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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
WHEN DOES LABOR SCARCITY ENCOURAGE
INNOVATION?
Daron Acemoglu
Working Paper 09-07
March ll, 2009
Room
E52-251
50 Memorial Drive
Cambridge, MA02142
This paper can be downloaded without charge from the
Paper Collection
http://ssrn.com/abstract=1 369706
Social Science Research Network
at
When Does
Labor Scarcity Encourage Innovation?*
Daron Acemoglu
Massachusetts Institute of Technology
March
2009.
Abstract
This paper studies the conditions under which the scarcity of a factor (in particular, labor)
encourages technological progress and technology adoption. In standard endogenous growth
models, which feature a strong scale
effect,
an increase in the supply of labor encourages techno-
famous Habakkuk hypothesis in economic history claims that
technological progress was more rapid in 19th-century United States than in Britain because
logical progress. In contrast, the
of labor scarcity in the former country.
for
Similar ideas are often suggested as possible reasons
why high wages might have encouraged
rapid adoption of certain technologies in conti-
nental Europe over the past several decades, and as a potential reason for
regulations can spur
of these questions.
function of the
9,
and
I
more rapid innovation.
define technology as strongly labor saving
economy
is
tion exhibits increasing differences in 9
and
labor.
The main
if
the production func-
if
is
paper shows that
strongly labor saving. In
technology
is
strongly labor
provide examples of environments in which technology can be strongly labor
saving and also show that such a result
These
if
result of the
technology
contrast, labor scarcity will discourage technological advances
models.
the aggregate production
strongly labor complementary
labor scarcity will encourage technological advances
I
if
exhibits decreasing differences in the appropriate index of technology,
labor. Conversely, technology
complementary.
why environmental
present a general framework for the analysis
I
is
not possible in certain canonical macroeconomic
results clarify the conditions
under which labor scarcity and high wages enwhy such results were obtained or conjectured
courage technological advances and the reason
in certain settings,
JEL
but do not always apply in
many models used
in
the growth literature.
Classification: O30, 031, 033, C65.
Keywords: Habakkuk
hypothesis, high wages, innovation, labor scarcity, technological
change.
*I
thank Philippe Aghion, Dan Cao, Joshua Gans, Andres Irmen, Samuel Pienknagura, and participants at
presentations at American Economic Association 2009 meetings, Harvard and Stanford for useful comments.
Financial support from the National Science Foundation
is
gratefully acknowledged.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/whendoeslaborscaOOacem
Introduction
1
There
is
widespread consensus that technological differences are a central determinant of pro-
ductivity differences across firms, regions, and nations. Despite this consensus, determinants
and adoption of new technologies are poorly understood.
of technological progress
A
basic
question concerns the relationship between factor endowments and technology, for example,
whether the scarcity of a
factor,
technological progress. There
is
and the high factor prices that
Wages, John Hicks was one of the
invention,
economists to consider this possibility and argued:
to invention of a particular kind
of a factor which has
Similarly, the
first
become
ment
for
production
is itself
a spur to
—directed to economizing the use
relatively expensive..." (1932, p. 124).
famous Habakkuk hypothesis
(1962). claims that technological progress
in Britain
In his pioneering work, The Theory of
in the relative prices of the factors of
and
induce
currently no comprehensive answer to this question, though
a large literature develops conjectures on this topic.
"A change
this leads to, will
in
economic
was more rapid
history,
proposed by Habakkuk
in 19th-century
United States than
because of labor scarcity in the former country, which acted as a powerful induce-
mechanization, for the adoption of labor-saving technologies, and more broadly for
Habakkuk quotes from
innovation. 1 For example,
"...
it
was
Pelling:
scarcity of labor 'which laid the foundation for the future continuous
progress of American industry, by obliging manufacturers to take every opportunity
of installing
new
types of labor-saving machinery.'
"
(1962, p. 6),
and continues
"It
and
seems obvious
inelasticity of
entrepreneur
...
—
it
certainly
seemed so to contemporaries
American, compared with British, labour gave the American
a greater inducement than his British counterpart to replace labour
by machines." (1962,
p.
17).
'See Rothbart (1946), Salter (1966), David (1975), Stewart (1977) and
discussions of the
— that the dearness
Habakkuk
hypothesis.
Mokyr
(1990) for related ides
and
Mantoux's (1961)
classic history of the Industrial
emphasizing how the changes
in labor costs in spinning
Revolution also echoes the same theme,
and weaving have been a major impetus
to innovation in these industries. Robert Allen (2008) extends
and proposes that the
relatively high
wages
and strengthens
in 18th-century Britain
this
argument,
were the main driver of
the Industrial Revolution. Allen, for example, starts his book with a quote from T. Bentley in
1780 explaining the economic dynamism of British industrial centers:
"...
up
all
Nottingham, Leicester, Birmingham, Sheffield
hopes of foreign commerce,
if
etc.
must long ago have given
they had not been constantly counteracting
the advancing price of manual labor, by adopting every ingenious improvement the
human mind
could invent."
Similar ideas are often suggested as possible reasons
why high
wages, for example induced
by minimum wages or other regulations, might have encouraged
technologies, particularly those
among
others,
More
complementary
faster adoption of certain
to unskilled labor, in continental
Beaudry and Collard, 2002, Acemoglu 2003, Alesina and
recently, these ideas
The famous Porter
tal technologies
have become relevant
in
Europe
(see,
Zeira, 2006).
the context of environmental regulation.
hypothesis applies Hicks's (1932) ideas to the development of environmen-
and claims that
and increase productivity.
tighter environmental regulations will spur faster innovation
While
this hypothesis plays a
of environmental policy, just like the
Habakkuk
major
hypothesis,
its
role in various discussions
theoretical foundations are
unclear.
