Managing a Polarized Structural Change
Sang Yoon (Tim) Lee∗

Yongseok Shin†

April 29, 2016
Preliminary and Incomplete

Abstract
We propose a novel framework that integrates an economy’s distribution of skills
with its occupation and industrial structure. Individuals with heterogeneous skills
choose to become a manager or worker by comparative advantage, and those who
become workers are again sorted into occupations that differ in terms of skill contents.
Industrial sectors differ in how they combine occupational outputs (or tasks) for
production. In equilibrium, occupation-specific TFPs and individuals’ occupational
choices endogenously determine sector-level and aggregate TFPs. Using this model,
we show faster TFP growth among middle-skill worker occupations relative to therest leads not only to job and wage polarization across worker occupations (horizontal
polarization), but also to a higher employment share and relative wage for managers
over workers (vertical polarization). Moreover, the speed of both types of polarization
is faster in sectors that use more middle-skill worker tasks and less managerial tasks,
which also leads to endogenously higher TFP growth for those sectors. We documentthat such predictions are supported empirically. If sectoral outputs are complementaryin producing final output, the faster TFP growth among middle-skill occupations
endogenously shifts capital and labor toward sectors that are less dependent on the
middle-skill task, resulting in structural change. In the limiting balanced growth path,the middle-skill occupations vanish, but all sectors coexist and grow at the same rate.
A quantitative analysis shows that even in the presence of sector-specific TFP growth,
occupation-specific TFP growth alone can account for more than half of structuralchange.

∗†University of Mannheim and Toulouse School of Economics: sylee.tim@uni-mannheim.de.
Washington University and Federal Reserve Bank of St. Louis: yshin@wustl.edu.

1Introduction

We develop a novel model that integrates an economy’s occupation and industrial
structure with the individual skill distribution. The model jointly determines how heterogeneous individuals sort into different occupations and how much of each occupation
is employed by different industrial sectors. In the model, the changes in wage and employment shares across occupations, and across industrial sectors, are interrelated andreinforce each other.
We use this model to connect labor market polarization and structural change.
Structural change—the shift of economic output and employment across broadlydefined sectors—is a well-established economic fact. Job and wage polarization—theemployment shares and wages of low- and high-skill occupations increasing relative to
middle-skill occupations—is a distinct feature of advanced economies (Acemoglu andAutor, 2011; Goos et al., 2014). In addition, we establish two new facts in the U.S.
data: (i) the divergence in the relative employment shares and wages between managers and workers, which we call vertical polarization, and (ii) the faster speed of bothpolarization across workers, and between managers and workers, in sectors that use
middle-skill occupations more intensively. We then show in our model that all these
phenomena can be generated by one common cause: relatively faster growth of the
TFP of middle-skill worker occupations, which is not sector-specific.
In our model, individuals are heterogeneous in two dimensions, managerial talent
and worker human capital, which are not sector-specific. Individuals decide whether tobecome a manager or a worker based on comparative advantage, and if they become aworker, also whether to work in a low-, middle- or high-skill occupations based on theiroccupation-neutral worker human capital. Notably, they only choose an occupation (ortask) and are indifferent across sectors. The only difference across sectors is how
intensively different occupational outputs or tasks are used in production, and sectorlevel TFP is endogenously determined by the equilibrium distribution of occupation
choices.
An exogenous improvement in the productivity of the middle-skill worker occupation, relative to the managerial and other worker tasks, leads to job and wage polarization across workers, i.e., the employment shares and wages of the low- and high-skill
occupations increase relative to the middle-skill occupation, assuming complementarity across worker tasks. Part of such an productivity improvement can be viewed
as routinization, or technological advancement among routine jobs (which tend to be
middle-skill jobs in the data), reducing the need for workers working in such jobs,
and recent empirical studies have established that routinization has played a dominantrole (Autor and Dorn, 2013; Goos et al., 2014) in polarization. When distinction is2

necessary, we refer to this as horizontal polarization.
A novel feature of our model is that TFP growth among middle-skill jobs leads
to an increase in the employment share and wages of managers relative to all workers(vertical polarization), if managerial and worker tasks are also complementary. We
also show that the speed of both horizontal and vertical polarization is faster in sectorsthat uses more middle-skill tasks and less managers. In the U.S. data, this sector is
manufacturing, and we document that horizontal and vertical polarization has indeed
progressed faster in manufacturing than in services. This implies faster endogenous
TFP growth for the manufacturing sector, which is also supported by the data. Then,if manufacturing and service outputs are complementary, it leads to structural change
from manufacturing to services. We also show that structural change, although originating from occupation-specific TFP growth, reinforces polarization.
An implication of structural change driven by occupation-level TFP growth is that,in the limiting balanced growth path of our model, some tasks may vanish but both
sectors must coexist, unlike many existing theories of structural change. The intuition is straightforward: because most theories rely on sector-specific forces to explain
structural change, the shift of resources from one sector to another must continue aslong as the same forces exist. In contrast, in our model of endogenous sector-level
TFP, occupation-level TFP growth is sector-neutral and only affects sectors indirectly
through how they combine task outputs. Once certain occupation shares become negligible, structural change must cease even as TFP continues to grow at the occupationlevel.
Our model is a first attempt to provide a framework that links the occupational
structure of an economy to sectoral aggregates. In particular, we can use microestimates of occupational employment and wages, which have been studied extensively
in labor economics, to study how occupational choices aggregate up to macro-levelsectoral shifts. The main finding in this regard is that micro-level productivities andelasticities in the labor market are important for understanding sectoral shifts at theaggregate level.
This is of particular empirical relevance for the U.S. The 1980s marks a starting
point of rising labor market inequality, much of which can be attributed to polarization.
It was also the starting point of a clear rise in manufacturing productivity (Herrendorf
et al., 2014) and the rise of low-skill service jobs (Autor and Dorn, 2013). As of yet,polarization and structural change have been treated separately, and vertical polarization has not been considered. We provide a unified explanation with occupation-levelproductivity enhancements among middle-skill jobs as the single driving force.
When we quantify the model also allowing for exogenous sector-specific TFP
growth, we find that occupation-specific TFP growth alone can still account for more3

than half of structural change. Conversely, we find that exogenous sector-specific TFP
growth is mostly important for accounting for vertical polarization, but not horizontal
polarization.

However, sector-specific TFP growth cannot generate within-sector

polarization, horizontal or vertical. Lastly, we document that while the routinizationindex commonly used in the polarization literature is positively correlated withoccupation-specific TFP growth, the correlation is about 0.5, indicating that otherforces may be at work.

Related Literature Task-based models are found in Acemoglu and Autor (2011)
and Goos et al. (2014) to explain polarization in employment shares, but neither
addresses wage polarization. None of them relate polarization to structural change
across macroeconomic sectors, nor treat managers as an occupation that is qualitatively
different from workers.
The manager-level technology in our model is an extension of the span-of-control
model of Lucas (1978), in which managers hire workers to produce output. However,unlike all existing variants of the span-of-control model, in our model managers organizetasks instead of workers. That is, instead of deciding how many workers to hire, theydecide on the quantities of each task to use in production, and for each task, howmuch skill to hire (rather than how many homogeneous workers). Moreover, rather
than assuming a Cobb-Douglas technology between managerial talent and workers, we
assume a CES technology between managerial talent and tasks.1
Our model is closely related to the rapidly growing literature in international tradethat use assignment models to explain inequality between occupations and/or industries (Burstein et al., 2015; Lee, 2015). The majority of such models follow in the
tradition of Roy: all workers have as many types of skills as there are available industry/occupation combinations, and select themselves into the job they in which they
have a comparative advantage. To make the model tractable, they typically employ aFréchet distribution which collapses the model into an empirically testable set of equations for each industry and/or occupation pair. While the manager-worker division in
our model is also due to Roy-selection, but the horizontal sorting of workers into tasks
is qualitatively different. In addition, we assume only 2-skill types, which is arguably
more suitable for studying the endogenous formation of skills.
Since Ngai and Pissarides (2007), most production-driven models of structuralchange rely on exogenously evolving sectoral productivities. Closer to our model is
Acemoglu and Guerrieri (2008), in which the capital-intensive sector (in the sense of1

Starting with the standard span-of-control model, we incorporate (i) non-unitary elasticity between
managers and workers, (ii) heterogeneity in worker productivity as well as in managerial productivity, (iii)
multiple worker tasks or occupations, and (iv) multiple sectors.

4having a larger capital share in a Cobb-Douglas technology) vanishes in the limitingbalanced growth path. While sectors in their model differ in how intensively they use
capital and labor, in our model they differ in how intensively they use different tasks.
By contrasting different types of labor, rather than capital and labor, we can connect
structural change—which happens across sectors—to labor market inequality acrossoccupations. Moreover, as mentioned above, unlike any other existing explanation of
structural change, our model implies that it is certain occupations, not broadly-defined
sectors, that may vanish in the limit.
While we are the first to build a model in which individuals with different skills
sort themselves into different occupations, which in turn are used as production inputs
in multiple sectors,2 there have been recent attempts such as Buera and Kaboski(2012) and Buera et al. (2015), in which multiple sectors use different combinationsof heterogeneous skills as production inputs. One important distinction is that we
separate worker human capital (continuous distribution) and skill levels of a task or
occupation (discrete). More important, our driving force (i.e., routinization) is specificto a task or occupation, not to workers’ human capital levels. This way, not only can
we address broader dimensions of wage inequality and use micro-labor estimates todiscipline our model,3 but also represent sectoral TFPs endogenously by aggregatingover equilibrium occupational choices, rather than relying on exogenously evolving
worker-skill specific productivities.
The rest of the paper is organized as follows. In section 2, we summarize threegroups of facts: horizontal polarization, structural change, and the rise of managers.
In section 3, we present the model and solve for its equilibrium allocation. Section 4
performs comparative statics on the equilibrium, which demonstrate that routinization
leads to horizontal and vertical polarization, and ultimately structural change. Section
5 characterizes the BGP of the dynamic economy and studies local stability. Section
6 calibrates the model to data from 1980-2010, and also quantifies how importantroutinization may have been in explaining recent trends in structural change. Section
7 concludes.

