technology adoption, skills, and heterogeneous firms · 2012. 8. 19. · technology adoption,...
TRANSCRIPT
Technology Adoption, Skills, and Heterogeneous Firms
Werner Barthel Christian Bauer
LMU Munich∗
June 5, 2010
Preliminary
Abstract
Firms in laggard countries frequently improve productivity by adopting existing technologies, and must
bear significant labor costs increases to implement advanced technologies. We develop a new equilibrium
model to analyze the optimal production structure of firms, account for both firm characteristics (such
as technology intensity) and country characteristics (such as the availability of technology-complementary
inputs like skilled labor). We allow firms to be heterogeneous in the scope for technology in production. In
equilibrium, more technology intensive firms choose better technologies, employ a larger fraction of high-
skilled labor, and consequently are more productive. We show that technology differences are stable across
firms while productivity differences increase if more productive technologies become available. In this model,
firm heterogeneity explains larger wage gaps at higher levels of technology: with heterogeneous firms, but
not with homogeneous firms, access to better technologies causes a reallocation of labor that increases the
wage gap.
1 Introduction
Firms in laggard countries frequently operate below the world technology frontier because they have limited
access to the most advanced technologies1 and because they do not find it profitable to engage skilled workers at
high wages who are usually needed to accommodate new technologies.2 Thus, rather than conducting original
∗We thank, without implicating, Theo Eicher, Jorg Lingens, and Lugan Bedel for valuable comments and discussions. Com-
ments are welcome: [email protected], [email protected]. We gratefully acknowledge financial
support from the German Research Foundation (DfG) through GRK 801.
1Limited access to advanced technologies, as in Parente and Prescott (1994), emerge in Parente and Prescott (2000) as vested
interests, in Parente and Prescott (2004) as “society-imposed constraints on the technology choices” and in Coe et al. (1997) or
Harding and Rattso (2005) as trade barriers. See Comin and Hobijn (2004) and Caselli and Coleman (2006) for recent contributions.
2Cf. Bresnahan et al. (2002)
1
R&D, productivity gains are achieved by adopting technologies from world technology leaders. Naturally, firms
are more prone to adopt new technologies if, given their accessibility, skilled labor is less scarce and available
at lower costs.
In this paper, we study the optimal technology choice and organization of production by profit-maximizing
entrepreneurs, accounting for both firm and country characteristics, in general equilibrium. We develop a new
model based on increasing returns to specialization at the firm-level to analyze how differences in productivity
emerge endogenously when firms with heterogeneous technologies respond optimally to an enhanced access to
more advanced technologies. A new result is that firms who choose to be more productive initially benefit more
than less productive firms from the accessibility of new technologies.3 That is, in contrast to the bulk of existing
models on technology adoption, where the identity of who adopts, or what constitutes the corresponding costs,
is often treated as exogenous,4 we explain technology and workforce composition disparities by differences in
firms’ scope for technology in production as well as relative costs of engaging skilled or unskilled workers. The
model predicts more pronounced technology differences across firms in economies where, whenever firms choose
to operate below the technology frontier, skilled labor is relatively abundant. If, on the other hand, technology
barriers bind, technology differences are more pronounced if skilled labor is scarce. An implication of our analysis
is that differences in firms’ scope for technologies are an important determinant of the difference in high- and
low-skilled wages.
We strive to assess the potential productivity gains and impact of newly accessible technologies in an en-
vironment where firms optimize over technology, taking into account that technologies differ across firms and
that the costs of implementing new technologies endogenously depend on the environment. That is, we want to
account for the endogenous change in the wage differences, especially as low- and high-skilled labor supply is
often limited within a country at a given time, when new technologies become accessible. Specifically, we aim at
understanding which firms have the most pronounced incentive to hire skilled workers that are necessary to cope
with new technologies and how the ensuing reallocation of resources across firms interacts with relative wages
in the determination of differences in productivity levels. Our goal is to coherently address these interrelated
questions.
In parts, the exercise is explorative.
On the one hand, in a concave world without externalities, relatively backward firms should want to improve
their technology by more than more productive firms. On the other hand, there is (albeit relatively scarce)
evidence for technology adopting countries that suggests that operating more advanced technologies facilitates
the implementation of better technologies. This is precisely because using better technologies presumes the
availability of workers with the ability to cope with more advanced technological knowledge. In line with this
3Our finding of more pronounced adoption of frontier-technologies by technological leading firms is reminiscent of Comin and
Hobijn (2004)’ s findings. Their empirical country-level analysis reveals a positive effect of the preceding technology’ level on the
adoption of the current technology.
4See, e.g. Eeckhout and Jovanovic (2002) and Barro and Sala-i Martin (1997).
2
reasoning, Nakamura and Ohashi (2008) show for example that more productive plants in the Japanese steel
industry were more likely to adopt new technologies than technological laggards as barriers to technology adop-
tion fell in the 1950’s and 60’s.5 Moreover, relatively more productive firms may also dispose of an inherent
structure that is conducive to adoption of advanced technologies. In order to account for heterogeneous re-
sponses to the accessibility of new technologies, we introduce thus the notion of firm heterogeneity in the scope
for technology. This allows us to study the effects of inter-firm labor reallocations induced by heterogeneous
technology adoption decisions on the relative wage of high and low skilled labor.
In general, the effects of new technologies on the optimal organization of production are profound. As new
technologies become accessible, the differences in the skill intensity among existing production tasks naturally
shrink: previously very skill-intensive tasks decrease in skill-intensity by more than tasks that can be handled
without much skills to begin with; thus making the reorganization of production towards more specialized tasks
profitable. However, this optimal organization differs not across firms. More technology intensive firms rather
additionally dispose of divisions needed for their technological advantage.
To characterize the optimal technological response of heterogeneous firms to changes in barriers to technology
adoption, we construct a model where heterogeneous firms optimally determine the level of technology and the
organization of production. We adopt from Acemoglu et al. (2007) the notion that firms endogenously choose
the optimal production structure by choosing the division of labor within the firm. Crucially, we allow firms
to simultaneously choose the optimal structure of intermediate tasks in production, taking into account that
different tasks differ in the productivity of skills. In line with available evidence (see, e.g., Bresnahan et al.
(2002)), we assume that skills and technology are complements: more advanced production techniques also
require tasks in which high-skilled workers are relatively more productive than low-skilled workers. Hence, the
aggregate demand for skills and its supply impacts on, and interacts with, the technology choice of firms.
In the model, barriers to technology adoption limit firms’ decisions to an exogenously given set of technologies
that excludes the most advanced technologies. We introduce firm heterogeneity in productivity by assigning each
firm a technology intensity index that captures the extent to which it may benefit from the division of labor
given the relative productivity of skills in each intermediate production step. In equilibrium, more technology
intensive firms find it optimal to operate more advanced technologies, which involves a gap in technology that
increases in the difference of technology intensities. As a result, more technology intensive firms engage a more
skilled workforce and exhibit higher productivity levels. Similarly, when barriers to technology adoption are
reduced, more technology intensive firms choose to increase technology by more than less technology intensive
firms. The ensuing reallocation of labor impacts on the relative demand of high and low skilled labor, and
5Two related empirical studies have major caveats in this respect: Bustos (2010) reports that firms with an intermediate
productivity level are those who adopt new technologies, but as the measure of technology is binary, high productive firms already
adopted the new technology. Schor (2004) reports no differences in productivity effects of barriers to technology adoption but uses
aggregated firm level data.
3
dampens the incentives to adopt new technologies (as the implementation of skill-complementary technology
becomes more expensive).
We specify our model of technology adoption and the organization of production to match two main
empirical regularities with equilibrium outcomes. First, lower barriers to technology adoption imply a greater
aggregate demand for high skilled labor and a higher wage gap; second, firms with a higher intensity of
technology in production demand relatively more high-skilled workers. To generate these outcomes we
use structural ingredients that are standard. However, we allow firms to adjust the organization of single
intermediate tasks in production, and thus the entire production process. It is thus the combination of the
accessibility of more advanced technologies and the complementary organizational change implemented by
firms that determines firm productivity. Most notably, this combination implies that following a reduction in
technology barriers, more technology intensive firms become relatively more productive than less technology
intensive firms and hence increase output by more than less technology intensive firms.6 Consequently, their
share in labor demand increases, driving up the aggregate demand for high-skilled labor and thus the wage gap.7
Our paper contributes to the literature on technology adoption by bringing together the idea of international
knowledge spillovers and technology-complementing skills at the firm-level. We follow the lead taken by Parente
and Prescott (2000), as well as Klenow and Rodrıguez-Clare (2005), who analyze how barriers to international
knowledge spillovers shape a country’s production structure. Klenow and Rodrıguez-Clare (2005) argue that
geographically hindered knowledge spillovers may explain a significant fraction of long-term differences in
TFP across countries. On the firm level, Parente and Prescott (2000) introduce adoption barriers by limiting
the ability to improve a firm’s quality through intentional investment. As the respective returns decrease in
plant-level quality, firms with higher quality levels (and hence higher productivity) have relatively less incentives
to implement new technologies. We further explore firms’ incentives and model the optimal organization of
production. In contrast to previous work, and guided by empirical evidence, we outline non-concave technology
adoption and exploit firm-level heterogeneity to create asymmetric feedback effects. Furthermore, following
Acemoglu and Zilibotti (2001) and Caselli and Coleman (2006), we account for the necessity of appropriate skills
in the technology adoption process. In their contribution, Acemoglu and Zilibotti (2001) argue that Northern
technologies are most efficiently used within a Northern-like skill structure. A mismatch of technologies and
skills can thus lead to sizable productivity differences even if there are no restraints to technology adoption.