In contrast to these conjectures
els
and hypotheses, most commonly-used macroeconomic mod-
imply that labor scarcity should discourage technological progress.
models,
when new
technologies are embodied in capital goods, predict that labor scarcity or
technologies. 3
Endogenous growth models
These models typically have a
single factor of production,
high wages should discourage the adoption of
also
make
labor,
Neoclassical growth
the same prediction.
and exhibit a strong
scale effect.
An
new
increase in the size of the labor force will induce
"See Porter (1991) and Porter and van der Linde (1995) for the formulation of this hypothesis. Jaffe, Peterson,
Portney and Stavins (1995) review early empirical evidence on this topic and Newell, Jaffee and Stavins (1999)
provide evidence on the effects of energy prices on the direction of technological change. Recent work by Gans
(2009) provides a theoretical explanation for the Porter hypothesis using the framework presented here.
3
See Ricardo (1951) for an early statement of this view. In particular, with a constant returns to scale
production function F (L, K), an increase in the price of L or a reduction in its supply, will reduce equilibrium
K, and to the extent that technology is embedded in capital, it will reduce technology adoption.
more rapid technological
progress,
and
either increase the
growth rate
(in the first-generation
models, such as Romer, 1986, 1990, Segerstrom, Anant and Dinopoulos, 1990, Aghion and
Howitt, 1992, Grossman and Helpman, 1991), or the level of output (in the semi-endogenous
growth models such as Jones, 1995, Young, 1998, Howitt, 1999).
The
directed technological change literature
characterization of
how
Acemoglu, 1998, 2002, 2007) provides a
(e.g.,
the bias of technology will change in response to changes in the supplies
of factors. In particular,
Acemoglu (2007)
establishes that, under
in the supply of a factor always induces a
an increase
change
weak
regularity conditions,
towards
in technology biased
that factor. 4 This result implies that labor scarcity will lead to technological changes biased
against labor. Nevertheless, these results do not address the question of whether the scarcity
of a factor will lead to technological advances
to changes in technology that increase
(i.e.,
the level of output in the economy). Reviewing the intuition for previous results
explain
why
variable 8
this
is
useful to
In particular, suppose that technology can be represented by a single
is.
and suppose that
6
and
of labor induces an increase in
if
6
and L are
and now because
6,
Then, an increase
and because of 8-L complementarity,
0,
biased towards labor. Conversely,
leads to a decline in
labor, L, are complements.
the marginal product of labor and
is
this
in the
supply
induced change
is
substitutes, an increase in the supply of labor
of 6-L substitutability, the decrease in 6 increases
again biased towards this factor.
This discussion not
only illustrates the robust logic of equilibrium bias in response to changes in supply, but also
makes
it
clear that the
abundant
they
factor, could take the
may correspond
The question
advances
induced changes in technology, though always biased towards the more
is
of
form of more or
less
advanced technologies;
in other
words,
to the choice of equilibrium technologies that increase or reduce output.
whether scarcity of a particular factor of production spurs technological
arguably more important than the implications of these induced changes on the bias
of technology. This paper investigates the impact of labor scarcity on technological advances
(innovation and adoption of technologies increasing output).
and a comprehensive answer
4
A
change
in
technology
is
to this question.
The key
biased towards a factor
if
it
given factor proportions. Conversely, a change in technology
is
I
present a general framework
the concept of strongly factor (labor)
increases the marginal product of this factor at
is
biased against a factor
if it
reduces
its
marginal
product at given factor proportions.
In a dynamic framework, changes towards less advanced technologies would typically take the form of a
slowdown in the adoption and invention of new, more productive technologies.
saving technology. 6 Suppose aggregate output (or net output) can be expressed as a function
Y (L, Z, 9),
where L denotes labor,
of technologies. Suppose that
Y
Z is
is
a vector of other factors of production, and 9
supermodular
Suppose
the vector 9 do not offset each other.
9 increases output.
differences in 9
Then technology
and L.
is
Intuitively, this
technological advances
change
is
if
is
changes in two components of
an increase
if
in (any
Y (L, Z, 9)
is
strongly labor
result of the
complementary
component
of)
exhibits decreasing
if
Y (L, Z, 9)
exhibits
paper shows that labor scarcity induces
strongly labor saving. In contrast,
when
technological
strongly labor complementary, then labor scarcity discourages technological advances.
This result can be interpreted both as a positive and a negative one.
it
a vector
means that technological progress reduces the marginal
The main
technology
also that
strongly labor saving
product of labor. Conversely, technology
increasing differences in 9 and L.
in 9, so that
is
characterizes a wide range of economic environments
where labor
On
the positive side,
scarcity can act as a force
towards innovation and technology adoption, as claimed in various previous historical and
economic analyses.
On
the negative side, most models used in the growth literature exhibit
increasing rather than decreasing differences between technology
typically an implication of functional form assumptions).
To
highlight
settings,
I
what decreasing
differences
this
is
between technology and labor means
satisfied.
change can be strongly labor saving
(1998, 2006),
and
is
also related to the
is
An
is
7
in specific
consider several different environments and production functions, and discuss
the decreasing differences condition
logical
and labor (though
important
class of
when
models where techno-
developed by Champernowne (1963) and Zeira
endogenous growth model of Hellwig and Irmen (2001).
