparative advantage (CA, hereafter) of routine skills over robots, of nonroutine skills over routine skills in task non-routineness at task ω is defined respectively23

as 33
CAL,R (ω) ≡da(ω)/dω,a(ω)

CAH,L (ω) ≡da(ω)/dω da(ω)/dω
−
.
a(ω)
a(ω)

Analogously, CA of domestic skills over foreign skills in task non-routineness at sub-task
ω is defined as′

F (ω )/dωda!
(ω ) ≡.
F (ω ′ )
a!
′

F,D

CA

′

′

Intuitively, CA of factor 1 over factor 2 is a productivity growth rate of factor 1 relative
to factor 2 when an assigned task is marginally more non-routine. Now, the elasticity oftechnological adjustment with respect to factor prices is directly tied to CAs as follows.
Proposition 4. [Comparative Advantage and Extensive-margin Technology Adjustment]
Extensive-margin technology adjustment to a marginal factor price change hinges onthreshold-specific comparative advantage (CA). Specifically, the automation intensity is adjusted asdθdθ=−= αCAL,R (θ)−1 > 0.
d ln qd ln q R
Routine-/non-routine skill adjustments over tasks and ethnic skill adjustments over sub-tasks
are given bydψdψ=−= αCAH,L (ψ)−1 > 0,d ln qd ln qdφ
dφ=−= CAF,D (φ)−1 < 0.
!
!
DF
d ln qd ln qElasticity of technological change in response to factor price shifts are shown to be linked
to CAs at respective thresholds across factors. These links are important because any equilibrium shifts including task reallocation involves information of CAs. The intuition of Proposition 4 is simple: when two factors are well-diﬀerentiated, adjustments become smaller inmagnitude. CA at the task thresholds governs how sensitive task reallocation is in responseto changes in factor prices. Next, I provide a sketch of Proposition 4 separately in two
adjustments.
33

Recall the normalization of task productivities of robots and domestic skills such that′′
D (ω ) = 1 ∀ω ∈ [0, 1].
aR (ω) = 1 ∀ω ∈ [0, 1], a!

24Automation adjustment
The elasticity of automation intensity with respect to a routine skill price and rental priceof robot is formulated.
⎧
q
p(θ)
⎪
dθ
dθ
⎪
⎪
= q=
=
> 0 (wage-driven automation)⎨d ln qdq−ṗ(θ)Aa(θ)−ṗ(θ)
p(θ)
dθ
dθ R
qR
⎪
⎪
⎪
=
q
=
=
< 0 (technology-driven automation)⎩ d ln q RRRdqṗ(θ)Aṗ(θ)

The first one shows a mechanism of automation adoption, driven by changes in skill prices,while the second one expresses technology-driven automation via declining rental price ofrobots, though both are mathematically isomorphic. The second equality in each equation
comes from graphical relationships in Figure 7, showing the response of θ in the marginal
move of q R , q, respectively. The third equality comes from single-crossing of marginal costsqRof routine skills and robots, p(θ) = R . To save space, define a notation of ẋ(ω ∗ ) ≡Adx(ω)|ω=ω∗ . Thus, the elasticities of automation with respect to factor prices are
dω
1
2−1
/
0
˙
p(θ) −1
dθ
a(θ)
dθ
=
=α
= αCAL,R (θ)−1 > 0=−
R0
d ln qd ln q R−ṗ(θ)a(θ)
holds.34
Labor adjustment
The elasticity of occupation threshold with respect to a routine skill price and rental priceof robot is formulated as
⎧
p(ψ)
q
dψ
dψ
⎪
⎪
=
⎪
⎨ d ln q = dq q =˙ − ṗ(ψ)}a(ψ)˙ − ṗ(ψ)
{p(ψ)
p(ψ)
dψ
q
dψ
−p(ψ)
⎪
⎪
=
⎪
⎩ d ln q = dq q =˙ − ṗ(ψ)˙ − ṗ(ψ)}a(ψ)−{p(ψ)p(ψ)<0>0

The second equality in each equation comes from graphical relationships in Figure 8, showing
the response of ψ in the marginal move of q, q, respectively. The third equality comes from

34

The third equality comes from some algebra, using the formula of p(ω);˙a(ω). Derivation is analogous in paragraph of labor adjustment below.
−αp(ω)
a(ω)

25

dp(ω)
d(a(ω))−α= p(ω)=dωdωFigure 7: Marginal response of θ in routine skill price change (left), in rental price of robots
change (right)
single-crossing of marginal costs, of non-routine and routine-skills p(ψ) = p(ψ). Thus, theelasticities of labor adjustment with respect to factor prices are
dψ
dψ
=
=−
d ln qd ln q

1

˙
˙
p(ψ)
p(ψ)
−
p(ψ) p(ψ)

2−1

= −α

1

˙
˙
a(ψ)
a(ψ)
−
a(ψ) a(ψ)

2−1= αCAH,L (ψ)−1 > 0.
R0,R2!
F and q with q
D , the same argument holds in the lower level
Replacing ψ with φ, q with q!
sub-task reallocation. Analogously,

holds.

3.6

dφ
dφ
=−
= CAF,D (φ) < 0
!
!
R0,R3
F
D
d ln qd ln qComparative Statics of Immigration Inflow

Using the task-based mechanics, this section analytically explores endogenous adjustmentof automation and labor market dynamics in response to immigration inflow. To make themodel tractable, I consider the steady state without frictions for adjusting automation. The
timeline of events is depicted in Figure 9. In the short run, immigration inflow triggers
labor adjustments including ethnic skill adjustment, while automation technology remains
unchanged. In the long run, automation adjustment is jointly induced by short-run routine
skill price shifts and by technological drops of imputed rental prices of robots, accompanied byfurther labor adjustments. Social welfare is evaluated based on the consequent labor market
26

Figure 8: Marginal response of ψ to routine skill price change (left), of φ to domestic andforeign skill price change (right)
outcome in the long run, which will be evaluated in Section 6.1. The detailed analyticalsteps are in the Model Appendix.
Guided by empirical distribution patterns of immigrants of diﬀerential skills across sectors35 , I consider the labor market impact of low-skilled and high-skilled immigration separately in manufacturing (or plant) and information service (or oﬃce), separately. Let us
presume that low-skilled immigrants provide foreign routine skills S F and engage in routine
occupations in the plant sector, and high-skilled immigrants provide foreign non-routine skills
F
S , and engage in non-routine occupations in the oﬃce sector. Mirrored by the foreign skill
dispersion across sectors, sectors are symmetrically treated, albeit distinct in nesting foreignand domestic skills in technology; routine (or non-routine) occupation is foreign-intensive inthe plant (or oﬃce) sector, thus, routine- (or non-routine-) skill is a composite of distinct
foreign and domestic skills. To save space, I redefine the notations for symmetric formulation
across sectors, wherever applicable as follows. For any variable z ∈ {S, q, A, a(ω), · · · , etc.}
in each sector, define ethnic-variable z)e for e ∈ {F, D}, immigration-intensive aggregate
35

Empirically, plant and oﬃce sectors correspond with manufacturing and information service sectors, respectively. (See Section 4.1) In terms of education attainment, a dominant part of immigrants are low-skilled
in manufacturing, and high-skilled in information service. Moreover, when occupations are aligned with routineness, proxied by automatability index from Frey and Osborne (2017), I found prominent concentrationof immigrants in routine occupations in manufacturing, and non-routine occupations in information service.
(See Data Appendix for details.) Immigrants in non-routine occupation in plant, and routine occupation inoﬃce are modeled equivalently as native counterparts.

27

variable z), and non-immigrant intensive aggregate variable z3 as follows⎧⎨z)e ≡ z e , z) ≡ z, z3 ≡ z⎩z)e ≡ z e , z) ≡ z, z3 ≡ z(plant)(14)(oﬃce).

Figure 9: Timeline of events
Notes: Timeline is common across sectors. adj. denotes adjustment.

3.6.1

Short runLabor market response to immigration inflow At the beginning of the economy,
suppose that immigration inflow enters island l. (l is implicit throughout this section). The
immigration inflow expands foreign skill supply curves by self-selection into each occupation
class. In conceptual model, suppose for simplicity that low-skilled immigration enters aroutine occupation in manufacturing, while high-skilled immigration enters a non-routine
occupation in information service.36 Suppose in the short run, an establishment is only
allowed to run labor adjustments over tasks by shifting a pair of thresholds ψ and φ.
When non-routine skills are suﬃciently diﬀerentiated from routine skills (i.e., CAH,L (ψ)
is suﬃciently large) at the occupation threshold, a series of gross equilibrium shifts areshown to occur by the following order in the plant (or oﬃce, respectively, italics in square
brackets) as depicted in Figure 10. Step 1: Low-skilled [High-skilled] immigration inflow
drops a routine skill price q [non-routine skill pri ce q]. Step 2: Both routine skill price q andnon-routine skill price q are upshifted due to production shift. Step 3: Occupation threshold
36

This assumption is relaxed and self-selection rule into each occupation class is estimated in 3.3.

28

Figure 10: Short run (upper-left: tasks in plant, upper-right: tasks in oﬃce, bottom: sub-tasks
in both sectors)

ψ expands [shrinks]. Step 4: Routine skill price q rises [falls], while non-routine skill price q
falls [rises]. Step 5: Immigration threshold φ expands. Step 6: Domestic skill price q)D falls,
while foreign routine skill price q)F rises. Formerly, under the fixed automation intensity,
short-run transition of skill prices are shaped by three eﬀects of immigration defined asfollows.
Definition 3. [Labor supply, Scale, and Substitution Eﬀects of Immigration]
Labor supply eﬀect is mechanical impacts of immigration shocks on skill prices, such∂q occ. Scale eﬀect is impacts of immigration shocks on skill prices from production
that
∂ S)F
∂q occ ∂Y ∂)qF
level change from labor supply eﬀect, such that
. Occupational and foreign∂Y ∂)q F ∂ S)F
substitution eﬀect is impacts of immigration shocks on skill prices from task reallocation∂q occ ∂ψand factors between foreignacross factors between non-routine and routine skills,∂ψ ∂ S)F
29

and domestic skills,∂)
q e ∂φ, respectively.
∂φ ∂ S)F

Immigration and labor adjustments The immigration inflow mechanically diminishes
the foreign skill price, downshifting the aggregate foreign-intensive skill price in each sector.
In particular, the routine skill price drops in plant, and non-routine skill price drops in oﬃce.
(Step 1, labor supply eﬀect) On the other hand, the mechanical drops of skill prices lead tocost advantage for establishments, expanding production. This induced production growth
expands demand for both routine and non-routine skills, augmenting their skill prices inturn. (Step 2, scale eﬀect) Suppose θ is fixed, and R0-R3 are imposed. Then, induced by
the forces, establishments run a pair of labor adjustments by changing the occupation andimmigration thresholds as follows.
(
(

dψ
3 χ +
)short = ±αCAH,L (ψ)−1 { Φα)#$%&F)d ln Sdφ)short = CAF,D (φ)−1 [F)
d ln S

1
#$%&

)F
}φΦ

scale eﬀectlabor supply eﬀect

)F
φΦ
#$%&

)F − Φ
) D ){ α)
)F ±
+ (Φ
χΦ
# $% &

labor supply eﬀectscale eﬀect

1dψ()shortχ)S)F&# d ln$%}]occupational substitution eﬀect

where ± applies to plant and oﬃce, respectively. Φ is defined as a function of uncompensatedlabor supply elasticity of any factor of interest, ϵU , such that Φ ≡ (ϵU + 1)−1 .37 Φ captures
the reactions of labor supply side for each factor. χ is a task interval such that χ ≡ ψ − θand χ ≡ 1 − ψ for both sectors. Recall the notation rule (14) also applies to the two.
Intuitively, in the higher task level, demographic eﬀect lowers the immigration-intensive
skills, while scale eﬀect augments the non-immigration intensive skills, expanding the task
range of immigration-intensive occupation. In the sub-task level, mechanical labor supply
shift from immigration inflow dominates the sub-task allocation, expanding the task rangeof foreign skills.38
Immigration and routine skill prices After this task reallocation, factor demands shift
to utilize relatively more foreign-intensive skills. Then, the occupation manager reallocates
sub-tasks to exploit the relative cost advantage of foreign skills by expanding the foreign
skill coverage, φ. The short-run labor market consequence is characterized by routine skill
37) ≡ {φΦ) F + (1 − φ)Φ
) D } in plant.
Define Φ38
Eﬀects from the higher task-level, scale eﬀect and occupation substitution eﬀect, aﬀect both foreign anddomestic skills, thus, are second order, regulated by diﬀerential labor supply propensities of foreign and)F − Φ
)D.
domestic groups, Φ30

price changes as follows.
Suppose that θ is fixed and CAH,L (ψ) is suﬃciently large. Then, the elasticity of routine
skill price with respect to immigration inflow is given by(d ln qχφΦF)short = Φ{ α)# $% &
d ln S)F

scale eﬀectdψ
1)short+ (χ d ln S)F
#
$%
&

}

(15)

occupational substitution eﬀect⎧⎨< 0 (plant)dφ+ (ΦF − ΦD )()}(16)+ I (G = p) {
−φΦF
short
# $% &⎩> 0 (oﬃce)
d ln S F#$%&
labor supply eﬀect
foreign substitution eﬀect

where I(G = p) is an indicator function of belonging to the plant sector. The task allocationin favor of routine occupations (larger ψ) and scale eﬀect would upshift the short run routine
price. Only in the plant sector, mechanical eﬀect from immigration inflow (labor supply
eﬀect) is added.39
When CAH,L (ψ) is suﬃciently large, low-skilled immigration inflow lowers the routine
skill price in the plant sector, while high-skilled immigration inflow raises the routine skill
price in the oﬃce sector. Intuitively, when non-routine skills are suﬃciently diﬀerentiatedfrom routine skills at the occupation threshold, labor supply eﬀect in the plant (or scaleeﬀect in oﬃce) cannot be completely oﬀset by the induced second-order substitution eﬀects
from task reallocation.
3.6.2

Long run: Unintended Consequence of Immigration Inflows

In the long run, an establishment re-optimizes the automation intensity, guided by thechange of routine skill prices relative to the current robot rental price. Suppose the shortrun predictions (15), a series of gross equilibrium shifts are shown to occur by the following
order in the plant (or oﬃce, respectively, italics in square brackets) sector.
The following Step 1-3 occurs in the long run as depicted in Figure 11. Before Step 1,
robot rental price q R drops and q has changed in the short run. The relative price changeqdrives the automation adjustment in Step 1. Step 1: Automation intensity θ falls [rises].
qR
Step 2: Routine skill price q upshifts [downshifts] by expansion [shrinkage] of task interval.
Step 3: A pair of skill prices q, q downshifts [upshifts] due to production shift. Note that
adjustment over sub-tasks are abstracted because it is second-order in magnitude.
39

Analogous to the previous footnote, the ethnic substitution eﬀect is of second-order.

31

Figure 11: Long run (left: plant, right: oﬃce)
Formerly, under the fixed automation adjustment, long-run transition of skill prices isshaped by two eﬀects defined as follows.
Definition 4. [Displacement and Productivity Eﬀect of Automation]dq
Displacement eﬀect is automation impacts to the routine skill prices, such that
. Prodθductivity eﬀect is automation impacts to skill prices from production level change, such thatdq occ dY dq dY dR dY dS(++).
dY dq dθ dR dθ
dS dθImmigration-induced automation The series of equilibrium responses illustrates unintended consequences of immigration via automation adjustment: in plant sector, if low-skilled
immigration inflow decreases the short-run routine skill price, and the automation intensity
falls. In oﬃce sector, high-skilled immigration inflow raises the short-run routine skill price,and the automation intensity rises (Step 1). Suppose θ is adjustable and (15) holds. Then,the automation intensity is adjusted as⎧⎨< 0 (plant)
d ln qdθ∂θ)long ((
)long = ()short⎩> 0 (oﬃce)∂ ln qd ln S)Fd ln S)F(17)

This is a key prediction in the benchmark model, distilling a hypothesis of immigrationinduced automation; low-skilled immigration inflow downshifts automation, while high-32

skilled immigration upshifts automation via the routine skill price change.40
Automation and skill prices As a result of automation shifts, routine skill price upshifts[downshifts]. (Step 2, displacement eﬀect) Then, establishments lower [raise] the routine
skills due to the upshifted [downshifted] routine skill price to shrink [expand] the production
as counteracting feedback. Moreover, due to the smaller [wider] task coverage of automation,
the establishments undergo [achieve] a productivity loss [gain], benefited from the newtechnological advance, embodied by a drop of quality-adjusted rental price of robots. As
a result, skill prices uniformly downshift [upshift] across both occupation workers. (Step3, productivity eﬀect) Combining Step 2 and 3, automation’s impact on skill prices areformulated across sectors as41
d ln q
= Φ(
dθ

−1
ψ−θ
# $% &

displacement eﬀect

+

Φα
#$%&

),

productivity eﬀect

d ln q=dθΦαΦ# $% &.

productivity eﬀect

Observe that the impact of induced-automation change to the skill prices are given as
follows, using a combination of displacement and productivity eﬀects. Finally, automation
adjustment is accompanied by labor adjustment to re-optimize to the updated menu of skill
prices. Iterate labor adjustments in Step 3-6 in the short run. The total impact in the longrun is an entirely empirical question and will be analyzed in the augmented quantitativemodel.
3.6.3

Aggregate Impact of Immigration via Automation Adjustment

So far, both drivers and outcomes of short-run and long-run technology adjustments areillustrated from the lens of task allocation. Counteracting these two time-horizons, the net
impact of immigration to automation and the economic consequence are characterized by
the change of robot adoption and routine skill prices in the regional level.
Proposition 5. [Three Elasticities with respect to Immigration]
Suppose R0-R3 are imposed. Then, the regional elasticity of robot deployment, routine
skill prices, and native premium with respect to immigration inflow in each sector is givenbyd ln q∂θ)long > 0
)short is negative [positive] from (15). Since (
F
)
∂
ln q
d ln S
across sectors, the results immediately hold.
40

If CAH,L (ψ) is suﬃciently large, (41
The formulation does not include eﬀects from further labor adjustments. To avoid complexity, further
labor adjustments are excluded here, but included in the simulation of the elasticities below.

33

d ln R
) F + 1 ( dθ )long)net = α)χφ Φ# $% & #$%&
θ d ln S)F
d ln S)F(

scale eﬀect(

d ln q)F + {
χφΦ)net = Φ[ α)# $% &
d ln S)F
scale eﬀect

αΦ
#$%&

+

productivity eﬀect

displacement eﬀect

−1
dθ
}(
) +
)F netψ−θ# $% & d ln S
displacement eﬀect)F − Φ
) D )( dφ )net }+ (Φ)F$%d ln S&#
labor supply eﬀect+ I (G = p) {(d ln()
q D /)
qF ))net =
d ln S)F)F−αφΦ
# $% &)FαφΦ# $% &

labor supply eﬀect

(18)

dψ
1
(
)net
ψ − θ d ln S)F
$%
&
#

]

occupational substitution eﬀect(19)

foreign substitution eﬀect)F − Φ
) D )(α)
)F +
− (Φ
χφΦ
# $% &
scale eﬀect

αΦ
#$%&

)(

productivity eﬀect

dθ
)long
d ln S)F)D)F − Φ
)D
)F
dθ
Φ
dφ
dψ
Φ
Φ
)net ∓ [)net − I(G = p)()long }]
+
)(
{(
−(
F
F
φ
1 − φ d ln S)χ)d ln S)d ln S)F
#
$%
& #
$%
&
foreign substitution eﬀect

occupational substitution and displacement eﬀect(20)∓ applies to plant and oﬃce, respectively. The first result associated with robot density
captures the degree of automation, illustrated by a pair of motivating graphics in Figure 1
and 2. The second result regarding routine-skill price serves both a cause (short-run shift)and consequence (long-run shift) of the automation. The third result of inequality in skill
prices across foreign and domestic groups are formulated as a function of task allocations.

4

Data

This section presents the primary data sources, introduces empirical counterparts for model
objects and summarizes key features of the data. In particular, a pair of driving forces of automation dynamics is featured: demographic change (Force A, in Figure 12) and technology
progress (Force B, in Figure 14). Further details are relegated to Data Appendix.