2

Facts
1. Jobs and wages have polarized, figure 1 Source: U.S. Census and ACS 2010,
replicated and extended from Autor and Dorn (2013) and extended to 2010.
Occupations are ranked by their 1980 mean wage.

2

Bárány and Siegel (2015) build a model in which occupations are tied to sectors.
Autor et al. (2006); Acemoglu and Autor (2011) show that residual wage inequality controlling for
education groups is much larger than between-group inequality.
3

5

.3

.5

Change in real log hourly wage
.1
.2

100 x Change in employment share
−.1
0
.1
.2
.3
.4

0

−.2
0

20
40
60
80
Skill percentile (1980 occupational mean wage)
1980−1990
1980−2010

100

0

20
40
60
80
Skill percentile (1980 occupational mean wage)

1980−2000

1980−1990
1980−2010(a) Employment Polarization

100

1980−2000(b) Wage Polarization.1

routine occupation share
.2
.3
.4
.5

.6

Change in manufacturing employment−.07 −.06 −.05 −.04 −.03 −.02 −.01 0.7

Fig. 1: Job and Wage Polarization, 30 years.

20
40
60
80
Skill percentile (1980 occupational mean wage)

100

0

0

0

20
40
60
80
Skill percentile (1980 occupational mean wage)

1980−1990
1980−2010

100(a) Routinization

1980−2000(b) Manufacturing Employment

Fig. 2: Routinization and Structural Change, 30 years.
2. Routinizable jobs correlate with structural change, figure 2. Source: U.S. Census
and ACS 2010. Left panel replicated from Autor and Dorn (2013), right panel
shows change in manufacturing employment by occupation cell.
3. Manufacturing has relatively higher share of intermediate occupations, of which
low-to-middle skill jobs have been relatively shrinking, figure 3. The right panel
includes mining and construction into manufacturing. Source: U.S. Census and
ACS 2010.
4. Structural change continues in the U.S., figure 4. Source: BEA (value-added),
NIPA Table 6 (persons involved in production). Manufacturing includes mining
and construction.
5. Both employment share and relative wages for managers are increasing: figure 5.
See appendix for definition of managers in the census.

6

.5
0

.1

Manufacturing Share
.2
.3
.4

.5
Manufacturing Share
.2
.3
.4
.1
0
0

20
40
60
80
Skill percentile (1980 occupational mean wage)
1980
2000

100

0

20
40
60
80
Skill percentile (1980 occupational mean wage)

1990
2010

1980
2000

(a) Manufacturing

100

1990
2010

(b) Manufacturing+Mining+Construction

Fig. 3: Manufacturing employment shares across skill percentiles.

100%

100%

90%

90%

80%

80%

70%

70%

60%

60%

50%

50%

40%

40%

30%

30%

20%

20%

10%

10%

0%
1970

1975

1980

Services

1985

1990

1995

Manufacturing

2000

2005

0%
1970

2010

Agriculture

1975

1980

Services(a) GDP, Value-Added

1985

1990

1995

Manufacturing

2000

2005

2010

Agriculture(b) Aggregate Employment Shares

1990

2000

12
13
14
employment share of management (%)

1.9
multiple of mean worker wage
1.5
1.6
1.7
1.8

2005

Year
non−management
ratio

11

1980

1.4

15

1.4

1.6

ratio

1.8

mean hourly wage (2005 USD)
20
25
30
35

2

40

Fig. 4: Structural Change, 1970-2013.

1980

1990

2000

2010

Year

management
Manager Wage(a) Wage Levels

Manager Employment Share(b) Relative Employment and Wages

Fig. 5: Managers vs Workers

7

2.4
1.4

multiple of mean worker wage
1.6
1.8
2
2.2

employment share of management (%)
8
10
12
14
16
1980

1990

2000

2010

1980

1990

Year

2000

2010

Year

servicesmanufacturingservices(a) Manager Employment Sharemanufacturing(b) Relative Manager Wage

Fig. 6: Managers by Sector
6. Moreover, this occurs faster in the manufacturing sector: figure 6.

3

Model

There are a continuum of individuals each endowed with two types of skill, (h, z).
Human capital, h, is used to produced tasks. Management, z, is a special skill fororganizing tasks. WLOG we assume that the mass of individuals is 1, with associateddistribution function µ.
There are 2 sectors i ∈ {m, s}.4 In each sector, goods are produced in teams. A
single manager uses her own skill and physical capital to organize three types of tasksj ∈ {0, 1, 2} (e.g., low-, medium-, high-skill occupations; or manual, routine, abstract
tasks). Each task requires both physical and human capital, and how much of each isallocated to each task is decided by the manager. Aggregating over the goods producedby all managers within a sector yields total sectoral output.
Within a sector, a better manager can produce more goods with the same amountof tasks, but task intensities may differ across sectors. Specifically, we assume that
 1
 ω
ω−1
ω−1 ω−11,yi (z) = ηiω xz (z) ω + (1 − ηi ) ω xh (z) ωxz (z) = Mz k α z 1−α ,


 σ
σ−1
2
X
1
σ−1
σ
xh (z) = 
νij τij (z) σ 

(1a)

(1b)

j=0

Z
τij (z) = Mj

tj (k, h)dµ,(1c)

hij (z)4

In our application, the two sectors stand in for ”manufacturing” and ”services,” respectively. However,I analytical model can be extended to incorporate any countably finite number N of sectors; we use thesubscripts m and s to avoid confusing them with the subscripts for tasks.

8t0 (k, h) = k α h̄1−α , t1 (k, h) = k α h1−α t2 (k, h) = k α (h − χ)1−α ,(1d)Pwith j νij = 1. The tj (·)’s are the amounts of task output produced by an individualwith human capital h and physical capital k, the latter of which is allocated by the
manager. Integrating over individual task outputs over the set of workers hired bya manager of skill z for task j in sector i, hij (z), yields a task aggregate τij (z). The
substitutability between tasks is captured by the elasticity parameter σ, and ω captures
the elasticity between all workers and managers.
For task (or occupation) 0, a worker’s own human capital is irrelevant for production: all workers’ effective skill input becomes h̄. This is to capture manual jobs that
do not depend on skills. For task 2, some of your skills become useless and effective
skill input becomes h − χ. This is to capture analytic jobs, for which lower levels ofskill are redundant. We will refer to the managerial task as “task z,” which is verticallydifferentiated from tasks j ∈ {0, 1, 2}, which are horizontally differentiated. The Mj ’s,
j ∈ {0, 1, 2, z}, capture task-specific TFP’s, which are sector-neutral.
Several points are in order. As is the case with most models of sorting workers
into tasks, the worker side of our model can be viewed as a special case of Costinot
and Vogel (2010). However, we model managers and have more than one sector. In
contrast to Acemoglu and Autor (2011), we have a continuum of skills rather thantasks, and a discrete number of tasks rather than skills. While the implications arecomparable, our formulation is more suitable for exploring employment shares acrosstasks (which are discrete in the data). The model is also comparable to Goos et al.
(2014), who show (empirically) that relative price changes in task-specific capital,
representing routinization, can drive employment polarization. However, they do notmodel skill and hence cannot explain wage polarization.
Now let Hij denote the set of individuals working in sector i ∈ {m, s} on taskj ∈ {0, 1, 2}. We can define- Hi = ∪j∈{0,1,2} Hij : set of individuals who work as workers in sector i ∈ {m, s},- Hj = ∪i∈{m,s} Hij : set of individuals who work as workers on task j ∈ {0, 1, 2},- H = ∪i∈{m,s} Hi = ∪j∈{0,1,2} Hj : set of all individuals who work as workers,- Z = Zm ∪ Zs : set of individuals who work as managers in sector i ∈ {m, s}.
Output in each sector is thenZYi =yi (z)dµ,(2)

Ziand a final good is produced by combining output from both sectors according to a
CES aggregator: 
 1 −1−1 −11.
Y = G(Ym , Ys ) = γm Ym + γs Ys

9(3)where γM + γS = 1 and we will assume  < 1.5

3.1

Planner’s Problem

We assume complete markets for solve a static planner’s problem. A planner allocatesaggregate capital K and all individuals into sectors i ∈ {m, s} and tasks j ∈ {0, 1, 2, z}.
The objective is to maximize current output (3) subject to (1))-(2) and
 



Z
Z

X
X

K = Km + Ks ≡kmj (z) + 
ksj (z) dµj∈{0,1,2,z}j∈{0,1,2,z}
Zm
Zs
Z
Z
Hij ≡
hdµ =
hij (z)dµ, j ∈ {0, 1, 2},
Hij

Zi

where Ki is the amount of capital allocated to sector i, Hij the total amount of humancapital allocated to task j in sector i, and (kij (z), hij (z)) is the amounts of physicaland human capital allocated to task j in sector i under a manager with skill z, wherej ∈ {0, 1, 2, z} for k and j ∈ {0, 1, 2} for h.
For existence of a solution, we assume that
Assumption 1 The population means of both skills are finite, that is,ZZ
zdµ < ∞,
hdµ < ∞.
andAssumption 2 There exists a strictly positive mass of individuals who do not lose allof their h-skill by working in task 2, i.e. µ(h > χ) > 0.
The following assumption is needed for uniqueness:
Assumption 3 The measure µ(z, h) is continuous and has a connected support on(h, z) ∈ [0, hu ) × [0, zu ), where xu ≤ ∞ is the upperbound of skill x ∈ {h, z}; i.e.
µ(h, z) > 0 on (0, 0) ≤ (h, z) < (hu , zu ) ≤ ∞.
Along with assumption 2, this implies χ < hu . Before showing existence and uniquenessof the solution, we first characterize the solution in the following order:
1. Characterize optimal physical capital allocations across tasks within a sector.
2. Characterize optimal human capital (h) allocations across tasks within a sector.
3. Characterize optimal labor (manager-worker) allocations within a sector.
4. Solve for optimal capital and labor allocations across sectors.
5

The estimated  between the manufacturing and service sector (broadly defined) is close to zero, as weshow in section 6.1.