In these models, technological change takes the form of machines replacing tasks previously
performed by labor.
I
show that there
is
indeed a tendency of technology to be strongly labor
saving in these models.
Most of the analysis
choices.
of
in this
paper focuses on the implications of labor scarcity on technology
Nevertheless, these results are also directly applicable to the question of the impact
wage push on technology choices
The
adjective "strongly"
is
added
in the context of a competitive labor
here, since the term
"labor saving"
is
market because, in
often used in several different
contexts, and in most of these cases, the "decreasing differences" conditions here are not satisfied.
7
The
change has been the key driving force of the secular increase in wages also
may be more likely than decreasing differences. Nevertheless, it should be
noted that in the context of a dynamic model, it is possible for the past technological changes to increase wage
levels, while current technology adoption decisions, at the margin, reduce the marginal product of labor. This
is illustrated by the dynamic model presented in subsection 5.1.
fact that technological
suggests that increasing differences
this context, a
minimum wage above
supply. 8 However,
I
also
wage push can be very
the equilibrium wage
is
equivalent to a decline in labor
show the conditions under which the implications
different
of labor scarcity
—particularly because the long-run relationship between labor
supply and wages could be upward sloping owing to general equilibrium technology
Even though the investigation here
is
effects.
motivated by technological change and the study
of economic growth, the economic environment
because
and
I
use
A
is static.
static
framework
useful
is
enables us to remove functional form restrictions that would be necessary to generate
it
endogenous growth;
it
for labor scarcity to
encourage innovation and technology adoption. This framework
thus allows the appropriate level of generality to clarify the conditions
on Acemoglu (2007). Section 2 adapts
this
framework
for the current
is
based
paper and also contains
a review of the results on equilibrium bias that are useful for later sections. The main results
are contained in Section
3.
These
results are applied to a
number
4.
Section 5 discusses various extensions. Section 6 concludes.
2
The Basic Environments and Review
This section
inclusion
is
is
based on, reviews, and extends some of the results
in
economy
N+
of production
is
labor, denoted by L,
for land, capital,
good and
and the
3,
in Section
Its
(2007).
but since
many
of
provide only a brief review. Consider a static
I
consisting of a unique final
models
Acemoglu
necessary for the development of the main results in Section
the results are already present in previous work,
and stand
of familiar
1
rest are
factors of production.
The
denoted by the vector
Z= (Z\,..., Zm)
and other human or nonhuman
factors. All agents' preferences are
defined over the consumption of the final good. To start with,
are supplied inelastically, with supplies denoted by
L
£
factor
first
R+
let
and
Z
us assume that
all
factors
£ IR^. Throughout
I
will
focus on comparative statics with respect to changes in the supply of labor, while holding the
supply of other factors, Z, constant at some
is
nothing special about labor). 9
level
The economy
Z
(though, clearly, mathematically there
consists of a
continuum of firms
(final
good
producers) denoted by the set T, each with an identical production function. Without loss of
The
implications of "wage push" in noncompetitive labor markets are more complex and depend on the
market imperfections and institutions. For example, Acemoglu (2003) shows that wage
push resulting from a minimum wage or other labor market regulations can encourage technology adoption when
there is wage bargaining and rent sharing.
"Endogenous responses of the supply of labor and other factors, such as capital, are discussed in subsections
5.2 and 5.3.
specific aspects of labor
this context, a
minimum wage above
supply. 8 However,
I
also
wage push can be very
the equilibrium wage
is
equivalent to a decline in labor
show the conditions under which the implications of labor
different
scarcity
—particularly because the long-run relationship between labor
supply and wages could be upward sloping owing to general equilibrium technology
Even though the investigation here
is
effects.
motivated by technological change and the study
of economic growth, the economic environment
because
I
use
is
static.
A
static
framework
useful
is
enables us to remove functional form restrictions that would be necessary to generate
it
endogenous growth;
it
for labor scarcity to
encourage innovation and technology adoption. This framework
thus allows the appropriate level of generality to clarify the conditions
on Acemoglu (2007). Section
2 adapts this
framework
for the current
are contained in Section
3.
These
results are applied to a
number
4.
Section 5 discusses various extensions. Section 6 concludes.
2
The Basic Environments and Review
This section
is
is
in
The main
economy
N+
of production
is
labor, denoted by L,
for land, capital,
good and
and the
1
rest are
in Section
Its
(2007).
but since
many
factors of production.
The
denoted by the vector
Z= (Z\,
and other human or nonhuman
defined over the consumption of the final good.
3,
results
of
provide only a brief review. Consider a static
I
consisting of a unique final
models
Acemoglu
necessary for the development of the main results in Section
the results are already present in previous work,
and stand
of familiar
based on, reviews, and extends some of the results
based
is
paper and also contains
a review of the results on equilibrium bias that are useful for later sections.
inclusion
and
To
are supplied inelastically, with supplies denoted by
factor
first
...,
Z/v)
factors. All agents' preferences are
start with, let us
assume that
all
factors
L £ R+ and Z £ R+. Throughout
I
will
focus on comparative statics with respect to changes in the supply of labor, while holding the
supply of other factors, Z, constant at some
is
nothing special about labor). 9
producers) denoted by the set
The
level
The economy
J7 each with an
,
Z
(though, clearly, mathematically there
consists of a
continuum of firms
identical production function.