4.1

Unit of Analysis

The two sectors, plant and oﬃce in the conceptual model are represented by a pair of
tradable sectors, manufacturing, and information service, respectively, classified according
to BEA (Bureau of Economic Analysis) code, generally following NAICS (North American
34

Industry Classification System). The manufacturing sector consists of three sub-industries(31-33), while the information service sector is comprised of sub-industries of Information(51), Finance and Insurance (52), Professional, Scientific, Technical Services (54), and Management of Companies and Enterprises (55).42 Islands (regional labor markets) are proxied
by Commuting Zones (CZs), developed by Tolbert and Sizer (1996), best regarded as asemi-independent unit of regional labor markets, shaped by commuting patterns of local
workers. In contrast to the traditionally-used Metropolitan Statistical Areas (MSAs), CZs
cover the entire U.S. mainland, including less-developed rural areas, and maintain the consistent boundary definitions throughout the periods, 1980-2015. 722 CZs in the U.S. mainland,
excluding Hawaii and Alaska, are on board. Urban areas comprise the largest 37 CZs in thelabor force out of 722 CZs, accounting for the half the nationwide country labor force in1980. (See Data Appendix for a map)

4.2

Demography and Labor Markets

Demographic change The analysis draws on repeated cross-section datasets from the
Census Integrated Public Use Microdata Samples (IPUMS (2019)) for years 1980, 1990,
2000, and American Commuting Survey (ACS) for 2005 to 2015. The analysis is restricted
to working-age individuals from 16 to 64, excluding students.43
The demographic projection draws on 2017 National Population Projections by Census Bureau (2017), oﬀering nationwide projections of population by nativity, sex and single
age every future year from 2016 to 2050. The U.S. underwent a remarkable rise in immigration inflow in 1980-2015, and their presence is projected to grow, albeit it is less stark.
Net immigration inflow represented 39.5% of growth in the working-age population in 19802015. The working-age population is forecast to grow until 2050, and under the status quo
immigration policy, 26.1% of future population growth relies on net immigration inflow.
Figure 12 illustrates the demographic change in the target period 1980-2015, and future
projections in 2016-2050, of young-aged (16-35), middle-aged (36-50) and old-aged (aged50-64) domestic labor and foreign labor, separately in two skill groups.44
42

Non-tradable local service industries including Wholesale Trade (42), Retail Trade (44-45), Real Estate
Rental and Leasing (53), Education Services (61), Health Care and Social Assistance (62), and Accommodation and Food Services (72) are excluded due to scarce density of high-skilled immigrants.
43

Incorporating investments into education would substantially complicate the analysis, because college
enrollment is potentially guided by the future prospects of acquiring high skills. The demographic or technological impact to education attainments is relegated to future work.
44

As Census does not oﬀer projections of educational attainment, skill level is extrapolated from the trendof 2010-2015, which exhibits stably rising trend in high-skilled ratio (college enrollments) across age groups.

35

Figure 12: Driving force A: demography transition in the U.S. throughout 1980-2050. (left:
low-skilled, right: high-skilled. 1980-2015 is target period, while 2016-2050 is projection period.)
Notes: Data in 1980-2015 from Census and ACS. 2016-2050 are based on 2017 National Population Projections from the Census. Old natives aged from 50 to 64, middle natives: aged from 35 to 49, young
natives: aged 16 to 34) For smoother connection of data and projection, the graphics include students,
and cover Alaska and Hawaii.

In the low-skilled population, the presence of foreign labor is expected to grow from 19.0%
in 2015 to 23.1% in 2050 (6.5% in 1980), while the young group of natives slightly declines
from 38.5% in 2015 to 39.5% in 2050 (48.3 % in 1980). In the high-skilled population, thepresence of foreign labor is smaller but steadily growing from 14.7% in 2015 to 16.7% in 2050
(6.7 % in 1980). The young group of natives slightly declines from 34.4% in 2015 to 32.9%
in 2050 (55.0 % in 1980).
Next, I turn to a regional perspective. To map the CZs onto the geographic units in Census, and ACS, I draw on crosswalks with connection weights from Dorn (2009), respectively
bridging 1990 CZs and Census County Groups in 1980, Public Use Micro Areas (PUMAs) in1990 and after. Immigrants are defined as foreign-born, non-citizen (including naturalized
citizen), and natives are the rest of the individuals in the US. (Borjas (2003)) Figure 13
illustrates that immigration inflow is highly dispersed across CZs. Figure 13 illustrates theimmigration inflow across skill-levels into CZs from 1980-2015 jointly by graphs and maps.
The graphs show remarkably biased inflow of foreign labor, constituting a large variation of
immigration-driven demographic change. Measured by absolute headcount terms, only the
top 12 and 10 megacity CZs out of 722 accounts for 50% of the net low-skilled and highskilled immigration inflow in 1980-2015, respectively. Moreover, it is intriguing to observethat high-skilled immigrants are more likely than low-skilled immigrants to be concentrated
in urban areas (shown in blue bars).
To construct demography-driven labor supply shocks across CZs from 1980-2015, labor
force participation with occupation choice (routine/non-routine across two sectors, plus a36

Figure 13: Immigration inflow across CZs in 1980-2015 (left: low-skilled immigration, right:
high-skilled immigration)
Notes: The inflow is proxied by net inflow of working-age immigrants between 1980 and 2015 relative to)F − N)F
N
80, computed from Census (1980) and ACS (2015).
regional total working age populations in 1980, 15
N 80 + N 80
In graphs: blue bars capture urban areas, and red bars show rural areas.

group of alternative occupations, including service occupations in non-tradable service) byage, sex, nativity, education attainment, non-labor income and other observables (including
the number of children) in 2015, are estimated by logit model. Then, this self-selection rule isapplied to 1980-2015 data to construct the demography-driven labor supply shocks, serving
as inputs to the dynamic full-model.
Labor markets across sectors This paper explicitly distinguishes high-/low-skilled individuals and non-routine-/routine-skills: the former is characterized by education attainment
regardless of their labor market status, while the latter is based on occupation class as a result37

of self-selection to each occupation class in equilibrium.45 Guided by the benchmark model,
production workers in the manufacturing sector are classified into routine occupations (e.g.,laborers) and non-routine occupations (e.g., frontline managers and mechanical engineers).
Symmetrically, non-production workers in the information service sector are also split into
routine occupations (e.g., administrative clerks) and non-routine occupations (e.g., managers and engineers).46 Real wages are computed by annual labor income with top-coding
adjustments following the Census and ACS and inflation adjustments by consumer priceindex from FRED (2019b), divided by working hours.

4.3

Technology and Capitals

As extensively employed in macroeconomics, capital investments and stocks are taken from
National Income and Product Accounts (NIPA) from BEA. It covers physical capitals and information capitals, in consistent classification of industries (BEA industries based on NAICS)
and capitals throughout the years of interest, from 1980 to 2015.
As automation is unobservable per se, it is measured by automation-embodying capitals.
The model presumes that R is skill-replacing robots, and K is skill-augmenting capitals;
however, empirically sorting out the two is a non-trivial task. In the manufacturing sector,
industrial robot data from the World Robotics from the International Federation of Robotics(IFR (2015)) is a well-established source for skill-replacing machines.47 To connect BEA andIFR datasets, I explore which combination of capital categories in NIPA best correspondwith industrial robots from IFR. Through a cross-validation exercise, a pair of metalworking
machines and special industry machines is found to match best the industrial robots fromthe IFR. (See Data Appendix) In the information service sector, based on definitions of BEA
capital classification, I proxy autonomous IT capitals with a set of mainframe computers,
system integrators, custom software, and own-account software.48 Investment prices are
all quality-adjusted, following indices and methods from Cummins and Violante (2002),
45

Of course, these two measures, however, are strongly correlated; low-skilled individuals enter routine
occupations with substantially higher probability than high-skilled ones.
46
The number of non-production workers in manufacturing are relatively small compared to production
workers. The occupation class could be interpreted as managerial capital included in TFP terms. Production
workers in information sector is negligible and thus, excluded.
47
Recent empirical works on robotization (e.g. Acemoglu and Restrepo (2017) and Graetz and Michaels(2017)) uses IFR (2015). Since industrial robots are rigorously defined, the proxies only cover a very tiny
fraction (less than 10%) of the total physical capital stocks in BEA, not representative enough to draw
aggregate economic implications, sought in this paper.
48
Unlike industrial robots, to the best of my knowledge, no external dataset is available for capitals,
accurately embodying IT automation. The results remain robust to several alternatives of categories.

38

deflated by the GDP deflator from FRED (2019a), and normalized to 1980 as unity. Figure
14 illustrates investment-specific technical change, sharply contrasting paths of investment
prices of physical capitals and information capitals. Quality-adjusted investment prices ofphysical capital prices are relatively stable, while those of information capitals rapidly drop
over time.

Figure 14: Driving force B: Technological progress: quality-adjusted capital investment priceNotes: Denominated by GDP deflator from FRED (2019a), and normalized to unity in 1980.

The rental price for capital is computed based on the canonical perpetual inventory
method.
M
qtM = (δtM + RealInterestRatet − E(pMt ))ptwhere δtM is a depreciation rate, E(pMt ) is expected growth rate of investment price(proxied by a three-year moving average trend of the previous year), and pM
t is an investment
price of capital M , at year t, respectively. Both robots and equipment are treated as qualitycontrolled capital units, nominal values denominated by the normalized quality-adjusted real
investment prices.
Founded by the categorization, regional capital deployments are imputed using the benchmark model. As the benchmark model involves neither automation friction nor labor adjustment frictions, automation intensity and skill thresholds are common within regions at
the baseline year such that49
θi,t0 = θl,t0 ψi,t0 = ψl,t0 ∀i ∈ l.
49

Acemoglu and Restrepo (2017) adopts a similar imputation strategy of regional industrial robot deployments in CZ level.

39′For all regions l, l in sector G, the following relationships hold:

KlG′ ,t0qlR′ ,t0 RlG′ ,t0
W l′ ,t0 + W l′ ,t0 + VlK
qlK′ ,t0 KlG′ ,t0RlG′ ,t0
W l′ ,t0 + W l′ ,t0
′ ,t
0
,
=
=
=
=′
G (1)
R
G
G
K
G
K
′
Rl,t0
ql,t0 Rl,t0 (2) W l,t0 + W l,t0 + Vl,t0
Kl,t0 (1 ) ql,t0 Kl,t0 (2 ) W l,t0 + W l,t0
M(1), (1′ ) comes from the equalized rental prices of the national capital market, ql,t=0
M′ql′ ,t0 (M ∈ {R, K}). (2), (2 ) hold from Proposition 3. Thus, the region-level capital in eachsector is computed as a sum of industry-level capital stocks, weighted with a factor ratio of
each region within industry g in sector G (3 digit and 4 digit BEA codes in manufacturing
and in information service, respectively) such thatGKl,t= Σg∈G04.4

W l,t0 + W l,t0
g

W gt0 + W t0

G
Kg,t0 , Rl,t
= Σg0

W l,t0 + W l,t0 + Vl,tK0
g

W gt0 + W t0 + +Vtg,K0

Rg,t0 .

Establishment Size Distribution

Plant-level literature (e.g. Dinlersoz, Wolf et al. (2018)) documents that remarkable heterogeneity exists for technology adoption. To identify diﬀerential propensity for automation
adoption, establishment-level robot investments would be the ideal source. However, thedataset with this granularity is unavailable. Instead, I view that evolution of establishment
employment size is informative to diﬀerential automation adjustments. Specifically, I relyon establishment-level labor size, from County Business Patterns (CBP). CBP covers thesemi-universe of the establishments in county × industry-level (SICs from 1980 to 1997, and
NAICS from 1998-2015) throughout the periods of interest. Cross-validating the aggregatedlabor from CBP and Population Census, establishment i’s recorded employment in CBP, denoted by NiCBP ,is found to include both routine- and non-routine occupations. CBP classifies
each establishment by 12 bins, but I approximate NiCBP by taking middle points of each bin
CBP
except the largest bin so that Ni,l∈ {3, 7, 15, 35, 75, 175, 375, 750, 1250, 2000, 3750, 5000}.50
To translate counties in each year into CZs, I connects David Dorn’s crosswalk between
county in 1990 and 1990 CZs with a series of inter-temporal county crosswalks from the
Census Bureau. The sample is narrowed down to include at least 10 establishments acrosssectors throughout the years, because distributional outcome gets very noisy in very small
samples. Three statistical moments of size distribution, mean, standard deviation per mean,
50

The results are insensitive to alternative methods, such as random sampling. A lower bound 5000 of
the largest bin is taken without top-coding adjustments. This is a conservative choice so that the eﬀect ofdisappearance or emergence of this class of establishments is limited to minimum.

40and skewness of employment shares in addition to Herfindahl-Hirschman index (HHI)51 arefeatured. Each moment represents magnitude, dispersion, asymmetry and concentration of
the size distribution.

Figure 15: Evolution of size distribution across CZs in 1980-2015 (top: manufacturing, bottom:
information service, urban (blue) vs. rural (red) areas)
Notes: Blue lines capture a set of urban regions, while red lines represent a set of rural regions. Dotted lines
are 25 and 75 percentiles weighted by establishment size each year, respectively. Shaded areas are NBER
recession periods.

5

Identification and Estimation

This section delivers an identification strategy, elaborates estimation procedures, and reports
estimation results, separately in two stages. In Stage 1, intensive-margin of technology,′aF (ω )} for tasks (and sub-tasks)
namely, continuous productivity functions of {a(ω), a(ω), )!F
and regional factor-augmenting productivities of {ARl , Al , Al } are recovered. In Stage.2,
given the flexible productivity pattern of factor assignment over the task spectrum, dynamic
parameters of automation adjustment are estimated from cross-sectional variation of size
distribution evolutions of establishments.
NiCBP)2 . HHI index is influenced by the number of estabΣi∈l NiCBP
lishments in a region. To neutralize the eﬀect of entry-exits, I compute adjusted HHI index by normalizing
the size distribution such that for years after t > 1980. The number of establishments recorded in Mtbin isΣ M bin′′bin bin l,tadjusted to Ml,tbin such that Ml,tbin = Ml,t
. Adjusted HHI index is reported in the main text.
binΣbin Ml,80
51

HHI index in region l is defined as Σi∈l (41

Stage 1: Recover the Intensive-margin Technology
In Stage.1-1, guided by the model, region-specific elasticities of three economic outcomeswith respect to immigration inflow are estimated by reduced-form specifications. In Stage
1-2, drawing on the analytical mappings derived in Proposition 5 as moment conditions, I
semi-parametrically estimate the continuous productivity functions, consistently generating
the reduced-form elasticities by General Method of Moments (GMM).
Stage 1-1 Estimate Reduced-form Region-specific Elasticities with respect to
Immigration Inflow
The benchmark model generates CZ-specific elasticities on a trio of outcomes in the production economy, namely, robot deployment, routine skill price and native premium in skill
prices. To obtain the empirical counterparts, run the following regression across CZs indexed
by l throughout the decade from the baseline year t = 1980 to t = 1990.52 In each sector,
Dq)l,tfor each dependent variable zl,t ∈ {Rl,t , q l,t , F }, specify
q)l,t
F
z
F
ln zl,t = β0z + β1z ln S)l,t+ β2,lI(l) ln S)l,t+ εzl,t(21)

where I(l) is an dummy variable for CZ l.53 Then, region-specific elasticity with respect toimmigration is a common shifter plus a CZ-specific coeﬃcient. Formerly, for each zl,t , regionl∗ -specific elasticity with respect to immigration inflow is obtained by(d ln zl,tz)l=l∗ = β1z + β2,l∗.
F)d ln Sl,t

As is often evoked in the literature, the issue of endogeneity of immigration inflow intoF
region l arises; ln S)l,t
and unobservable terms εzl,t , which potentially includes regional business climates, could be systematically correlated, plausibly due to immigrants’ selection ofregions. To address this issue, I exploit historical ethnic settlement patterns in t−1 = 1940
to construct an instrument for immigration entry in the following periods. Formally, following the immigration literature54 , the instrument for the immigration inflows into l from the52

I deliberately focus on the decadal change from the baseline year 1980; automation intensities do notcoincide with cost shares of robots, because the quantitative simulation presumes frictional automation
adjustment in the subsequent years.
53

A decent part of CZs in these years have no immigrants recorded in the Census. In that case, I set

FS)l,t= 1 to include all regions. The magnitude of this value is not sensitive to the results.
54

See Jaeger, Ruist and Stuhler (2018) for a review and an extensive list of literature, using this strategy.

42

Table 2: Quasi-experiment results
Notes: The common eﬀect captures β1z in the specification above. A bracket is a standard error. (***
p<0.1 %) State fixed eﬀects are included in all specifications. Sample attrition occurs due to degenerationof dummy coeﬃcients or lack of valid instruments.

reference year t−1 to year t is given by′

IVl,t = Σe′

e
Nl,t
−1

′

′
Nte−1

# $% &′

settlement share of e in l

∆Nte
# $% &a new immigration wave of e′

where e is an ethnic group in the reference year t−1 = 1940 and ∆ captures the change from
e′
Nl,t
−1
55
captures the settlement ratio of athe reference year t−1 = 1940 to year t of interest.
′
Nte−1
region l of ethnic group e′ population in 1940. The instrument is a weighted sum of allocationof a new wave of immigrants in each ethnic group, following the historical distribution of thesame group.
As is common in the instrumental variable strategy, the two identification conditions,exogeneity and exclusion restriction are imposed. The exogeneity necessitates that the unobserved factors attracting immigrants to region l in t−1 are orthogonal to changes in therelative economic incentives oﬀered by the two zones in the following periods. Exclusion restriction requires that the only channel through which immigrant distribution in t−1 aﬀects
robot adoption after t0 is its eﬀect on the actual distribution of immigrants across regions.
Reassuringly, median values of reduced-form estimates of routine skill prices and robots′

More generally, background of Bartik-type shift-share instruments is discussed in Borusyak, Hull and Jaravel(2018).
55
Ethnic groups in the 1970 Census are: Mexico, Central America, Central Europe, South America,
Caribbean, India, China, Philippines, Africa, Vietnam, Japan and East Asia, Canada and Other North
America, the UK and Ireland, Southern Europe, Cuba, Middle East, Other Southeast Asia, Korea, Other
Southwest Asia, Western Europe, Australia and New Zealand and Northern Europe. (ordered by the size ofworking-age population)

43

Figure 16: Histograms of a trio of region-specific elasticities with respect to immigration
across CZs (left three: low-skilled immigrants, manufacturing (599 CZs), right three: high-skilled
immigrants, information service (552 CZs), blue: urban areas, red: rural areas)
Notes: To exclude outliers, samples are restricted to CZs with foreign income share with 0.5% and above inrespective occupation class.

are negative in manufacturing and positive in information service, consistent with the working hypothesis. It is not surprising that immigration inflow augments the native premiumby labor supply eﬀect, measured by a ratio of domestic and foreign skill prices.
One could find remarkable regional heterogeneity of elasticities with respect to immigration inflows. To highlight the geographical divergence, the regions are split into urban andrural areas, where urban areas are defined as a group of largest CZs, accounting for the top
half of the labor force in 1980. The rest of the area outside the urban CZ is classified asrural. (See Data Appendix for the map) Strikingly, a pair of elasticities of routine skill priceand robot deployments are systematically right skewed in urban areas relative to rural areas
across sectors. In particular, across sectors, immigration inflow upshifted routine skill prices
and robot deployments in urban areas but downshifted both in rural areas.56
Stage 1-2: Semi-parametrically Identify and Estimate a Set of Comparative Advantage Schedules from Region-specific Elasticities
The discrepancies of regional reduced-form estimates are exploited as a source of identification by the task-based framework with flexible substitution patterns. In each region, givena threshold-specific CA of foreign skills to domestic skills, a pair of threshold-specific CAsof routine skills to robots and of routine skills to non-routine skills, are just-identified fromelasticities of robots and of routine skill price with respect to immigration inflow under some56

In Miami, FL, the elasticity of routine skill price is weakly negative (-0.013), which is more consistent
with recent reappraisal of Borjas (2017) than the classical no significant negative results in Card (1990).

44

regularities of elasticities.

Figure 17: Identify a pair of CAs of routine skill vs. robots, non-routine skills vs. routine
skills
Notes: A red curve is an iso-quant of elasticity of robots, and a blue curve is an iso-quant of elasticity ofroutine skill price. Signs outside of brackets are for plant sector, those inside brackets are for oﬃce sector.
Suppose that CAF,D (φ) is given and CAH,L (ψ) is suﬃciently large as Proposition 7 presumes.