This approach is modified by Caselli and Coleman (2006) to account for the skill-biased technology differences
uncovered in the data. In Caselli and Coleman (2006)’s theoretical model, barriers to technology adoption
6This result is roughly in line with the sparse available evidence. Konings and Vandenbussche (2008), studying the effects of
import restricting anti-dumping politics on European firms, show that initially more productive firms are hurt by those politics while
less productive ones benefit. Ozler and Yilmaz (2009) show for Turkish plant-level data that, while all firms increase productivity
from removing trade barriers, firms with the highest output levels profit the most.
7In our model, this does not occur if firms are homogeneous.
4
shape production structures at the country level. We add to this by highlighting the impact of barriers to
technology adoption on the organization of production at the firm level, taking differences in the scope for
technology and the costs of technology-complementary inputs (skilled labor) into account.
The remainder of the paper proceeds as follows. Section 2 introduces the model. Section 3 characterizes
optimal firm behavior and establishes comparative static results. Section 4.3 studies how homogeneous firms’
decisions in a general equilibrium environment interact. Section 4.5 introduces firm-level heterogeneity and
analyzes feedback effects of technologies and wages. Section 5 concludes.
2 The Model
We begin by introducing a firms’ production with respect to technology and organization and subsequently
define its maximization problem.
2.1 A Firm’s Production Structure: Technology and Organization
Each firm i produces output Yi according to the generalized C.E.S. production function
Yi := Nκi+1i
[1Ni
∫ Ni
0
xαi,jdj
] 1α
. (1)
A firm can choose the degree of technological specialization (i.e., the “number” of different production tasks),
Ni as well as the input quantity of each task, xi,j . We refer to a firm’s optimal choice of Ni, which is limited
to the set of available technologies Ni ∈ (0, T ], as its level of technology in production. As the ”number effect”
of tasks is captured by specialization, tasks contribute as an average, 1Ni
∫ Ni0
xαi,jdj, to production. Its optimal
composition is ensured through the organization of the production process.
An important determinant of N is a firms’ technology type κi > 0,8 which we coin as intensity of technology
in production. κ captures the extent to which a firm may benefit from the division of labor, and differs across
firms. In particular, a fraction γ of firms is of type κh and a fraction 1 − γ is of type κl, where κh > κl. h
characterizes more and l less technology intensive firms. If possible, we abstract from the firm index i to save
on notation.
As in standard models with C.E.S. production, α ∈ (0, 1) determines the elasticity of substitution between
different tasks in production, 11−α > 1. The elasticity of output with respect to N depends on the production
function of tasks. If tasks were produced homogeneously, this elasticity is κ + 1. In our model, tasks are
8The separation of gains from specialization/technology and concavity was introduced by Ethier (1982). κi = 0 would leave the
decision whether to increase N or x undetermined in the homogeneous task case.
5
produced with heterogeneous technologies, reflecting different productivities of skills in different tasks. In this
case, the elasticity becomes καα−1 + 1. For reasons detailed below, we impose κ < 1−α
α9 (which implies that
output reacts less to a change in the level of technology when tasks are produced heterogeneously10). With
heterogeneous tasks, gains from technology and the elasticity of substitution between different tasks cannot be
separated additively as in the case with homogeneous tasks. However, they are still governed by two distinct
parameters, κ and α.
Each task j is produced within the firm. Whereas technology governs the array of different tasks, the
simultaneous organization of the production process determines the relative quantities of different tasks. The
production of task j follows a generalized Cobb-Douglas production function
xj(Lj , Hj) := zjL1−ζjj H
ζjj (2)
where zj = ζ−ζjj (1 − ζj)−(1−ζj). Hj denotes the input quantity of high-skilled and Lj of low-skilled labor.
Total employment in each firm is given by L ≡∫ N
0Ljdj and H ≡
∫ N0Hjdj. We denote by
kj = wζjHw
1−ζjL (3)
the minimum unit cost of producing one unit of j, derived from (2) in Appendix 6.4. Defining w ≡ wHwL
as
the wage gap between high- and low-skilled labor leads to kj = wLwζj .
The above formulation of the Cobb-Douglas production function is adopted from Antras (2005). The
modification zj corrects for the fact that standard Cobb-Douglas functions imply a change in productivity
(measured by the inverse of the minimum unit costs) as ζj varies even if both inputs were equally expensive
(which will not be the case in equilibrium). We adopt (2) to ensure that minimum unit costs are the same for
different tasks if both inputs were to be paid identical prices.
We impose a plausible relation between the productivity of skills and the division of labor. In particular, we
impose that Lj (Hj) is relatively less (more) productive in producing task j + 1 as it is in producing task j. As
a consequence, a production process with a higher level of technology requires the use of increasingly high-skill
intensive tasks. Put differently, the absolute technical rate of substitution |TRS(Hj , Lj)| = | dLjdHj| = ∂xj
∂Hj/∂xj∂Lj
is
increasing in j:
9The elasticity is evaluated at the optimum. If κ increases the elasticity has to be dampened at the profit maximizing level of
technology. The restriction than ensures a positive marginal productivity of N ( ∂Y∂N
=(
1 + καα−1
)YN
).
10Homogeneous production of tasks and an endogenous choice of technology would however require to impose additional costs to
technology adoption. See Appendix 6.1 for further details.
6
∂|TRS(Hj , Lj)|∂j
=∂(∂xj∂Hj
/∂xj∂Lj
)∂j
=LjHj
∂ζj∂j
(1− ζj)2> 0 ⇐⇒ ∂ζj
∂j> 0.
A simple formulation for ζj that satisfies the above restriction and ensures that ζj ∈ (0, 1) is
ζj ≡j
T. (4)
For the highest accessible technologies, i.e., for N → T , ζN → 1. N > 0 ensures the use of some technology
in production which necessitates the employment of a minimum amount of high-skilled labor. T is the most
sophisticated accessible technology and constitutes the upper bound of available technologies within a country.
It is important to note that this setup applies for countries without major domestic innovations. Our model
hence applies to laggard countries where firms gain by adopting technologies from world technology leaders.
Importantly, however, not all of the new and most efficient production techniques are available within all
countries. Quite regularly, political and technical barriers to technology adoption prevent the use of the most
advanced technologies11. Major changes as trade liberalization, sanction relief, or improved international
mobility and communication lead to a reduction in barriers and, in our model, to an increase in T . In this
model, we focus on the consequences, not the causes, of barriers to technology adoption.
Note that accessibility to more advanced technologies naturally and simultaneously re-shapes the organizational
structure of the production process: previously advanced technologies and high-skill intensive tasks become
technical ”laggards” and relatively cheaper in production (for low skilled wages fall short of high skill wages):
the relative costs of producing one unit of task j + 1 relative to the production costs of task j (kj+1kj
= w1T )
decrease in T . For a more detailed analysis of the complementary nature of technology, skills, and organization
we refer to Chapter 3.
2.2 The Firm’s Problem
Each firm maximizes
Π(N, {Lj}N0 , {Hj}N0
)= A1−βY β −
∫ N
0
[wHHj + wLLj ] dj,
subject to N ∈ (0, T ]12. A1−βY β = pY is firm’s revenue, derived in section 4.1 (Households) from CES
preferences for horizontally differentiated products, A is an endogenously determined measure of market
size, and β determines the elasticity of demand (1/(1 − β). We neglect a specific technology cost function
because our specification for tasks imposes an organizational structure such that differences in factor prices
naturally lead to a bounded N . This holds if the technology intensity is not too large what is ensured by κ < 1−αα .
11Cf. Parente and Prescott (1994), Parente and Prescott (2000), and Klenow and Rodrıguez-Clare (2005)
12This constraint is neglected in the optimization process. Lemma 2 states when the constraint binds.
7
Profit maximization requires the firm to choose the optimal quantity of Hj and Lj in each task. This allows
us to write the labor costs for each task as
wHHj + wLLj = kjxj ,
where kj is the minimum unit cost of producing one unit of j. Using the minimum unit cost kj , the firm’s
problem becomes
maxN,{xj}N0
Π(N, {xj}N0
)= maxN,{xj}N0
{A1−βY (N, {xj}N0 )β −
∫ N
0
kjxjdj
}. (5)
3 Optimal Production Structure and Technology Choice
In this section, we analyze the optimal organizational structure and the level of technology in a firm’s
production process given wages (wH , wL) and market size (A). We impose eκiT > w > eκi for i ∈ {l, h} to
ensure positive high- and low-skilled labor demands.