10

Capital allocation within sectors Thanks to the HD1 assumptions, we canwrite sectoral technologies as ω
 1 ω−1ω−1 ω−11,
Yi = ηiω Xizω + (1 − ηi ) ω Xihω
α 1−α
Xiz = Mz Kiz
Zi ,

(5a)


 σ
σ−1
X 1 σ−1σσ
Xih = νij Tij 
,

(5b)

j

where Xih is a sectoral task aggregate and

1−α
α
α 1−α
Ti0 = M0 Ki0h̄µ(Hi0 ), Ti1 = M1 Ki1
Hi1 ,α
Ti2 = M2 Ki2[Hi2 − χµ(Hi2 )]1−α .

Given sectoral capital Ki , the planner equalizes marginal product across tasks:
M P Ki0 = M P Ki1 = M P Ki2
M P Ti1 · αTi1
M P Ti2 · αTi2
M P Ti0 · αTi0
=
=
⇒
Ki0
Ki1
Ki2
 1 
 σ−1
M P Ti1 · Ti1
Ki1
νi1 σ
Ti1 σ
⇒
=
≡ πi1 =·
M P Ti0 · Ti0
Ki0
νi0
Ti0
 1 
 σ−1
M P Ti2 · Ti2
Ki2
νi2 σ
Ti2 σ
=
≡ πi2 =·,
M P Ti1 · Ti1
Ki1
νi1
Ti1

(6a)
(6b)

where M P Tij is the marginal product of Tij w.r.t. Xi , and πij is the capital input ratio
in tasks j ∈ {1, 2} and j − 1. Due to the Cobb-Douglas assumption, πij divided bytask output ratios is the marginal rate of technological substitution (M RT S) between
tasks j and j − 1; furthermore, πij divided by either factor input ratios in tasks j andj − 1 measures the M RT S of that factor between tasks j and j − 1. (For capital, thisis equal to 1.) Given (6) we can write1σ

Xih = νi0σ−1 (1 + πi1 + πi1 πi2 ) σ−1 Ti0 .
|
{z
}

(7)

≡Πih

Of course, M P K must also be equalized across the managerial task and the rest:
M P Kiz = M P Ki0
M P Xiz · αXiz
M P Xih · M P Ti0 · αTi0
⇒
=
Kiz
Ki0
⇒

M P Xiz · XizKiz=
M P Xih · M P Ti0 · Ti0
Ki0


≡

πiz =

ηi
1 − ηi

1 
 ω−1
ω
Xiz ω
·
· Πih ,
Xih
(8)

which then allows us to write, using (7),
1

ω

1

ω

Yi = (1 − ηi ) ω−1 [1 + πiz /Πih ] ω−1 Xih
1

σ

σ−1
= (1 − ηi ) ω−1 [1 + πiz /Πih ] ω−1 νi0σ−1 Πih
Ti0

11

(9)

Sorting skills across worker tasks within sectors Given the sectoral production function (5), we can now decide how to allocate worker skills h to worker tasksj ∈ {0, 1, 2}. Since skill doesn’t matter in task 0 and some becomes irrelevant in task
2, there is positive sorting of workers into tasks; i.e. there will be thresholds (ĥ1 , ĥ2 )s.t. all workers with h ≤ ĥ1 work in task 0 and those with h > ĥ2 work in task 2. Notethat these thresholds must be equal across sectors, hence are not subscripted by i.
For each threshold, it must be that the marginal product of the threshold worker
is equalized in either task:(1 − α)Ti1(1 − α)Ti0· h̄ = M P Ti1 ·· ĥ1 ,Hi1h̄µ(Hi0 )(1 − α)Ti1(1 − α)Ti2
M P Ti1 ·· ĥ2 = M P Ti2 ·· (ĥ2 − χ)
Hi1
Hi2 − χµ(Hi2 )M P Ti0 ·using assumption 3, soĥ1 =h̄1 Li1,
πi1 Li0

1−

χ
ĥ2

=

(h̄2 − χ)Li2,
πi2 h̄1 Li1(10)

where Lij ≡ µ(Hij ) and h̄j ≡ Hij /Lij ; that is, we are assuming
Assumption 4 The means of skills in tasks j ∈ {0, 1, 2, z}, that is, (h̄j , z̄), are equal
across sectors i ∈ {m, s}.
This is an assumption is needed because we assume discrete tasks; it can be thoughtof as the limit of vanishing supermodularity within segments of a continuum of tasks.
Assumption 1 also guarantees that all objects are finite and well-defined. Using (10)we can reformulate (6) as!1−α σ−1νi1  M1 ĥ1

πi1 =
·
,
νi0
M0 h̄

"
#σ−1


M2
νi2
χ 1−α
·
πi2 =
1−
.
νi1
M1
ĥ2

(11)

Sorting managers and workers within a sector Now we know how to allocate
Ki , Hi within a sector, but we still need to know how to divide individuals into managers
and workers; that is, determine Zi ∪ Hi given a mass of individuals within a sector.
Since individuals are heterogeneous in 2 dimensions, the key is to get a cutoff rulez̃j (h) s.t. for every h, individuals with z above z̃j (h) become managers and below
become workers. Since the h-skill is used differently across tasks, we need to get 3 suchrules for each sector; however the rule must be identical across sectors.
For h ≤ ĥ1 , this rule is simple. For these workers, h does not matter, so z̃0 (h) = ẑ,i.e., is constant. The constant is chosen so that the marginal product of the threshold
manager is equalized in either task:
M P Xiz ·(1 − α)Ti0(1 − α)Xiz· z̃0 (h) = M P Xih · M P Ti0 ·· h̄
Zih̄µ(Hi0 )12⇒z̃0 (h) = ẑ =

z̄Liz
Zi
=
,
πiz Li0πiz Li0(12)

where Liz = µ(Zi ) and z̄ = Zi /Liz (which is equal across sectors by assumption 4).
Then from (8) we can write
"
#ω−1
1−ω
 
σ−ω
ηi νi0σ−1
Mz ẑ 1−α
σ−1
πiz =
·
.
· Πih
1 − ηiM0 h̄

(13)For h ∈ (ĥ1 , ĥ2 ], the rule is linear:(1 − α)Ti1(1 − α)Xiz· z̃1 (h) = M P Xih · M P Ti1 ·
·h
Zi
Hi1
z̃1 (h)πi1 z̄Liz
πi1 Zi=.
= φ1 =hπiz Hi1
πiz h̄1 Li1

M P Xiz ·⇒and finally for h > ĥ2 , the rule is affine:(1 − α)Xiz(1 − α)Ti2· (h − χ)· z̃2 (h) = M P Xih · M P Ti2 ·Zi
Hi2 − χµ(Hi2 )z̃2 (h)πi1 πi2 Ziπi1 πi2 z̄Liz= φ2 ==.
h−χπiz (Hi2 − χLi2 )πiz (h̄2 − χ)Li2

M P Xiz ·⇒Observe thatẑ = φ1 ĥ1 ,1 − χ/ĥ2 = φ1 /φ2 ,(14)

so (ĥ1 , ĥ2 , ẑ) completely determine the φj ’s, and all objects are well defined given
assumption 2, since all tasks are essential.

Sectoral production function and allocation across sectors Equations(10)-(13) completely describe the task thresholds. What is important here is that allthese thresholds are determined independently of the amount of physical capital. To
see this more clearly, rewrite (9) to obtain
ω

σ

σ−1
α 1−α
M0 Ki0Li0
Yi = ψi · [1 + πiz /Πih ] ω−1 Πih
1

1

ψi ≡ (1 − ηi ) ω−1 νi0σ−1 h̄1−αand furthermore since
Ki = Ki0 (Πih + πiz )| {z }

(15)

ΠKi

"
Li = Li0 1 + (ĥ1 /h̄1 )πi1 +|1 − χ/ĥ2(h̄2 − χ)/ĥ1
{z
Π Li

13

#
· πi1 πi2 + (ẑ/z̄)πiz ,}(16)

we obtain
ω−σω(ω−1)(σ−1)ω−1ΠK
Yi = M0 ψi · Πih
i
{z
|

−α

Φi : TFP

α 1−α
Πα−1
.
Li Ki Li}

(17)

Note that sectoral TFP, Φi , can be decomposed into 3 parts: M0 , that is common
across both sectors, ψi , which is sector-specific but exogenous, and the parts determineby (Πih , ΠKi , ΠLi ), which is sector-specific and endogenously determined by (ĥ1 , ĥ2 , ẑ).
Furthermore, since the thresholds depend only on the relative masses of individuals
across tasks within a sector, they do not depend on the employment size of the sector(nor capital). Hence even as Ki or Li changes, these thresholds do not as long as the
distribution of skills remains constant.
Sectors only differ in how intensely they use each task, i.e., the mass of individuals
allocated to each task. As usual, these masses are determined so that the M P K and
M P L are equalized across sectors:
Ks
=
κ≡
Km



γs
γm

1 


Ys
Ym

 −1


=

Ls
Lm

(18)

where κ is capital input ratios between sectors m and s.