Figure 17 best summarizes an idea for identification of CAs of routine skills vs. robots andbetween non-routine vs. routine- skills. Using the equations derived from the model, given
a CA of foreign skills to domestic skills at the immigration threshold, a pair of elasticities ischaracterized by a pair of iso-quants of CAs, and for some regularities of elasticities, a singlecrossing point is shown to exist. The impact of routine skill price on automation, embodied
by robot adoption. and the impact of automation on routine skill price hinges on a pairof CAs, {CAL,R (θ), CAH,L (ψ)}, where CAH,L (ψ) regulates labor adjustment of immigrants,
while CAL,R (θ) governs the automation adjustment. The intuition of the proposition directly
stems from comparative statics of elasticities in Proposition 5. Next, the following procedure
is designed to recover task productivities.
Semi-parametric identification and estimation
Each region l is characterized by a tuple of (θl , ψl , φl ) and the number of regions is N . 6N
moment conditions (1)-(6) are set up as suﬃcient conditions for identifying the parameter′set Θ = {CAL,R(ω), CAH,L(ω), CAF,D
(ω ),
l
l
l

45

)F 57ARl , Al , Al }

⎧
′
⎪
⎪
ϵR
= m1l (CAL,R(ω), CAH,L(ω), CAF,D(ω ) | θl , ψl , φl )
⎪
l
l
l
l
⎨
q
′
(ω), CAH,L(ω), CAF,D(ω ) | θl , ψl , φl )ϵl = m2l (CAL,R
l
l
l
⎪
⎪
⎪
⎩ϵq#F /#qD = m3 (CAL,R (ω), CAH,L (ω), CAF,D (ω ′ ) | θ , ψ , φ )
l

l

l

l

l

l

l

l

(1)
(2)
(3)

where m1l , m2l , m3l are analytical mappings from Proposition 5. (See Model Appendix for
explicit function forms.) Rewriting single-crossing conditions of marginal costs in Proposition
1 yields
⎧
ql
xR
⎪
l,t0
L,R
−1
⎪
(4)
a ( R R | CAl (ω)) = θ =
⎪
⎪
⎪
q /Al
1 − xK
⎪
l,t0
⎪
⎨
xH
q /Al
l,t0
| CAH,L
(ω))
=
ψ
=
1
−
(5)
a−1 ( l
l
K
⎪
q
1
−
x
⎪
l,t
0
l
⎪
⎪
F )F
⎪
−1 q
)
/A
⎪!
′
l
⎪
⎩aF ( D l | CAF,D(ω )) = φ = x)Fl,t0
(6)
l
q)l
H
K
where xR
l,t0 , xl,t0 , xl,t0 is a factor share of robots, non-routine skills and capital equipment,
respectively, and x)Fl,t0 is a factor share of foreign skills within the immigration-intensive skills
in region l in year t0 ; formerly, they are defined by

xR
l ≡

F
F!
ql S lql S lq R Rlq K KlK !
q!
H
K
L
l Sl
F
,
x
≡
,
x
≡
x
≡
,
x
≡
W total l
Wltotal l
Wltotal l
Wltotal lq)l S)lwhere the total factor cost, W total ≡ q R Rl + q l S l + q l S l + q K KlK .
The second equality in (4)-(6) comes straightforward from the links between factor shareand task thresholds from Proposition 2 such that

58

R
K
F!
F
!
q l S l = α(ψl − θl )Yl , q l S l = α(1 − ψl )Yl , q!
l Sl = φl Wl , Vl = αθl Yl , Vl = (1 − α)Yl .

Without task productivities specified, iso-elastic task aggregation to produce the final goodyields that task thresholds (extensive margin technology) are mapped with labor/capitalincomes and foreign/domestic labor incomes.
57

Observe that moment conditions exceed the number of parameters: 3M < 3N (N =721 in manufacturing, N = 720 in information service. Thus, the parameters are generically identifiable.
58

This system of equations belongs to Seemingly Unrelated Regression Equations (SURE) in that)F{AR
l , Al , Al } does not enter in (1)-(3). (4)-(6) are jointly included as moment conditions, because stan′
L,R)F
dard errors of{AR(ω), CAH,L(ω), CAF,D(ω )} derived
l , Al , Al }systematically aﬀect the estimates of {CAlllfrom the information in (1)-(3).

46To facilitate the estimation, task productivity functions (and their diﬀerentiated CA functions) are common across regions inside a sector. (thus, a subscript l is dropped hereafter)59
In addition, CA functions are polynomially parameterized by a trio of M dimensional
vectors β = {β, β, β!F } such that
′

′

m
H,L
m
F,D
F
m
(ω) = ΣM(ω ) = ΣM
CAL,R (ω) = ΣM
m=0 β m (ln ω) , CA
m=0 βm (ln ω )m=0 β m (ln ω) , CA
(22)
) minimizes the weighted distance of
where M is a large positive integer. The estimator Θ
violations of moment conditions:1 N
Σ g (Θ)θ
N l=1 lwhere W is a weight matrix and g l (Θ) is 6N vector-valued function of 3(M + N ) vector ofΘ such that g l (Θ) = [g 1 g 2 g 3 g 4 g 5 g 6 ] and′min g N (Θ) W g N (Θ)⎧
1⎪⎪gl1 = ϵR⎪l − ml (β | θl , ψl , φl )⎨q
gl2 = ϵl − m2l (β | θl , ψl , φl )⎪⎪⎪⎩g 3 = ϵq!F /q!D − m3 (β | θ , ψ , φ )
l

l

l

l

l

l

s.t. g N (Θ) =

⎧
ql
m
4
⎪
) − ΣMg=ln(⎪
m=0 β m (ln ω)
l
R /AR
⎪
q
⎪
l
⎪
⎪
⎨
q /Al
m
) − ΣM
gl5 = ln( lm=0 β m (ln ω)
q
⎪
l
⎪
⎪
F )F
⎪
⎪
q
/
A
′ m
⎪
F
⎩gl6 = ln( l D l ) − ΣK
m=0 βm (ln ω )qlΘ is jointly estimated by two-step GMM. On the first stage, set W = I and estimate the) 1 . On the second stage, using Θ,) update the weight as inverse of covariance ofconsistent Θ1) 1 ))(g l (Θ) 1 ) − g N (Θ) 1 ))′ )−1 and) 1 ) − g N (Θthe moment conditions W = S −1 = ( ΣN(g (ΘN l=1 l) 2 .60 The set of CA patterns across factors is consistently recovered
reestimate the eﬃcient Θto generate the distribution of regional elasticities with respect to immigration.
59

Note that realized CAs diﬀer across regions due to heterogeneous task thresholds.

60

Estimation was implemented by simulated annealing algorithm. Only non-zero coeﬃcientsin√)M dimensional vector, β, is used for computation. GMM estimator converges in distribution as N (Θ−dΘ0 ) → N (0, (G′ S −1 G)−1 ), and a variance-covariance matrix is estimated as

) ′S
) 1)
)−1 G)) −1 /N where S) = 1 ΣN (g (Θ) 1 ))(g (Θ) 1 ) − g (Θ) 1 ))′ , G) = 1 Σl ∂ g l ( Θ) 1 ) − g N (ΘV)Θ = (G
l
N
l
l
N
N ∂Θ′

47

Figure 18: Estimated comparative advantage schedules (top: manufacturing, bottom: information service)Notes: A band bounded by solid gray curves is bootstrapped standard errors, computed from the estimatedF }. Each bubble represents CZ
CA functions based on standard errors of non-zero estimates of β = {β, β, β!
used for the semi-parametric estimation of the CA’s task position. The size of the bubbles is within-sector
employment share in 1980. Outliers of too small or large shares are excluded from the figure, but includedin estimation. Four cities with largest industry employment in each sector is depicted in the figures.

The next graphics show the fit of elasticities with respect to immigration, simulation versus reduced-form estimates. The CZs are dispersed along the 45-degree line, which indicatesa good fit.
Once β is estimated, using polynomial parameterization of (22), the three CA curves
within each task (and sub-task) domain in 1980 are recovered. Then, each CA curve isparametrically extrapolated outside each task domain to cover the entire task (or sub-task)
spectrum.61 Using the entire CA curve and the definition of CAs (Definition.2), one could
61
This is not so restrictive for the ensuing analysis because the most of factor shares in later periods
overlap with the factor share coverage in 1980. Linear approximation is used for the extrapolation of thecurves.

48

Figure 19: Simulated elasticities from estimated CAs vs. reduced-form estimates (top: manufacturing, bottom: information service)Notes: A dashed 45 degree line shows a perfect fit. The size of the bubbles is within-sector employment
share in 1980. Triangles capture urban areas as defined above.
F (ω ′ )}, serving as critical inputs
recover the entire series of task productivities {a(ω), a(ω), a!
to shape the dynamic analysis in Stage 2.
This proposed semi-parametric estimation procedure could make a methodologically contribution to the long-lasting controversies, occurring in nearly separate communities, about
the elasticity of substitutions of foreign vs. domestic labor (Ottaviano and Peri (2012)), andof labor vs. capital (Raval and Oberfield (2017)). By applying this task-based model with its
non-linear cross-factor substitution patterns into more than 700 regional and 35 years of richvariation of immigration inflows, could serve as a potential remedy by integrating a seriesof reduced-form estimates to empirically recover the entire comparative advantage schedule.
The ensuing quantitative analysis is tightly disciplined by cross-factor substitutions.

Stage 2: Recover the Adjustment Costs of Automation
Four parameters, dynamic parameters of adjustment costs (non-convex and convex costs),
market competition parameter, stationarity of TFP shocks Θ = {ζ, τ, σ, ρ} are identified
49

and estimated by 4N (N is a number of available CZs) regional variation of size distribution evolution of establishments and volatility of sectoral robot investment series moments.
The moments of size distribution are mean, standard deviation per mean, skewness, andHerfindahl index. The idea of identification follows the sensitivity of moments to the parameters, summarized in Section 5.2. The sectoral shifter ξt is calibrated to match the sectoral
production trend.
Simulate method of moments (SMM) For an arbitrary set of parameters, Θ = {ζ, τ, σ, ρ},the discrete-continuous dynamic choice dynamic programming problem is solved and a policy function is generated. Using the policy function, generate the model-simulated data.
Using the model-simulated data, compute simulated moments Ψs (Θ), as opposed to data
moments, Ψd . All model and data moments are constructed equivalently; given size distribution in 1980 as a starting point, the model generates outputs in 140 quarter data points in 35
years. Outputs in the end quarter of the year are used as the annual moments, since the earlier quarters involve an adjustment process initiated from the demographic and technology
shocks at the first quarter. Regional data moments are weighted median with the numberof establishments across CAs in 1980.62 To capture dynamic variation of moments, mean isdenominated by the 1980 mean size, and other three unit-free moments are diﬀerenced from1980. For each distributional moment, squared distances between model moments and data
moments are summed up across 35 years, crafting a degree of deviation in CZ-level.
) minimizes the weighted distance between the data moments Ψd (Θ) and
The estimate Θ
the simulated moments Ψs (Θ). Formally, I solve
1 R1′Σr=1 Ψs (Θ)] W [Ψd − ΣRΨs (Θ)].
R
R r=1
The number of simulations is set to R = 100. The optimal weight matrix W is set
as inverse of variance-covariance matrix of the data moments, weighted with the initialestablishment number.63
minΘ [Ψd −62

The model abstracts from entries and exits of establishments, so that every model-implied moment isconstructed weighted with the number of regional establishments in 1980.
63

In general, the variance-covariance matrix of the SMM estimator is given by[GT W G]−1 GT W ΣW T G[GT W G]−1

where
G=

∂gn (Θ),∂Θgn (Θ) = Ψd −1 RΣ Ψs (Θ)
R r=1

If I choose Σ = W −1 , a variance-covariance matrix is collapsed to (GT W G)−1 . Therefore, the asymptotic

50

Estimation results
The estimated parameters across sectors are summarized as follows.
Table 3: Estimates of the parameters

Notes: The brackets are bootstrapping standard errors with 1000 iterations.

To illustrate the workings of the model and understand the intuition behind the estimatesacross sectors, the simulated upgrading policy given the parameter estimates is depictedacross the initial productivity class A0 as follows. In manufacturing, upgrading frequency isrelatively unsynchronized compared to information service so that intervals of less productive plants are much longer than productive plants (larger non-convex costs ζ), generating
dispersion of vintages of robots. However, each upgrading is relatively more drastic in larger
extended intensity (smaller convex costs τ ). Information service exhibits a contrasting behavior in that upgrading frequency is synchronized, but wider gap in intensities, resulting in
more equalized access to the latest vintage.
External validity
To ensure the external validity of the model, I compare the model’s predictions of nationwide
labor share and labor productivity against the data. I stress that neither of these two aretargeted in the estimation. Nationwide labor share is labor income of both routine and nonroutine occupations divided by real GDP, denominated by GDP deflator normalized in 2015,
Σi W i,t + W i,tformerly. Labor productivities are GDP per labor hours across occupations,Σi Pi,t Yi,tdistribution of Θ is given by√d) − Θ) −→ N (0, (GT W G)−1 )n(ΘSee McFadden (1989) and Pakes and Pollard (1989) for the statistical properties of the SMM estimator.

51

Figure 20: Simulated automation upgrading in equilibrium (top: manufacturing, bottom:
information service)Notes: The blue lines indicate the top 4 productive bins, green ones show the middle 4, and the red
ones represent the 4 least productive bins. Average across 100 simulations, given parameter estimatesΘ∗ = {ζ, τ, σ, ρ}.

Σi Pi,t Yi,t. Across sectors, labor share is steadily declining, while labor productivity is
Σi Li,t + Li,trising, less stark after the Great Recession. Figure 21 shows the model reproduces thispattern well, assuring the quantitative assessments in the next section.

Figure 21: Non-targeted economic aggregates: model: solid line vs. data: dotted line (lefttwo: manufacturing, right two: information service)Notes: Shaded years are NBER recession periods.

52

6

Quantitative Assessments

Armed with the dynamic model with non-linear comparative advantage (CA) schedules,
it is now ready to assess quantitatively the economic consequences of immigration. The
analysis follows along with the flow chart in Figure 3. To start, Section 6.1 evaluates theaggregate outcomes of immigration on wages and welfare (defined below). Then, Section 6.2
investigates the aggregate eﬀect on wages by three channels, and decomposes each channel
by competing forces shaping the task reallocation mechanics, formulated in Section 3.6.

6.1

Aggregate Economic Outcomes of Immigration

Aggregate labor market outcomesFirst, I explore the aggregate eﬀect of immigration inflow on wages. To see this, I suppose
the U.S. underwent no net immigration growth since 1980. I assess the contribution of
immigration by comparing the wage shifts during 1980-2015 in the counterfactual case withthe baseline.64 Figure 22 shows the simulated wage of the no immigration growth case.
The top two graphs show the simulated routine native wages and non-routine wages
in manufacturing, relative to the baseline. The discrepancies from the baseline indicatethat low-skilled immigration exhibits an aggregate positive role on routine native wages
and a negative role on non-routine wages. The bottom two graphs show the simulated
counterfactual routine native wages and non-routine wages in information service, relative
to the baseline. They show that high-skilled immigration has positive eﬀects in both routine
wages and non-routine native wages. Across sectors, I narrow down the simulation within
the top 30 CZs with highest rise in immigration density in the proxy of Figure 13, as depictedin dashed curves. The top 30 CZs absorbed 46% and 68% of low-skilled, and high-skilled
immigration inflow. Reassuringly, the top 30 CZs exhibit steeper curves than all the CZs
with positive immigration inflow.
As simulation algorithm closely follows the timeline in Section 3.6, it allows for isolating
the direct impact of immigration separately and the indirect eﬀect via automation adjustment. To exclude the contribution of automation adjustment, I measure the shifts of wages
exclusively from immigration in the simulation algorithm and analogously take a diﬀerence
between no immigration case and the baseline.
Figure 23 shows the immigration’s impacts on wages of diﬀerent occupation classes with
64
In no immigration growth case, both baseline and counterfactual cases are limited to the CZs receiving
positive immigration inflow.

53

Figure 22: Simulated wage paths compared to the baseline by occupation class (top two:
manufacturing, bottom two: information service)Notes: The mean value across CZs of interest, weighted with labor hours of each occupation class. The value
is the detrended average of 100 simulations.

and without automation adjustment across sectors. Without automation adjustment, lowskilled immigration has negative impacts on routine wages, implying that low-skilled immigration stifled the automation; the gap between with and without automation adjustment
implies that weakened adoption of automation augments the routine native wages and harms
non-routine wages in manufacturing via automation adjustment. In contrast, high-skilled
immigration alone has positive impacts on routine wages, implying that high-skilled immigration heightens the adoption of automation; the gap between with and without automation
adjustment is negligible in the impact on routine wages, but sizable in non-routine wages.
This implies that increased automation upshifts the non-routine native wages in information
service via automation adjustment.

54

Figure 23: Immigration’s impacts on wages: with vs. without adjustments of automation(top two: manufacturing, bottom two: information service)Notes: The mean value across CZs, weighted with labor hours in each occupational class. The value is adetrended average of 100 simulations.

Welfare assessments
To systematically evaluate previous and future immigration policies, I introduce social welfare as a weighted sum of consumption utilities of all native citizens.6566
Definition 5. [Social Welfare]
The social welfare function (SWF) in year t is a weighted average of curved utilities
across citizens such thatSW F (t) ≡ Σj ξ j,t Υ(U (Cj,t ); υ) (ej = D)
65

Only the utilities of U.S. citizens are incorporated to avoid an ideological issue on how much utilities ofimmigrants should be evaluated.
66
The social welfare is more comprehensive proxy to capture national benefit than wages on three points.
First, the social welfare discounts inequality under distributional preference. Second, the welfare covers nonworking individuals as well as labor force. Third, consumption utility depends on total earning, including
non-labor income.

55

where an individual utility is curved as
Υ(U (Cj,t ); υ) ≡{expU (Cj,t )}υ − 1.
υυ ≤ 0 captures inequality preference, and Cj,t = Wj,t + N LIj,t , where Wj,t and N LIj,t islabor income and non-labor income, respectively, for all j at t, excluding the students andother occupations outside the model economy. ξ j,t is a welfare weight of the planner suchthat Σj ξ j,t = 1, proxied either by identical weights or by marginal utilities of consumption.
γ
Cj,t
−1
with a CRRA parameter γ = 0.1 isParameterize a utility function by U (Cj,t ) =γ
externally calibrated. I try ν ∈ {−0.03, −0.01, 0} in utility transformation by the planner.
ν = 0 corresponds with the linear case without any utility transformation. Figure 24 shows

Figure 24: Baseline welfare transition (computed from the data)a data-based welfare path in 35 years across 6 (3×2) parameter specifications. The marginalutility weights punish inequality more severely than identical weights with more progressive
preference, as does negatively larger γ. In every welfare protocol, the overall welfare trend isdeclining after 2000, mirrored by exacerbating total earning inequality, as shown in occupational decomposition of welfare in the right graph in Figure 24. In both sectors, non-routine
occupations exhibit rising welfare shares, and routine occupations display declining shares.
Founded by the baseline welfare, I examine immigration’s impacts on the aggregate domestic welfare. By interacting implied regional change of labor incomes across occupation
classes by labor income data, I compute model-implied labor incomes, serving as inputs tothe welfare for 1980-2015.
Then, I assess the welfare contribution of immigration by comparing the baseline with
non-immigration growth case. Both low-skilled and high-skilled immigration during 19802015 played a welfare-enhancing role in 2015. I examine the sensitivity of the welfare impacts
with respect to substitution between foreign and domestic skills, CAF,D (φ). To see this, I
consider counterfactual simulations where the substitutions are set to be uniformly minimum
56

Figure 25: Simulated welfare transition (additive weight case with γ = −0.01. left: low-skilled
immigration, right: high-skilled immigration)

Note: red (or blue) dashed line is smallest (or largest, respectively) diﬀerentiation between foreign anddomestic skills among CZs. The value is a detrended average of 100 simulations.

(red dashed line) and maximum (blue dashed line) across regions. The ethnic substitution
pattern is more sensitive to substitution estimates in manufacturing; in the CZ with largest
substitution67 , low-skilled immigration is welfare-harming. In the information service sector,
the welfare implication is relatively stable with respect to substitution estimates.

6.2

Understanding the Task-based Mechanics

To diagnose the aggregate simulation results in Section 6.1, I decompose the aggregate eﬀect
into three channels. Each channel is formulated by competing forces, coursing the task
reallocation mechanics (Definition 3 and 4).
Channel 1: The Impact of Immigration without Automation Adjustment
Firstly, I decompose the direct impact on skill prices without automation adjustment, shownin Figure 23, into the four forces, formerly defined in Definition 3. Namely, the impactof immigration shocks dS!F on the skill price in each occupation class is split by as follows.
Recall that q)e is ethnic skill price of e ∈ {F, D} and q3 is non-immigration intensive skill pricein each sector.
"dq)e """ =dS!F "
θ

67

dq)e
dS!F
#$%&

labor supply eﬀect

′

′

dq)e
dq)e
dW ei dψdW ei dφdYidq)e)short +)short }
Σi∈l
(
(
+
+
′ Σi∈l′ Σi∈le!F dW edYl
dψ dS!Fdφ dS!F
dW
d
S
l
l
$%
& #
#
$%
& #
$%
&
scale eﬀect

occupational subtitution eﬀect

foreign substitution eﬀect

This turns out to be the highest φ, inferred by foreign income shares. See the CA curves in Figure 18.