The first order maximization conditions derived from (5) determine the optimal choice of N as well as the
optimal input quantity for each task j ∈ [0, N ] :
κ =1− αα− 1− α
α
kαα−1N
1N
∫ N0k
αα−1j dj︸ ︷︷ ︸
=(N∂KN∂N
)/KN
, (6)
xj = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N k− 1
1−αj . (7)
where KN ≡[
1N
∫ N0k
αα−1j dj
]α−1α
is an index of average unit costs (corresponding to the tasks’ average in
the production function (1)) that increases in N . The derivation of these conditions is somewhat tedious and
therefore delegated to Appendix 6.2. The optimal level of technology in production (6) is given by a trade-off
between the degree of gains from technology, κ, and the elasticity of average unit costs needed to implement the
optimal level of technology,(N ∂KN
∂N
)/KN . If production profits more from higher levels of technology, higher
cost elasticities can be supported.
The optimal organization of production, characterized by the relative use of tasks
xj+1
xj=(kj+1
kj
)− 11−α
, (8)
depends on respective unit costs as well as on the elasticity of substitution between tasks. In contrast to the
level of technology, optimal organization is not affected by the intensity of technology in production. Precluding
the most skill intensive task, xN , the level of technology has no effect on organization. Furthermore, both, the
8
level of technology in production as well as the organization of the production process are independent of the
size of the market (A) and of the degree of “market competition” as measured by the elasticity of substitution
between different products (β).
Given the optimal output ((50) in Appendix 6.2) and the demand of a representative household, Y = Ap1
β−1 ,
we show that the firm’s price
p =KN
Nκβ(9)
depend directly and indirectly (through KN ) on N . We can express production costs C(Y )13, evaluated at
equilibrium, in terms of output by using (18) and (19) and rearranging:
C(Y ) = wLL+ wHH =Y KN
Nκ. (10)
Remark that, as in a standard monopolistic framework, the price is determined as marginal costs over β.
Productivity is the most important efficiency measure of producing output from inputs. Here, the use of two
inputs, L and H, necessitates the consideration of relative prices. More precisely, we define productivity as the
inverse of the costs to produce one unit of output:
φ ≡ 1C(Y )Y
= NκK−1N = Nκk−1
N
(κα
α− 1+ 1)α−1
α
(11)
where the last equation uses the optimal value of N (6).
In order to get a fully-fledged view of how barriers to technology adoption effectively interact with the
(endogenous) wage gap in affecting firm’s choice of technology in production, we use our specific production
function of tasks where kj = wLwjT . With this, the index of average unit cost can be rewritten as
KN = wL
(1N
∫ N
0
wαα−1
jT dj
)α−1α
= wL
(w
αα−1
NT − 1
NT
αα−1 ln w
)α−1α
. (12)
Using kN = wLwNT together with (12) in (6), we obtain the optimal choice of N as a function of the wage
gap:
κ =1− αα−
NT ln w
wα
1−αNT − 1
(13)
In fact, (13) actually determines N/T , i.e technology in production relative to the upper bound of available
technologies. Optimal organization, characterized as the relative use of tasks, consequently implies that
13Remark the parallelism of (10) to a standard Dixit-Stiglitz economy with definitions as in Section 4.1 where PIu = E at the
optimum. Here, KNN[
1N
∫N0 xαj dj
] 1α
= C(Y ). That is, the benefits of average unit costs times average tasks times their number
equals costs. Other things equal, with KNN1−1/α decreasing in N , a higher N implies more production keeping costs constant.
9
xj+1
xj= w−
1T (1−α) . (14)
The internal composition of the average task is determined through the wage gap, the amount of accessible
technologies, and the elasticity of substitution between tasks. Clearly, as mentioned above, the level of technology
does not interfere with organization. Moreover, productivity in (11) becomes
φ = Nκw−1L w−
NT
(κα
α− 1+ 1)α−1
α
(15)
and displays, in contrast to technology in production, level effects as it decreases in the low-skilled wage.
Labor demands for each task, factor demands in each firm as well as relative high- to low-skilled labor demands
are derived in the Appendix 6.3. Labor demands for each task j read
Hj = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N w1
α−1L w
jT
αα−1−1 j
T(16)
Lj = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N w1
α−1L w
jT
αα−1
(1− j
T
)(17)
while factor demands in each firm are given by
H = β1
1−βANβκ
1−β K− β−α
(1−α)(1−β)
N w1
α−1L w
NT
αα−1−1 κ
ln w(καα−1 + 1
) (18)
L = β1
1−βANβκ
1−β K− β−α
(1−α)(1−β)
N w1
α−1L w
NT
αα−1
ln w − κ
ln w(καα−1 + 1
) . (19)
Relative labor demands within a firm can be summarized as
H
L=
1w
κ
ln w − κ(20)
and are characterized by the wage gap (w) and the intensity of technology (κ) in production. While the
endogenous level of technology has no direct effect on relative labor demands, w and κ jointly determine
technology and relative factor demands. Consequently, the complementary nature of technology and skills has
to be understood as a positive correlation through common determinants.
In the following, we establish comparative static results of firm-level responses to changes in barriers to
technology adoption, wage gaps, and basic production function parameters.14
14Note that in this section we do comparative statics in partial equilibrium, i.e. there will be no feedback effects through wages.
10
3.1 Optimal Production Structure
Lemma 1 The availability of more advanced technologies, i.e. a higher T , reduces the high-skill intensity,HjLj
= w−1 jT−j , of all previously available tasks. Ex-ante more high-skill intensive tasks are subject to stronger
reductions in high-skill intensity than ex-ante less high-skill intensive ones.
The proof is given in Appendix 6.5.1. The availability of more advanced technologies, i.e. a decrease in barriers
to technology adoption, affects all previously employed tasks: they become technical ”laggards” and consequently
their production is transformed into a less high-skill intensive one. However, the technical downgrade of previous
high-tech tasks is relatively stronger than that of more ordinary tasks. As a consequence, high-skill intensity of
task j + 1 relative to task j is decreased.
Proposition 1 Organization of the production process xj+1xj
= w−1
T (1−α) :
• The availability of more advanced technologies shifts the production process towards the employment of
higher-j tasks as do lower wage gaps and lower elasticities of substitution between tasks.
• Consider two firms (h, l) with Nh > Nl. Than, the organization of h equals that of l for all tasks j ∈ [0, Nl].
The proof is given in Appendix 6.5.2. A firm optimally adjusts its organizational structure to changes in
the wage gap and its production possibilities. Adjusting its organization means essentially to shift relative
quantities of tasks in production to minimize total production costs. Increasing high-skill intensity imposes that
production of task xj+1 requires more skilled labor than production of xj . Consequently, the relative employment
is optimally reduced when relative high-skill renumeration increases. When the range of available technologies
grows, production costs of each intermediate task decrease as their high-skill intensity declines (cf. Lemma 1).
However, as the high-skill intensity of task j + 1 relative to task j decreases relatively more (cf. Lemma 1),
higher-j tasks become relatively less costly. Consequently, their relative employment is increased. A higher α
implies a greater elasticity of substitution between tasks. The complemented organizational structure involves
an allocation of tasks’ quantities from more ”expensive” higher-j tasks to ”cheaper” lower-j tasks.
Optimal organization depends on parameters and variables that contemporaneously affect the choice of
technology in production. However, the level of technology has no direct effect on the optimal composition
of tasks15. Note that adopting Nh instead of Nl requires the additional implementation of more high-skill
intensive tasks xNl+1 . . . xNh , but leaves the structure of tasks x0 . . . xNl unchanged.
3.2 Optimal Technology Choice
We begin by abstracting away from the technology constraint T and consider the unrestricted maximization
problem, verifying the validity of our approach later.
15Note that whereas κ increases the level of technology, it has no effect on the organizational structure.
11
Lemma 2 There exists a unique N∗(w).
The proof is given in Appendix 6.5.3. Given technological intensity, the elasticity of substitution between
tasks, and the availability of technologies, the wage gap uniquely determines the level of technology in production.
Proposition 2 The availability of more advanced technologies induces firms to increase their choice of tech-
nology in production: dNdT = N
T > 0. A larger wage gap decreases their choice of technology in production:dNdw = − N
ln w w < 0.
The proof is given in Appendix 6.5.4. An enhanced access to the most advanced technologies enables firms
to increase their level of technology in production. Intuitively, a higher T transforms the previously chosen level
of technology in production relatively less advanced and its organizational structure less high-skill intensive.
The adoption of a higher N becomes profitable. Furthermore, a higher initial level of technology embodies an
inherent structure (a higher κ) that favors a greater increase in N . Note that the fraction of implemented to
available technologies, N/T , remains unaffected.
A firm’s choice of technology decreases in the wage gap as the employment of higher-j tasks which are
more high-skill intensive becomes more costly. In our model, endogenous technology choice allows firms to react
more decisively to a wage gap increase than without the capability of a technological downgrade. The impact
of changes in the wage gap on N can be characterized further by ∂N∗
∂wL> 0 or ∂N∗
∂wH< 0. Consequently, lowering
the high-skilled wage entails a decrease in the wage gap, high-skilled labor becomes relatively cheaper and the
adoption of more high-skilled intensive tasks less expensive. However, lowering the low-skilled wage increases
the wage gap, high-skilled labor becomes relatively more expensive and technology in production is decreased
favoring the provision of less high-skill intensive tasks. Note that production structures embodying higher N ’s
have to be re-organized and to decrease N more decisively to a wage gap decrease.
The following Lemma characterizes the conditions under which the firm’s technology choice is not restricted
by barriers to technology adoption.