3.2

Existence and Uniqueness

Having characterized the optimal allocation (which is identical to the equilibrium allocation), we can now establish existence and uniqueness:
Theorem 1 Under assumptions 1-4, the solution to the planner’s problem exists andis unique.
Proof: First define a renormalization of ΠLi :Π̃Li ≡ (1 − ηi )νi0 M0σ−1 h̄(σ−1)(1−α) ΠLi =

X

Vij ,

j=0,1,2,z

where the weights Vij areh̄α+σ(1−α),
h̄
α+σ(1−α)ĥVi1 ≡ (1 − ηi )νi1 M1σ−1 · 1,h̄1hiα+σ(1−α)ĥ1 (1 − χ/ĥ2 )Vi2 ≡ (1 − ηi )νi2 M2σ−1 ·,h̄2 − χσ−ωẑ α+ω(1−α)σ−1Mzω−1 ·
Viz ≡ ηi Π̃ihz̄
Vi0 ≡ (1 − ηi )νi0 M0σ−1 ·

14

(19a)
(19b)

(19c)
(19d)

and Π̃ih is a renormalization of Πih :Π̃ih ≡ νi0 M0σ−1 h̄(σ−1)(1−α) · Πih .
Note that the only differences in the Vij ’s across sectors i comes from the task intensity
parameters νij , ηi (since Πih is also a function only of the νij ’s in equilibrium). The
total amount of labor in each task j can be expressed as
X Vij· Li , where Lm = 1/(1 + κ), Ls = κ/(1 + κ)Lj =Π̃i∈{m,s} Li(20)for j ∈ {0, 1, 2, z}. This system of equations that solves the planner’s problem arealso the equilibrium market clearing conditions; the LHS is the labor supply and RHS
Pdemand for each task j. Since h̄j Lj = Hj , z̄Lz = Z, j Lj = 1 and κ = κ(ĥ1 , ĥ2 , ẑ) isa function of (ĥ1 , ĥ2 , ẑ) from (18), the solution to (ĥ1 , ĥ2 , ẑ) is found from the systemof three equations"(1 − ω) log Mz + log Z − log
log ẑ =σ−ω#P ηi Π̃ihσ−1
i

· Li

Π̃Li

(21a)

α + ω(1 − α)(1 − σ) log M1 + log H1 − log

log ĥ1 =
P (1−ηi )νi1
i

Π̃Li


· Li
(21b)

α + σ(1 − α)log ĥ2 = log(ĥ2 − χ)
(1 − σ) log



M2
M1



+ log



H2 −χL2
H1

−





− log

P (1−ηi )νi1
i

Π̃Li

· Li

.P

(1−ηi )νi1
· Li
i
Π̃Li

α + σ(1 − α)
(21c)

where
"Z Z
ĥ1
Z(ĥ1 , ĥ2 , ẑ) =Z ĥ2 Z

Z

+
ẑ

Z ĥ2
H1 (ĥ1 , ĥ2 , ẑ) =Zĥ1H2 (ĥ1 , ĥ2 , ẑ) =

Z

+
ĥ1

φ1 (ẑ,ĥ1 )·h#
zdF (z|h)dG(h)ĥ2φ2 (ĥ1 ,ĥ2 ,ẑ)·(h−χ)

hF φ1 (ĥ1 , ẑ) · h|h dG(h)hF φ2 (ĥ1 , ĥ2 , ẑ) · (h − χ)|h dG(h)ĥ2and G(h) is the marginal distribution of h, and F (z|h) the distribution of z conditional
on h; that is
Z h̃ Z z̃µ(h̃, z̃) =dF (z|h)dG(h).

Existence is straightforward. Note that the domain of ẑ ∈ [0, zu ), ĥ1 ∈ [0, ĥ2 ), andĥ2 ∈ [ĥ1 , hu ). Hence holding other variables fixed,

15



z

H s1

Zs
Zm

H m1

H s2
Hm2

H s0
H m0

ẑ

ĥ2

ĥ1

(0,0)

h

Fig. 7: Equilibrium

1. as ẑ → 0, LHS of (21a) approaches −∞, while RHS approaches ∞. Conversely,as the LHS approaches log zu , RHS approaches −∞.
2. as ĥ1 → 0, LHS of (21b) approaches −∞, while RHS remains finite. Conversely,
as LHS approaches log ĥ2 , RHS approaches −∞.
3. as ĥ2 → max{ĥ1 , χ}, LHS of (21c) remains finite, while RHS approaches −∞.
Conversely, as LHS approaches log hu , RHS becomes larger than LHS.
Assumption 3 ensures that all RHS’s are continuous; hence a solution exists. The
assumption also ensures that the mapping in the RHS is monotone; so for any guessof the two other thresholds, each threshold is found uniquely as a function of theother two. That is, the RHS of system (21) with respect to (log ẑ, log ĥ1 , log ĥ2 ) is acontraction; so the fixed point to the system is unique.

The equilibrium skill allocation is depicted in figure 8. The thresholds determine thetasks, and employment is split across sectors while preserving the means for each task.
The different masses of sectoral employment across tasks are due to the task intensity
parameters νij , ηi .

3.3

Equilibrium wages and prices

Since there are no frictions, the planner’s allocation coincides with a competitive equilibrium. Hence, the solution (ĥ1 , ĥ2 , ẑ) gives all the information needed to derive equilibrium prices. The price of the final good can be normalized to 1:
 1
1− 1−P = 1 = γm p1−,m + γs p s1

pi = [Yi /γi Y ]−  .

(22)Sectoral output prices can be obtained by applying the sectoral output in (15)-(17) in(22). The interest rate R is given either by the dynamic law of motion for aggregate

16capital, or fixed in a small open economy. So w0 is
w0 =1 − α Ki0 R··
α
Li0 h̄

where the capital-labor ratio can be found from (15)-(16).
Given w0 , indifference across tasks for threshold workers imply
w1 ĥ2 = w2 (ĥ2 − χ)

w0 h̄ = w1 ĥ1 ,⇒w0 /w1 = ĥ1 /h̄,(23a)

w1 /w2 = 1 − χ/ĥ2 .

(23b)and likewise, the threshold manager implies a “managerial efficiency wage”
wz ẑ = w0 h̄

⇒

w0 /wz = ẑ/h̄.

(23c)

Hence, relative wages for task j are simply the inverse of the thresholds.

4

Comparative Statics

The sectoral technology representation (17) implies that this model has similar implications as Ngai and Pissarides (2007): the sector with the larger TFP shrinks. The
major difference is that these TFP’s are endogenous.
What is more interesting is the implications of growth in task-specific TFP’s—
this is equivalent to the price of task-specific capital falling in Goos et al. (2014)—orchanges in the distribution for skill. In particular, we are interested in the effect ofroutinization, which we model as an increase in the task 1’s TFP, M1 . This is illustrated
in a series of comparative statics, which is possible since the equilibrium is unique andskill distribution continuous (under assumption 3). To simplify notation, define theelasticities of the thresholds w.r.t. M1 :∆h1 ≡d log ĥ1,
d log M1∆h2 ≡ χ̃ ·d log ĥ2,
d log M1∆z ≡d log ẑ,d log M1

where χ̃ ≡χĥ2 − χ> 0.

Similarly define ∆x as the elasticity of any variable x with respect to M1 . Given(∆h1 , ∆h2 , ∆z ) we know what happens to all the other variables of interest since∆φ1 = ∆z − ∆h1 ,∆φ2 = ∆φ1 − ∆h2 ,∆W1 = −∆h1 ,∆W2 = −∆h2 ,∆Wz = −∆z .

where Wj ’s are the wage ratios
W1 = w1 /w0 ,W2 = w2 /w1 ,Wz = wz /w0 .

We proceed as follows:171. approximate the change in thresholds (ĥ1 , ĥ2 , ẑ) within a sector, taking the distribution of skill in sector i, µi , as given;
2. given the comparative statics in the thresholds, approximate the change in employment shares across tasks within a sector, taking µi as given;
3. approximate the differences in polarization across sectors holding Li constant;
4. approximate the change in employment shares across sectors.

4.1

Wage and Job Polarization

To approximate the change in thresholds, we will first focus on the within sector
allocation of skill implied by (10) and (12):.
ĥ1 · πi1 (ĥ1 ) = Hi1 (ĥ1 , ĥ2 , ẑ) Li0 (ẑ, ĥ1 )hi.
1 − χh /ĥ2 · πi2 (ĥ2 ) = Hi2 (ĥ1 , ĥ2 , ẑ) − χLi2 (ĥ1 , ĥ2 , ẑ) Hi1 (ĥ1 , ĥ2 , ẑ).
ẑ · πiz (ĥ1 , ĥ2 , ẑ) = Zi (ĥ1 , ĥ2 , ẑ) Li0 (ẑ, ĥ1 )

(24a)
(24b)
(24c)

where (πij , πiz ) are defined in (11) and (13), and (φ1 , φ2 ) are defined in (14). The
masses and skill aggregates are defined over a sector-specific distribution µi , which istaken as given.
For the approximation, we will assume that ∆Lij → 0 for j ∈ {0, 1, 2, z}. Thisimplies that the density function is sufficiently small everywhere, which we assume
to ignore the indirect effects of M1 on (Lij , Hij , Zi ) that arise from changes in thethresholds. This can be thought of a limiting case of either when skills are discrete(Goos et al., 2014),6 or when there are both a continuum of tasks and skills which arematched assortatively (Costinot and Vogel, 2010). Within the context of our model, it
can be understood as approximating the equilibrium response using only the responseof labor demand (the LHS’s), while keeping labor supply (the RHS’s) fixed. We canthen show that
Proposition 1 (Routinization and Polarization) Suppose there is an increase in
M1 , and that ∆Lij → 0 for j ∈ {0, 1, 2, z}. Then
1. ∆h1 ≈ −∆h2 > 0 iff σ < 1, and2. ∆φ1 < {∆h1 , ∆z ≈ ∆φ2 } < 0 if ω < σ < 1.
This implies that capital and labor flow out of task 1 (job polarization), relative wages
decline in task 1 (wage polarization), and both the employment share and wages of
6

In fact, they assume that wages are fixed and labor is inelastically supplied.