57

"
4i dψ
d3
q ""
d3
q
dW
d3
q
dYi
+
Σi∈l
)short
Σi∈l
(
" =
4l
dψ dS!F
dS!F "θ #dYl $% dS!F& #dW$%&
scale eﬀect

occupational substitution eﬀect

As defined in Definition 3, labor supply eﬀect is a mechanical impact of immigration
shocks on immigration-intensive skill prices. Scale eﬀect is immigration’s impacts on skill
prices induced from production gains via more aﬀordable skill prices. Immigration inflow
would lower the foreign skill prices, and producers produce goods aﬀordably and expand sales
in the product market. This production expansion augments skill prices across occupations.
Substitution eﬀect is immigration’s impacts on skill prices induced from reallocating the tasks
across factors. In each layer of the task spectrum, this eﬀect is two-fold: between routineand non-routine skills and between foreign and domestic skills. Low-skilled (or high-skilled)
immigration inflow would partially substitute both non-routine (or routine) skills in a higher
layer and routine (or non-routine) domestic skills in a lower layer, lowering their skill prices.
Recall that all three forces are defined excluding the indirect eﬀects from induced automation adjustment. I record the eﬀect from each force in simulation algorithm, all of which
are immune from the impacts of automation upgrading. Figure 26 shows decomposition byeach force. Across sectors, labor supply eﬀect dominates scale eﬀect and substitution effect within impacts on similarly-skilled wages of immigration inflow. Regarding the impacts
on less immigration-intensive occupation wages (3q ), scale eﬀects and substitution eﬀects
approximately cancel each other.
Channel 2: Immigration-induced Automation
Why does automation occur? In the quantitative model, automation is adjusted in thefollowing accounting formula from mutually exclusive drivers.68
dθ dq
dθ dq
dθ = ( )()dS)F + ( )()dS)D +
dq dS)F
dq dS)F
$%
& #
$%
&
#
immigration

domestic demographydθ( R )dpR
dp# $% &

technological progress (> 0)+(dθ)f ric
# $% &

adjustment friction (< 0)Using the estimated full-model, I assess the role of each driver for automation upgrading,by turning oﬀ each driver compared to the baseline, as depicted in Figure 27. To turn oﬀ
technological progress, the quality-adjusted investment price is assumed to be stable in 1980;
68

However, these are not exhaustive; extra factors outside the paper’s focus including rental price ofcapitals, TFP growth, and other equilibrium interactions aﬀect automation upgrading.

58

Figure 26: The direct eﬀect of immigration on wages without automation adjustment (toptwo: low-skilled immigration, bottom two: high-skilled immigration)
Notes: The value is an average of 100 simulations, of mean across CZs weighted with labor hours acrosseach occupation class. Other eﬀects from equilibrium interaction are excluded from the graphs.
RpR
t = p0 . To eliminate the friction for automation, all adjustment cost parameters are set
to zero; ζ = 0 and τ = 0.69 To exclude the circular impacts from automation on wages,
aﬀecting future automation, I exclude the labor market impacts of the automation from thesimulation.70
Perhaps unsurprisingly, technological progress is the prime driver for automation upgrading. The top two graphs in Figure 27 depict simulated paths of the nationwide automation
intensity across sectors from 1980-2015 under two counterfactual scenarios compared to thebaseline. In the stationary fertility case, I suppose that fertility rate has been hypotheticallystationary so that age composition of working-age natives remains fixed at the level of 1980
until 2015, with the total working-age population unchanged. Both low- and high- skilled
immigration plays an outsized role in the labor-driven automation upgrading across sectors.
In manufacturing, low-skilled immigration’s impact on automation is larger than age composition change on average. The eﬀect is much larger when regions are limited to the top 30

69

Without adjustment friction, every productivity class of establishments run the automation adjustment,
exclusively driven by annual factor price changes.
70

Automation typically lower routine wages, serving as a counterbalancing force to deter the further
automation upgrading.

59

Figure 27: The role of immigration for adjustments of automation (left: manufacturing, right:
information service)Notes: Automation intensity is nation-wide weighted average of establishment-level θ, weighted by real
production value. The value is the average of 100 simulations.
CZs with the highest low-skilled immigration inflow. In information service, the labor supply into routine occupation due to the age composition change is larger than immigration’s
impact, but immigration still played a sizable role relative to their headcount share.
Channel 3: Automation’s Impacts on Wages
Finally, I examine the labor market outcome of automation. The impact of automation
adjustment dθ on the skill price in each occupation class is given by the forces defined in
Definition 4 such that
/

dq)e
dθ

/

0

′

dq el

dW ei
Σ
= I (G = p)
+
i∈l
′
dθ
dW el
long
#
$%
&
displacement eﬀect<0 (only plant)

0
dq i
dW i
d3
q
= I (G = o)+Σi∈ldθ long
dW l
dθ
$%
&
#
displacement eﬀect< 0 (only oﬃce)

dq)e
dYi
Σi∈l
dY
dθ
&
# l $%

productivity eﬀect> 0

d3
q
dYi
Σi∈l
dY
dθ
&
# l $%

productivity eﬀect>0

where I(G = g) is an indicator function of the sector g ∈ {p, o}. Recall that displacementeﬀect is automation’s impacts on routine skill prices, induced by routine skill demand shifts
60

from reallocating tasks from routine skills to robots. Automation upgrading would slashroutine skill demands, thus, reduce the routine skill prices. Productivity eﬀect is automation’s
impacts on skill prices induced from production gains. In the heterogeneous establishment
model, this eﬀect takes place in two-fold: if a part of establishments upgrade automation, the
new technology allows for producing goods with lower marginal costs. Moreover, automation
upgrading would suppress routine skill prices by displacement eﬀect, and this would allow forcost-eﬀective production for all producers, including non-upgraders.71 Consequently, both
upgraders and non-upgraders expand sales, augmenting the skill demands and their skill
prices.
Then, what is the eﬀect of automation upgrading on wages? To see this, I assess thecontribution of automation upgrading by comparing the baseline case relative to the nonautomation case, where automation upgrading is mechanically halted. Then, I isolate thewage shifts exclusively from automation upgrading in the simulation algorithm and decompose the eﬀect into each force, recorded in the simulation algorithm. Figure 28 shows theeﬀect of automation on wages; automation reduces routine wages and augments non-routine
wages, expanding the wage inequality in both sectors. As displacement eﬀect dominates
productivity eﬀect, the eﬀect on routine wages is negative. In contrast, positive productivity
eﬀect drives the growth of non-routine wages. The automation’s impact on wages are larger
in manufacturing than in information service.

7

Counterfactuals and Policy Experiments

This section extends the simulation to the future period 2016-2050 and reports the policy
experiments. Section 7.1 and 7.2 explores the eﬀect of restricting future immigration of eachskill level. Section 7.3 explores the economic consequence of a universal basic income policy
to native citizens across sectors, fixing the current immigration policy.

7.1

Baseline Prediction from 2016-2050

To extend the simulation toward the future period, demographic transition in CZ-level in
2016-2050 is needed as a future baseline, in which demography transition proceeds as predicted in the Census projection (baseline case, hereafter). To obtain this, I draw on statelevel decomposition projections by Weldon Cooper Center (2018). It provides a projection
71

The model predicts that non-upgraders with more labor-intensive technology would typically absorb thedisplaced labor with lower wages, and thus, turn larger.

61

Figure 28: The eﬀect of automation on wages (top: manufacturing, bottom: information service)Notes: Percentage change of wages from 1980. The value is a detrended average of 100 simulations, of meanacross CZs weighted with labor hours across occupation classes.
of state-level working-age population by ten age groups (15 to 19, 20 to 24, 25 to 29, 30 to34, 35 to 39, 40 to 44, 45 to 49, 50 to 54, 55 to 59, 60 to 64)× sex cells in three decadalsnapshots of 2020, 2030, and 2040. To construct CZ-level demography-driven labor supply
shocks, datasets in the future periods are needed. Of course, nobody has future population
datasets, and I construct the hypothetical datasets in 2020, 2030, 2040 and 2050 as follows.
In the following, all the extrapolations are run under linear and spline approximation, chosen
based on the internal validity of the target period.
To begin, I construct a table of shifts of statistical population sample weights across CZs
in 2020, 2030, 2040 and 2050 relative to 2015 by demographic bins: age group, sex, nativity
and five education attainment groups72 . The state-level demographic decadal trend from the
projections by Weldon Cooper Center (2018) in 2030-40 is extrapolated to the final decadeof 2040-50 by 10 age groups× sex cells. This state-level demographic trend is applied to
all CZs belonging to each state. Immigration inflow to each CZ is extrapolated based on
immigration inflow trends across CZs from 2008 to 2015, scaled by the nationwide projectionof immigration growth from the nationwide projection from Census Bureau (2017). A trend
72
Education attainments are classified into five groups: below high-school degrees, high-school degrees,
some college years, four-year college degrees, and above four-year college.

62

of education attainments conditional on the demographic characters is also extrapolated fromthe data across CZs from 2000 to 2015. By tweaking statistical population sample weightsof the 2015 ACS data by the future population weight shifts across demographic bins, thehypothetical population datasets in 2020, 2030, 2040, and 2050 are constructed.
After compiling the future datasets, the remaining is analogous to 1980-2015. I recover
the self-selection rule of labor force participation and occupation choice from 2008-2015,
estimating a logit model in the stacked 2010-2015 ACS repeated cross sectional datasets.
Then, this self-selection rule is applied to the hypothetical population datasets obtained in
the first step to construct a projection of demography-driven labor supply shocks across CZs
in 2016-2050.73

7.2

Welfare Implication of Restricting Immigration

Suppose that the U.S. government restricts the net future immigration inflow at the beginning of 2020 to keep the foreign labor mix in each skill class at the level of 2020. Presume
that the immigration policy only covers foreign-born individuals without citizenship andnaturalized citizens, excluding descendants of immigrants with citizenship. Other protocol
follows the future baseline as stated in 7.1. Also, recall that the welfare incorporates only
utilities of the native individuals.

Figure 29: Simulated welfare transition by immigration regulation after 2020 (left: low-skilled,
right: high-skilled)
Notes: Use additive weights and ν = −0.01. The value is the average of 100 simulations in 2015, 2020, 2030,
2040, and 2050.
Then, the counterfactual simulation shows that this regulation policy of low-skilled andhigh-skilled immigration would deteriorate the welfare across sectors in the long run in 2050
73
The quality-adjusted capital investment price is extrapolated from the trends in 2008-2015 after theGreat Recession.

63

by 0.5% and 2.0%, respectively, relative to the future baseline immigration policy projectedby the Census. (See Figure 29) The result mirrors the welfare analysis during the baseline
period, 1980-2015. As low-skilled immigration weakens the adoption of automation in thelong run, alleviating the inequality and enhancing the domestic welfare. As high-skilled immigration benefits the non-routine native workers without harming the routine occupations,
inclusive of the adjusted technology, the restriction in the future would more unambiguously
diminish the domestic welfare, compared to restricting low-skilled immigration.

7.3

Economic Consequences of a Universal Basic Income (UBI)

Suppose the U.S. government launches a universal basic income (UBI) policy in the beginningof the year 2025, granting an unconditional stipend to every U.S. citizen above 25 years
old. Overviewing the recent policy debates, two plausible plans of unconditional granting(A) 1,200 USD per month and (B) 800 USD per month are on board.74 I suppose that
immigration policies are fixed to the baseline.75
Granting the unconditional non-labor income floor to U.S. citizens is predicted to alters
substantially the self-selection of natives in each occupation class, including non-working.76
I compute the UBI-induced labor supply shocks for 2025 to 2050 from the estimated 20102015 self-selection rule. Within-CZ estimation shows sensitivity of labor force participation
with respect to non-labor income is systematically negative; UBI typically drives a regional
shrinkage of domestic labor force by inducing voluntary quasi-retirements.
Rising automation and wage response As automation and wages mutually serve as
causes and consequences of each other, I explore how a UBI reshapes the automation adjustment in parallel to wages in each occupation class. The counterfactual simulation predicts
that a UBI substantially boosts automation, especially by raising routine occupation wages(shown below). Plan A (or B) would raise automation intensity by 0.62% (or 0.45%). Observe that the automation upgrading is projected to take a decade to reach a ceiling, since
automation upgrading is frictional and less productive establishments delays upgrading au74
From the administration viewpoint, a long political process is required until it is implemented. In thatcase, the early announcement of UBI might form the expectation, which might induce wage rises and generate
the similar results. The current simulation should be viewed as an extreme exercise.
75
The primary obstacle to implement a UBI is its massive expenditure, and I suppose that this would
be compensated by heavier income tax. It might be worth considering to adjust immigration policies tocompensate for the necessary expenditure.
76

To simplify the simulation, I abstract from labor supply response of high-income earners to heavier
income tax.

64

tomation. Furthermore, as shown later, the routine wage would be harmed by the surging
automation, serving as a counteracting force to further automation. This would generate
further lags of the adoption of automation for less productive establishments.

Figure 30: Nationwide dynamics of automation (left: manufacturing, right: information service)Notes: The value is the average of 100 simulations.

Next, I simulate changes in wages in response to a UBI policy. Right after the announcement of the UBI, both routine and non-routine wages start to elevate because someindividuals exit from the labor force. This eﬀect is more distinct in low-skilled individuals (typically engaged in routine occupations) whose exit sensitivity to non-labor income islarger than high-skilled individuals (typically engaged in non-routine occupations). Thus,
non-routine wages also rise from their skill demand rise by partially substituting tasks previously done by routine workers. Routine immigrant wages are relatively immune from theinitial labor supply shock.
Due to delayed automation upgrading, routine native wages are gradually harmed, but
after automation reaches a ceiling, routine native wages are on the constant rise, consistent with non-routine wages, benefitting from aging demography and technological progress.
However, routine immigrant wages are on the constant decline due to immigration inflow
and rising automation. The egalitarian-oriented UBI has the potential to exacerbate thewage inequality by substantially accelerating automation.
A natural corollary of a UBI is a widening gap between native and immigrant wages.
As a UBI provides an unconditional stipend exclusively to native citizens, the policy shockprimarily hits domestic labor. As a result, establishments expand their use of foreign labor.
Fixing the national immigration policy, nationwide foreign dependency, proxied by employment share across sectors, would consistently rise. The provision of a UBI not only raises
reliance on automation, but heightens economic dependency on foreign labor in parallel toshrinking domestic labor force.

65

Figure 31: Simulated wage paths relative to the baseline by UBI regimes
Notes: Top three: manufacturing. Bottom three: information service. The value is the average of 100
simulations.

Figure 32: Dependency on foreign labor by UBI regimes (left: manufacturing, right: information service)Notes: The value is the average of 100 simulations.

8

Concluding Remarks

This paper explores the impact of immigration inflow on automation adoption by establishments across various regions, and their labor market consequences. The dynamic model
predicts that immigration’s impact on native wages during 1980-2015 could vary and even bereversed by including automation adjustments and their indirect eﬀects on native wages. I
find that low-skilled immigration stifled robotization in manufacturing, and thus, augmented
routine native wages in the long run. In contrast, strong use of computerization driven by
high-skilled immigration in information service sector further upshifted non-routine native66

wages. Consequently, I find there is a long-run welfare-enhancing role for both skill types ofimmigration.
Methodologically, to the best of my knowledge, this paper is the first to empiricallyestimate the cross-factor substitution schedule, and task-specific productivities despite thesizable body of literature employing a task-based approach, which is prominently used inlabor economics and macro economics. By analyzing more than 700 geographical units and35 year inter-temporal variations, I exploit consistently asymmetric inflows of ethnicallydriven immigration inflow into U.S. regional labor markets as instruments used to infer thesubstitution patterns of routine skills vs. robots, non-routine skills vs. routine skills andforeign vs. domestic skills, under the perspective of optimal task allocation. As stressed inthis paper, the substitution schedule dictates the ensuing quantitative simulation. The joint
investigation of foreign labor and automation at the regional level is shown to be helpful
for understanding the labor market consequences of immigration and constructing a theorygrounded, more nuanced policy prescriptions.

67

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71Appendices
December 3, 2019

Contents
A Model Appendix
A.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 Optimal task allocation . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.2 Dynamic task allocation equilibrium . . . . . . . . . . . . . . . . . .
A.2 Dynamic Task-Based Model with Heterogeneity . . . . . . . . . . . . . . . .
A.3 Proof of Proposition 3 [Automation Upgrading Policy] . . . . . . . . . . . .
A.4 Equilibrium Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.6 Analytical Formula of Elasticities with Respect to Immigration . . . . . . .

2
2
2
4
6
8
10
15
24

B Simulation Appendix
B.1 Preparation at the Baseline Year . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Computation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
28

C Data Appendix
C.1 Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.1 Foreign labor distribution across sectors . . . . . . . . . . . . . . . .
C.1.2 Marshallian labor supply elasticity . . . . . . . . . . . . . . . . . . .
C.1.3 Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2.2 Robots vs. equipment . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2.3 Investment price . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Establishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.1 Urban and rural areas . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.2 Size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.4 Corroborating Reduced-Form Results . . . . . . . . . . . . . . . . . . . . . .

34
34
34
37
37
38
38
38
39
40
40
41
42

1

A

Model Appendix

A.1

Equilibrium

A.1.1

Optimal task allocation

Proposition 1 [Optimal Task Allocation over Production Factors]
I show that regularity conditions in the main paper yields corner conditions, ensuring singlecrossing of marginal cost curves. The proof largely follows Acemoglu and Restrepo (2018),
and common in the literature of task-based approach.
R1 [Routine skills vs. Robots]ψ − θ ARS<
< (1 − ψ)AR
θ a(1)RIfψ − θ ARSAR<
< (1 − ψ)holds, thenθ a(1)
R
a(0)
q
q
qR
< R <
a(1)
A
a(0)

holds. Given factor demand functions S =(A.1)(ψ − θ)/q(ψ − θ)αYθαY Sand R = R ,
=
=
q
q
R
θ/q R

ψ − θ qR
holds. Then, one getsθq⎧qψ − θ qR
qR
ψ − θ AR
⎪
⎪
<
⇒
< R
⎨
θ a(1)
θ
q
a(1)
A
R
R
R
q
q
q
A
q
θ
ψ
−
θ
⎪
⎪
⎩
< (1 − ψ)⇒ R < (1 − ψ)
<
.
θ
q
a(0)
A
ψ − θ a(0)a(0)

The two equations yield (A.1). Together with R0, it assures single-crossing of the two
marginal costs: there exists θ ∈ (0, 1) equalizing the task price such that p(θ) =R2 [Non-routine skills vs. Routine skills]S(ψ − θ)Aa(θ)(ψ − θ)Aa(1)< <(1 − ψ)a(θ)(1 − ψ)a(1)
S

2

qR
.
AR

If

ψ − θ Aa(1)ψ − θ Aa(θ)
S< <holds, then1 − ψ a(θ)
1 − ψ a(1)
S
q
q
q
q
,
<
>
a(θ)
a(1)
Aa(θ)
Aa(1)

holds. Given factor demand functions S =(A.2)(ψ − θ)/q(ψ − θ)αY(1 − ψ)αY S
and S =, ==qq
(1 − ψ)/qS

ψ−θqholds. Then, one gets1−ψq⎧⎪ψ − θ Aa(θ)
Aa(θ)
q
ψ−θq
⎪
⎪
>
⇒
>
⎨
1 − ψ a(θ)
1−ψq
a(θ)
q
q
Aa(1)
ψ − θ Aa(1)⎪ψ − θ q⎪⎪⎩ 1 − ψ q < 1 − ψ a(1) ⇒ q < a(1)

q
q
<
a(θ)
Aa(θ)
q
q
.
⇒
>
a(1)
Aa(1)

⇒

The two equations yield (A.2). Together with R0, it assures single-crossing of the two
marginal costs: there exists ψ with θ < ψ < 1 equalizing the task price such that p(ψ) = p(ψ).
R3 [Foreign skills vs. Domestic skills]φ%F(1 − φ)A

If

<

φ
S%F
<
%
%F a%
D
F (1)
S
(1 − φ)Aφ
S%F
φ<holds, then
<
%
%
%F a%
%
F
F
F (1)
D
(1 − φ)A a (0)(1 − φ)A
S

F
F
q%
q%
D,
D
< q%
> q%
%
%
%
%
F
F
F
F
A a (0)A a (1)

holds. Given factor demand function S%F =

(A.3)

F
& q%
φS&
(1 − φ)S& S%F
φS/=
and S%D =
,
=
D
F
D
D
& q%
(1 − φ)S/
q%
q%
S%

D
φ q%
holds. One gets
1 − φ q%
F
⎧
D
D
⎪
φ
1
φ q%
q%
⎪
⎪
⇒
<
<
⎪
⎨
F
%F a%
%F a%
1−φq
F (0)
F (0)
F
(1 − φ)A
A
q%
%
%
⎪
qD
φ
1
φ qD
⎪
⎪
⇒
<
<
⎪
⎩1 − φ %
%F a%
%F a%
F (1)
F
F (1)
(1 − φ)A
qF
q%
A

qF
D
< q%
%
%
F
F
A a (0)Fq%⇒> qD .
%%FF
A a (1)⇒

The two equations yield (A.3). Together with R0, it assures single-crossing of the two′%
F (φ) = p
D (φ).
marginal costs: this assures that there exists ω = φ with θ < φ < 1 such that p%
In sum, the regularities of R0-R3 assures single-crossing of marginal cost curves; eachtask is implemented by robots, routine-skills, or non-routine skills, and each sub-task is run
3

by domestic skills or foreign skills. !
A.1.2

Dynamic task allocation equilibrium

Proposition 2 [Equilibrium Outcomes]
Proof.
Impose the regularities R0-R3. Then, the perfect specialization equilibrium emerges from
Proposition 1. The task-level technology is reduced to⎧⎪⎪{AR r(ω)}α k(ω)1−α⎪⎨y(ω) = {Aa(ω)s(ω)}α k(ω)1−α⎪⎪⎪⎩{Aa(ω)s(ω)}α k(ω)1−α(ω ∈ [0, θ), low-end tasks)(ω ∈ [θ, ψ)), medium tasks)(ω ∈ [ψ, 1]), high-end tasks).