(final
Without
good
loss of
more complex and depend on the
market imperfections and institutions. For example, Acemoglu (2003) shows that wage
push resulting from a minimum wage or other labor market regulations can encourage technology adoption when
there is wage bargaining and rent sharing.
Endogenous responses of the supply of labor and other factors, such as capital, are discussed in subsections
5.2 and 5.3.
implications of "wage push" in noncompetitive labor markets are
specific aspects of labor
any generality
let
normalized to
1.
I first
J-, \J-\,
to
1.
The
price of the final
good
is
also
describe technology choice in four different economic environments. These are:
Economy D
1.
us normalize the measure of
a decentralized competitive economy in which technolo-
(for decentralized) is
gies are chosen
by firms themselves. In
this
economy, technology choice can be interpreted
and the entire analysis can be conducted
as choice of just another set of factors
in
terms
of technology adoption.
2.
Economy E
(for externality) is identical to
nality as in
Romer
M
Economy
3.
(for
for
a technological exter-
(1986).
monopoly)
be the main environment used
will
remainder of the paper. In
in the
Economy D, except
for
much
of the analysis
economy, technologies are created and supplied by
this
a profit-maximizing monopolist. In this environment, technological progress enables the
creation of "better machines," which can then be sold to several firms in the final
sector.
Thus, Economy
M
incorporates Romer's (1990) insight that the central aspect
distinguishing "technology" from other factors of production
Economy O
4.
(for oligopoly) generalizes
Economy D
2.1
In the
first
for
Economy
Economy D,
6
T has
l
U G R+
,
Z
l
l
E R^, and 9 G
denote output, since
M
are then extended to this environment.
Y
is
l
,
Z
1
).
The
decided
introduced as a benchmark.
will
= G(L\Z\F),
C RK
is
(1)
the measure of technology.
be defined as net aggregate output below.
by assuming, throughout, that the production function
in (L
is
access to a production function
y
where
M to an environment including oligopolis-
markets are competitive and technology
all
by each firm separately. This environment
i
the non-rivalry of ideas.
—Decentralized Equilibrium
environment,
Each firm
Economy
is
competitive firms supplying technologies to the rest of the
tically (or monopolistically)
economy. The main results
good
cost of technology 9 e
©
in
terms of
G
final
is
I
I
use lower case y 1 to
simplify the exposition
twice continuously differentiable
goods
is
C (9).
Each
good producer maximizes
final
profits; thus,
solves the following problem:
it
'N
max
where u>l
by the
L
i
,Z
i
,e
= G(L ,Z\9 )-w L L -S" w Zj Z) - C
i
i
)
the wage rate and wzj
is
The
firm.
of labor
Tr(L
Vdi < L and
/
Ji<kT
JieT
Definition
prices
I
An
1
equilibrium in
refer to
%
any 8 that
l
1
T
}
N,
1, ...,
<
Z)di
is
Zj for j
=
(2)
,
taken as given
all
a total supply
is
...,N.
1,
a set of decisions {l/,
(3)
Z
1
x
,
„
8 }.
{w^,wz) and
solve (2) given prices
part of the set of equilibrium allocations,
is
)
denoted by wz- Since there
is
Economy D
{wl,wz) such that {L ,Z ,8
l
Z
l
(6
market clearing requires
total supply Zj of Zj,
/
=
the price of factor Zj for j
is
vector of prices for factors
and a
i
,
{L\ Z
and factor
(3) holds.
l
,
8
1
as equi-
}
librium technology. For notational convenience let us define the "net production function":
F{L\ Z\
Assumption
F{L\ Z\
1
Assumption
9
l
)
is
{
)
= G{L\ Z\
concave in {L\
restrictive, since
1 is
6
below
will relax this
Proposition
D
is
Economy
- C
l
(9
)
(4)
.
l
).
D
and technology. Such an assumption
to exist; the other
is
necessary for
economic environments considered
assumption.
Suppose Assumption
1
9
)
requires concavity (strict concavity or constant returns
it
to scale) jointly in the factors of production
a competitive equilibrium in
Z\
1
holds.
1
Then any equilibrium technology
8 in
Economy
a solution to
max F( I,
2,8'),
(5)
0'ee
and any solution to
Proposition
1
this
problem
is
is
we can simply focus
implies that to analyze equilibrium technology choices,
on a simple maximization problem.
equilibrium
an equilibrium technology.
a Pareto
An
optimum (and
important implication of
vice versa)
this proposition
is
maximum
of
and corresponds
to a
that the
F
in the
l
entire vector (L ,Z\9').
It is
also straightforward to see that equilibrium factor prices are equal to the marginal
products of the
G
or the
F
functions.
That
is,
the wage rate
is
u>l
= dG(L, Z,9)/dL =
dF{L, Z, 9)/dL and the prices of other
=
for j
1,...,N,
where 9
is
by
factors are given
w Zj =
dG{L, Z, 9)/dZj
= dF(L
Z, 9)/dZj
:
the equilibrium technology choice (and where the second set of
equalities follow in view of equation (4)).
An
important implication of
technology
nology,
(5)
should be emphasized at this point.
a maximizer of F(L, 2,9), this implies that any induced small change in tech-
is
cannot be construed as "technological advance" since
9,
output at the starting factor proportions. In particular,
Y
notation
2.