18

managers increase (vertical polarization).
Proof: Under the assumption (or, holding labor supply fixed), the comparative static
is identical across sectors. System (24) becomes∆h1 + ∆πi1 ≈ 0,

∆h2 + ∆πi2 ≈ 0,

∆z + ∆πiz ≈ 0,

where
∆πi1 = (σ − 1) [(1 − α)∆h1 + 1] , ∆πi2 = (σ − 1) [(1 − α)∆h2 − 1] ,σ − ω πi1 (1 + πi2 )∆πi1 + πi1 πi2 ∆πi2
∆πiz = (ω − 1)(1 − α)∆z +
·
σ−1
Πih

(25)

Hence we obtain that∆h1 ≈ −∆h2 ≈

1−σ
> 0,
α + σ(1 − α)∆W1 < 0,∆W 2 > 0⇔

σ < 1.

Furthermore if ω < σ < 1,
πi1
σ−ω
·
· ∆h1 < 0,(σ − 1) [α + ω(1 − α)] Πih∆φ1 < 0, ∆φ2 ≈ ∆z < 0, and ∆Wz > 0.
∆z ≈

(26)


The change in thresholds makes it easier to analyze what happens to employment sharesby task. If σ < 1, and holding management employment shares constant, employment
and payroll in task 1 shrinks while they increase in tasks 0 and 2. Hence, similarly as in
Goos et al. (2014), we get employment polarization only when tasks are complementary,i.e. σ < 1; we also get wage polarization even with endogenous choice of tasks. At thesame time, capital flows out to the other tasks as well.
Furthermore if ω < σ, we also find that the mass and wage of managers increase
relative to all workers. But while it is clear that the thresholds move in a direction that
continues to shrink Li1 , it is unclear what happens to Li0 and Li2 , since both tasks 0
and 2 gain employment from task 1 but lose employment to managers.
So let us think about the (supply side) changes in Lij , j ∈ {0, 1, 2, z}, arising fromthe change in thresholds within a sector i, still taking the sectoral distribution µi as
given. To sign the ∆Li0 , ∆Li2 , we need additional parametric restrictions for sufficiency:

Lemma 1 Suppose the skill distribution in sector i is uniform and that ω < σ < 1. A
sufficient condition for employment in tasks 0 and 2 to rise isσ − ω < (1 − σ) [α + ω(1 − α)] .

19

So ifσ − (1 − σ)α< ω < σ,
1 + (1 − σ)(1 − α)all employment shares except task 1’s increase. This also implies that the average skillof task 1 workers rises.
Proof: Using the approximations from Proposition 1 and (24), we can approximate∆Li0 − ∆Li1 ≈ ∆h̄1 ,∆Li0 − ∆Li2 ≈ ∆h̄2 −χ ,∆Li0 − ∆Li1 ≈ ∆z̄ .

Since {∆z , ∆h2 } <0, we know that {∆z̄ , ∆h̄2 −χ } <0, that is, the average skill of managers, and workers in task 2, become diluted. We cannot sign ∆h̄1 ; however, under theuniform distribution assumption


σ−ω
πi1
∆Li0 = ∆z + ∆h1 ≈ 1 −·∆h1(1 − σ) [α + ω(1 − α)] Πihusing (26). Since πi1 /Πih is a fraction bounded above by 1, the condition in the lemma
guarantees that ∆Li0 > 0, so∆Li1 < 0 < ∆Li0 < {∆Li2 , ∆Liz } ,although we can still not order the last two.

This is intuitive. If tasks were substitutes, task 1 would crowd out all other tasks,
including managers. As task 1 becomes the dominant occupation, wages also increase.
However, when tasks are complements, workers need to flow to the other tasks, and forthis to happen relative wages must decline in task 1. Moreover, if management is more
complementary with tasks than tasks are among themselves, more individuals must
become managers—and in equilibrium, manager wages must increase. The withinsector comparative static is depicted in figure 8.

4.2

Structural change

Previous models of structural change either rely on a special non-homogeneous formof demand (rise in income shifting demand for service products) or relative technology differences across sectors (rise in manufacturing productivity relative to services,
combined with complementarity between the two types of goods, shifting production
to services). Our model is also technology driven, but transformation arises from askill neutral increase in task productivities, or routinization. Most importantly, in contrast to recent papers arguing that sectoral productivity differences can explain theskill premia or polarization (Bárány and Siegel, 2015; Buera et al., 2015), we argue

20

z

z

Z

Z

ẑ

H2

H2

ẑ

H1

H1

H0

H0

(0,0)

(0,0)

h
ĥ1

h

ĥ2

ĥ1

(a) Equilibrium wt low M1ĥ2(b) Equilibrium wt high M1

Fig. 8: Comparative Static, Within-Sector

exactly the opposite—that routinization can explain sectoral productivity differences
and structural change.
To begin this analysis, note that from (25),∆Πih ≡

d log Πihd log Π̃ih
(1 − σ) [α + ω(1 − α)]· ∆z ,=≈
d log M1
d log M1σ−ω

(27)

and using Proposition 1, the ∆Vij ’s can be approximated from (19) as
∆Vi0

= 0,

∆Vi2 ≈ −∆h̄2 −χ > 0,

∆Vi1 ≈ −∆h̄1 < 0∆Viz ≈ ∆z̄

by Lemma 1,

(28a)

> 0.

(28b)

So the ∆Vij ’s are sector-neutral and can be ordered as∆V1 < {0 = ∆V0 } < {∆V2 , ∆Vz }.
Also note thatd log Π̃Lid log ΠLi==
∆ Π Li ≡d log M1
d log M1P

j=0,1,2,z Vij ∆Vj

Π̃Li

=

X
j=0,1,2,zLij· ∆Vj .
Li

(29)

Decomposing Polarization The change in the total amount of labor in eachtask, expressed in (20), can be decomposed similarly as in Goos et al. (2014):7i
X Lij h∆Lj =· ∆Vj − ∆ΠLi + ∆Li
Lji∈{m,s}7

However, our decomposition differs from theirs. Their thought experiment is to separate the effects fromkeeping industry output fixed and when it is allowed to vary. Ours is to separate the effect from keeping
sectoral employment fixed and when allowing it to vary.

21

=

i
X Lij hX Lij 0· ∆ Vj −· ∆Vj 0 +∆LiLj
Li
j 0 =0,1,2,zi∈{m,s}|{z}by (29),j ∈ {0, 1, 2, z}.

Bij

(30)
A change in the Vij ’s occurs even holding Li ’s constant, shifting the term Bij . This leads
to “within-sector polarization,” as we saw in the previous subsection. In particular,from (28), the ∆Vj ’s are sector-neutral and common across sectors. So any differencein how the share of task j employment evolves across sectors depends on the weighted
average of the ∆Vj ’s by the employment shares of all tasks within a sector, Lij /Li .
Holding Li ’s constant, we know from Lemma 1 that task 1 is shrinking and othertasks growing within-sectors. Now we can compare the ∆ΠLi ’s across sectors, which is
the weighted average of within-sector employment shifts as seen in (29). Thus
Lemma 2 The weighted average of within-sector employment share changes, ∆ΠLi , issmaller in the sector with a larger within-sector employment share in task 1, and larger
in the sector with larger shares in all other tasks. That is,
Ls1 /Ls < Lm1 /Lm⇒∆ΠLs > ∆ΠLm .

(31)

This implies that, holding sectoral employment shares constant, manufacturing polarizes more compared to services.8
The term ∆Li in (30) captures structural change. To compute ∆Li , rewrite capital
input (or employment) ratios in (18) using (17):
 ω−σ
 1 
 1 
 " 
Πsh (ω−1)(σ−1)
1 − ηs ω−1 νs0 σ−1
γs
·
κ=
γm
1 − ηm
νm0
Πmh
#−1

 ω


ΠKs ω−1 −α ΠLs α−1
×
.
ΠKm
ΠLm

(32)

So relative employment is completely determined by the relative endogenous TFP ratio
between the two sectors. Since the elasticities of Πih are sector-neutral (both change
at the negative rate of ẑ), we obtain





ω
∆ΠKs − ∆ΠKm + (1 − α) ∆ΠLs − ∆ΠLm . (33)∆κ ≈ (1 − ) α −ω−18Of course, the assumption in the lemma is a condition on employment shares, which are endogenous.
However, the condition holds throughout our observation period in the data, so our analysis is valid. Alternatively, we could assume νm1 >> νs1 and ηm << ηs . The astute reader would have already noticed
that what the task-specific TFP’s effectively do is shift the relative employment shares over time as if theparameters νij , ηi were changing.

22

So if (∆ΠKi , ∆ΠLi ) are larger in services, employment shifts to services; that is, routinization (a rise in M1 ) leads to structural change. We have already seen that ∆ΠLiis smaller in the manufacturing when (31) holds. Now from (27),(1 − σ)[α + ω(1 − α)]∆z< 0,· πiz + Πih ·∆ΠKi ≈
ΠKiσ−ωso under the assumption in Lemma 1, ∆ΠK2 > ∆ΠK1 if
πsz
πmz
>
Πsh
Πmh

⇔

1−ω
1−ω
ηm
ηs
· (νs0 Πsh ) σ−1 >· (νm0 Πmh ) σ−1 ,1 − ηs1 − ηmwhich holds when the manager share of capital is larger in services, or η1 << η2 . Hence,both because of shifts in labor and capital, structural change occurs toward services.
To understand why capital reallocation matters for structural change, note that we
can write change in sectoral employment shares as∆Lm = −Ls · ∆κ < 0,

∆Ls = Lm · ∆κ > 0,(34)

or plugging in ∆κ from (33),
"


X
0
Li ∆ Π L 0∆Li ≈ (1 − ) (1 − α) ∆ΠLi −
i

i0

|

+ α+

ω
1−ω

{z

}

CLi



∆ΠKi −

X

Ki0 ∆ΠK 0

i0

{z

|

CKi

i



#
.

}

This makes clear that structural change in our model is due to a reallocation of bothlabor and capital, in contrast to Goos et al. (2014). The reason that capital mattersin our model is because labor in our model is in skill units, which is different from
employment shares. However, given sectoral capital Ki within a sector, physical capitaldoes not affect employment shares as it is simply allocated to equalize its MRTS with
the MRTS of skills across tasks; only when we let factors move across sectors does its
effect appear in the model. Also note that the term CLi can be written asX
X Li0 j 0
X Lij
· ∆ Vj −Li · 
∆ Vj 0  ,
CLi =
Li
Li0
0
0
j

i

j

which is the “between-sector” counterpart to the within-sector component Bij : that is,
CLi captures the average change in employment in sector i compared to the weighted
average across sectors. The contribution from capital, CKi , is additional.