By Cobb-Douglas property of task-level technology,q R r(ω) = qs(ω) = qs(ω) = αp(ω)y(ω), q K k(ω) = (1 − α)p(ω)y(ω)

By the unit-elastic task aggregation of tasks, the manager spends equalized expenditure
to each task such thatp(ω)y(ω) = P Y∀ω ∈ [0, 1].

Then, factor demand functions arer(ω) =

αp(ω)y(ω)αP Yαp(ω)y(ω)αP YαP Yαp(ω)y(ω)=, s(ω) == R , s(ω) =
=
R
q
q
q
q
q
q

and
k(ω) =(1 − α)P Y(1 − α)p(ω)y(ω)
=
.
qK
qK

Task price is equalized to factor price in eﬃciency unit such that⎧ R
q
q K 1−α⎪⎪( R )α (
)
⎪
⎪
⎪
A α
1−α
⎪
⎨
q
q K 1−α)α ()p(ω) = (Aa(ω)α
1−α
⎪
⎪
⎪
⎪
q
q K 1−α
⎪
⎪
)
)α (
⎩(
1−α
Aa(ω)αThen, task demand for ω is given by4(ω ∈ [0, θ))(ω ∈ [θ, ψ))(ω ∈ [ψ, 1])).

⎧ R⎪⎪( A α )α ( 1 − α )1−α P Y
⎪
⎪
⎪
qR
qK
⎪
⎨
Aa(ω)α α 1 − α 1−αPY) ( K )PY
y(ω) == (qqp(ω) ⎪⎪⎪⎪⎪
Aa(ω)α α 1 − α 1−α⎪⎩() ( K )PYqq

(ω ∈ [0, θ])(ω ∈ (θ, ψ])(ω ∈ (ψ, 1]).

Thus, factor demand is given by⎧⎪⎨ θαP YqRr(ω) =
⎪
⎩0

⎧
⎪
0
⎪
⎪
⎪
⎨ (ψ − θ)αP Y(ω ∈ [0, θ])s(ω) =q⎪⎪(ω ∈ (θ, 1])
⎪
⎪
⎩0

and
k(ω) =

⎧
⎪
⎪
0
⎪
⎪
⎨
(ω ∈ (θ, ψ]) s(ω) = 0⎪⎪(1 − ψ)αP Y⎪⎪⎩(ω ∈ (ψ, 1])q(ω ∈ [0, θ])(ω ∈ [0, θ])(ω ∈ (θ, ψ])(ω ∈ (ψ, 1])(1 − α)P Y(ω ∈ [0, 1]).
qK

Since each skill follows a law of one price, factor demand is equal within task intervals
such thats(ω) =SRS, s(ω) =, r(ω) = , k(ω) = K.
ψ−θ
1−ψ
θ

Plugging these back into the technology, the aggregate production is formulated as
' ψ
' θ
R
[α{ln s(ω) + ln Aa(ω)} + (1 − α) ln k(ω))]dωY = exp[ [α(ln r(ω) + ln A ) + (1 − α) ln k(ω)]dω +
θ
0
' 1
+
[α{ln s(ω) + ln Aa(ω)} + (1 − α) ln k(ω)]dω]ψRS ψ−θ S (1−ψ) α 1−α= B(θ, ψ, α){( )θ (
)
(
)
} K
θ
ψ−θ
1−ψ

where

' ψ
' 1
ln a(ω)dω +ln a(ω)dω}].
B(θ, ψ, α) = {(AR )θ (A)1−ψ }α exp[α{θψNested sub-task allocation between domestic and foreign groups is analogous to the upper
level. Given the skill demand S& in each sector, the occupation manager further allocatesdomestic and foreign skills to the sub-task spectrum to produce the aggregate upper level′skill demand. R3 and Proposition.1 implies that there exists φ ∈ (0, 1) in ω spectrum such
that
F
D
q%
q%
=
%F a%
(
%
F (φ)
Da
D (φ)
A
A

5

By iso-elastic task aggregation, the aggregate price is given by
ln q& =

' 1
0

′

′

ln p&(ω )dω .

The skill production function is reduced to′s&(ω ) =

⎧
%F a%
%
⎨A
F (ω ′ )sF (ω ′ )

⎩A
(
%
%
Da
D (ω ′ )s
D (ω ′ )′(0 ≤ ω < φ)′(φ ≤ ω ≤ 1).

Then, domestic and foreign skill demand is given by
⎧
&
⎪
⎪
⎨ q&S
′
F (ω ) =
F
s%
q%
⎪
⎪
⎩0

⎧
⎪
⎪
⎨0
′
%
D
s (ω ) = q&S&
⎪
⎪
⎩%
qD

′

(0 ≤ ω < φ)′(φ ≤ ω ≤ 1)

Plugging this into the aggregation of prices, one obtains q& =′(0 ≤ ω < φ)′(φ ≤ ω ≤ 1).
F )φ (q%D )1−φ(q%′
where B (φ) =′B (φ)%F )φ exp[) φ ln a%
F (ω ′ )dω].1 Analogous to the upper level tasks, the equilibrium splits the(A0aggregate factor income by a ratio of sub-task interval of domestic and foreign groups such((
F = φW
D = (1 − φ)W
%, W
% .!
that W

A.2

Dynamic Task-Based Model with Heterogeneity

Dynamics and heterogeneity of unobservable productivities are added to the benchmark
model. In each sector G, the aggregate final good is diﬀerentiated by σ G > 0, elasticity of
substitution in the global tradable market in sector G. (G is dropped in σ below)σ1&t = {Σi∈G (Pi )1−σ } 1−σ&t = {Σi∈G (C&i,t ) σ−1
σ } σ−1 , P.
CA baseline production Xi,t is augmented by unobservable stochastic productivity shock
Ai,t such that
Y&i,t = (Ai,t )η (Yi,t )1−η
(A.4)
where Ai,t is a serially correlated unobservable productivity in the AR1 process:ln Ai,t = ξ + ρ ln Ai,t−1 + ϵtwhere ρ ∈ (−1, 1) and ϵt is i.i.d. standard normal productivity shock. A baseline production is non-stochastically governed in market competition,
1(%
D = 1, a
D (ω ) ∀ω ∈ [0, 1].
Recall the normalization A′′

6

Yi,t = (Pi,t −σ %G
) Yt
P&t

Thus, the transition of production in response to price change follows a relative priceshift and sector-level growth such that
′

Yi,t
Y
+, * i,t−1=

individual growth′

where Pi,t ≡Pi,t( ′ )−σP* i,t−1
+, -

G
Y%
t
G
Y(
t−1
*+,-

relative price changesectoral growthPi,tis relative price in the sector. Rewriting this, one gets2&
Pt′
Pi,t 1−σ ′Y%GPi,t Yi,t = ( ′ ) Pi,t−1 Yi,t−1 t .
GPi,t−1
Y(
′

(A.5)

t−1

34

To produce the baseline production XiG , an establishment solves a Hicksian problem ofcost minimization such that
min

' 1

r(ω),k(ω),s(ω),s(ω) 0

s.t. exp(' 1

pi (ω)yi (ω)dωln yi (ω)dω) ≥ Yi , where yi (ω) = {AR r(ω) + Aa(ω)s(ω) + Aa(ω)s(ω)}α k(ω)1−α

0

' 1
0

s(ω)dω = S i ,' 1s(ω)dω = S i ,

0

' 1
0

r(ω)dω = Ri

' 1

k(ω)dω = Ki

0Lower-level immigration-intensive occupation manager’s cost minimization problem isanalogous to the benchmark model. Impose the R0-R3 and Proposition 1 (task specializationof each factor) holds. The task price is equalized to factor price in eﬃciency unit as in
Proposition 2. Plugging the marginal cost functions for each task into the aggregated price2

Yi,t = (

′
′
′
′
(
(
G
G
Pi,t −σPi,tPi,t −σ
Pi,t
Y
Y
t
)
Y
Y
=
(
) Yi,t−1 t
=⇒
i,t−1
i,t
′
′
′
′
G
G
Pi,t−1Pi,t−1Pi,t−1 Pi,t−1
Y(
Y(

t−1

t−1

′

Pi,t 1−σ)> 1 so that baseline production value increases with′
Pi,t−1
respect to quality-adjusted price advantage.
4
Each establishment is small enough so that their automation choice does not aﬀect the global industry
price.
3′′

When σ > 1, Pi,t−1 > Pi,t implies (7equation, the final price good for i is given by
Pi = exp[' 1

ln pi (ω)dω]0= exp[α{ln(qiR )θi + ln(q)ψ−θi + ln(q)1−ψ } + (1 − α) ln q K − ln B(θi , ψ, α) − ln αα (1 − α)1−α ]=A.3{(qiR )θi q ψ−θi q 1−ψ }α (q K )1−ααα (1 − α)1−α B(θi , ψ, α)(A.6)

Proof of Proposition 3 [Automation Upgrading Policy]

As the form of threshold rule is common in investment literature (e.g. Adda and Cooper(2000)), this section clarifies the suﬃcient state variables for automation upgrading policy,
featuring the analytical form of a periodic profit function. A periodic profit function is
′

Pi,t
R R
Πi (X t , Ai,t , θi | Θ) = αθi,t Ai,t ( ′ )1−σ Pi,t−1 Yi,t−1 ∆G
t−1 − pi,t Ii,t (θi,t | θi,t−1 )Pi,t−1*+,+,*
investment
gross profit′− I(θi,t > θi,t−1 )c(X, A, θ | ΘA )Pi,t−1 Y&i,t−1*+,-

adjustment cost

5Standardized by the previous production value, the periodic profit function is reducedtoΠi (X t , Ai,t , θi | Θ)Pi,t−1 Yi,t−1′
R
pR
Pi,t 1−σ G
i,t Ii,t (θi,t | θi,t−1 )= αθi,t Ai,t ( ′ ) ∆t−1 −− I(θi,t > θi,t−1 )c(X t , Ai,t , θi | ΘA ).
Pi,t−1 Yi,t−1
Pi,t−1πi (X t , Ai,t , θi | Θ) ≡The optimal automation θi is chosen to maximize πi .
The relative price index is
5

Gross profit excluding investment and adjustment cost isPi,t Yi,t −

K
Vl,t
*+,-

rental price for equipments− (W l,t + W l,t ) = αθi,t Pi,t Y&i,t = αθi,t Ai,t Pi,t Y i,t+,*labor compensation

Arranging (A.5), one gets′

Pi,t Yi,t = (Pi,t 1−σ
P GY G)Pi,t−1 Yi,t−1 Gt tG .
′Pi,t−1
Pt−1 Yt−1* +, ≡ ∆G
t−1

Substituting this into the equation above yields the gross profit function.

8′Pi,t =R θi,t{(qi,t) (q t )ψ0 −θi,t (q t )1−ψt }α (q K )1−α

PtG αα (1 − α)1−α B(θi,t , ψt , α)R
/AR αθi,t
qi,t
)
exp(α
=(
qt
+,
*

' θi,t0depend on θi,tln a(ω))dω Qi,t-where automation-independent price bundle is(q t )αψt (q t /A)α(1−ψt ) (q K )1−αQi,t ≡)ψ)1
PtG αα (1 − α)1−α exp{α( 0 t ln a(ω)dω + ψt ln a(ω)dω)}
Recall that labor adjustment, implemented by task allocation between routine skills andnon-routine skills and sub-task allocation between foreign-domestic skills is set frictionless.
Thus, the thresholds ψ, φ is set every period to reflect the skill price inequality to satisfy
the single crossing conditions. Since regional skill prices are equalized as a law of one price,
ψi,n = ψl,n , φi,n = φl,n n for every i ∈ l. Taking a ratio change of price index,
R
)θ
qi,t
/AR αθ
) i,t exp(α 0 i,t ln a(ω))dω
′
R
qt
Pi,t
qi,t
Qi,t
, ∆Qi,t−1 , θi,t ) = R= f (θi,t−1 ,
′
R
)
q
Q
Pi,t−1qi,t−1 /A αθ
i,t−1
θ
t
) i,t−1 exp(α 0 i,t−1 ln a(ω))dω * +, (q t−1
≡ ∆Qi,t−1
⎞α
⎛
R
qi,t
) θi,t)θi,t ⎟
(
⎜
⎟
⎜
qt
exp(α
ln a(ω))dω⎟ (AR )θi,t−1 −θi,t∆Qi,t−1=⎜) θ0i,t−1⎟⎜ qR
exp(α 0ln a(ω))dω⎝ ( i,t−1 )θi,t−1 ⎠
q t−1
+,
*

(

automation-dependent

6

Thus, observable state variables suﬃcient to characterize the automation policy at t are
reduced to
Xt = {

θi,t−1* +, -,

automation intensity6Rqi,t−1
q t−1
* +,
-

current vintage,∆Qi,t−1* +, -

growth of price aggregates

If not upgrading, the automation-dependent term is reduced to (

9

}

qt

q t−1

)−θi,t−1 .

A.4

Equilibrium Shifts

This section delivers building blocks of equilibrium shifts, applied in comparative statics in
Section A.5. The impact of demographic changes, including immigration, on automation
adoption or their join impact to the labor market is formulated by combination of thefollowing analytical linkages.
A.1.2.1 Drivers of Factor demand curve shift
In response to task reallocation, the aggregate factor demand curve shifts. As shown inequilibrium above, a factor demand curve is characterized by an iso-quant of its factor
income. For notational convenience, regional factor incomes are proxied by the following
notations.
Definition A.1. [Factor Incomes]%e and the capital income V R , V K in region l as
Define the labor income W l , W l , Wlll
W l ≡ ql S l ,W l ≡ ql S l ,(e ≡ q&e S
%e ,
W
l

VlR ≡ q R Rl ,VlK ≡ q K Kl(
F
VlR , W l , W l , VlK , W
l are factor incomes for robots, routine skills, non-routine skills, equip-

ment and foreign skills in region l, respectively. As shown in Proposition 1, demand curves
are proxied by factor incomes. Factor income is changed either by extensive-margin technology adjustments or production shifts. The factor demand curve shifts from task reallocation
are formulated as
d ln Wd ln Wd ln(ψ − θ)−1d ln Wd ln(1 − ψ)−1d ln V R
d ln θR
1
=−
=
=
,
=
=
,
=
= R
dθ
dψ
dθ
ψ−θ
dψ
dψ
1−ψ
dθ
dθ
θ

and

1 d ln W D−1
d ln W F
= ,
=
.
dφ
φ
dθ
1−φ

The elasticity of factor income with respect to production is unity such that∂ ln V R∂ ln W∂ ln Y= 1.
==∂ ln Y∂ ln Y∂ ln S

A.1.2.2 Consequences of Factor demand curve shift
The factor demand shifts, in turn, alter the equilibrium factor prices and allocations. The
elasticity of factor prices and allocations with respect to factor incomes are given by10

d ln S4d ln q4
= Φ,
=1−Φ
5
5
d ln Wd ln W

where Φ is a function of uncompensated labor supply elasticity, ϵU , such that Φ =(ϵU + 1)−1 for corresponding skills. The skill demand curve shift is split to changes in skill
price and allocation by Φ. I show this statement below.

Figure A.1: Aggregate skill demand curve shifts, skills (left) / capital (right)′

As a result of task reallocation, demand curve shifts from W to W such that qS = W′
and (q + dq)(S + dS) = W . Thus, dropping the second-order term,
qdS + Sdq = dW

(A.7)

holds. Define the slopes of labor demand, labor supply curves, as v d , v s , respectively as
vd ≡

dq
q
=− ,
S
dS d

vs ≡

dq
q
= ϵ−1
U
dS s
S

where S d , S s are skill demand, and supply, respectively. Introduce Φ as a function of uncompensated labor-supply elasticity, ϵU .
Φ≡
7
7

vs
= (ϵU + 1)−1
vs − vd(A.8)

Therefore, under the demand system, the elasticity of skill prices or allocations with

Eliminate the knife-edge case such that v s = −q= vd .
S11

respect to skill demand is reduced to Φ and 1 − Φ, respectively.
⎧d ln q
vs W
vsS
⎪
⎨
=
=
d ln W
q + vsS q
q + vsS1
Wqd ln S⎪⎩==d ln W
q + vsS S
q + vsS

= Φ

A.8

= 1−Φ

A.8

dq
The first equality comes from by plugging dS = s into A.7.8
v
Domestic and foreign skill demand shifts are analogously formulated. If the immigrationintensive skill demand rises given any domestic and foreign skill supply curve, domestic andforeign skill demand shifts change the domestic skill price and allocation as
⎧
(e
%e
⎪
W
vs
d ln q&e
vsS
⎪
⎪
=
=
⎨
%e q&e%e(eq&e + v s S
q&e + v s S
d ln W
e
e
e
(
%
&
⎪
W
d ln S
1
q
⎪
⎪
=
=
⎩
e
s
e
e
e
e
%
%e
(
%
q& + v S S
q& + v s S
d ln W

%e
= Φ

A.8

%e .
= 1−Φ

A.8

The following corollary is used in simulation algorithm in Simulation Appendix B.2.
Corollary A.1. [Demand-driven Equilibrium Shifts]
The response of q and S to q ′ and S ′ from the factor income shift from W to W ′ is givenbyq′ = q +Proof.

(W ′ − W )Φ(W ′ − W )ϵU Φ, S′ = S +
S
q

⎧
1 dq
1 vs
1Φ
dW
dq
⎪
⎨
=
dW =
dW =
dW = Φ
sq
q dW
qq+v S
qS
W
1 dS
1
1
1 ϵU Φ
dW
dS
⎪
⎩
=
dW =
dW =
dW = ϵU ΦsSS dW
Sq+v S
S q
W

(A.9)

dW
> 0) unambiguously raises q, but moves
Observe that the increase of factor demand (W
S conditional on the sign of ϵU . Each third equality holds because
⎧
⎪
⎨

vs
Φ
= S −1 (ϵU + 1)−1 =
s
q+v S
S
1
ϵU
ϵU Φ
⎪
−1
⎩
=q
=
.
q + vsS
ϵU + 1q

q
dq
To write concretely, one gets ( s + S)dq = dW and (q + v s S)dS = dW . Manipulating these,
=
v
dW
s
dS
1
v
and
=
holds.
q + vs S
dW
q + vs S
812

Rewriting A.9 yields
⎧ ′
dq
dW
W′
q
⎪
⎨ =
+1=Φ
+1=(
− 1)Φ + 1
q
q
W
W
′
′
⎪
⎩ S = dS + 1 = ϵU Φ dW + 1 = ( W − 1)ϵU Φ + 1.
S
SWW

Therefore, one gets⎧
(W ′ − W )Φ⎪⎨q ′ = q +S(W ′ − W )ϵU Φ⎪′⎩S = S +.
q!

Due to perfect elasticity of capital supply, the capital demand curve shift on capital
allocation is unity. Since the equilibrium capital deployment is M =1 for M ∈ {R, K}.

VM
d ln M, one gets=qM
d ln V M

Figure A.2: Aggregate factor demand curve shifts, skills (left) / capital (right)
Note: M ∈ {R, K}.

A.1.2.4 Consequences of Factor supply curve shifts
Equilibrium shift from skill supply curve shifts are characterized by labor supply propensity.
Formerly, regardless of the slopes of the labor supply curve,
d ln Sd ln q= −Φ < 0,=Φ>0sd ln Sd ln S sholds.
Consider that the skill supply shifts rightward by dS s . Recall that v d , v s are slopes ofdemand and supply curves, respectively.
13

Figure A.3: Aggregate skill supply curve shifts (Case.1, left, Case.2, right)• Case.1: v d < 0 < v s (skill supply curve is upward -sloping)
dq(

1
1
1
1
+ d ) = dq( s − d ) = dS s > 0
s|v | |v |vv• Case.2: v s < v d < 0 (skill supply curve is downward -sloping)−dq −dq11− d = dq( s − d ) = dS s > 0|v s |v|v |v

In both cases, skill supply upshifts lower the skill price, and raise the allocation such that⎧d s⎪
⎨ dq = v vsddSv − sv s
−v
dS
⎪
⎩
= d
s
dS
v − vsTherefore,holds. Thus, one gets

<0
> 0.