(L, 2, 9) to denote net output in the
Clearly, in
that
Y
is
Since equilibrium
Economy D,
differentiable in
Y
(1,2,9)
L and
9
= F
have no
effect
on net
for future use, let us introduce the
economy with
(1,2,9)
will
it
= G
factor supplies given by
(1.2,9) -
L and
C (9).
Then, assuming
9*
differentiable in
and that the equilibrium technology
is
L, the change in net output in response to a change in the supply of labor, L, can be written
as
dY{L,2,9")
dY{L,2,9*)
_
==
dl
dl
where the second term
is
the induced technology
+
dY [1,2,9*)
d9
effect.
d9*
dl''
When
this
term
is
strictly negative,
then a decrease in labor supply (labor scarcity) will have induced a change in technology that
increases output
second term
is
— that
is,
a "technological advance". However, by the envelope theorem, this
equal to zero, since 9*
is
Therefore, there
a solution to (5).
is
no
on
effect
net output through induced technological changes and no possibility of induced technological
advances because of labor scarcity in this environment.
Economy E
2.2
The
—Decentralized Equilibrium with Externalities
discussion at the end of the previous subsection indicated
why Economy D does
not enable
a systematic study of the relationship between labor scarcity and technological advances (and
in fact,
why
economy).
there
A
first
is
no distinction between technology and other factors of production
approach to deal with
this
problem
is
to follow
Romer
in this
(1986) and suppose
that technology choices generate knowledge and thus create positive externalities on other
firms. In particular,
suppose that output of producer
l
y
where
9
is
simplicity,
some aggregate
we can take
= G(V,Z
i
is
be the sum of
all
given by
l
,9\9),
of the technology choices of
9 to
now
(7)
all
other firms in the economy.
firms' technologies.
In particular,
if
9
For
is
a
= fie:F 9 k di
A'-dimensional vector, then 9 k
1,2,...,
K). The remaining assumptions are the same as before
L Z and 9
l
jointly concave in
Let us
each component of the vector
for
first
l
l
)
,
and a
(in particular,
slightly modified version of Proposition
1
(i.e.,
for
with
G
=
k
being
can be obtained.
now becomes
note that the maximization problem of each firm
N
max
z\z\e
ir{U,
Z\
9\
9)
U
Vw
Zj Z)
- C
(9
l
)
(8)
,
*r?
t
and under the same assumptions
in equilibrium,
= G{L\ Z\ 9\ 9) - w L U -
=L
as above, each firm will hire the
Z =Z
l
and
for all
£
i
!F.
Then the
same amount
of
all
factors, so
following proposition characterizes
equilibrium technology.
Proposition 2 Equilibrium technology
in
Economy E
is
a solution to the following fixed point
problem:
£cirgmaxG{L,Z,9',9
9
= 9)-C(9')
(9)
.
e'ee
Even though
this
is
a fixed point problem,
its
used in the same way for our analysis. However,
no longer applies to the equivalent of equation
as
Y(L, Z,
9)
—
G(L, Z, 9,9) —
C (9). Then
dY(L,Z,9*)
is
and induced
is, if
once again assuming
can be
us define net output again
differentiability,
dY{l,Z,9*)
dY{L,Z,9")
_
~
+
if
we have
d9*
dV
89
C (9)
G is increasing in the vector 9),
increases in 9 will raise output
it
envelope theorem type reasoning
(6). In particular, let
not equal to zero. In particular,
externalities are positive (that
very similar to (5) and
is
crucially, the
dl
dL
but now the second term
structure
is
increasing in 9
then 8Y/89
and thus correspond
will
and
if
be positive
to induced technological
advances.
2.3
Economy
M—Monopoly Equilibrium
The main environment used
technologies to final
for the analysis in this
good producers. There
is
paper features a monopolist supplying
a unique
final
good and each firm has access to
the production function
1
y
= a~ a (1 - ay G(L\ Z\ 9) a q
1
{
(9)'- a
,
(10)
q
with a G
This
(0, 1).
is
similar to (1), except that
production function, which depends on technology
G(L ,Z
l
8.
used by firm
9.
is
i
denoted by q
This intermediate good
is
{6)
9.
The more important
a subcomponent of the
The quantity
—conditioned on 8 to emphasize that
supplied by the monopolist.
as a convenient normalization.
now
,8) is
The subcomponent
bined with an intermediate good embodying technology
x
l
role of the
needs to be com-
of this intermediate
embodies technology
it
The term a~ a
G
— a)"
(1
parameter a
is
included
to determine
is
l
the elasticity of aggregate output with respect to the quantity of the intermediate q (8)
the subcomponent G. In particular, the higher
the less responsive
it is
more responsive
output to
is
is
models of endogenous technology
similar to
and Howitt,
1991, Aghion
1992), but
is
Romer, 1990,
(e.g.,
somewhat more general since
does not impose that technology necessarily takes a factor-augmenting form.
the vector
The monopolist can
8
C (8)
is
increasing in 8 (that
is
create (a single) technology 8 £
created,
denoted by
embodying
\-.
it is
increasing in each
component of
It
can then
The
Q
at cost
C (8)
from the technology
can produce the intermediate good embodying technology 8 at
it
constant per unit cost normalized to
normalization).
is,
Let us also
that greater 8 corresponds to "more advanced" technology.
8), so
menu. Once
(and
l
Grossman and Helpman,
focus on the case where
G
to q (8)).
This production structure
it
a, the
is
and
1
— a
unit of the final
good
(this is also a
set a (linear) price per unit of the intermediate
fact that the
monopolist chooses the technology 9 and
this technology to all final
good producers explains why
8
convenient
good of type
sells
8,
intermediates
was not indexed by
i
in
and each firm takes the available technology,
8,
(10).