23

(c) High M1

Service Sector

Lz
L2
L0

Manufacturing Sector

Lz
L2
L0

Service Sector

Lz
L0 L1 L2

Lz
L0 L1 L2(b) High M1 , κ fixed

Manufacturing Sector

Service Sector

L0 L1 L2

Lz

Manufacturing Sector

Lz
L0

L1

L2

Service Sector

L0 L1 L2

Lz

Manufacturing Sector

Lz
L2
L1
L0

(a) Low M1(d) BGP m1

Fig. 9: Comparative Statics, Across-Sectors

Of course, from (30), structural change also contributes to polarization. To see this,rewrite (30) using (34) as


X Lij
Lsj
Lmj
∆ΠLi +
Lm −
Ls ∆κ
Lj
Lj
Lj
i∈{m,s}


X

Lsj
Lmj
∆Lj − ∆Vj = −Lij ∆ΠLi +−Lm Ls ∆κ .
Ls
Lm

∆Lj = ∆Vj −
⇒ Lj

(35)

i∈{m,s}Thus,
Lemma 3 Suppose lemma 4.2 holds. Then structural change also contributes to polarization.
Proof: The term in the square brackets in (35) is negative for j = 1, and positive forall other tasks, under lemma 4.2.

This is intuitive. Manufacturing has a larger within-sector employment share in task
1 (that is, if it is more routine-intense), employing more for that task. So in addition
to task 1 shrinking in both sectors, if sector 1 also shrinks (structural change), thereis even more polarization.
Lemmas 4.2 and 3 are depicted in the first 3 subplots in figure 9. In figure (a),manufacturing is depicted as having a higher share in task 1, and services in task z.
As we move from (a) to (b), sectoral employment shares are held fixed, and task 1
shrinks in both sectors. The change in employment shares is larger in manufacturing
due to lemma . This leads to structural change in (c), according to lemma 3. Because
manufacturing has a higher share in task 1, shrinking its size contributes to polarization.

Polarization or Structural Change? One may argue that it is not task productivities that lead to structural change, but advances in sector-specific productivities
that lead to polarization. While it is most likely in reality that both forces are in play,24

in the context of our model, as long as technologies are either task- or sector-specific(that is, there are no task- and sector-specific technologies), sector-specific productivity
shifts does not lead to polarization within sectors.
Suppose that in addition to the Mj ’s, there were Ai sectoral TFP’s, so that Yi =Ai Φi Kiα L1−α
. As in Ngai and Pissarides (2007), a rise in Am changes κ at a rate of 1−,ithat is, manufacturing shrinks. But it is easily seen that none of the thresholds change,and hence neither do the Φi ’s (the endogenous sectoral TFP’s). So polarization can
only arise by the reallocation of labor across sectors that use different mixes of tasks.
To be precise, from (30),d log Lj
Lmj
Lsj
d log Lj
= (1 − ) ·= (1 − )Lm −Ls < 0.
d log Amd log κ
Lj
Lj

(36)

Note that d log Lj /d log κ is equal to the term in square brackets in (35), and negative
under assumption (31). Hence, polarization only occurs because manufacturing shrinks.
The reason is that In our micro-founded model, tasks are aggregated up into sectoral
output, not the other way around.
Equation (36) also puts a bound on how much sectoral shifts alone can accountfor job polarization. For example, in the data, manufacturing employment fell fromapproximately 33% to 19% from 1980 to 2010. If this were solely due to a change in
Am , this means that (denoting empirical values with hats):
d log κ̂
≈ 14/67 + 14/33 ≈ 0.63
d log Am
which means that
#
"
h
i
L̂mj
L̂sj
dL̂j
0.33 −
0.67 .
≈ 0.63 L̂sj L̂m − L̂mj L̂s = 0.63
d log AmL̂j
L̂j
In Section 6, we measure the employment share of routine, manufacturing jobs androutine, service jobs (as a share of total employment; that is, Lm1 and Ls1 ) in 1980
were 26% and 33%, respectively (refer to Table 2). SodL̂j= 0.63 [0.33 · 0.33 − 0.26 · 0.67] = −0.04,
d log Am
that is, a change in Am alone would imply an approximately 4 percentage point drop inroutine jobs from 1980 to 2010. As shown in Table 2, the actual drop was 13 percentage
points.

5

Dynamics

The above result implies that on a dynamic path in which M1 grows at a constant rate,
polarization happens faster than structural change. This implies that in the limit, task25

1 vanishes, structural change ceases, but both sectors still employ non-trivial amountsof labor, unlike previous models of structural change.
Such a dynamic version of the model is a straightforward extension of the neoclassical growth model. Assume that aggregate labor L grows at rate n, and a representative
household with CRRA preferences
Z ∞c(t)1−θ − 1
exp(−ρt) ·dt
1−θ
0where ct = Ct /Lt , and a law of motion for aggregate capitalK̇t = Yt − δKt − Ct ,and for simplicity let us assume that Ṁ1 /M1 = m1 and M0 = M2 = Mz , Ṁ0 /M0 = m.
Then from (18), we can also write the aggregate production function as"1

Yt = Yst · γm



Ymt
Yst

 −1


1



# −1

+ γs

= Φst · Lst Ktα Lt1−α ·

 1
 
−1
−1

γs Lst1= Φst · (Lst /γs ) 1− · Ktα L1−αtwhere Φst , Lst , the endogenous sectoral TFP and employment share of services at timet, are functions of (ẑt , ĥ1t , ĥ2t ).
Now define1Φ1−α≡ Φst · (Lst /γs ) 1−tthe endogenous aggregate (Harrod-neutral) TFP and its growth rate gt ≡ Φ̇t /Φt . As
in the RCK model, define the normalized consumption and capital per efficiency unitof laborĉt ≡ Ct /Φt Lt

k̂t ≡ Kt /Φt Lt ,so output per efficiency unit of labor isŷt ≡ Yt /Φt Lt = f (k̂t ) ≡ k̂tα .
The dynamic equilibrium is characterized byi
1 hĉ˙t = · f 0 (k̂t ) − (n + δ + ρ + gt θ) · ĉtθ˙k̂t = f (k̂t ) − (n + δ + gt )k̂t − ĉt


Φ̇t
gt ≡
= g ĥ1t , ĥ2t , ẑt .
Φt26

So instead of having sectoral shares as in Acemoglu and Guerrieri (2008), we haveendogenously evolving TFP which pins down the sectoral shares at every instant. Using(17) and (33), the endogenous growth rate gt becomes
#
"


X
Π̇ih
Π̇Li
ω−σ
ω
Π̇Ki
.
(1 − α)gt = m +
Lit ·
·
+
−α ·
+ (α − 1) ·(ω − 1)(σ − 1) Πih
ω−1
ΠKi
ΠLi
i

On a BGP, gt must be constant. Hence it must be that (ĥ1 , ĥ2 , ẑ) no longer evolve:
Clearly this happens when ĥ1 = ĥ2 , or from (10)-(11),!(1−α)(1−σ) ĥ2 − χ(h̄2 − χ)Li2  νi0··ĥ2 − χ =
Li0
νi2
h̄
⇒

νi0 Li2(ĥ2 − χ)σ+α(1−σ)=,νi2 Li0h̄2 − χassuming h̄ = 1. Then ẑ is determined by (12), sectoral-task employment masses are
determined by (ΠKi , ΠLi ) according to (32), and on a BGP
g∗ =

m
.
1−α

The long-run dynamics is depicted in figure 9(d), where both polarization and structural change continue until task 1 vanishes.

6

Quantitative Analysis

In the quantitative analysis, we assume that there are 10 rather than 3 (horizontallydifferentiated) worker tasks. All worker tasks j ≥ 2 are characterized by a skill lostparameter χj . (The characterization of the equilibrium is exactly the same as before.)
Each task broadly corresponds to a 1-digit occupation category in the census, as shownin Table 2. The 10 occupation groups can broadly be further be broadly grouped into
low-, medium- and high skill tasks, or manual, routine, and abstract jobs, according
to the mean wages of each occupation group and routinization indices.
The objective is to quantify how much of the observed changes in employment andwage shares from 1980 to 2010 can be explained by task-level productivity growth. To
do so, we first calibrate the skill distribution over (h, z) to 1980 data. Then we choose
all other model parameters, except for the elasticities of substitutions between tasks,(σ, ω), to fit the 1980 data exactly. Lastly, the elasticities are calibrated jointly with thegrowth rates of Mj ’s, j ∈ {0, 1, . . . , 9, z}, to the time trends in employment and wage
shares from 1980 to 2010. When doing so we also include a manufacturing-specific,
exogenous sectoral TFP Am , and calibrate its growth rate from 1980 to 2010. This
allows us to quantify how the growth in the Mj ’s contributes to explaining the data,

27

(1)
γ

AIC
RMSE1
RMSE2

(2)
∗∗

(3)
∗∗

∗∗

0.371
(0.003)∗∗0.003(0.000)0.346(0.005)∗∗0.002(0.000)0.258(0.004)0.003(0.004)-550.175
0.106
0.039

-551.264
0.106
0.039

-550.866
0.106
0.039

Standard errors in parentheses†∗∗∗
p < 0.10, p < 0.05,
p < 0.01

Table 1: Aggregate Production Functionthat is, the role of task-level productivity growth in explaining polarization, structuralchange, and sectoral and total output growth.
When obtaining our empirical moments, we subsume mining and construction into
manufacturing, and government into services, and the agricultural sector is dropped.
That is, aggregate output and capital correspond to the moments computed without
agriculture, as is the case for employment shares.9

6.1

Aggregate Production Function

The aggregate production function is estimated outside of the model. For the estimation, we only look at the manufacturing and service sectors, where manufacturing
includes mining and construction, and government is included in services. We estimate
the parameters (γ, ) from the system of equations


 1−

pm Ym
log
= log γ + (1 − ) log pm − log γpm+ (1 − γ)ps1− + u1
PY


−1
−1
1
1





+ u2
log(Y ) = c +log γ Ym + (1 − γ) Ys
−1where γ ≡ γm , using non-linear SUR (seemingly unrelated regression), on all years ofreal and nominal sectoral output observed in the BEA/NIPA.
Real production by sector is computed by a cyclical expansion procedure as in(Herrendorf et al., 2014) using production value-added to merge lower level industries(as opposed to consumption value-added, as in their analysis).10 The price indices are
implied from nominal versus real sectoral quantities. We try different base years as
9Agriculture shares are only about 2% of both total output and employment, and has been more or lessflat since the 90s.
10
The constant c is included since it is not levels, but relative changes that identifies .