⎧
dq
vdvs
q
q 1
⎪
⎨
=
=− Φ
=−
dS s
S ϵU + 1S
v d − sv s
1⎪
⎩ dS = −v=Φ=dS sϵU + 1vd − vs⎧⎪⎨ d ln q = −Φ < 0
d ln S sd ln S⎪⎩= Φ > 0.
d ln S sGiven the fixed demand curves, the skill price and allocation changes from supply curve
shifts are characterized by Φ. In both cases, upshifts of skill supply curves decrease skill
14

prices.
Perfect elastic factors
Under imposed perfect elasticity of immigration-intensive aggregate skills and capital supply, supply curve shifts are characterized by changes in foreign-intensive skill price andcapital rental price. The elasticities of aggregate factor allocations with respect to thed ln Md ln S&== −1. With production P Y kept,
d ln q&d ln q M
α&
χP Y
θαP Y
(1 − α)P Y
log-diﬀerentiate each factor demand function, S& =,R =and K =
Rq&q
qK

rental prices are negative unity such that

immediately yields the results.

Figure A.4: The eﬀect of downshifts in immigration-intensive aggregate skill supply curve (left)
and aggregate capital supply curve (right)

A.5

Comparative Statics

The equilibrium response in response to immigration inflow is mostly symmetric across thesectors, but distinct in treatment of domestic and foreign skills.
Short runStage 1: immigration shock
The order of events is indexed by 10 and 11.
• As immigration flows in, an aggregate skill price drops. (labor supply eﬀect of immigration)(Fd ln q%d ln q&%F < 0)10 = φ()10 = −φΦd ln S%F
d ln S%F

15

• As the aggregate skill price decreases, a manager demands more skills, and the allocation rises.
(• Production shifts.
(

d ln S&d ln q&)10 = −(
)10%F
d ln Sd ln S%F

d ln Sd ln Y)11 = α&χ(
)10%Fd ln Sd ln S%F• Factor demand curve shifts from the production change.
(

d ln W d ln Yd ln Yd ln Wd ln W d ln Yd ln Yd ln W)11 = (
)11 = (
)11 , (
)11 = (
)11 = (
)11
)(
)(
%
%
%
%
%
d
ln
Y
d
ln
Y
F
F
F
F
F
d ln Sd ln Sd ln Sd ln S%F* +, - d ln S
* +, - d ln S
=1

(

=1

d ln V M
d ln V M d ln Yd ln Y)11 = (
)11 = (
)11 (M ∈ {R, K})
)(
%
%
d
ln
Y
F
F
d ln Sd ln S%F* +, - d ln S
=1

• A factor demand shifts change the equilibrium, interacted with skill supply curve witha rate of Φ, Φ. (scale eﬀect of immigration)⎧d ln W⎪⎪)11⎨= (1 − Φ)(d ln qd ln qd ln Wd ln Wd ln Sd ln Sd ln Wdln S F)()(()11 = (
)11 = I (G = o) Φ()11 ()11 = (
)11 =d ln W
⎪
!
!
F
F
F
F
F
d ln Wd ln W⎪d ln S!
d ln S!
d ln S!
)⎩= (" #$ % d ln S
" #$ % d ln S
F 11
d
ln S
=Φ
=1−Φ
⎧
d ln W⎪⎪)11⎨= (1 − Φ)(d ln Wd ln Wd ln qd ln Wd ln qd ln Sd ln Sdln S F()11 = (
)11 = I (G = p) Φ()11 ()11 = (
)11 =
)(
)(
⎪
!
!
d ln W
F
F
F
F
F
d ln Wd ln W⎪d ln S!
d ln S!
d ln S!
⎩= (" #$ % d ln S
" #$ % d ln S
)
F 11
d
ln S
=Φ
=1−Φ

(

(plant)(oﬃce)(plant)(oﬃce)

d ln V M
d ln M)11 = (
)11 (M ∈ {R, K})FF
d ln S!
d ln S!

where I (G = g) is an indicator function of sector g ∈ {p, o}.
Stage 2 Labor adjustment
The order of events is indexed by 20, 21, 22 and 23.

• In the response of factor price changes, the task reallocation between routine- andnon-routine skill occurs.
(d ln qd ln qd ln qd ln qdψdψdψd ln q)(){() =()10 +()11 }+()11 = CAH,L (ψ)−1 {−()10 +(
)11 }
F 20
F
F
F
d ln qd ln q d ln Sd ln Sd ln Sd ln Sd ln S F
d ln S F
*
+,
=0

16

(

dψ

dψ
d ln S

) =(
)(
F 20
d ln qd ln qd ln S

dψ

) +(
){(
F 11
d ln qd ln qd ln qd ln SS* d ln+,

) +(
F 10

) } = CAH,L (ψ)−1 {(
F 11

=0

-

d ln qd ln qd ln Sd ln S

) −(
F 10

) }
F 11

• As a result of task reallocation, skill demand curve shifts.
(

d ln(ψ − θ)1d ln Wd ln(1 − ψ)−1d ln Wdψdψ
)20 = (
)20 = (
)20 , (
)20 = (
)20 =
)20
(
%
%
%
%
%
χ
χ
F
F
F
F
F
d ln Sd ln Sd ln Sd ln Sd ln Sd ln S%F• The skill price and allocation shifts. Also, the capital allocation shifts. (occupational
substitution eﬀect of immigration)⎧d ln W⎪⎪)20⎨= (d ln qd ln Sd ln Wd ln Wd ln Wd ln Sd ln Sd ln S F)()()()20 = (
)20 = I (G = o) Φ(
)
=
(
)
(
20
20
20
F
d ln W
⎪
!
!
F
F
F
F
d ln Wd ln W⎪d ln Sd ln S!
d ln S!
)⎩= (1 − Φ)(" #$ % d ln S
" #$ % d ln S
F 20
d ln SΦ=1−Φ⎧d ln W⎪⎪)20⎨= (1 − Φ)(d ln Wd ln Wd ln qd ln Wd ln Sd ln qd ln Sdln S F()20 ()20 = (
)20 = I (G = p) Φ()20 = (
)20
)(
)(
F
⎪
!
!
d ln W
F
F
F
Fd ln Sd ln Wd ln W⎪d ln S!
d ln S!
⎩= (" #$ % d ln S
" #$ % d ln S
)
F 20
d
ln S
=Φ
=1−Φ

(plant)(oﬃce)(plant)(oﬃce)• Production changes from factor employment changes due to task reallocation.
(d ln Yd ln Yd ln Sd ln Sd ln Yd ln Sd ln S)20 = (
)20 +()20 = α{(ψ−θ)()20 +(1−ψ)()20 } = 0
)20 (
)20 (
%
%
%
%
d
ln
S
F
F
F
F
d
ln
S
d ln Sd ln Sd ln Sd ln Sd ln S%F• Equipment shifts as(

d ln K d ln V K
d ln Yd ln K)()20 = (
)20 = (
)20%%F
d ln S F
d ln S%F* d ln+,V - d ln S
=1Domestic and foreign skill adjustment• The domestic and foreign skill demand curve shifts, because the demand of aggregate
routine (non-routine) skills changes.
((ed ln Wd ln S&d ln S&)21 = (
)11 + (
)20 > 0d ln S%F
d ln S%F
d ln S%F• The domestic and foreign skill prices and allocations shift.
(

(e
%e
(e
d ln q&e%e ( d ln W )21 , ( d ln S )21 = (1 − Φ%e )( d ln W )21)21 = Φd ln S%F
d ln S%F
d ln S%F
d ln S%F

17

(

F
D
d ln S&d ln S&d ln q&d ln q%d ln q%%F + (1 − φ)Φ
(D }{()11 + (
)20
)21 = φ()21 + (1 − φ)()21 = {φΦF+,
- d ln S*
d ln S F
d ln S%F
d ln S%F
d ln S%F!
=Φ%F ± 1 ( dψ )20 }& χφΦ= Φ{α&
χ& d ln S%F± applies to plant and oﬃce sector, respectively.
• The occupation manager reallocates the tasks between domestic and foreign groups in
response to the current skill prices.
(

D
F
F
dq%
dφ
d ln q%dφ
d ln q%dφ()21 =)10 + ()21 } +
)
{(
%F
%F
%F 21
D d ln S
d ln S%F
d ln S%F
d ln q%*d ln+,S - d ln S
* +, <0>0F
d ln S&d ln S&d ln q%%F − Φ
(D ){()10 + (Φ)11 + (
)20 }]= CAF,D (φ)−1 [(d ln S%F
d ln S%F
d ln S%F• The domestic and foreign skill demand curve shifts, because of the task reallocation.
(

(
(
D
F
1
d ln W−1dφ
dφd ln W)22 = (
)21 > 0, ()22 =)21 < 0
(
%
%
%
φ
1
−
φ
F
F
F
d ln Sd ln Sd ln Sd ln S%F• The equilibrium skill prices and employments change. (foreign substitution eﬀect ofimmigration)(%e%e d ln W(e(ed ln q&e d ln Wd ln Sd ln Sd ln q&e)22 =)22 , (
)22 =
)
(
(
%F
%F 22
(e
(e
d ln S%F
d ln S%F*d ln+,W - d ln S*d ln+,W - d ln S
"e
=Φ

(

"e
=1-Φ

F
D
d ln q&d ln q%d ln q%dφ%F − Φ
(D )()22 = φ()22 + (1 − φ)()22 = (Φ)21%%%
d ln S F
d ln S F
d ln S F
d ln S%F

Immigration and Labor adjustments

Following the analytical steps so far, the intermediate results in the short run are derived.
Task reallocation in the short run is given as follows:
(

&
dψ
dψ
d ln q&d ln q4%F + Φα&4 χ( d ln S )10 })short = ()20 = ±αCAH,L (ψ)−1 {−()10 + (
)11 )} = αCAH,L (ψ)−1 {φΦd ln S%F
d ln S%F
d ln S%Fd⎧ln S%F
d ln S%F⎨
H,L(plant)(ψ)−1 {Φα(ψ − θ) + 1}φΦF%F = αCA4 χ + 1}φΦ= ±αCAH,L (ψ)−1 {Φα&⎩−αCAH,L (ψ)−1 {Φα(1 − ψ) + 1}φΦF (oﬃce)

18

(

F
dφ
dφ
d ln q%d ln S&d ln S&%F − Φ
(D ){()short = ()21 = CAF,D (φ)−1 [−()10 + (Φ)11 + (
)20 }]
d ln S%F
d ln S%F
d ln S%F
d ln S%F
d ln S%F
1dψ&F − Φ
& D ){ α&
&F ±
&F
}]
+ (Φ
χΦ
(
= CAF,D (φ)−1 [φΦ)short* +, *+,χ
& d ln S&F+,*
labor supply eﬀectscale eﬀect

⎧
⎪
F
F
D
F
⎪
⎨CAF,D (φ)−1 [φΦ + (Φ − Φ ){α(ψ − θ)φΦ +occupational substitution eﬀectdψ1()20 }]
ψ − θ d ln S F
=
dψ
−1
F
F
D
F
⎪
⎪
(
)20 }]⎩CAF,D (φ)−1 [φΦ + (Φ − Φ ){α(1 − ψ)φΦ +1 − ψ d ln S F(plant)(oﬃce)± applies to plant and oﬃce sector, respectively.
Immigration and Routine skill price

The impact of immigration to routine skill price without automation adjustment is given as
follows:⎧
d ln qd ln qd ln q
⎪
⎪
) +(
) +(
)22 (plant)
⎨(
F 10
F 21
d ln q
d
ln
S
d
ln
S
d
ln S F()short =⎪⎪( d ln q )11 + ( d ln q )20
d ln S%F(oﬃce)
⎩
%F
%F
d
ln
S
d
ln
S
⎧
dφ
1
dψ
⎪
⎪
+ Φ{α(ψ − θ)φΦF +−φΦF)20} + (ΦF − ΦD )(
)21
(
⎪
F
⎪
* +, *+,
ψ − θ d ln Sd ln S F
⎪
⎪
⎪
+,
+,
*
*
scale eﬀect⎨labor supply eﬀect
occupational substitution eﬀect
foreign substitution eﬀect
=
dψ
1
F
⎪
⎪
Φ
(
− ψ)φΦ +)20⎪⎪Φ α(1+,*ψ−θ⎪d ln S F⎪⎪+,*⎩ scale eﬀect
occupational substitution eﬀect%F += Φ{ α&χφΦ* +, -

1dψ
)20(
χ d ln S F*+,-

scale eﬀect}

occupational substitution eﬀectdφ+ (ΦF − ΦD )()short }d ln S F*+,
labor supply eﬀect+ I (G = p) {−φΦF* +, -

foreign substitution eﬀect

Long runStage 3: Automation adjustment
The order of events is indexed by 30 and 31.
• The task reallocation between routine-skill and robots occurs.
(

dθ
)30 =
d ln S%Fdθ)d ln q* +, (= CAL,R (θ)−119(

d ln q)shortd ln S%F(plant)(oﬃce)

• The skill/robot demand curve, characterized by factor income, shifts.
(

d ln(ψ − θ)−1d ln V R
d ln θ
1d ln Wdθdθ
)30 = (
)30 =
)30 , (
)30 = (
)30 = (
)30
(
%
%
%
%
%
ψ
−
θ
θ
F
F
F
F
F
d ln Sd ln Sd ln Sd ln Sd ln Sd ln S%F• The skill price, allocation and capital allocation shifts. (displacement eﬀect of automation)⎧d ln W⎪⎪)30⎨(
d ln qd ln Wd ln Wd ln Wd ln Sd ln Sd ln SF
d ln S!
()()()30 = (
)30 = I (G = o) Φ()30 ()30 = (
)30 =d ln W
⎪
!
!
F
F
F
F
F
d ln Wd ln W⎪d ln S!
d ln S!
d ln S!
)30⎩(1 − Φ)(" #$ % d ln S
" #$ % d ln SFd ln S!
Φ(plant)(oﬃce)=1−Φ(d ln Rd ln Rd ln V R)()30 = (
)30
R!
F
F
d ln Vd ln S!
" #$ % d ln S
=1

• Production changes from factor employment changes via task reallocation. (productivity eﬀect of automation)(d ln Yd ln Yd ln Sd ln Rd ln Yd ln Sd ln R)30 = (
)30 + (
)30 = α{(ψ − θ)()30 + θ(
)30 }
)30 (
)30 (
%
%
%
%
d
ln
S
d
ln
R
F
F
F
F
d ln Sd ln Sd ln Sd ln Sd ln S%F−11dθdθ= α{(ψ − θ)
)30 + θ ()30 } = 0(
ψ − θ d ln S%F
θ d ln S%F

Due to interior optimality of task reallocation, this eﬀect is set to 0 in analytical formula by
the envelope theorem.
Domestic and foreign skill adjustment• In plant sector, the domestic and foreign skill demand curve shifts, because the demandof aggregate routine skills changes. In oﬃce sector, automation does not directly aﬀect
the non-routine skill demands.
e(d ln Sd ln Sd ln Wd ln W e) =() = 0 (oﬃce)) =() > 0 (plant) (
F 31
F 30
F 31
F 30
d ln Sd ln Sd ln Sd ln S• The domestic and foreign skill prices and allocations shift.
(

(

(e
%e
(e
d ln q&e%e ( d ln W )31 , ( d ln S )31 = (1 − Φ%e )( d ln W )31)31 = Φd ln S%F
d ln S%F
d ln S%F
d ln S%FF
D
d ln q&d ln q%d ln q%d ln S&−Φdθ%F + (1 − φ)Φ
(D }()31 = φ()31 +(1−φ)()31 = {φΦ
)30 = I (G = p)
)30
(
%
%
%
%
*
+,
ψ
−
θ
F
F
F
F
d ln Sd ln Sd ln Sd ln Sd ln S%F!
=Φ• As routine skill price drops, a manager demands more routine (or non-routine) skills,
and the equilibrium skill allocation rises and production shifts.
20⎧d ln qd ln S⎪⎪) = −()31⎨(
F 31
d ln Sd ln S F
d ln qd ln S⎪⎪) = −(
)
⎩(
F 32
F 30
d ln Sd ln S(plant)(oﬃce)⎧
d ln S⎪⎪)31⎨αχ(d ln Yd ln S F)32 =(d ln S⎪⎪d ln S%F
)
⎩αχ(
F 32
d ln S(plant)(oﬃce)• Demand curve shifts from the production change.
(

d ln W d ln Yd ln Yd ln Wd ln W d ln Yd ln Yd ln W)32 = (
)32 = (
)32 , (
)32 = (
)32 = (
)32
)(
)(
%
%
%
%
%
d
ln
Y
d
ln
Y
F
F
F
F
F
d ln Sd ln Sd ln Sd ln S%F* +, - d ln S
* +, - d ln S
=1

(

=1

d ln V M
d ln V M d ln Yd ln Y)32 = (
)32 = (
)32 (M ∈ {R, K})
)(
%
%
d
ln
Y
F
F
d ln Sd ln S%F* +, - d ln S
=1

• A rising demand of routine or non-routine skills changes the equilibrium, interactedwith skill supply curve with a ratio of Φ, Φ. (productivity eﬀect of automation)⎧d ln W⎪⎪)32⎨= (d ln qd ln qd ln Sd ln Wd ln Wd ln Wd ln Sd ln S F)()()()32 = (
)32 = I (G = o) Φ(
)
=
(
)
(
32
32
32
F
d ln W
⎪
!
!
F
F
F
F
d ln Wd ln W⎪d ln Sd ln S!
d ln S!
)⎩= (1 − Φ)(" #$ % d ln S
" #$ % d ln S
F 32
d ln S=Φ=1−Φ⎧d ln W⎪⎪)32⎨= (1 − Φ)(d ln Wd ln Wd ln qd ln qd ln Wd ln Sd ln Sdln S F
)(
)(
(
)32 (
)32 = (
)32 = I (G = p) Φ()32 = (
)32F⎪!
!
d ln W
F
F
F
Fd ln Sd ln Wd ln W⎪d ln S!
d ln S!
⎩= (" #$ % d ln S
" #$ % d ln S
)
F 32
d
ln S
=Φ
=1−Φ

(

(plant)(oﬃce)(plant)(oﬃce)

d ln Md ln V M)32 = (
)32 (M ∈ {R, K})!
Fd ln S F
d ln S!

Foreign skill adjustment• In both sector, the foreign and domestic skill demand curve shifts, due to the changing
demand of aggregate routine skills.
((ed ln S&d ln W)33 = (
)32%F
d ln Sd ln S%F• The foreign and domestic skill prices and employments are adjusted.
(

(

(e
%e
(e
d ln q&e%e ( d ln W )33 , ( d ln S )33 = (1 − Φ%e )( d ln W )33)33 = Φd ln S%F
d ln S%F
d ln S%F
d ln S%FFD(ed ln Wd ln q%d ln q%d ln q&%F + (1 − φ)Φ
(
D }(
)33
) = φ() + (1 − φ)() = {φΦ
F 33
F 33
F 33
+,
- d ln S%F*
d ln Sd ln Sd ln S
!
=Φ

Φ
dθ
dθ
&
&
= Φα(ψ
− θ)()30 = ΦαΦ()30Fψ − θ d ln Sd ln S F

21

Following the analytical steps so far, the intermediate results in the long run are derived
as follows.
Automation and skill prices
Proof.
Though the formulation of the process is asymmetric across sectors, the final consequences
are same. In plant, the eﬀect to the routine foreign and domestic skill price is given byd ln q e=dθd ln q e()31+, * dθ+displacement eﬀect

d ln q e()33+, * dθ= Φe (

d ln Sd ln S)30 + Φe αχ(
)31
dθ
dθ

productivity eﬀect1−1−1+ α(ψ − θ)Φ
} = Φe {+ αΦ}.
=Φ {
ψ−θ
ψ−θ
ψ−θ
e

Thus, the eﬀect to the aggregate routine skill price is given byd ln q F
d ln q D
d ln q−1=φ+ (1 − φ)= (φΦF + (1 − φ)ΦD ){+ αΦ}.
*
+,
- ψ−θ
dθ
dθ
dθ

(A.10)=ΦThe eﬀect to the non-routine skill price is given byd ln q=dθ

d ln q
(
)32
* dθ
+, -

= Φ(

d ln Wd ln Yd ln S)32 = Φ()32 = Φα(ψ − θ)(
)31
dθ
dθ
dθ

productivity eﬀect= Φα(ψ − θ)(−(d ln q1)31 ) = Φα(ψ − θ)Φ
= ΦαΦ.
dθψ−θ(A.11)

In oﬃce, the eﬀect to the aggregate routine skill price is given byd ln q=dθ

d ln q
(
)31
* dθ
+, -

+

displacement eﬀect

d ln q
)32
(
* dθ
+, -

= Φ(

d ln Sd ln Wd ln Sd ln S)30 + Φ()32 = Φ()30 + Φαχ(
)30
dθ
dθ
dθ
dθ

productivity eﬀect

d ln q−1−1=Φ+ Φα(ψ − θ){−()30 } = Φ(+ Φα).
ψ−θ
dθ
ψ
−θ
+,
*
=

(A.12)1ψ−θThe eﬀect to the non-routine foreign and domestic skill price is given byd ln q e=dθd ln q e
(
)33
* dθ
+, -

e

=Φ (

d ln Sd ln S1eee)32 = Φ α(ψ − θ)()32 = Φ α(ψ − θ)Φ
= Φ αΦ.
dθdθ
ψ−θ

productivity eﬀect22

Thus, the eﬀect to the aggregate non-routine skill price is given byd ln q F
d ln q D
d ln q=φ+ (1 − φ)= ΦαΦ.
dθdθdθ

(A.13)(A.10),(A.11),(A.12) and (A.13) yield the results in the main text.
Stage 4:Labor adjustment• The task reallocation between routine- and non-routine skill occurs.
(d ln qd ln qdψ−() = CAH,L (ψ)−1 { − (
)
)32 +
F 40
F 31
d ln Sd ln Sd ln S F+,+,**
displacement eﬀect productivity eﬀect(dψ
d ln S) = CAH,L (ψ)−1 {−F 40

d ln q
(
)
F 30
d
ln
S
+,
*

} (plant)

foreign substitution eﬀect

d ln q
−(
) +
F 32
d
ln
S
+,
*

displacement eﬀect productivity eﬀect

d ln q()22
d ln S F+,*

d ln q
(
)
F 22
d
ln
S
+,
*

} (oﬃce)foreign substitution eﬀect• As a result of task reallocation, skill demand curve, characterized by labor income,shifts.
(dψdψd ln(ψ − θ)1d ln Wd ln Wd ln(1 − ψ)−1d ln V R)40 = 0
) =(
)40 =
(
)40 , ((
)40 = (
)40 =
)40 , (
F 40
F
F
F
F
F
ψ − θ d ln S
1 − ψ d ln Sd ln Sd ln Sd ln S%F
d ln Sd ln S• The equilibrium skill price and employment shifts. Also, the equilibrium capital level
shifts.