Lemma 3 For all upper bounds of all available technologies there exists a wage gap such that the constraint
N ≤ T does not bind and barriers to technology adoption do not effectively restrain the level of technology in
production. Furthermore, the optimal level of technology (N) is bigger than one.
The proof is given in Appendix 6.5.5. A small skill premium combined with a high technological intensity
would induce firms to choose a higher level of technology than available. Barriers to technology adoption would
restrain the level of technology to N = T . Sufficiently high wage gaps consequently induce firms to choose levels
of technology below the upper bound. If not otherwise stated we assume in the following that the constraint
N ≤ T does not bind.
The second property stems from the assumption eκT > w and ensures that a more technology intensive
production structure implies a higher level of technology in production combined with a higher productivity.
12
Proposition 3 A more technology intensive production structure, i.e. a higher κ, involves a higher level of
technology in production. More technology intensive firms increase their level of technology in production by
more when more advanced technologies become available. The level of technology in production is higher for a
higher elasticity of substitution between tasks (α) if and only if a threshold holds16.
The proof is given in Appendix 6.5.6. κ is the degree of gains from technology where a higher degree
characterizes a more technology intensive production process. Naturally, a higher value of κ leads to a greater
chosen level of technology in production. Firms with κh consequently choose a higher level of technology in
production than firms with κl.
An increase in α implies a higher elasticity of substitution between tasks. The complemented organizational
change involves reallocations of tasks’ quantities from more ”expensive” higher-j tasks to ”cheaper” lower-j
tasks. Thus, a higher level of technology in production entails lower costs of implementing more advanced
technologies (if a threshold holds).
3.3 Implications of Optimal Behavior
The optimal choice of technology in production and the organization of tasks has far-reaching effects on relative
labor demands and productivity.
Proposition 4 Relative labor demands (HL ) are higher for firms with a more technology intensive production
structure. Furthermore, higher wage gaps imply lower relative factor demands.
The proof is given in Appendix 6.5.7. A more technology intensive production structure implies the choice of
a higher level of technology in production. Its implementation requires, in turn, more high-skilled intensive tasks.
However, the relative use of task, i.e. the organizational structure, does not depend on technology intensity. As
a consequence, relative demand of high-skilled labor is higher for more technology intensive firms.
A higher wage gap dampens the level of technology in production and, simultaneously, provokes a re-
organization of the production structure. In other words, not only the number of relatively high-skill intensive
tasks is reduced, but also their relative employment. Consequently, relative demand of high-skilled labor de-
creases.
Proposition 5 Productivity (φ) is higher for firms with a more technology intensive production structure: ∂φ∂κ =
lnNφ > 0. The availability of more advanced technologies has a positive effect on productivity: ∂φ∂T = φ κT > 0.
Larger wage gaps decrease productivity: ∂φ∂w = − φ
ln ww . .
The proof is given in Appendix 6.5.8. As more technology intensive firms use a higher level of technology
in the production process they profit more from technological externalities (Nκ), and thus exhibit a higher
16More precisely, if and only if NT
ln w > κ
(1 + 1−α
α1
καα−1 +1
).
13
productivity. Furthermore, the availability of more advanced technologies increases the level of technology in
production. Complemented by an appropriate re-organization of the production structure, firms’ productivities
benefit from lower barriers to technology adoption. A larger wage gap leads to a lower level of technology in
production that, complemented by a re-organization of the production structure, decreases productivity.
3.4 The Effect of a Lead in Productivity
In our environment, a lead in technology will be increased once more advanced technologies become accessible.
Proposition 6 The availability of more advanced technologies induces more technology intensive firms to in-
crease their productivity by more: ∂2φ∂κ∂T = φ
T (1 + κ lnN) w.
The proof is given in Appendix 6.5.9. More productive firms dispose of a more technology intensive
production process and a higher level of technology in production. They consequently operate with a relatively
more high-skilled workforce and an inherent structure allowing more high-skilled tasks in production. This gives
them an advantage of adopting advanced technologies (cf. over less productive firms) and to rise productivity by
more. Intuitively, a more skilled workforce operating at a high technology level facilitates the implementation
of advanced technologies.
4 General Equilibrium
We embed a firm’s choices of the level of technology and the organizational structure of production in a general
equilibrium (GE) framework. We start by setting up a Dixit-Stiglitz economy with a representative household
that has a taste for variety and supplies low- and high-skilled labor inelastically. Products are differentiated and
supplied by a mass Q of firms.
4.1 Household Preferences
There exists a continuum of final goods Yi, with i ∈ [0, Q], where Q represents the number (measure) of final
goods. All consumers have identical preferences,
u =
(∫ Q
0
Y βi di
) 1β
, 0 < β < 1, (21)
where 11−β > 1 is the elasticity of substitution between final goods. These preferences imply the demand
function
Yi =(piPI
)− 11−β E
PI
14
where pi is the price of good i, E is the aggregate spending level, and
PI ≡
(∫ Q
0
p− β
1−βi di
)− 1−ββ
(22)
is the price index of final goods and we define PI as the numeraire (PI ≡ 1). Ap− 1
1−βi , the implied demand
function for each firm, is identical to the demand function used in the previous sections (with A = E).
4.2 Characterization of GE with Symmetric Firms
In this section we present a benchmark economy with symmetric firms. Naturally, symmetry among firms implies
that all firms have the same intensity of technology in production (∀i, κi = κ). Let Q denote the number of
firms i. Technologies are given by
Yi = Nκ+1i
[1Ni
∫ Ni
0
xαi,jdj
] 1α
xi,j = zjL1− j
Ti,j H
jTi,j .
Maximizing behavior of a firm is organized in two steps, as described above. The optimal relative labor
demand is given by
Hi,j = Li,jwLwH
jT
1− jT
.
and the f.o.c’s for the j’s and N imply
βA1−βY β−1i
∂Yi∂Ni
= kNixNi ,
βA1−βY β−1i
∂Yi∂xi,j
= kj
kj = wLwjT is the minimal unit cost to produce one unit of xj . Firms enter until the expected profit becomes
equal to a fixed entry cost, PIf ,
Π = 0 ⇐⇒ A1−βY βi −∫ N
0
kjxi,jdj = PIf. (23)
Demand of one firm’s good of the representative household is given by
Yi = Ap− 1
1−βi
Labor markets clear,
15
∫ Q
0
∫ N
0
Li,jdjdi =∫ Q
0
Lidi = LS (24)∫ Q
0
∫ N
0
Hi,jdjdi =∫ Q
0
Hidi = HS . (25)
LS and HS are the inelastic supplies of low- and high-skilled labor. Labor/the household is renumerated as
follows
wLLS + wHH
S = A (26)
Since there are no savings, but firms have to pay fixed entry costs, product market clearing implies
A =∫ Q
0
(piYi − f)di. (27)
4.3 Technology and Wages in Symmetric General Equilibrium
Suppose there exists an exogenously given number of firms, Q. Symmetry among firms and labor market clearing
imply Q∫ N
0Ljdj = QL = LS . Plugging in (19) for Li, solving for w
1α−1L , and substituting for this expression in
(18) we get an expression of the wage gap w:
HS
LS=
1w
κ
ln w − κ. (28)
The above equation implicitly and uniquely determines w depending on the relative scarcity of high-skilled
labor, and κ. In the Appendix 6.7 we provide an intuition that w > eκ holds for all combinations of HS/LS
and κ.
Proposition 7 The wage gap is lower for higher relative skill endowments (HS
LS) and larger for a greater
intensity of technology in production (κ).
The proof is given in Appendix 6.8.1. If high-skilled labor is relatively more scarce, given other things equal,
its relative renumeration must increase to equilibrate the labor market. A greater intensity of technology in
production benefits the adoption of higher levels of technology in production. This implies the employment of
higher-j tasks which are more high-skill intensive. As a result, the relative demand of high-skilled labor, and
thus the wage gap, increases.
Lower wage gaps (caused e.g. through an increase in skilled labor supply) may or may not increase the
level of technology in production, depending on whether the restriction N ≤ T binds. In order to study GE
feedback effects of changing wage gaps on technology we need an approximation of the (implicitly given) level
of technology in production:
16
Proposition 8 The optimal level of technology in production, given w, can be approximated from (13) by
N ≈ 2κT(καα−1 + 1
)ln w
(29)
Then, given the wage gap equation (28), there exists an approximated threshold such that N ≤ T :
HS
LS≤
καα−1 + 1
2κ+ καα−1 + 1︸ ︷︷ ︸<1
καα−1 + 1κα
1−α + 1︸ ︷︷ ︸<1
.
The proof is given in Appendix 6.6. The approximated optimal level of technology in production exhibits
qualitatively the properties found by conducting comparative statics with the exact solution. A relative skill
endowment greater than the threshold implies that increasing its value further does not increase technology as,
in this case, technology adoption is constraint by barriers.
If firms are more technology intensive, high-skilled workers have to be relatively more scarce to ensure
that N ≤ T : a higher κ increases N as a first effect; however, a higher κ also implies a higher wage gap that
decreases N as a second effect. In the above approximation result, the first effect dominates the second and con-
sequently a higher technology intensity would have to be compensated by a greater scarcity of high-skilled labor.