All factor markets are again competitive,
and the
max
price of the intermediate
tt(L\
Z\
l
(9)
\9,x)
good embodying
= oT a
this technology, \, as given
Q
1
- a)" G{L\ Z\
(1
9)
l
q>
-a
(9)
and maximizes
N
-w L L'-Y w Z] Z)- X q
l
(9)
(ID
which gives the following simple inverse demand
its price, x,
and the factor employment
l
q
{\,L\Z
for intermediates of
type 8 as a function of
levels of the firm as
=a" G{L\Z\8)x' l/a
l
l
\8)
10
-
(12)
,
The problem
of the monopolist
to
is
maximize
its profits:
n = (x-(l-a))/
max
g* (*,'
L\ Z i
9) di
\
- C (9)
subject to (12). Therefore, an equilibrium in this economy can be defined
Definition 2
An equilibrium in Economy M is
technology choice
9,
and factor
prices
(wl,wz) and technology
(11) given
(wl,wz) such that {L ,Z
l
{.Z7,
l
l
,q
as:
Z
l
l
L Z
l
,q
(x,
l
[\,L ,Z
l
\
l
,
9)j.
\
9)}
_ solve
and the technology choice and pricing deci-
9; (3) holds;
maximize
sions for the monopolist, {9,x),
a set of firm decisions
(13)
(13) subject to (12).
This definition emphasizes that factor demands and technology are decided by different
agents (the former by the final good producers, the latter by the technology monopolist).
This
is
an important feature both theoretically and as a representation of how technology
determined
Assumption
demands and technology
in practice. Since factor
1
Assumption
are decided by different agents,
can now be relaxed and replaced by the following.
2
G(L\ Z\
concave in (L\
9) is
To characterize the equilibrium, note that
Zi
over marginal cost and
for all
i
is
£
equal to
T
.
x
—
(for all 9
)
G G).
demand
(12) defines a constant elasticity
so the profit-maximizing price of the monopolist
a~ l G(L, 2,9)
is
is
given by the standard
Consequently, q
1-
l
(9)
=
monopoly markup
1
q
curve,
(\
=
1,
L,
Z
\
&)
=
Substituting this into (13), the profits and the maximization
problem of the monopolist can be expressed as maxj e e
n (^) —
G{L, Z,9) —
C (9). Thus we
have established the following proposition:
Proposition 3 Suppose Assumption
M
is
2 holds.
Then any equilibrium technology
9 in
Economy
a solution to
maxF{L,Z,9') = G(L,
2,9')
- C (<?')
(14)
.
0'ee
and any solution
to this problem
is
an equilibrium technology.
This proposition shows that equilibrium technology
identical to that in
Economy D,
in
Economy
that of maximizing F(L, 2, 9)
Naturally, the presence of the
monopoly markup introduces
These distortions are important
in
=
M
is
G(L, 2,6) —
C (9)
as in (4).
distortions in the equilibrium.
ensuring that equilibrium technology
11
a solution to a problem
is
not at the level that
„
maximizes net output. In particular,
price
is
=
x
us use the fact that the profit-maximizing monopoly
let
and substitute (12) into the production function
1
C (9),
of technology choice,
and the
Y (L, 2,8) =
Clearly, since the coefficient in front of
[L, Z, 9) will
that
C (8)
Finally,
wl =
(1
—
it
q)
output function
as
Y~G
(L, 2, 6)
- C {6)
G [L, 2, 9)
,
8),
(15)
.
than
strictly greater
is
x
1,
as in
Economy
E,
a solution to (14) (recall
is
9).
dG(L, 2, 8)/dL and
Y
G
or the
=
u<zj
F
(1
- a)~ dG(L,
Z, 9)/dZj, which are proportional
function defined in (4) as well as the derivatives of the net
defined in (15). In what follows,
Economy O
It is also
economy
a~ 1 G(L\ Z
a)
can be verified that in this economy, equilibrium factor prices are given by
_1
to the derivatives of the
2.4
in this
be increasing in 9 in the neighborhood of 8* that
increasing in
is
—
cost of production of the machines, (1
from gross output. This gives net output
Y
and then subtract the cost
(10),
I
take
M
Economy
as the baseline.
— Oligopoly Equilibrium
straightforward to extend the environment in the previous subsection so that tech-
nologies are supplied by a
Let 9 be the vector 9
=
number
of competing (oligopolistic) firms rather than a monopolist.
(#i, ...,9s),
and suppose that output
is
now
given by
s
y>=a~ a (l-
1
a)' G(L\ Z\
8)^^ ^s)
X
~a
(16)
,
s=l
where 9 S 6
q\ (9 S ) is
Q s C R hs
is
a technology supplied by technology producer
s,
used by
final
good firm producer
supplied
demand
functions
intermediates as
gl (v s
,
V, Z
l
|
8)
= cT x G{L\ Z\ 9)x7 l/a
Equation (16) implicitly imposes that technology 8 S
This can be relaxed by writing
8S
,
and
Factor markets are again
i.
competitive, and a maximization problem similar to (11) gives the inverse
1
1,...,5,
the quantity of intermediate good (or machine) embodying technology 9 S
by technology producer
for
=
s
=
1
(<?3
(9 S
>
0)) 8 3
,
y'
= q" q
will
impact productivity even
1
(1
- a)' G(L\ Z*
(17)
,
~6')
,
a
£f=]
q' (e s
B
1_Q
)
,
if
firm
where
so that the firm does not benefit from the technologies that
it
i
chooses
¥ s
(e u
q' (S)
s
...,~0 s
\
=
0.
with
does not purchase. Let
8-s be equal to 9 with the 5th element set equal to 0. Then, provided that G(L,Z,9) — G(L, Z,6- s ) is not
too large, in particular, if G(L,Z,6) - G{L,Z~6- a ) < q(1 - a) G(L,Z,6)/ (S - 1), then the analysis in the
text applies. This latter condition ensures that no oligopolist would like to deviate and "hold up" final good
producers by charging a very high price.