28

Ranked by mean wage
(except management)

SOC
Code

Low Skill Services

1980

Employment Shares
2010
Manufacturing

1980

Total Wage Shares
2010
Manufacturing

400

10.44

13.92

0.59

0.23

6.75

7.60

0.52

0.16

Middle Skill
Administrative Support
Machine Operators
Transportation
Sales
Technicians
Mechanics & Construction
Miners & Precision Workers

300
700
800
240
200
500
600

59.09
16.57
9.81
8.73
7.87
3.23
7.91
4.97

46.48
14.13
3.75
6.64
9.37
3.86
6.02
2.71

25.86
3.47
8.79
3.80
0.79
1.00
4.44
3.58

12.93
1.53
3.02
2.28
0.62
0.57
3.19
1.73

53.43
12.90
8.21
7.73
7.40
3.35
8.40
5.43

35.90
9.60
2.39
4.15
8.45
4.33
4.88
2.10

24.76
2.90
7.37
3.37
1.06
1.13
4.91
4.03

10.02
1.15
1.91
1.46
0.85
0.66
2.61
1.38

High Skill
Professionals
Management Support

40
20

19.22
11.02
8.20

26.16
16.51
9.65

3.87
1.73
2.14

3.64
1.45
2.20

24.20
13.36
10.84

33.98
20.78
13.20

6.07
2.59
3.48

5.51
2.12
3.39

Management

1

11.26

13.44

2.47

2.59

15.62

22.52

4.22

5.81

Table 2: Occupation Groups used for Calibration
well: 1947, 1980, and 2005, corresponding to columns (1)-(3) in Table 1. As shown
there, the values are in a similar range as in Herrendorf et al. (2014), although  is not
significant with 2005 as a base year. For the calibration, we will use take the values of(γ, ) in column (1) values as a benchmark.
The capital income share α is computed as 1-(labor income/total income), and fixed
at 0.360. Since we do not model investment, for the calibration we also need the levelof total capital stock (for manufacturing and services) for each decade, which we takefrom the Fixed Asset Table and is directly plugged into the model.

6.2

Occupations

For the calibration, we broadly categorize all occupations in the census into 11 broad
categories, as summarized in Table 2. The left panel describes their SOC code witha short job description, and the middle and right panels their employment and total
wage shares in 1980 and 2010, respectively. For the employment and wage share panels,
the first two columns shows the size of each occupation in 1980 and 2010 as a fractionof total employment/wages. The next two columns show the size of each occupation
within manufacturing as a fraction of total employment/wages.
The RTI indices of each task is tabulated in Table 4, along with model parameters
calibrated from the data.

6.3

Calibration

Given the aggregate production function, we find the rest of the parameters by fitting
an initial equilibrium to 1980 data, while finding (σ, ω), the growth rate of task-TFP
growth, mj , and the growth rate of exogenous manufacturing-specific sectoral TFP
growth, am , to fit time trends in aggregate employment and wage shares by task andsector. All parameters have been specified except for the skill distribution, which we

29

From
data

Fit to1980

ParameterValue

kt (2)
α

0.361

γ


0.348
Estimated in section 6.1
0.004

BEA/NIPA

Mj ≡ MAm
νij (20)ηi (2)

1.054 Output per worker, normalization
1.015 Manufacturing employment share
Employment shares by task/sector
Table 4
Manager share by sector

χj (8)aγhTable 48.171 Wage shares by task, 1980
0.289

γz
h̄
Fit to2010

Target

1.000
1.000

σ
ω
mj
am

Normalizations;
Not separately identified from Mj

0.261 Output per worker growth,
0.150
income/employment
Table 4
shares by task/sector
0.044
Table 3: Parameters
All parameters valued 1 are normalized.

assume to be a type IV bivariate Pareto (Arnold, 2014), with the c.d.f. given by:i−ah.
µ(h, z) = 1 − 1 + h1/γh + z 1/γz
We normalize γz = 1, since we cannot separately identify both skills from the skillspecific TFP’s. For the same reason, we normalize h̄ and Mj ≡ M0 for all j ∈{0, 1, . . . , z}. All model parameters are summarized in table 3.
The calibrated skill loss parameters χj and employment weights (ηi , νij ) for each
occupation are tabulated in Table 4, along with the task-level productivity growth
rates (mj ) implied by the model. The RTI indices and productivity growth rates arealso visualized in Figure 14.

Calibrating the distribution The model is fitted exactly to 1980 moments. For
any guess of (γh , a, χ), we can find (ĥjt , ẑt ), j ∈ {0, . . . 9} for t = 1980 that exactly
match observed employment shares, by integrating over the guessed skill distribution.
Given the thresholds, we then compute the model-implied income shares using (23),
w1 H1
H1
=
,
w0 H0
ĥ1 L0w2 (H2 − χ2 L2 )H2 − χ2 L2
=
,
w1 H1
H1 (1 − χ2 /ĥ2 )

30

wz Z
Z
=
,
w0 H0
ẑL0

Ranked by mean wage
(except management)

χj

Emp Wgts (νij , ηi )
Manu.
Serv.

mj

RTI

Low Skill Services

-

0.021

0.177

0.000

-0.426

Middle Skill
Administrative Support
Machine Operators
Transportation
Sales
Technicians
Mechanics & Construction
Miners & Precision Workers

0.005
0.012
0.018
0.024
0.031
0.037

0.817
0.090
0.259
0.119
0.026
0.034
0.157
0.131

0.508
0.171
0.015
0.078
0.119
0.038
0.061
0.026

0.015
0.043
0.021
0.002
-0.001
0.014
0.031

2.428
0.609
-0.5860.494-0.074-0.6370.642High Skill
Professionals
Management Support

0.044
0.050

0.162
0.068
0.095

0.317
0.182
0.135

-0.0070.001-0.671-0.657

-

0.058

0.098

0.000

-1.122

Management

Table 4: Calibrated Employment Weights and Growth Rates
and similarly for j ∈ {3, . . . 9}. The LHS is the ratio of total wagebills by task, which
we observe from the data, while the RHS is a function only of the thresholds, whichthemselves are functions of (γh , a, χj ). Hence, we iterate over (γh , a, χj ) so that themodel-implied ratios match observed average wage ratios exactly. Then for the rest of
the calibration, we can fix these three parameters and ensure that the other parameters
generate an equilibrium that yields the implied thresholds.

Calibrated within the Model The rest of the model must be calibrated jointly,
since (σ, ω) are only identified over time trends (as should be apparent from our comparative statics). Hence, they are calibrated together with the growth rates as follows:
1. Guess (σ, ω, mj , am ). For each year 1980, 1990, 2000, and 2010, take kt , capitalper worker, directly from the data.
2. Given the guess, calibrate the parameters in the second panel of table 3 to exactly
fit 1980 moments.
3. Given the 1980 equilibrium, calibrate the parameters in the third panel of table
3 to fit time trends.
When we calibrate the 1980 equilibrium, we take the distributional parameters, inaddition to the implied thresholds, as given. Then given a guess for M0 , we can plugin the observed employment shares from 1980, for each task by sector using (10)-(13)0 s, in addition to A . We thenand (18). From this, we can obtain all the ηi ’s and νijmiterate over M0 so that output per worker in 1980 is exactly equal to 1.
Given values for (σ, ω, mj , am ), all the other parameters are now fixed to 1980.

31

o Data, x Modelo Data, x Model
0.3

0.9

0.6

0.25

0.8

0.98

0.55

0.2

0.7

0.96

0.5

0.15

0.6

0.94

0.65
Routine (Left)
Manual (Right)

0.45
1980

1990

2000

0.5
1980

0.1
2010

(a) Aggregateo Data, x Model

0.15

0.75

0.1

0.7

0.52

0.05

2000

0.92
2010

0.55

Abstract (Right)
Manager (Right)

0.8

1990

2000

o Data, x Model
0.2

0.65
1980

1990

Manager (Left)
Manual (Right)(b) Services Share By Task

0.85
Routine (Left)
Manual (Right)

1
Routine (Left)
Abstract (Left)

Abstract (Right)
Manager (Right)0
2010

Abstract (Right)
Manager (Right)

0.26

0.49

0.22

0.46

0.18

0.43

0.14

0.4
1980

(c) Manufacturing

0.3
Routine (Left)
Manual (Right)

1990

2000

0.1
2010

(d) Services

Fig. 10: Data vs. Model, Employment Shares by Task
We then fit these 14 parameters to fit the trends in output per worker growth andemployment/wage shares by task and by sector, taking the value of capital per worker
for each year, kt , t = 1990, 2000, 2010, as given and fixed from the data.