To study wage level effects we endogenize the number of firms. Assume that there exists free entry according
to condition (23) which essentially states that firms enter until profit becomes equal to fixed entry costs, f .
Combining (9) and (10) we know that∫ N
0kjxi,jdj = piYiβ and the free entry condition in (23) becomes
piYi − piYiβ = f which directly determines output in general equilibrium as
Yi =f
pi(1− β). (30)
Replacing Yi with the above expression in the condition for product market clearing (27) leads to the number
of firms,
Q =A
f
1− ββ
. (31)
Naturally, the number of firms increases in market size measured by A and decreases in entry costs, f .
Knowing the number of firms allows us to determine wage levels through labor market clearing. Using low-
skilled labor demand (19), the number of firms (31), household’s renumeration (26), and the wage gap (28) in
LQ = LS , we get the following expression for the low-skilled wage:
wL = wNT
β1−2β β
β2β−1
(1− βf
) 1−β2β−1
LS1−β2β−1
(1 + w
HS
LS
) 1−β2β−1
Nβκ
2β−1
(κα
α− 1+ 1) β
1−2β
(32)
Proposition 9 Low- and high-skilled wages are higher for an increased access to advanced technologies (∂wL∂T =wLT
βκ2β−1 > 0) if and only if 2β − 1 > 0.
17
The proof is self-evident. A standard value of the elasticity of substitution between differentiated products
is one third which implies a β of two-third. In this case, 2β − 1 > 0 and the low-skilled wage increases in T . As
the equilibrium high-skilled wage is given implicitly by wL and w, it increases also in T .
Suppose there are two countries (A, B) that differ only in their access to technologies (TA > TB). Whereas
the wage gap is identical in both countries, wage levels will be higher in A. Define the average level of technology
in production within a country as N = QN/Q = N . Given symmetry among firms, the average technological
level within a country equals the choice of technology in production of a single firm. Assume a symmetric
negative shock to the relative skill endowment in both countries. As a consequence, the average technology
level in A declines more sharply than in B: its better access to advanced technologies led to a higher higher
technology average; furthermore, the size of technology adjustment (cf. Prop. 2) depends positively on the
technological level.
However, an increased access to advanced technologies has no effect on aggregate relative factor demands
or the skill premium within a country. In the following, we thus introduce firm heterogeneity with respect to
productivity to allow for endogenous factor reallocations across firms that will affect aggregate skill demand
and the wage gap.
4.4 Characterization of the Asymmetric General Equilibrium
As in the symmetric case firms enter the market until profit equals fixed costs of production. Subsequent to
fixed costs payment, each firm learns about its technology intensity. More precisely, each firm draws κh (κl)
with probability γ (1− γ). Suppose that a sufficiently large number of firms enters. Then, according to the law
of large numbers, a fraction γ of firms operates at a high technology intensity and a fraction 1 − γ at a low
technology intensity. As established in partial equilibrium analyses, h-firms use a higher level of technology in
production, employ a more skilled labor force, and display a higher productivity than l-firms17.
General equilibrium conditions change for labor - and product market clearing. Labor market clearing now
demands aggregating different firm-level demands of high- and low-skilled labor are aggregated as18
∫ γQ
0
∫ N
0
Lz,j,hdjdz +∫ Q
γQ
∫ N
0
Lz,j,ldjd =∫ γQ
0
Lz,hdz +∫ Q
γQ
Lz,ldz = LS (33)∫ γQ
0
∫ N
0
Hz,j,hdjdz +∫ Q
γQ
∫ N
0
Hz,j,ldjdz =∫ γQ
0
Hz,hdz +∫ Q
γQ
Hz,ldz = HS . (34)
Product market clearing with heterogeneous firms implies
17Up to now, we assume that entry costs f are sufficiently small ensure a positive production of both kinds of firms. Further
analysis is clearly needed.
18We replace the firm level index i by z to distinguish different firms within a category of technology intensity.
18
A =∫ γQ
0
(pzYz − f)dz +∫ Q
γQ
(pzYz − f)dz. (35)
Calculating the skill premium reveals to be tricky with heterogeneous firms and manipulations are referred
to Appendix 6.8.2 where we show that
HS
LS=
1w
γκhφβ
1−βh + (1− γ)κlφ
β1−βl
ln w(γφ
β1−βh + (1− γ)φ
β1−βl
)−(γκhφ
β1−βh + (1− γ)κlφ
β1−βl
) . (36)
The implicit equation of the wage gap with heterogeneous firms thus has close structural similarities to the
equation determing the wage gap with homogeneous firms (28). Moreover, technology intensities as well as ln w
are weighted by firm categories (γ) that are in addition adjusted by firms’ productivities and market elasticity.
Lowering barriers to technology adoption and thus increasing the availability of more advanced technologies
affects firm specific productivities heterogeneously. This may trigger labor reallocations that extend the scope
of feedback effects between technologies and the wage gap.
4.5 Skill Premia and Differences in Technologies and Productivities in Asymmet-
ric General Equilibrium
Proposition 10 The availability of more advanced technologies
• has no impact on the wage gap if firms are homogeneous.
• implies a higher wage gap if firms are heterogeneous.
The proof is given in Appendix 6.8.3. The skill premium with homogeneous firms does not depend on the
access to advanced technologies (see (28)). Regardless the level of technology chosen by all firms, technology
intensity in combination with the wage gap captures all effects on relative labor demands. Given firm
heterogeneity, relative demands differ across firms and thus allow for labor reallocations through a shift in
productivity differences φd ≡ φh/φl.
Proposition 11 The availability of more advanced technologies with heterogeneous firms
• does not alter technology differences, NhNl
, across firms.
• increases the productivity differences across firms: T∂φd∂T
φd= (κh − κl)(1 − T (dw/dT )
w ln w ) = −(κh −
κl)T∂ ln (w1/T )
∂T
ln (w1/T )> 0.
19
The proof is given in Appendix 6.8.4. An increase in the availability of more advanced technologies has no
effect on technology differences as the fraction of technology (N/T ) in production is unchanged within all firms.
Nevertheless, two distinct effects emerge: Firstly, h-firms as well as l-firms increase their respective level of
technologies proportionally to T ; this leaves their technology differences (Nh/T )/(Nl/T ) unchanged. Secondly,
wage gap increases induce h-firms to decrease their level of technology not only by more, but also proportionally
to their initial level of technology in production; thus, Nh/Nl is not altered.
The elasticity of productivity differences with respect to T equals technology intensity differences times the
(negative) elasticity of log-relative task production costs, ln (kj+1/kj) = ln (w1/T ). The availability of more
advanced technologies re-organizes the production of tasks through reducing relative skill-intensities and thus
relative costs. Consequently, h-firms that employ more high-skilled intensive tasks than l-firms benefit by more.
The index κ determines how much a firm profits from its level of technology in production and how much
more h-firms use technology in production than l-firms. Consequently, the difference in κ’s measures how much
more h-firms gain from higher technology levels made available by a reduction in barriers to technology adoption.
For illustration purposes, we simulate GE results with the following parameters: β = 0.75, α = 0.5, κ =
0.2519, κh = 0.4, κl = 0.2, γ = 0.75, and HS/LS = 1/320. Technology intensities in the case of heterogeneous
firms are chosen such that its weighted average equals technology intensity of homogeneous firms. In Figure 1,
differences in productivities increase considerably when more advanced technologies become accessible:
Figure 1: Relative productivity of a more (h) vs. a less (l) technology intensive firm
The above results deliver a natural explanation of the wage increase with heterogeneous firms: An increased
access to advanced technologies increases the productivity differences, φd. Thus, κh gains more weight in the
determination of the weighted average of κ’s in (36). As a consequence, a parallel mechanism to the symmetric
firms case is triggered: A higher (average) technology intensity increases the wage gap. Furthermore, as we can
observe in (Figure 2), the wage gap seems to be considerably higher when firms are heterogeneous.
However, the silent, but decisive mechanism of a wage gap increase consists in the reallocation of labor from
less to more productive firms when more advanced technologies become accessible (shown in Figure 3).
19See Acemoglu et al. (2007) for β and κ, α in the homogeneous task case.
20This implies a fraction of high-skilled in the population of 0.25.
20
Figure 2: Wage gap for different levels of available technologies; red: homogeneous firms; blue: heterogeneous
firms;
Figure 3: Relative employment increases in available technologies; red: relative employment of high-skilled (h-
firm/l-firm); blue: relative employment of low-skilled (h-firm/l-firm);
21
Proposition 12 A relatively more skilled workforce
• does not alter technology differences, NhNl
, across firms if barriers to technology adoption do not bind.
• decrease technology differences if barriers bind for h-firms.
The proof is given in Appendix 6.8.5. The reasoning of the first part is analogous to that in Prop. 11.
The second part is even more simple: Other things equal, a more skilled workforce implies a lower wage gap
(analogous to the homogeneous firms case). As l-firms increase their level of technology while h-firms cannot,NhNl
= TNl
decreases. The divergent behavior of technology differences can also be observed in (Figure 4).