12
.
where x s
s=
the price charged for intermediate good embodying technology 9 S by oligopolist
1S
1,...,5.
C
Let the cost of creating technology 9 S be
each unit of any intermediate good
An
Definition 3
<Ll ,Z
l
<L ,Z
r
such that
!
l
technology vector
,
\_q\
j
l
\
9)] s=1
1,
...,
—
(0i,
...,
maximize firm
\
=
1, ...,
5, (9 S x s ),
,
S. Consequently, with the
producer
will solve the
and factor
9s).
profits given
maximize
producing
prices
(wi,wz)
(wl,wz) and the
same
for
subject to (17).
its profits
=
a profit-maximizing price for intermediate goods equal to x s
s
cost of
a.
maximization problem of each technology producer
profit
The
and the technology choice and pricing decisions
(#i, ...,#s); (3) holds;
=
1
1,..,,S.
a set of firm decisions
is
technology choices
,
\
(x s ,L\Z
technology producer s
The
#)]
|
=
(9 S ) for s
again normalized to
Economy
equilibrium in
(x s ,£\Z
l
[q s
,
is
s
is
similar to (13)
1
f° r
an Y
@s
and implies
G ©s and each
steps as in the previous subsection, each technology
problem:
max n s
es
(6„)
=
ee,
G(L, Z, 6 lt
...,9 ai
...,
- Cs
9S)
(9„)
This argument establishes the following proposition:
Proposition 4 Suppose Assumption
is
a vector
(9\, ...,9 S )
such that 9*
2 holds.
is
max G(L,Z
each
for
s
=
1, ...,
S,
Then any equilibrium technology
in
Economy
solution to
>
and any such vector
9* ,...,9 ,...,9*
S
1
S
)-G
S
(9 S )
gives an equilibrium technology.
This proposition shows that the equilibrium corresponds to a Nash equilibrium and thus,
as in
Economy
results
E,
below and
it is
all
given by a fixed point problem. Nevertheless, this has
little effect
on the
of the results stated in this paper hold for this oligopolistic environment. 11
2
= for all s and s' is identical to the
It is also worth noting that the special case where d G/d9 s dd s
product variety models of Romer (1990) and Grossman and Helpman (1991), and in this case, the equilibrium can again be represented as a solution to a unique maximization problem, i.e., that of maximizing
G(L, Z, B\, ..., 9 S ..., 9s) — ^2 3 ~iC s {9 S )- Finally, note also that, with a slight modification, this environment
can also embed monopolistic competition, where the number of firms is endogenous and determined by zero
profit condition (the technology choice of non-active firms will be equal to zero in this case, and the equilibrium
problem will be max^ge,, G{L,Z,9\, ...,9 3 ..., 9*s 0, ..., 0) — C s (9 3 ) for 1 < s < 5", with S' being determined
endogenously in equilibrium.
1
'
i
,
,
,
,
13
Review of Previous Results on Equilibrium Bias
2.5
I
now
briefly review the previous results concerning the bias of technology in response to
These
changes in factor supplies.
though
results apply to all of the
for concreteness, the reader
may
vector and
let
Acemoglu
I
9 £ Q,
and
is
impose the following assumption, which
also
of the results in the next section.
Assumption
Recall that 9
(2007).
details
3 Let
for
G = R+ C (9)
.
each k
dC(9—k ,9. k
)
=
>
1, 2,
K -dimensional
a
a9 k
fl
Moreover, G(L,Z,9)
is
fc
,
be relaxed
for
twice continuously differentiable and strictly convex in
is
we have
=
and
o8 k
-.o
will
To
"
dC(9—k ,9. k )
hm
0,
K
...,
1
far,
and the
us denote the equilibrium technology at factor supplies (L, Z) by 6* (Z, Z).
simplify the discussion here,
some
Economy M. Further
wish to consider
proofs of these results can be found in
environments discussed so
dC(9 k ,9^ k
—
hm
fc
)
=
oc for
all 9.
a9 k
-too
continuously differentiable in 9 and L, and concave in 9 £ 0, and
satisfies
dG(L,Z,9 k ,9_ k )
v
'
=i g (0,+oo) for
'
all 9,
L,
and Z.
o9k
The most important
above and
will
part of this assumption, that
C
is
increasing,
was already mentioned
ensure that we can think of higher 9 as corresponding to "more advanced"
technology.
Assumption 3 enables us
exists (for k
=
1.
We
...,K).
theorem and ensures that d9*k /dL
to use the inverse function
say that there
is
weak (absolute) equilibrium
bias at [L,
Z)
if
^dwLd9l
2In other words, there
is
d9 k 8L ~
weak equilibrium
bias
if
the combined effect of induced changes in
technology resulting from an increase in labor supply
at the starting factor proportions
(i.e., it
'
is
to raise the marginal product of labor
"shifts out" the
shows that an increase
in labor supply (or the supply of
weak equilibrium
Conversely,
bias.
it
also
demand
for labor).
any other factor)
The next
will
result
always induce
shows that labor scarcity (here corresponding to a
decline in the supply of labor) will induce technological changes that are biased against labor.