⎧d ln W⎪⎪)40⎨= (d ln qd ln Sd ln Wd ln Wd ln Sd ln Wd ln Sd ln S F()()()40 = (
)40 = I (G = o) Φ(
)
=
(
)
)
(
40
40
40
F
d ln W
⎪
!
!
F
F
F
F
d ln Wd ln W⎪d ln Sd ln S!
d ln S!
)⎩= (1 − Φ)(" #$ % d ln S
" #$ % d ln S
F 40
d ln SΦ=1−Φ⎧d ln W⎪⎪)40⎨= (1 − Φ)(d ln qd ln Wd ln Sd ln qd ln Sd ln Wd ln Wd ln S F()()40 = (
)40 = I (G = o) Φ(
)
=
(
)
)(
)
(
40
40
40
F
⎪
!
!
d ln W
F
F
F
Fd ln Sd ln Wd ln W⎪d ln S!
d ln S!
⎩= (" #$ % d ln S
" #$ % d ln S)
F 40
d ln S=Φ=1−Φ(plant)(oﬃce)(plant)(oﬃce)• Production changes from factor allocation changes from task reallocation.
(d ln Yd ln Yd ln Sd ln Sd ln Yd ln Sd ln S)40 = (
)40 +()40 = α{(ψ−θ)()40 +(1−ψ)()40 } = 0
)(
)(
%
%
%
%
d
ln
S
F
F
F
F
d
ln
S
d ln Sd ln Sd ln Sd ln Sd ln S%F• Equipment shifts.

(

d ln K d ln V K
d ln Yd ln K)()40 = (
)40 = (
)40%%
FF
d ln Sd ln S%F* d ln+,V - d ln S
=123

Foreign skill adjustment• The domestic and foreign skill demand curve shifts, because the aggregate routine skill
demand gets smaller as a feedback.
((ed ln Wd ln S&)41 = (
)40 < 0d ln S%F
d ln S%F• The domestic and foreign skill prices and allocations upshift.
(

(e
%e
(e
d ln q&e%e ( d ln W )41 , ( d ln S )41 = (1 − Φ%e )( d ln W )41)41 = Φd ln S%F
d ln S%F
d ln S%F
d ln S%F• The occupation manager reallocates the sub-tasks in response to skill prices.

(

F
F
D
D
dφ
d ln q%d ln q%dφ
d ln q%d ln q%dφ)41 ={()31 + (
)41 } +{(
)31 + (
)41 }
F
D
%F
%F
d ln qd ln qd ln S%F
d ln S%F
d ln S%F* +, - d ln S
* +, - d ln S
<0

>0

(
(
(
(
F
D
D
F
d ln Wd ln W(%F {( d ln W )31 + ( d ln W )41 } + ΦD {()31 + (
)41 }]= −CAF,D (φ)−1 [−Φd ln S%F
d ln S%F
d ln S%F
d ln S%F&&(%F ){( d ln S )30 + ( d ln S )40 }D −Φ
= −CAF,D (φ)−1 (Φd ln S%F
d ln S%F• The domestic and foreign skill demand curve shifts by sub-task reallocation.
(

(
F
dφ
d ln W1)42 = (
)41 > 0,%φFd ln Sd ln S%F• The skill prices and allocations change.
(

(
D
dφ
d ln W
−1
(
)42 =
)41
%
1
−
φ
F
d ln Sd ln S%F%e%e d ln W(e(ed ln q&e
d ln q&e d ln Wd ln Sd ln S)42 =)42 , (
)42 =)(((e d ln S%F(e d ln S%F 42
%F
d
ln
W
d
ln
W
d ln S%FdlnS* +, * +, "e
=Φ

A.6

(

"e
=1−Φ

Analytical Formula of Elasticities with Respect to Immigration

Proposition 4 [Three Elasticities with respect to Immigration]
Proof.
• elasticity of robots24⎧⎨α(ψ − θ)φΦF
d ln Rd ln R
&F =)short = ()11 = α&χφΦ(⎩α(1 − ψ)φΦF
d ln S%F
d ln S%F(plant)((oﬃce)

Summing up the pair above, one gets
(

1
dθ
dθ
d ln R
1)long = ()30 = ()long
θ d ln S F
θ d ln S F
d ln S%F

d ln Rd ln Rd ln R& F + 1 ( dθ )long)net = ()short + ()long = α&χφΦ*+,
- *+,%%%
θ d ln S&F
d ln S F
d ln S F
d ln S F
scale eﬀect

displacement eﬀect• elasticity of routine skill prices(

d ln qd ln S)= Φ{
F shortα&χ ΦF +* +, -dψ
1()shortψ − θ d ln S F*+,-

scale eﬀect(

d ln qd ln S)= Φ[{
F long}+I (G = p) {occupational substitution eﬀect

+

αΦ
*+,-

productivity eﬀect−1
ψ−θ* +, -}(&F − Φ
& D )( dφ )short }+(Φd ln S&F+,
labor supply eﬀect *&F−αφΦ* +, -dθ)long +
d ln S F

displacement eﬀect

foreign substitution eﬀect

1
dψ
(
)long
ψ − θ d ln S F+,*]

occupational substitution eﬀect&F − Φ
& D )( dφ )long+ I (G = p) (Φ&F*+,d ln S
foreign substitution eﬀect

Summing up the pair above, one gets(d ln q&F + {)net = Φ[ α&
χφΦ
+,
*
%
F
d ln S
scale eﬀect

αΦ
*+,-

+

productivity eﬀect−1dθ)long +}(
ψ − θ d ln S&F* +, displacement eﬀect&F&F − Φ
& D )( dφ )net ).
+ I (G = p) (
−αφΦ
+ (Φ
&F
* +, *+,d ln S
labor supply eﬀect

1
dψ
)net
(
ψ − θ d ln S&F
*
+,
-

]

occupational substitution eﬀect

foreign substitution eﬀect• elasticity of native premium

(

(

F
D
F /q%
D)
d ln q%d ln q%d ln(q%)short = ()short − ()short%%%d ln S F
d ln S F
d ln S FFFFDDd ln q%d ln q%d ln q%d ln q%d ln q%=()10 + {(
)21 + (
)22 } − {()21 + (
)22 }d ln S%F
d ln S%F
d ln S%F
d ln S%F
d ln S%F%F(
D%F + (Φ%F − Φ
(%F ± 1 ( dψ )short } + ( Φ + Φ )( dφ )shortD ){α&= −αφΦ
χφΦχ
& d ln S%Fφ
1 − φ d ln S%F

F /q%
D)
F
D
F
F
F
F
d ln q%d ln q%d ln q%d ln q%d ln q%d ln q%d ln(q%)long = ()long − ()long = {()31 + (
)33 + (
)41 + (
)42 }d ln S%F
d ln S%F
d ln S%F
d ln S%F
d ln S%F
d ln S%F
d ln S%FF
D
DF
d ln q%d ln q%d ln q%d ln q%)31 + (
)33 + (
)41 − ()42 }− {(d ln S%F
d ln S%F
d ln S%F
d ln S%F&F − Φ
& D )I (G = p) 1 ( dθ )long − αφΦ
&F
= −(Φ
χ
& d ln S&F

25

Summing up the pair above, one gets(d ln(&
q F /&
qD ))net =
d ln S&F&F−αφΦ* +, -

labor supply eﬀect&F − Φ
& D )(α&& F +αΦ( dθ )long )+ (ΦχφΦ* +, d ln S&F
scale eﬀect&F − Φ
&D
dθ
Φ
Φ
dφ
dψ
Φ
)net ± [)net − I(G = p)()long }]
+
)(
{(
+(
F
&
φ
1 − φ d ln S&Fχ&d ln Sd ln S&F
*
+,
- *
+,
&F

&D

foreign substitution eﬀect

occupational substitution and displacement eﬀect!

Moment conditions
Plugging the elasticities of technology adjustment in Proposition 3 into the results in Proposition 4 yields the following closed-form analytical formulas in each sector.
Plant sector⎧ d ln R
⎪
(
)net
⎪
⎪
⎪
d ln S F⎪⎪⎪
d ln q
⎪
⎪
⎪
(
)
⎪
F net
⎪
⎪ d ln S⎨⎪⎪⎪
d ln(q D /q F )
⎪
⎪(
)net
⎪
⎪
⎪
d ln S F
⎪
⎪
⎪
⎪
⎪
⎩

1
q
= α(ψ − θ)φΦF + CAL,R (θ)−1 ϵshort
θ
1
Φ
q
q
{Φ + CAF,D (φ)−1 (ΦF − ΦD )2 }[(CAH,L (ψ)−1− 1)CAL,R (θ)−1 ϵshort= ϵshort +ψ−θ
ψ−θ
−CAH,L (ψ)−1 (ΦF − ΦD )ϵφshort ]ΦFΦD φ
1 (Φα(ψ − θ) + 1)φΦF
}−(
+
)ϵ
H,L
ψ−θ
CA
(ψ)
φ
1 − φ shortqϵshort(ΦF − ΦD )ϵφshortΦFΦD1
Φ F − ΦDΦ−{1 + (
+
)
}
[{
−
1}
−
]
φ
1 − φ CAF,D (φ)ψ−θ(ψ − θ)CAH,L (ψ)
CAL,R (θ)
CAH,L (ψ)= ΦF − (ΦF − ΦD ){α(ψ − θ)φΦF +where a trio of short-run elasticities are defined as

⎧
dq
q
⎪
⎪
⎪
ϵshort ≡ ()short =⎪⎪d ln S F⎪⎨
dφ)short =ϵφshort ≡ (⎪d ln S F
⎪
⎪
⎪ ψ
dψ
⎪
⎪
)short =⎩ϵshort ≡ (d ln S F(ΦF − ΦD ){(ΦF − ΦD )(ψ − θ)φ − 1} F
(ΦF − ΦD )2 ψ
1
+
+
]Φ
{Φ
}ϵ
CAF,D (φ)ψ−θ
CAF,D (φ) short
1 ψ
CAF,D (φ)−1 [(ΦF − ΦD ){(ψ − θ)φΦF +ϵ} − ΦF ]ψ − θ short[−φ + Φ(ψ − θ)φ +−CAH,L (ψ)−1 (Φ(ψ − θ) − 1)φΦF .

Oﬃce sector⎧d ln R
⎪
⎪
)
(
⎪
F net
⎪
⎪
⎪ d ln S⎪
d ln q
⎪
⎪
⎪
)
⎪
F net
⎨(
d ln Sd ln(q D /q F )⎪⎪()net =⎪⎪
F⎪⎪d ln S
⎪
⎪
⎪
⎪
⎪
⎩

1
q
F
= (1 − ψ)Φ + CAL,R (θ)−1 ϵshort
θ
Φ
Φ
Φ
q
D
F
(−1 + CAH,L (ψ)−1)CAL,R (θ)−1 + 1}ϵshort +CAH,L (ψ)−1 (Φ − Φ )ϵφshort
={
ψ−θ
ψ−θ
ψ−θ
F
F
D
{Φ(1 − ψ) + 1}φΦ
Φ
Φ
F
F
D
F
Φ − (Φ − Φ ){(1 − ψ)φΦ +
}+(
+
)ϵφ
(1 − ψ)CAH,L (ψ)φ
1 − φ short
q
F
D
F
D
F
D
ϵshort(Φ − Φ )ϵφshort
Φ
Φ
Φ −Φ
1
Φ
+
)
}
{
−
}
+{1 − (φ
1 − φ CAF,D (φ)
1−ψ(ψ − θ)CAH,L (ψ) CAL,R (θ)
CAH,L (ψ)

26

where a trio of short-run elasticities are defined as
⎧
⎪
dq
q
⎪
⎪
)
ϵshort ≡ (⎪F short⎪⎪⎪d ln S⎨dφϵφshort ≡ ()
F short⎪⎪d ln S
⎪
⎪
⎪
dψ
ψ
⎪
⎪
)
⎩ϵshort ≡ (F shortd ln S

BF= Φ{(1 − ψ)φΦ +

ϵψ
short
}
ψ−θ

F

F

D

F

= CAF,D (φ)−1 [−Φ + (Φ − Φ ){(1 − ψ)φΦ −

ϵψ
short
}]
1−ψ

F

= CAH,L (ψ)−1 {Φ(1 − ψ) − 1}φΦ .

Simulation Appendix

This problem is solved by the value function iteration approach. One period is set to a quarterin 1980-2015. Demographic change (including immigration inflow, reallocation, retirement)
and technology changes (changes of quality-adjusted investment price of robots and rental
price for equipment) are given in the first quarter. Labor force participation rate is computedin each decade by each demographic class, and interacted by the demographic change, thedemography-driven labor supply shocks are computed every year.
As is shown in Rust (1987) and the ensuing literature, a discount factor β is hard to
identify, thus, is exogenously set as β = 0.98. Assume Markov transition of the statematrix and a transition matrix is computed as follows. The evolution of θ is straightforwardRqi,t, comes from (B.3) andfrom the upgrading policy. Evolution of the relative vintage,
qt
rational expectation of routine prices. After discretizing A into 100 grids, AR (1) evolution isincorporated into a transition matrix. An initial TFP, A0 is inferred after allocating aggregate
production and factor inputs into each establishment in the ratio of the twelve size bin in CBP
under the isomorphic technology. ∆Qi,t−1 is computed from rational expectation of a priceF (ω ′ )} andtaker, specified at each region, given intensive-margin of production {a(ω), a(ω), a%%F{ARl , Al , Al }, estimated in Section 6, Stage 1. The simulation algorithm closely follows theadjustment process delineated in Section 3.

B.1

Preparation at the Baseline Year%e , S4i , Ri , Ki and initial unobservable productivity
This section recovers initial factor levels S
i
CBP
Ai,t0 , using employment size Ni,t0 from CBP as an informative allocation weight.
Initial factor allocation at the baseline year At the baseline year, every establishment
has no friction for automation and skill prices are homogeneous within regions due to the lawof one price, technology is isomorphic across establishments: θi,t0 = θl,t0 , ψi,t0 = ψl,t0 φi,t0 =φl,t0 for all i ∈ l. Then, introducing the scaling ratio for establishment i in region l at the

27

CBP
Ni,l
baseline year, τi,l ≡ CBP , every factor allocation to establishment is scaled by τi,l .
Nl%e , S4i , and capital allocations Rl,t0 , Kl,t0 (computed
Given the regional-level skill employments Sl%e =in Section 4.3), the individual establishment i’s factor allocation is computed by Si%e , S4i = τi,l S%e , Ri = τi,l Rl Ki = τi,l Kl . Individual working hour for one employτi,l S
llF
4
S%D + S%i + Si. hCBPis only used to translate
ment from CBP is computed as hCBP
= i
i,l
i,l
CBP
Ni,l
model-implied factor skill hours into CBP-based employment head counts to get comparable
moments of size distributions to data moments from CBP.

Recover the baseline establishment TFP By Proposition.3, labor incomes are
W i,t0 = αl (ψl,t0 − θi,t0 )Pi,t0 Y&i,t0 , W i,t0 = αl (1 − ψl,t0 )Pi,t0 Y&i,t0

At the baseline year, no friction across establishments is imposed as θi,t0 = θl,t0 (∀i ∈ l),
W i,t0 + W i,t0 = αl (1 − θl,t0 )Pi,t0 Y&i,t0 holds. Plugging Pi,t0 = Pl,t0 (∀i ∈ l) computed by (A.6),
individual-level production is inferred by
Y&i,t0 =W i,t0 + W i,t0
Pi,0 (1 − θl,t0 )αl(B.1)

The initial TFP of the establishments are recovered by Ai,t0 =%F
Yi,t0 = B(θi,t0 , ψl,t0 , αl | AR
i , Ai , {a(ω), a(ω), a (ω )})Y&i,t0, where
Yi,t0

′

× {(

S i,t0Ri,t0 θi,tSi,t0 1−ψl,t α 1−α
0} K
) 0(
)ψl,t0 −θi,t0 (
)
i,t0
θi,t0
ψl,t0 − θi,t0
1 − ψl,t0(B.2)

The following figure summarizes ln(Ai,t0 ) across the twelve size bins in CBP. Reassuringly,
CBP
the recovered Ai,t0 is positively associated with the establishment size, Ni,tfrom CBP, in0
dicating an empirical regularity that large establishments exhibit larger unobservable TFPs.

B.2

Computation StepsR
R
Period t starts with initial thresholds, {θi,0 , ψ0 , φ0 }, robot deployment Ri,0 , vintage qi,0
= q5
i,s ,
K
e 5
(
where s < t is the last time for upgrading, production Yi,0 , factor demand {Wi,0 , Wi,0 , Vi,0 },%0e , W
50 , V K }, which are inherited fromand aggregated factor demand functions proxied by{W0%FD.
the previous period. If no immigrants are observed, I set q = q%
The one iteration consists of Stage 1-4, crafted as a simulated version in the comparative28

statics in Section 3. Stage 1, 2 corresponds with short-run responses from immigration
inflow, and Stage 3, 4 does with long-run responses from induced automation adjustment in
the timeline of events.
Stage 1 Immigration inflow The order of events is indexed by 10 and 11.
e 5• Suppose the skill equilibrium shifts to{S%10 , S10 } by immigration inflow (and/or demographic change).