Figure 4: Non-monotonic technology gap with increasing relative high-skilled endowment; # of available tech-
nologies is set to 500; red: level of technology chosen by a l-firm; blue: level of technology chosen by a h-firm
Facing a low technology intensity leads firms to use less technologies than are available. In contrast to the
homogeneous firms case there exists an interesting feedback effect: More available technologies increases the
wage gap which leads to a decrease in the level of technology of the less technology intensive firm. An increase
in the technology intensity difference (κh/κl) naturally leads to a higher difference in technologies. In Figure
5 we provide an intuition for this relationship for κh/κl ∈ [1, 2] where we keep κl at 0.2. When κh/κl reaches
a certain threshold, more technology intensive firms operate with all available technologies. Furthermore, a
greater difference in technology intensity implies that less technology intensive firms are driven to use less
sophisticated techniques: the enhanced use of advanced techniques by h-firms drives-up the wage gap and
consequently discourages l-firms to employ high-skill intensive tasks. Note that this constitutes a pure wage
gap effect as κl is not changed.
22
Figure 5: Differences in technologies increase in the technology intensity difference (κh/κl); # of available
technologies is set to 500; red: level of technology chosen by a l-firm; blue: level of technology chosen by a
h-firm
5 Conclusion
Existing literature on technology adoption emphasizes country-level effects of relative skill endowments (e.g.
Caselli and Coleman (2006)) or firm-level effects of specific production structures (e.g. Acemoglu et al. (2007)).
In contrast, we have proposed a model where a firm’s technology choice not only depends on specific production
structures, but also on a country’s relative skill endowments and its access to advanced technologies. Moreover,
we shed light on the mechanism of how endogenous technological choice affects productivity. We made a cru-
cial, but plausible, assumption: To increase productivity, the adopted technologies have to be complemented
by organizational change. The organization of the production structure regulates the relative employment of
intermediate tasks. As more advanced technologies involve intermediate steps that are relatively more high-skill
intensive, advanced technologies and high-skilled labor are complements.
With homogeneous firms, relative skill endowments and a country’s general technology intensity determine
the skill premium of high- to low-skilled workers. A relatively scarce high-skilled labor supply implies a large wage
gap that transforms the adoption of more advanced technologies less profitable. The complementary optimal
organization structure involves the employment of primarily low-skilled labor intensive intermediate tasks and
leads to a low level of productivity.
With heterogeneous firms, the availability of more advanced technologies downgrades previously used
intermediate production steps; to some extend, they become out-dated. Their high-skill intensity and hence
their implementation costs decrease. More high-skill intensive intermediates are integrated in the production
process and allow the adoption of higher technological levels. Firms with a more technology intensive production
structure adopt higher levels of technology and they increase productivity by more. Consequently, production
and labor demand shares are shifted to more productive firms. As they demand relatively more skilled labor,
aggregated relative skill demand increases. Fixed labor supplies then imply a larger wage gap that consequently
increases technology gaps as well as productivity gaps.
23
6 Appendix
6.1 A benchmark case of homogeneous tasks and endogenous degree of special-
ization
This benchmark case follows Acemoglu et al. (2007). Suppose all tasks j are homogeneous and consequently,
kj = k ∀j. Then, concavity implies symmetry: x (j) = x (j′) for all j, j′ ∈ [0, N ] . Hence, Y = Nκ+1x. Profits
(from (5)) are given by
Π (N,x) = A1−βY β −Nkx− C (N) .
Here, C (N) is the cost of running technology N . If all tasks are equally expensive, it is necessary to obtain a
finite choice of N (cf. Acemoglu et al. (2007)). In our model, we are able to set C = 0 as the costs of technology
adoption are integrated fully in the tasks. Comparing the two models’ resulting elasticities of output with respect
to N is only feasible if we consider post-maximization elasticity for our model. Consequently, our elasticity is
smaller than Acemoglu et al. (2007)’s ante-maximization as ours incorporates conditions for a finite choice of
N (κ < 1−αα ). The first-order maximization conditions in the homogenous task benchmark imply
β1
1−βAN∗β[κ+1]−1
1−β κ = kβ
1−βC ′ (N∗) and x∗ =C ′ (N∗)κk
.
The first equation determines the optimal technology choice N∗. It can be shown that N∗ is decreasing in
k which itself increases in both wages. The latter observation is our starting point: if either wH and/or wL
decreases, N∗ will increase.
6.2 Derivation of the firm’s optimality conditions
The first order maximization conditions derived from (5) comprise a system of N + 1 equations:
βA1−βY β−1 ∂Y
∂N= kNxN , (37)
βA1−βY β−1 ∂Y
∂xj= kj ∀j ∈ [0, N ] . (38)
Combining (37) and (38) for some j yields
∂Y∂N∂Y∂xj
=kNxNkj
∀ j ∈ [0, N ] . (39)
24
From the definition of Y we calculate
∂Y
∂N=
Y
N
[κ+ 1− 1
α+
NxαN
α∫ N
0xαj dj
], (40)
∂Y
∂xj=
xα−1j Y∫ N
0xαj dj
. (41)
Using (41) in (38), we obtain, for each pair (j, j′),
xjxj′
=[k′jkj
] 11−α
. (42)
Plugging (42) back in (41) allows us to write
∂Y
∂xj=
x−1j Y
kα
1−αj
∫ N0k
αα−1j dj
. (43)
Similarly, combining (42) and (40) yields
∂Y
∂N=Y
N
κ+ 1− 1α
+NxαN
αxαj kα
1−αj
∫ N0k
αα−1j dj
. (44)
Plugging the last two expressions in (39), we obtain a non-linear expression for xj :
xj
(κ+ 1− 1
α
)k
α1−αj
1N
∫ N
0
kαα−1j dj︸ ︷︷ ︸
≡Kαα−1N
+x1−αj xαN
1α
=xNkNkj
. (45)
(45) again holds for all j ∈ [0, N ]. For j = N we obtain an implicit expression for the optimal choice of N :
(κ+
α− 1α
)k
α1−αN K
αα−1N =
α− 1α
(46)
Simple manipulations lead to equation (6) in the main text.
Consider next the optimal choice of xj . Dividing (45) by xN and using (42) enables us to rewrite xj as
xj = k1
α−1j kNxNK
α1−αN
1καα−1 + 1
. (47)
Using this expression in the production function yields
Y = Nκ+1kNxNK−1N
1καα−1 + 1
.
In order to use this in the FOCs, we rewrite ∂Y∂xj
in (41) using (47):
∂Y
∂xj= Y N−1kjk
−1N x−1
N
[κα
α− 1+ 1].
Combining this expression with (38) leads to the optimal demand for xN ,
25
xN = β1
1−βANβ(κ+1)−1
1−β k−1N K
− β(1−β)
N
[κα
α− 1+ 1].
Substituting with this expression in (47), we obtain the optimal demand for each j,
xj = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N k1
α−1j . (48)
This is (7) in the main text. The demand for the last, or marginal, task j = N can be simplified by rewriting
kN in terms of KN (using (6) and j = N in (7)):
xN = β1
1−βANβ(κ+1)−1
1−β K− 1
1−βN
(κα
α− 1+ 1) 1α
. (49)
Using (6) and (7) in the production function (1), we can express the partial equilibrium output of each firm
as
Y = β1
1−βANκ
1−β K− 1
1−βN . (50)
6.3 Factor demands
A firm’s demand for high- and low-skilled labor (H and L) can be derived as follows:
Demand of high-skilled labor to produce task j, Hj , is given through cost minimization (see Appendix 6.4) and
read
Hj = xjj
bw
jT −1 = β
11−βAN
β(κ+1)−11−β K
− β−α(1−α)(1−β)
N w1
α−1L
j
Tw
jT
αα−1−1
Taking the integral∫ N
0dj and using kj = wLw
jT leads to
H =∫ N
0
Hjdj = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N w1
α−1L
1Tw
∫ N
0
jwαα−1
jT dj
where integration by parts yields
H = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N w1
α−1L
1−wNT
αα−1
ln w 1T
α1−α−Nw
NT
αα−1
w ln w α1−α
. (51)
Substituting for wNT
α1−α − 1 using (13) simplifies the above equality further to
H = β1
1−βANβκ
1−β K− β−α
(1−α)(1−β)
N w1
α−1L w
NT
αα−1−1 (ln w)−1 κ
καα−1 + 1
. (52)
Similarly, Lj is given by
Lj = xj(1−j
T)w
jT = β
11−βAN
β(κ+1)−11−β K
− β−α(1−α)(1−β)
N w1
α−1L
(1− j
T
)w
jT
αα−1
26
and
L =∫ N
0
Ljdj = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N w1
α−1L
∫ N
0
wαα−1
jT dj − wH
where integrating and using the expression in (51) for H yields
L = β1
1−βANβ(κ+1)−1
1−β K− β−α
(1−α)(1−β)
N w1
α−1L
T(
1− wNT
αα−1
)(1− 1
ln w α1−α
)+Nw
NT
αα−1
ln w α1−α
(53)
Again, substituting for wNT
α1−α − 1 using (13) yields
L = β1
1−βANβκ
1−β K− β−α
(1−α)(1−β)
N w1
α−1L w
NT
αα−1
ln w − κ
ln w(καα−1 + 1
) . (54)
Relative factor demands:
Dividing (52) by (54) yields directly
H
L=
1w
κ
ln w − κ. (55)
6.4 Minimal unit cost function for the adjusted Cobb-Douglas Case
Minimize
wLL+ wHH s.t. zL1−ζHζ = x
where z = ζ−ζ(1− ζ)−(1−ζ). The Langrangian
L = wLL+ wHH − λ(zL1−ζHζ − x
)leads to the following FOCs
wL = λ(1− ζ)z(H
L
)ζwH = λζz
(H
L
)ζ−1
.