6.4

Model Fit

Figure 10 plots the model implied trends in employment shares across tasks, in aggregate and by sector. Aggregate task shares (in the bottom) are targeted, so it is
not surprising that we almost exactly fit the trend. Figures 10(a)-(b) were not targeted, and the model still performs well, although we underestimate the rise and fallof managerial and manual tasks in manufacturing, respectively. In services, the model
overstates the increase in managers.
Figure 11 plots the same graphs as in Figure 10 for income shares. Again, aggregate
shares are targeted, so it is not so surprising that Figure 11(a) fits well—although the32o Data, x Modelo Data, x Model

2

0.35

0.65
Routine (Left)
Manual (Right)

0.6

Abstract (Right)
Manager (Right) 0.3

0.55

0.25

0.5

0.2

0.45

0.15

0.4

0.1

0.35
1980

1990

2000

1.8

2.1

1.4

1.9

(a) Aggregate Shares

0.74

1.7
2010

0.35

0.6

Abstract (Right)
Manager (Right) 0.25

0.68

2000

o Data, x Model
0.3

Routine (Left)
Manual (Right)

1990(b) Average Wage Ratios

o Data, x Model
0.8

2.3

1.6

1.2
1980

0.05
2010

2.5
Routine/Manual (Left)Manager/Worker (Left)
Abstract/Manual (Right)

0.55

Routine (Left)
Manual (Right)

Abstract (Right)
Manager (Right)

0.29

0.2

0.62

0.15

0.56

0.1

0.5

0.05

0.44
1980

1990

2000

0
2010

0.5

0.23

0.45

0.17

0.4

0.11

0.35
1980

(c) Manufacturing Shares

1990

2000

0.05
2010

(d) Services SharesFig. 11: Data vs. Model, Incomes by Task
fit is not as good as employment shares. Surprisingly, the bottom two panels whichshow income shares within manufacturing and services, also fit well despite they were
not targeted. On the other hand, the model predicts that the average wage of manual
workers rise relative to abstract workers, contrary to the data.
Overall, the model targeted only to aggregate moments delivers a good fit by taskeven within sectors. Next, we turn to how much of each of these trends can explain
other outcomes.

6.5Counterfactuals

We do two counterfactuals. First, we set am = 0, which gives the model’s predictionsin the absence of any sector-specific TFP growth. Then we set mj = m and recalibrate
m to match output per worker growth, which gives the model’s predictions in theabsence of any task-specific TFP growth. Then we again set am = 0, but recalibrate

33

all the mj ’s to match the change in employment shares; this gives the model the bestchange to explain the data in the absence of any sector-specific TFP growth. The
results are plotted against the benchmark in figure 12. In the top panels, note thatstructural change alone can account for about 15–20 percent of horizontal and verticalpolarization. However, remember that with am , there are no changes in employment
shares within sectors. On the other hand, 12(c) and (e) shows that polarization alone
accounts for about 70 percent of structural change in aggregate, and also among routine
jobs. In management, polarization accounts for about 15% of structural change. In fact,with only sector-specific TFP growth, we overshoot structural change in management,
because the shift in management is much less than in routine jobs in the data.
Lastly in Figure 13, we show the benchmark and counterfactual (log) GDP per
worker, for aggregate and by sector. For both the aggregate and manufacturing, thebenchmark model lines up almost perfectly with the data; it is slightly less than thedata for services; this is also the case with only a sector-specific TFP growth. In
contrast, if there were only task-specific TFP growth, the model lines up better withservices but underestimates manufacturing.
In sum, task-specific TFP growth can account for 60-70 percent of the observed
structural change between 1980 and 2010. Due to the vertical and horizontal polarization induced by routinization, employment shifts to the sector that uses the routine
task less and management more intensively. Conversely, while sector-specific productivities are still required to fully account for structural change, we have shown bothanalytically and quantitatively that contrary to the data, it does not cause any polarization within sectors. However, it can also account for 15-30 percent of polarization
observed in the aggregate data.
How much is actually routinization? The correlation between the task-TFP growth
rates and RTI measure used in Autor and Dorn (2013) is above 0.4, and while not
perfectly in line, it is visually clear that there is a strong relationship between the twoin Figure 14. The rest we suspect comes from off-shoring and endogenously changes inthe distribution of skill, from which we have abstracted from.

6.6

Long-Run Growth Path

Lastly, we show the long-run dynamics of the model, assuming that the economy starts
in 1980. We assume that the CRRA coefficient θ = 2, as is standard in the data, andtarget an asymptotic interest rate of 2%, implying an approximately equal discountrate ρ. The depreciation rate is set to δ = 0.065, as computed from the NIPA accounts.
As can be seen, both routine and manufacturing continue to decline, with the former
at a faster rate. Likewise, managerial employment continues to rise, dominating the

34

0.6

0.15
Benchmark
am=0

0.145

mj=m, Recalibrated
am=0, Recalibrated

0.14

0.55
0.135
0.13
0.125

0.5
Benchmark
am=0

0.12

mj=m, Recalibrated

0.115

am=0, Recalibrated
0.45
1980

1990

2000

0.11
1980

2010

(a) Routine Employment Share

0.7

2000

2010

(b) Manager Employment Share

0.74
0.72

1990

0.88
Benchmark
am=0

0.87

Benchmark
am=0

mj=m, Recalibrated

0.86

mj=m, Recalibrated

am=0, Recalibrated

am=0, Recalibrated

0.85

0.68

0.84
0.66
0.83
0.64
0.82
0.62

0.81

0.6

0.8

0.58

0.79

1980

1990

2000

0.78
1980

2010

(c) Routine Services Share

1990

2000

2010

(d) Manager Services Share

0.84
0.82
0.8

Benchmarka =0m

mj=m, Recalibrated
am=0, Recalibrated

0.78
0.76
0.74
0.72
0.7
0.68
0.66
1980

1990

2000

2010

(e) Service Employment ShareFig. 12: Benchmark vs. Counterfactuals, Polarization and Structural Change

35

1
0.9
0.8

0.35

Data
Benchmark
a =0

Data
Benchmark
a =0

0.3

m

m

m =m, Recalibrated

mj=m, Recalibrated

j

0.7

0.25

am=0, Recalibrated

a =0, Recalibrated
m

0.6

0.2

0.5
0.4

0.15

0.3

0.1

0.2
0.05

0.1
0
1980

1990

2000

0
1980

2010

(a) Log GDP per Worker, Manufacturing

1990

2000

(b) Log GDP per Worker, Services

0.45
0.4

Data
Benchmark
a =0

0.35

mj=m, Recalibrated

0.3

am=0, Recalibrated

m

0.25
0.2
0.15
0.1
0.05
0
1980

1990

2000

2010

(c) Log GDP/Worker, Aggregate

Fig. 13: Benchmark vs. Counterfactuals, Output

36

2010

5

3
Growth Rate (Left)
Routinization Index (Right)

4

2

3

1

2

0

1

−1

0

−2

Correlation:
0.418
−1

400

300

700

800

240

200

500

600

40

20

Fig. 14: Task TFP growth and Routinization

37

1

−3

0.6

1.2
Routine Employment Share (Left)Management Employment Share (Left)
Manufacturing Employment Share (Left)
Log GDP/Worker (Right)

0.4

0.8

0.2

0.4

0
1980

2000

2020

2040

2060

0
2080

Fig. 15: Long-run Dynamics
economy. Finally, note that the first 150 years displays structural change and nearbalanced growth, consistently with the Kuznets and Kaldor facts.

7

Conclusion

We presented a new model which encompasses job polarization, structural change, anda modified span of control technology. We showed analytically and the quantitatively
that the model can be a useful tool for analyzing macroeconomic dynamics.

References
Acemoglu, D. and D. Autor (2011). Chapter 12 - skills, tasks and technologies: Implications for employment and earnings. Volume 4, Part B of Handbook of Labor
Economics, pp. 1043 – 1171. Elsevier.
Acemoglu, D. and V. Guerrieri (2008, 06). Capital Deepening and Nonbalanced Economic Growth. Journal of Political Economy 116 (3), 467–498.

38

Arnold, B. C. (2014). Univariate and multivariate pareto models. Journal of Statistical
Distributions and Applications 1 (11).
Autor, D. H. and D. Dorn (2013). The growth of low-skill service jobs and the polarization of the us labor market. American Economic Review 103 (5), 1553–97.
Autor, D. H., L. F. Katz, and M. S. Kearney (2006). The polarization of the u.s. labormarket. American Economic Review 96 (2), 189–194.
Bárány, Z. L. and C. Siegel (2015). Job polarization and structural change.
Buera, F. J. and J. P. Kaboski (2012, October). The Rise of the Service Economy.
American Economic Review 102 (6), 2540–69.
Buera, F. J., J. P. Kaboski, and R. Rogerson (2015, May). Skill Biased Structural
Change. NBER Working Papers 21165, National Bureau of Economic Research,
Inc.
Burstein, A., E. Morales, and J. Vogel (2015, January). Accounting for Changes
in Between-Group Inequality. NBER Working Papers 20855, National Bureau of
Economic Research, Inc.
Costinot, A. and J. Vogel (2010, 08). Matching and Inequality in the World Economy.
Journal of Political Economy 118 (4), 747–786.
Goos, M., A. Manning, and A. Salomons (2014). Explaining job polarization: Routinebiased technological change and offshoring. American Economic Review 104 (8),2509–26.
Herrendorf, B., R. Rogerson, and A. Valentinyi (2014). Growth and Structural Transformation. In Handbook of Economic Growth, Volume 2 of Handbook of Economic
Growth, Chapter 6, pp. 855–941. Elsevier.
Lee, E. (2015). Trade, inequality, and the endogenous sorting of heterogeneous workers.
Lucas, R. E. (1978, Autumn). On the size distribution of business firms. Bell Journalof Economics 9 (2), 508–523.
Ngai, L. R. and C. A. Pissarides (2007, March). Structural Change in a Multisector
Model of Growth. American Economic Review 97 (1), 429–443.

39