• Supply-side shocks shift prices from {q&0e , q40 }, with the initial demand curve, charactere%0e , W50 }, fixed. q%&eized by {W410 − q40 are demography (mechanical) eﬀects.
10 − q0 and q%
F φ0 %
D 1−φ050(eWe = W0 , q% = (q10 ) (q10 )
,
q
4
=
q%
10
10
′
10
e
B (φ0 )S410
S%
10

• The individual-level skill allocation shifts ase
S(i,10 =

e
K
(
5i,0
W
Vi,0
W
i,0
e
e
(
4
4
, S4i,10 =, Ki,10 = K , S%
10 = Σi Si,10 , S10 = Σi Si,10 , K10 = Σi Ki,10
e
(
q
4
q
10
q
i,10

• The price is updated as
Pi,10 =

R )θi,0 (q )ψ0 −θi,0 (q )1−ψ0 }α (q K )1−α{(qi,01010αα (1 − α)1−α B(θi,0 , ψ0 , α)

, Pi,10 Xi,10 = (YG,10
Pi,10 1−σ)
Pi,0 Yi,0Pi,0YG,0* +, ∆G,0• The factor demand is updated, given the initial task thresholdsK= (1 − α)PX,i,10 Xi,10
W i,10 = (ψ0 − θi,0 )αPX,i,10 Xi,10 , W i,10 = (1 − ψ0 )αPX,i,10 Xi,10 Vi,10
K
K
!
F
D
e
e
5
5
! !
! (
!
W
i,10 = φ0 Wi,10 , Wi,10 = (1 − φ0 )Wi,10 , W10 = Σi Wi,10 , W10 = ΣWi,10 , V10 = ΣVi,10• The skill price shifts from scale eﬀect (Corollary.A.1)

e
F φ0 %
D 1−φ0((e %e510 − W
50 )Φ
4
(W
(q%
(W
10 − W0 )Φ
11 ) (q11 )ee%, q411 = q410 +
, q%
q%
11 =
′
11 = q10 +e
B (φ0 )
S410
S%
10
*
+,
*
+,
scale eﬀectscale eﬀect• The establishment-level skill allocation shifts under the factor demand curveseS(i,11 =

K
e
!
510
W
Vi,10
W
i,10
e
e
(
4
4
, Ki,11 = K , S%
, S4i,11 =
11 = Σi Si,11 , S11 = Σi Si,11 , K11 = Σi Ki,11
e
q411
q
q%
11

29

Stage 2 Labor adjustment
Stage 2-1 Occupational adjustment• ψ is adjusted from ψ0 to ψ20 (Proposition 1)qa(ψ20 )= 11a(ψ20 )
q 11 /A• Factor demand curve shiftK= (1 − α)Pi,10 Yi,10
W i,20 = (ψ20 − θi,0 )αPi,10 Yi,10 , W i,20 = (1 − ψ20 )αPi,10 Yi,10 , Vi,20
K
K
!
F
D
e
e
5
5
! !
! (
!
W
i,20 = φ0 Wi,20 , Wi,20 = (1 − φ0 )Wi,20 , W20 = Σi Wi,20 , W20 = ΣWi,20 , V20 = ΣVi,20• The factor prices shift as skill substitution eﬀect (Corollary A.1)ee%q%
20 = q11 +

e
e %e
(
(
(W
20 − W10 )Φ
e
S%
11
+,
*

, q%
20 =

occupational substitution eﬀect

F φ0 %
D 1−φ0(q%20 ) (q20 ), q420 = q411 +′B (φ0 )

520 − W510 )Φ
4
(W
S411
+,
*

occupational substitution eﬀect• The establishment-level skill allocation shifts under the factor demand curveseS(i,20 =

e
K
!
5i,20
W
Vi,20
W
i,20
e
e
4
(
4
4
, Ki,20 = K , S%
, Si,20 =
20 = Σi Si,20 , S20 = Σi Si,20 , K20 = Σi Ki,20
e
q420
q
q(
i,20

• The establishment-level final good price is updated to

Pi,20 =

R )θi,0 (q )ψ20 −θi,0 (q )1−ψ20 }α (q K )1−α{(qi,02020

αα (1 − α)1−α B(θi,0 , ψ0 , α)

, Pi,20 Yi,20 = (YG,20
Pi,20 1−σ)
Pi,10 Yi,10
Pi,10
YG,10
* +, ∆G,10

Stage 2-2 Foreign/Domestic adjustment• Skill demand shifts
!
!
D
F
W
i,21 = (ψ20 − θi,0 )(1 − φ20 )αPi,20 Yi,20 , Wi,21 = (ψ20 − θi,0 )φ20 αPi,20 Yi,20
e
e
(
5
!
W
21 = Σi Wi,21 , W21 = (1 − ψ20 )αPi,20 Yi,20

30

• Skill prices/allocations shift from factor demand shifts in foreign substitution eﬀect(Corollary A.1)ee%q%
21 = q20 +

e
e
(
(
W
21 − W20 %e
)Φ
e
S%
20
+,
*

(

scale eﬀect from skill allocation

, q%
21 =

F φ0 %
D 1−φ0(q%
21 ) (q21 ), q421 = q420 +B ′ (φ21 )521 − W
520 )Φ
4
(W
S420
+,
*

scale eﬀect from skill allocation• φ is adjusted from φ0 to φ22 to satisfy (Proposition 1)
%F
F /A
q%
F (φ ) = 21
a%
.
22
D
q%
21

• Skill demand shifts
φ22 !
1 − φ22 !
!
!
D , W
F
e
e
D
(
5
5
!
Wi,21
WF , WW
22 = Σi Wi,22 , W22 = W21
i,22 =
i,22 =
1 − φ0φ0 i,21• Skill prices/allocations shift from factor demand shifts in foreign substitution eﬀect(Corollary A.1)ee%q%
22 = q21 +F φ22 (q%D 1−φ22
e
e
(
(
(q%
W
22 )
22 )
22 − W21 %e, q422 = q421
)Φ , q%
22 =
′
e
B
(φ
)
22
S%
21
+,
*

(

foreign substitution eﬀect• The establishment-level skill allocation shifts under the factor demand curveseS(i,22 =

e
K
!
5i,22
W
Vi,22
W
i,22
e
e
(
4
4
, S4i,22 =, Ki,22 = K , S%
22 = Σi Si,22 , S22 = Σi Si,22 , K22 = Σi Ki,22
e
q422
q
q%
22

′

Stage 3 Automation adjustment The equipment rental price changes to q K at the
beginning of the period.
5
R
R
R
R
• q6
t−1 = q0 drops to qt = q30 . θ is adjusted as
θi,30 = d(θi,0 ,Rqi,0, ∆Qi,0 , Ai,0 )q 30

where d(·) is a automation policy function. The vintage evolution is
R
qi,30
=

⎧
⎨q5
R
t

⎩q5
R
s

(i upgrades)(i does not upgrade, s < t)31• Factor demand curves shiftR
W i,30 = (ψ20 − θi,30 )αPX.i,20 Xi,20 , Vi,30= θi,30 αPX.i,20 Xi,20 , W 30 = ΣW i,30

• The factor prices shift by displacement eﬀect as (Corollary A.1)⎧
F )φ22 (q%D 1−φ22
e
e
⎪
(
(
⎪
(q%
⎪q%
30 )
e = q%e + ( W30 − W22 )Φ
%e , q%⎪= 30, q430 = q421
30
′
⎪
30
22
⎪
e
B (φ0 )
⎪
S%
⎪
22
⎪
+,
*
⎨

(plant)displacement eﬀect⎪
530 − W522
W⎪
e = q%e , q% = q% , q
⎪
4
⎪
)Φ
q%
421 + (
30
21 430 = q
⎪
30
21
⎪
⎪
S422
⎪
*
+,
⎪
⎩

(oﬃce)

displacement eﬀect• Robot and routine skill allocations shift from factor demand shifts
R30 =

R
Vi,30
R
qi,30

e
, S(
i,30 =

e
K
!
5i,30
W
Vi,30
W
i,30
, S4i,30 =, Ki,30 = ′ K
e
q430
q
q%
30

Investment occurs according to laws of motion:

R
Ii,30= Ri,30 − (1 − δ R )Ri,0• The establishment-level final good price is updated to
Pi,30 =

R )θi,30 (q )ψ20 −θi,30 (q )1−ψ20 }α (q K )1−α{(qi,03030

αα (1 − α)1−α B(θi,30 , ψ20 , α)

, Pi,30 Yi,30 = (YG,30
Pi,30 1−σ)
Pi,20 Yi,20
Pi,20
YG,20
* +, ∆G,20• A profit is collected with production after the shock is realized, and a new automation
intensity is chosen with updated investment price of robots.
RA
πi,30 = αθi,30 Pi,30 Y&i,30 − pRi,t Ii,30 − 1(θi,30 > θi,0 )c(X, Ai,30 , θi,30 | Θ )Pi,t−1 Yi,t−1

pR
i,t =

⎧
⎨p4R
t

⎩p4R
s

(i upgrades)(i does not upgrade)

Stage 4 Labor adjustment Stage 4 Labor adjustment is analogous as Stage 2.
• ψ is adjusted from ψ20 to ψ40 satisfyingqa(ψ40 )
= 30 .
a(ψ40 )
q 30 /A

32(B.3)

• Factor demand curve shifts.
K= (1 − α)Pi,30 Yi,30
W i,40 = (ψ40 − θi,30 )αPi,30 Yi,30 , W i,40 = (1 − ψ40 )αPi,30 Yi,30 , Vi,40
K
e
!
F
D
e
e
5
5
! !
! (
!
W
i,40 = φ21 Wi,40 , Wi,40 = (1 − φ21 )Wi,40 , W40 = Σi Wi,40 , W40 = ΣWi,40 V40 = ΣVi,40• The factor prices shifts. (Corollary A.1)ee%q%40 = q30 +

e
e
(
(
W
40 − W30 %e
)Φ
e
ΣS(
i,30
*
+,
-

(

, q%
40 =

occupational substitution eﬀect

F φ21 (q%D 1−φ21
(q%
40 )
40 )
, q440 = q430 +′B (φ21 )540 − W
530 )Φ4(W
Σi S4i,30*+,-

occupational substitution eﬀect• The establishment-level skill allocation shifts under the factor demand curves.
eS(i,40 =

K
e
!
5i,40
W
Vi,40
W
i,40
, Ki,40 = ′ K, S4i,40 =
e
q440
q
q%
40

• The establishment-level final good price is updated to′

PX,i,40 =R )θi,30 (q )ψi,40 −θi,30 (q )1−ψi,40 }α (q K )1−α{(qi,304040

αα (1 − α)1−α B(θi,30 , ψi,40 , α)

, Pi,40 = (1 1−η 1 η η) ( ) PY,i,40
1−ηηand the market competition governs the corresponding production as
Yi,40 = (Pi,40 −σYi,40 1
) Yi,30 , Xi,40 = ( 1−η ) η
Pi,30
A

Domestic (Foreign) skill adjustment• Skill demand shifts
!
!
D
F
W
i,41 = (ψ40 − θi,30 )(1 − φ21 )αPi,40 Yi,40 , Wi,21 = (ψ20 − θi,0 )φ21 αPi,40 Yi,40
e
e
(
5
!
W
41 = Σi Wi,41 , W41 = (1 − ψ40 )αPi,40 Yi,40• Skill prices/allocations shift from factor demand shifts in foreign substitution eﬀect(Corollary A.1)ee%q%41 = q40 +

e
e
(
(
W
41 − W40 %e
)Φ
e
S%
40
+,
*

(

scale eﬀect from skill allocation

, q%
41 =

F φ0 %
D 1−φ0(q%41 ) (q41 ), q441 = q440 +′B (φ41 )• φ is adjusted from φ21 to φ41 to satisfy
4
aF (φ41 ) =

33

F /A
4F
q440
D
q440

541 − W
540 )Φ
4
(W
S440
+,
*

scale eﬀect from skill allocation• Skill demand shifts.
φ41 !
1 − φ41 !
!
!
F
e
e
D
(
5
5
!
WD , W
WF , WW
42 = Σi Wi,42 , W42 = W41
i,42 =
i,42 =
1 − φ21 i,42
φ21 i,42• Skill prices shift from factor demand shifts in foreign substitution eﬀect. (Corollary
A.1)
e
e
%
q%
42 = q41 +F φ42 (q%D 1−φ42
e
e
(
(
W
(q%
42 − W41 %e
42 )
42 )
, q442 = q441
)Φ , q%42 =′eB (φ42 )
S%
41
+,
*

(

foreign substitution eﬀect• The establishment-level skill allocation shifts under the factor demand curves.
eS(i,42 =

e
K
!
5i,42
W
Vi,42
W
i,42
e
e
4
4
(
, Ki,42 = ′ K , S%
, S4i,42 =
42 = Σi Si,42 , S42 = Σi Si,42 , K42 = Σi Ki,42
e
q442
q
q%
42

C

Data Appendix

C.1

Labor

C.1.1

Foreign labor distribution across sectors

The model features the roles of low-skilled immigration in manufacturing, and high-skilled
immigration in information service. This section provides evidences to support this focus.
Foreign-domestic educational attainment Figure C.1 shows immigrants with with 12
year or less education (high-schools graduates and below) are over-represented in manufacturing, while immigrants with 17 year and more education (above bachelors) are overrepresented in in information service.
Immigration density across the occupational spectrum Immigration presence acrossthe spectrum of occupations along the non-routineness in 2015 for production and nonproduction workers in each sector is presented. Figure C.2 illustrates sharply contrastingpresence of foreign labor in the occupational spectrum across production and non-production
workers. To appropriately map the model into empirics, non-routineness in occupation is
proxied by one minus computerizable probability crafted in Frey and Osborne (2017). (usedin Figure C.2, C.1, C.2)9 Foreign labor is characterized by contrasting concentration along
9
Frey and Osborne (2017) cover both production workers and non-production workers across sectors. A
purely technical feasibility for automation is systematically codified in a form of numeric index, based on interviews to technology experts. Peri and Sparber (2009) document immigration presence across occupational
spectrum along task routineness, crafted from characteristics codified in ONET. They show that immigrants

34

Figure C.1: Educational attainment distribution: immigrants vs. natives (2015)(left: production workers in manufacturing, right: non-production workers in information service)Notes: Share of head counts within domestic and foreign group in each sector.

the occupational spectrum across sectors; among the production occupations, routine classof occupations are, on average, immigration-intensive. Among non-production occupations,non-routine class of occupations are on average, immigration-intensive relative to more routine class of workers.

Figure C.2: Occupational non-routineness and immigrant-native ratio (2015)(left: production workers in manufacturing sector, right: non-production workers in information service sector)
Note: Production workers, non-production workers are classified from occ2010 in ACS, 2015. The size ofthe bubble shows working hours in 2015. Shaded areas are 99% confidence intervals.

The following pair of tables list up the most and least non-routine occupations, and record
immigrant-native ratio, computed by working hours. In 92 production occupations, theaverage foreign-domestic ratio is 0.123 (top 10) vs. 0.211 (bottom 10). In 136 non-production
occupations, the average foreign-domestic ratio is 0.144 (top 10) vs. 0.120 (bottom 10).
agglomerate relatively in manual and analytically intensive occupations. Automation probability from Frey
and Osborne (2017) and the task characteristics from ONET are found to be strongly correlated.

35

Table C.1: Top/Bottom 10 non-routine occupations and foreign-domestic ratio (2015, production workers)

Notes: Ranking is made based on 92 production occupations. (2 occupations are excluded due to lack of
Frey-Osborne index)

Table C.2: Top/Bottom 10 non-routine occupations and foreign-domestic ratio (2015, nonproduction workers)

Notes: Ranking is made based on 136 non-production occupations. (5 occupations are excluded due to lackof Frey-Osborne index) Employment and working hours are taken and foreign-domestic ratio are computedfrom ACS, 2015.

36

C.1.2Marshallian labor supply elasticity

To get the uncompensated (or Marshallian) labor supply elasticity, regress working hours
by real wages, controlling for non-labor incomes.Using the cross-section individual datasets(Census/ACS), regress′ln Lj,l,t = γ0 + γ1 ln wj,l,t + γ1 ln wj,l,t × I(ej = F ) + γ2 ln N LIj,t + γ3 Xj,t + ϵj,twhere Xj is a vector of observable attributes of individual j, including age, gender,
marital status and number of children and N LIi is non-labor income. Years are takenfrom t ∈ {1980, 1990, 2000, 2007, 2015}. I(ej = F ) is a immigrant dummy for j. Then,
uncompensated labor supply elasticity of natives and immigrants are obtained by using a′common shifter γ1 and immigrant dummy coeﬃcient γ1 such that′FϵD
U = γ 1 , ϵU = γ 1 + γ 1

To control for endogeneity of the log real wage, it is instrumented by Bartik-type labordemand shocks. Following Diamond (2016), I define Bartik shock for each class of workers′
from year t to t as
Ll,g,t∆Bl,t = Σg∈G (ln wg,t′ − ln wg,t )Ll,tChanges in the productivity levels of the industries located within each city contributeto the city’s productivity change. Variation in productivity changes across industries will
diﬀerentially impact cities’ local skill productivity levels based on the industrial compositionof the city’s workforce. (Bartik (1991))
C.1.3

WelfareSome samples record very low total earning, the lower bound is set as 9600 USD (800 USD
per month) for every individual to ensure the finite marginal utility. No inflation is assumed
so that every earning is measured by CPI in 2015.
The realized welfare in 1980-2015 is directly computed from the dataset. The counterfactual welfare in 2016-2050 is computed based on the hypothetical datasets. Following
predictions from the model, average CZ-level labor income (incwage in ACS) in each occupation class is adjusted from the level in 2015. The self-selection rule in 2008-2015, 2020,
2030, 2040, 2050 delivers expected probability of entry of each individual into each occupation class. Interacting these two, the expected labor income of each sample is computed
as an ex-ante expected labor income every year. Non-labor income follows the time trend,
extrapolated from the one in 2008-2015. In UBI scenario, the social security payment is37

removed from non-labor income, and real wages are adjusted including income tax.

C.2

Capital

C.2.1

Classification

Guided by the model, physical capitals in manufacturing are classified into robots and equipment, while information capitals are split into autonomous IT capital, and IT equipment asfollows.
Table C.3: BEA capital classification

Notes: Categories in 3rd and 6th columns are my capital classification of the model, except
Transporation and Other equipment. Structure and IPP (intellectual property products) except
software are excluded.

C.2.2

Robots vs. equipment

To connect the BEA and IFR datasets, I explore which combination of capital categories in
NIPA best correspond with industrial robots from the IFR. By a series of machine learning
prediction algorithms (including random forest, SVMs), this combination of metal working
machinery and special industry machinery from NIPA, best minimizes the mean squared
prediction errors between these two.10 This chart indicates a correlation between the pair ofproxies in the datasets in 2015, indicating a very good fit.
10

The procedure is available on request.

38

Figure C.3: Cross-validating industrial robot data: the IFR vs. the BEA (2015)
Notes: Industries are from the IFR, connected to the BEA industries by an author. Shaded areas are 99%
confidence interval.

C.2.3

Investment priceInvestment prices of both robot and equipment in physical capitals and information capitals
from 1980 to 2015 are taken from taken from the BEA via FRED, deflated by GDP implicit
price deflator from FRED and they are normalized to one in 2000. All the prices are qualityadjusted, following indices and methods of Cummins and Violante (2002). The capital
investment price for robot is defined as the aggregate investment divided by aggregate capitalunits of metal working machinery and special industry machinery such thatprobot ≡I metal + I special(I metal /pmetal ) + (I spe /pspecial )where I is investment values and p is investment prices. Capital equipment price isindustrial equipment price from FRED. Superscripts indicate capital categories.

39

Figure C.4: Depreciation rate by capital category
Notes: Rental prices are imputed from investment prices, denominated by GDP deflator, and normalized to
one in 2000. (as computed in Section 4.3)

C.3

Establishment

C.3.1

Urban and rural areas

Urban areas are comprised of the largest CZs in the labor force, accounting for the half ofthe country labor force in 1980.

Figure C.5: Urban (blue zones) vs. rural (red zones)40

C.3.2Size distribution

Industry categories in the County business patterns are SICs from 1980 to 1997, and NAICS
from 1998-2015. To connect the two sub-industries, use David Dorn’s crosswalk from his
webpage. Manufacturing sector is SIC 19 (for 1980-86), 20 (for 1987-1997).
Two distributional moments, skewness and HHI, capture the feature of employment share
distributions, depicted as follows.

Figure C.6: Employment share by sector(top: 1980 vs. 2000, bottom: 2000 vs. 2015)
Figure C.7 shows the share of employments of establishments in four size class: establishments with 1000 employments and over is classified as mega size, ones with 250 to 1000
is called large, ones with 100 to 250 is called middle, and ones with employments below 100
is called small.

41

Figure C.7: Employment share by class size (1980-2015)
Notes: Mega size: 1000 and over. Large size: 250 to 1000. Middle size: 100 to 250. Small size: below 100.
Blue lines are urban, red lines are rural.

C.4

Corroborating Reduced-Form Results

The goal of this section is to test an immigration-induced automation hypothesis, graphicallysuggested in Figure 1. The robots in manufacturing (or autonomous IT capitals in information service) per routine working hours is regressed with respect to immigration inflow acrosscommuting zones in each sector. Formerly, for each sector G, run the following regression
across commuting zones l, stacked across years t ∈ {1980, 1990, 2000, 2004 − 2015} such that
(
F
N
Rl,t
l,t
G
) = α ln(
ln(
) + ϵl,t .
(
Ll,t
N
l,t

State-level and time fixed eﬀects are included. Identification conditions and instrument
construction is stated in Section 6. A pair of predictions, αp < 0 in manufacturing andαo > 0 in information service empirically backs up the hypothesis of immigration-induced
automation. For analysis featuring narrower industrial robots from the IFR, covering construction and agriculture in addition to manufacturing, see a companion paper of Yoshida(2019).

42

Table C.4: Immigration inflow and automation adoption (1980-2015)

Notes: years (1980,1990, 2000, 2004-2015) × 721 commuting zones. All models include an intercept andare weighted by employment hours in each period. Standard errors clustered at the state × year level arereported in parentheses.
*** p<1 %; ** p<5 %; * p<10 %References
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44