Combining the FOCs leads to the relative factor demand
H = LwLwH
ζ
1− ζ
and the factor demands depending on output and relative wages
27
H(x, wL, wH) = xζ
(wLwH
)1−ζ
L(x, wL, wH) = x(1− ζ)(wHwL
)ζ.
Define k as the minimum unit cost to produce x, total cost of producing x can be written as
kx = wLL+ wHH.
Plugging in H(x, wL, wH) and L(x, wL, wH), and rearranging provides the following simple formulation of
the minimum unit cost:
k = wL
(wHwL
)ζ.
6.5 Proofs for Partial Equilibrium Results and Comparative Statics
6.5.1 Proof of Lemma 1
Using factor demands of task j, (16, 17), gives the high-skill intensity
Hj
Lj= w−1 j
T − j
that decreases in T . Computing the relative high-skill intensity of task j + 1 to task j leads to
Hj+1Lj+1
HjLj
=j + 1j
T − jT − (j + 1)
what clearly decreases in T .
6.5.2 Proof of Proposition 1
The first derivative of relative task employment, xj+1xj
= w−1
T (1−α) , with respect to w is:
∂(xj+1xj
)∂w
= − 1T (1− α)w
xj+1
xj< 0.
Similarly, the first derivative of xj+1xj
with respect to T reads as:
∂(xj+1xj
)∂T
=ln w
T 2(1− α)xj+1
xj> 0.
28
6.5.3 Proof of the existence of a unique partial equilibrium N∗(w), Lemma 2
A firm’s optimal choice of the level of technology in production, N∗, given w is a solution to (13):
wα
1−αNT − 1︸ ︷︷ ︸
f(N,w)
=NT
α1−α ln wκαα−1 + 1︸ ︷︷ ︸g(N,w)
We start by showing that there exists optimal technology levels ∀w ∈ (1,∞) and subsequently establish
uniqueness. Applying L’Hopital’s Rule, the following holds ∀w ∈ (1,∞):
limN→0
f(N, w)g(N, w)
= limN→0
∂f(N,w)∂N
∂g(N,w)∂N
= 0 < 1
limN→+∞
f(N, w)g(N, w)
= limN→+∞
∂f(N,w)∂N
∂g(N,w)∂N
= +∞ > 1.
Consequently, ∀w ∈ (1,∞) ∃N∗ ∈ (0,∞) such that f(N∗, w) = g(N∗, w). It is immediate that f(N) is
a strictly convex and g(N) is a linear function. Consequently, f(N) and g(N) intersect at most twice. They
intersect at N∗ = 0 ∀w ∈ (1,∞) and thus N∗ ∈ (0,∞) is a unique partial equilibrium ∀w ∈ (1,∞).
6.5.4 Proof of Proposition 2
(13) determines implicitly a firm’s choice of technology in production. Assume that N is a solution to (13). How
that N reacts to changes in the wage gap is given by implicit differentiation: dNdw = −∂F∂w/∂F∂N . From (13), define
F =NT
α1−α ln wκαα−1 + 1
− wα
1−αNT + 1 = 0
and calculate partial derivatives. Firstly,
∂F
∂w=N
T
α
1− α1w
[1
καα−1 + 1
− wα
1−αNT
]and secondly,
∂F
∂N=
1T
α
1− αln w
[1
καα−1 + 1
− wα
1−αNT
].
Combining and rearranging leads to
dN
dw= − N
ln w w< 0.
How does the choice of technology in production react to a change in T is determined through:
29
∂F
∂T= − N
T 2
α
1− αln w
[1
καα−1 + 1
− wα
1−αNT
].
and, consequently,
dN
dT= −
∂F
∂b∂F
∂N
=N
T> 0.
6.5.5 Proof of Lemma 3
We partition the proof in two parts and employ the definitions of f and g from proof 6.5.3. First, we show that
∀T ∈ (0,∞) ∃!wT ∈ (1,∞) such that N∗ < T . Second, we show that under the assumption eκT > w, N∗ > 1
holds.
First Part: We begin by proofing that ∀T ∈ (0,∞) ∃wT ∈ (1,∞) such that ∀w > wT N∗(w) < T and establish
uniqueness subsequently.
Proof by contradiction: Assume that ∃T such that ∀w N∗(w) ≥ T (Assumption A1). Rearranging (13) and
taking the natural logarithm on both sides leads to
α
1− αN∗
Tln w = ln
(α
1−αN∗
T ln wκαα−1 + 1
+ 1
).
Without loss of generality we assume thatα
1−αN∗T ln w
καα−1 +1 ≤ 1 . A Taylor expansion of the natural logarithm
around −1 < x ≤ 1 implies ln (x+ 1) =∑∞n=1 (−1)n+1 xn
n . Consequently, the above equation can be reduced to
− κα
1− α= −1
2
α1−α
N∗
T ln wκαα−1 + 1
+13
(α
1−αN∗
T ln wκαα−1 + 1
)2
− 14
(α
1−αN∗
T ln wκαα−1 + 1
)3
+ ◦
(14
α1−α
N∗
T ln wκαα−1 + 1
)3
.
According A1 any w ∈ (1,∞) is admissible. Take lim w → 1. Consequently, g approaches zero and thus
f(lim w → 1) < g(lim w → 1) ∀T ∈ (0,∞) if ∞ > N∗ > T . However, from proof 6.5.3 we know that ∀N ≥ N∗
f ≥ g and thus N∗ ≥ T is not an optimal choice. This contradicts assumption A1. By contradiction, we have
established that ∀T ∈ (0,∞) ∃wT ∈ (1,∞) such that ∀w > wT N∗(w) < T .
Furthermore, from Prop. 2, N∗ is strictly monotone decreasing in w. Consequenty, ∃!wT ∈ (1,∞). QED.
Second Part: The assumption eκT > w can be rewritten as κTln w > 1. Taking derivatives of f and g with respect
to N leads to
30
∂f
∂N=
1T
α
1− αln ww
α1−α
NT
∂g
∂N=
1T
α1−α ln wκlαα−1 + 1
.
As ∂f∂N > 0 and ∂2f
∂N2 > 0, f is strictly convex in N . As ∂g∂N > 0 and ∂2g
∂N2 = 0, g is linear in N with a positive
slope. For N = 0, f and g would be zero. For N+ → 0, ∂f∂N < ∂g
∂N . Consequently, a necessary condition for an
equilibrium value of N (N∗) is that partial derivatives evaluated at N∗ satisfy: ∂f∂N (N∗) > ∂g
∂N (N∗). In the
following we show that ∂f∂N (N = κT
ln w ) < ∂g∂N (N = κT
ln w ) and consequently N∗ > κTln w .
Evaluating first derivatives at N = κTln w yields
∂f
∂N=
1T
α
1− αln we
κα1−α
∂g
∂N=
1T
α
1− αln w
(κα
α− 1+ 1)−1
and
∂f
∂N≶
∂g
∂N
⇐⇒ κα
1− α≶ − ln
(κα
α− 1+ 1).
As −1 < καα−1 < 0, the above Taylor expansion can be applied to − ln
(καα−1 + 1
)what leads to
κα
1− α<
κα
1− α+
12
(κα
1− α
)2
+13
(κα
1− α
)3
+ ◦
(13
(κα
1− α
)3).
and consequently ∂f(N = κTln w )/∂N < ∂g(N = κT
ln w )/∂N for all admissible parameter values. QED.
6.5.6 Proof of Proposition 3
How the choice of technology in production react to a change in κ can again be determined by the implicit
differentiation approach:
∂F
∂κ=N
Tln w
(α
1−α
)2
( καα−1 + 1)2
and
31
dN
dκ= −
∂F∂κ∂F∂N
= −N α
1−α
( καα−1 + 1)2
(1
καα−1 +1 − w
α1−α
NT
) .Replacing w
α1−α
NT from equation (13) leads to
dN
dκ=
N(καα−1 + 1
) (NT ln w − κ
) > 0
as Lemma 3 implies that NT ln w > κ. Taking the first derivative of dN
dT with respect to κ gives
d2N
dTdκ=
N
T(καα−1 + 1
) (NT ln w − κ
) > 0
as Lemma 3 implies that NT ln w > κ.
How the choice of N depends on α is determined as
∂F
∂α=N
T
ln w(1− α)2
καα−1 + 1 + κ(καα−1 + 1
)2 − wα
1−αNT
.and
dN
dα= −
∂F
∂α∂F
∂N
= − N
α(1− α)
καα−1 +1+κ
( καα−1 +1)2 − w
α1−α
NT
1καα−1 +1 − w
α1−α
NT
.
Substituting wα
1−αNT from equation (13) leads to
dN
dα= − N
α(1− α)
NT ln w − κ
(1 +
1−αα
καα−1 +1
)NT ln w − κ
.
dNdα > 0 if and only if κ < N
T ln w < κ(1 + 1−αα
1καα−1 +1 ). With N
T ln w > κ, dNdα > 0 if and only if
NT ln w > κ
(1 + 1−α
α1
καα−1 +1
).
6.5.7 Proof of Proposition 4
Taking the first derivative of (20) with respect to w directly gives:
∂(HL
)∂w
= −HL
1w
ln w − κ+ 1ln w − κ
< 0
Taking the first derivative of (20) with respect to κ directly gives yields:
∂(HL
)∂κ
=1w
ln w(ln w − κ)2
> 0
32
and taking the second with respect to w gives
∂2(HL
)∂κ∂w
= − 1w2
ln w(ln w − κ) + κ
(ln w − κ)2< 0.
6.5.8 Proof of Proposition 5
Taking the first derivative of (11) with respect to w and manipulating leads to
∂φ
∂w= − φ
ln ww.
Taking the first derivative of (11) with respect to κ and manipulating leads to
∂φ
∂κ= lnNφ > 0
Taking the first derivative of (11) with respect to T and manipulating leads to
∂φ
∂T= φ
κ
T> 0
6.5.9 Proof of Proposition 6
Taking the first derivative of ∂φ∂κ with respect to T and manipulating leads to
∂2φ
∂κ∂T=φ
T(1 + κ lnN) w > 0
which holds as Lemma 3 applies.
6.6 Proof of Approximation for Homogeneous Firms in General Equilibrium
The level of technology, N , as well as the wage gap,w , are given solely implicitly. We thus use second and first
order Taylor approximations21 to show under which parameter restriction the restriction N ≤ T does not bind.
A second order approximation of the level of technology leads to
N ≈ 2κT(καα−1 + 1
)ln w
where N exhibits qualitatively all partial equilibrium properties. The restriction N/T ≤ 1 does not bind if
and only if
w ≥ exp
(2κ
καα−1 + 1
). (56)
21Consequently, the presented restrictions are approximations.
33
The wage gap is determined as
HS
LSw︸ ︷︷ ︸
≡x(w)
=κ
ln w − κ︸ ︷︷ ︸≡y(w)
where x(w) is a linear and strictly monotone increasing and y(w) is a strictly decreasing function for w > eκ.
As for w+ → eκ x(w) < y(w), ∀w ≥ w∗ we have consequently x(w) > y(w). If we plug in the threshold value of
w (56) in the above equation such that x(w) < y(w) we can determine a parameter combination that ensures
w∗ > w. Consequently, the restriction N/T ≤ 1 does not bind if and only if
HS
LS≤
καα−1 + 1
2κ+ καα−1 + 1︸ ︷︷ ︸<1
καα−1 + 1κα
1−α + 1︸ ︷︷ ︸<1
.
6.7 Proof of Homogeneous Firms General Equilibrium Property
We provide a graphical intuition of the proof: Draw the left hand side and the right hand side of (28) with w on
the abcissa. Left of eκ, the right hand side stays below zero and right of eκ, the right hand side is monotically
decreasing between (0,∞).
6.8 Proofs for General Equilibrium Comparative Statics
6.8.1 Proof of Proposition 7
How the wage gap changes if the relative scarcity of high-skilled labor changes is determined through implicit
differentiation. Define
GHo =HS
LSw − κ
ln w − κ= 0.
and
dw
d(HS
LS
) = −
∂GHo
∂(HS
LS
)∂GHo∂w
= − w
1w
κln w−κ
(1 + 1
ln w−κ
) < 0.
shows that the wage gap is decreasing in the scarcity of high-skilled labor.
∂GHo∂κ
=− ln w
(ln w − κ)2
∂GHo∂w
=HS
LS+
κw−1
(ln w − κ)2
Thus, the wage gap clearly increases in κ:
34
dw
dκ= −
∂GHo∂κ
∂GHo∂w
=ln w
κw−1(ln w − κ+ 1)
6.8.2 Derivation of the equation that implicitly gives the wage gap with heterogeneous firms
As all h- and l-firms are identical within their category of technology intensity, we replace integrals, divide (34)
by (33), and cancel the number of firms:
HS
LS=γHh + (1− γ)Hl
γLh + (1− γ)Ll. (57)
Manipulating the above equation implies firstly, using (13) in (12), rewriting the latter using Kαα−1Ni
=
Niwαα−1L w
NiT
αα−1
(κiαα−1 + 1
)−1
. Plugging in (18) and (19) leads to the following firm-specific labor demands:
Hi = β1
1−βANκi
β1−β
i w1
β−1L w
NiT
ββ−1 w−1 κi
ln w
(κiα
α− 1+ 1) β(α−1)α(1−β)
(58)
Li = β1
1−βANκi
β1−β
i w1
β−1L w
NiT
ββ−1
ln w − κiln w
(κiα
α− 1+ 1) β(α−1)α(1−β)
(59)
where i ∈ {l, h}. Plugging into (57) leads to the following implicit equation that determines the wage gap:
LS
HS= w
ln wγφ
β1−βh + (1− γ)φ
β1−βl
γκhφβ
1−βh + (1− γ)κlφ
β1−βl
− 1
(60)
where φh and φl are the respective productivities of high- and low-technology intensive firms. The assumption
w > eκh ensures a positive solution. Inverting the above equation leads to
HS
LS=
1w
γκhφβ
1−βh + (1− γ)κlφ
β1−βl
ln w(γφ
β1−βh + (1− γ)φ
β1−βl
)−(γκhφ
β1−βh + (1− γ)κlφ
β1−βl
)what is (36) in the main text.
6.8.3 Proof of Proposition 10
The equation that implicitly determines the wage gap in the homogeneous firms case (28) is independent of T .
This is different for the heterogeneous firms case: We reduce (60) to
LS
HS= w
ln wγφ
β1−βd + 1− γ
γκhφβ
1−βd + (1− γ)κl
− 1
where
35
φd ≡ φh/φl = Nκhh N−κll w−
Nh−NlT
(κlαα−1 + 1κhαα−1 + 1
) 1−αα
(61)
is the productivity difference across firms of different technology intensities calculated from (15). Define
GHe ≡LS
HS
1w− ln w
γφβ
1−βd + 1− γ
γκhφβ
1−βd + (1− γ)κl
+ 1
where
∂GHe∂w
= −
LS
HS
1w2︸ ︷︷ ︸
>0
+1w
γφβ
1−βd + 1− γ
γκhφβ
1−βd + (1− γ)κl︸ ︷︷ ︸
>0
−∂φd∂w
ln wγβ
1− βφ
β1−β−1
d
(κh − κl)(1− γ)(γκhφ
β1−βd + (1− γ)κl
)2
︸ ︷︷ ︸>0
< 0
as ∂φd∂w = −φdw
(Nh−NlT + κh−κl
ln w
)< 0. Furthermore,
∂GHe∂T
=∂φd∂T
ln wγβ
1− βφ
β1−β−1
d
(κh − κl)(1− γ)(γκhφ
β1−βd + (1− γ)κl
)2
︸ ︷︷ ︸>0
> 0
as ∂φd∂T = φd
T
[(κh − κl)(1− T (dw/dT )
w ln w )]> 0 if ∂w/∂T = 0. This holds in ∂GHe
∂T as w is held constant given
the implicit differentiation approach. Thus, dwdT = −∂GHe∂T /∂GHe∂w > 0.
6.8.4 Proof of Proposition 11
Taking the first derivative of technology differences (61) with respect to T gives
∂φd∂T
= φd
[(κh − κl)(1−
T (dw/dT )w ln w
)]
where
36
∂φd∂T
≷ 0
⇐⇒ 1T
≷dw/dT
w ln w
⇐⇒ 1T
≷−∂GHe∂T /∂GHe∂w
w ln w
⇐⇒ 1 ≷
φd(κh − κl)γ β1−βφ
β1−β−1
d(1−γ)(κh−κl)(
γκhφβ
1−βd +(1−γ)κl
)2
LS
HS1w + γφ
β1−βd +1−γ
γκhφβ
1−βd +(1−γ)κl
+ φd(Nh−NlT + κh−κl
ln w
)ln wγ β
1−βφβ
1−β−1
d(κh−κl)(1−γ)(
γκhφβ
1−βd +(1−γ)κl
)2
⇐⇒ 1 ≷1
LS
HS1w + γφ
β1−βd +1−γ
γκhφβ
1−βd +(1−γ)κl
γ β1−βφ
β1−βd
(κh−κl)2(1−γ)(γκhφ
β1−βd +(1−γ)κl
)2
︸ ︷︷ ︸>0
+ (Nh−Nl) ln wT (κh−κl) + 1
and consequently ∂φd/∂T > 0.
Taking the first derivative of NhNl
with respect to T leads to
∂(NhNl )
∂T=Nl
(NhT −
NhdwdT
w ln w
)−Nh
(NlT −
NldwdT
w ln w
)N2l
= 0.
6.8.5 Proof of Proposition 11
Taking the first derivative of NhNl
with respect to HS
Ls leads to
∂(NhNl )
∂(HSLs )=
1N2h
Nl− dNh
dw︸ ︷︷ ︸=0ifNh=T
dw
d(HSLs )
−Nh−Nl dw
d(HS
Ls )
w ln w
.
37
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