'"See
Acemoglu (2007)
for versions of the results in this subsection
11
without this assumption.
.
Theorem
1
Suppose that Assumption
supplies (Z, Z) be 6* (L,Z). Then, there
K
and
3 holds
the equilibrium technology at factor
let
weak absolute equilibrium
is
bias at all (Z, Z),
i.e.,
a
Z%£ S>°tw(i,z),
dL
'd9 k
fc=i
with
strict inequality if
Note
^
d8*k /dL
for
also that while this result
and
some k
all
=
1,
of the other theorems below are stated for changes
changes are also equivalent to changes
in L, these
function (or equivalently the
F
function)
is
Finally, the next
change
in
in factor
proportions provided that the
Z
homothetic in L and
The
supplies of the other factors are held constant.
also endogenously respond to the
K.
...,
Z
or to
case in which
wage push
and, as assumed here, the
some
of these supplies
may
discussed in subsection 5.2.
is
theorem shows how general equilibrium technology choices can lead to an
increasing long-run relationship between the supply of a factor and
our analysis, this theorem
be analyzed
G
in a unified
clarifies
manner
its price.
In the context of
the conditions under which labor scarcity and wage push can
(see, in particular,
Corollary 3 and the analysis in subsection
5.3).
We
say that there
is
strong (absolute) equilibrium bias at (Z, Z)
dw L _ dw L
dL
dL
y^ dw L
89*
*-" d9 k
dL
if
k
'
.
fc=i
Clearly, here
holding 8
=
dw^/dL
9*
denotes the total derivative, while
(L,Z). Recall also that
Hessian with respect to {L,9),
negative semi-definiteness
Theorem
F
V 2 ^^^)^^),
is
is
F (L. Z,9ys
negative semi-definite at this point (though
not sufficient for local joint concavity).
.
Then
Hessian in [L,9),
serni- definite at (Z,
A number
Z)
denotes the partial derivative
jointly concave in (L, 9) at (L,8* (L\Z)), its
2 Suppose that Assumption 3 holds and
at factor supplies (L,
if
is
if
dw^/dL
there
V
2
is
let 9* (Z,
Z) be the equilibrium technology
strong absolute equilibrium bias at (Z, Z)
F(i^)(L,9) (evaluated at
(L,Z,9* (L,Z)),
is
if
and only
not negative
Z)
of implications of this
producer theory, where
all
demand
dogenous technology choices
theorem are worth noting.
curves are
downward
in general equilibrium
15
First, in contrast to basic
sloping, this result
shows that en-
can easily lead to upward sloping demand
^ 2 F^ L g^ L ^
curves for factors. In particular, the condition that
is
is
not negative semi-definite
not very restrictive, since technology and factors are being chosen by different agents (mo-
nopolists or oligopolists on the one
hand and
result also highlights that there cannot
Economy D, because
for
set
good producers on the
be strong bias in a
equilibrium existence in
production possibilities
Most importantly
final
Economy
and thus ensures that
D
fully
other). However, this
competitive economy such as
imposes convexity of the aggregate
V 2 F(^ m(£ g\
must be negative
our purposes here, this theorem implies that
when
V
2
semi-definite.
F( L g^ L
g)
is
neg-
ative semi-definite, then the equilibrium relationship (taking into account, the endogeneity of
technology) between wage and labor supply can be expressed by a diminishing function w*L (L).
We
will
make use
What do
of this feature in the next section.
the results presented in this subsection imply about the impact of labor scarcity
on technological progress? The answer
becomes more
is
almost nothing. These results imply that
scarce, either because labor supply declines or because
some regulation increases
wages above market clearing and we move along the curve w*L (L) to an employment
Z, technology will change in a
way
that
is
when labor
level
below
biased against labor. But as already discussed in the
Introduction, this can take the form of "technological regress" (meaning technology
is
now
less
of a contributing factor to output) or "technological advance" (change in technology increasing
output further). In a dynamic framework, these changes could take the form of the economy
foregoing
some
of the technological advances that
it
would have made otherwise or making
further advances, contributing to growth (see Acemoglu, 2002).
Therefore, these results are
not informative on the question motivating the current paper. Answers to this question are
developed in the next section.
3
Labor Scarcity and Technological Progress
In this section,
I
present the
main
results of the current
paper as well as a number of
relevant-
extensions.
3.1
Main Result
Let us focus on
that
= R+
,
Economy
so that 9
by the solution
is
M
in this subsection
and suppose that Assumption 3
a A"-dimensional vector
to the maximization
problem
and equilibrium technology choice
in (14).
10
holds. Recall
Assumption
is
given
3 also ensures equilibrium
technology 9* [L, Z)
uniquely determined and
is
dF(L,Z,9* (L.Z))
v
1
Y
Recall also that net output
C (9)
U =0fork =
'
(L,Z,9)
is
l,2,...,K.
given by (15), which, together with the fact that
strictly increasing, implies that
is
(L.Z))
dY(L,Z,e*
V
V
In light of this,
component
we say that
> oftxk =
LL
'
l,2,...,K.
there are technological advances
if
(18)
9 increases (meaning that each
of the vector 9 increases or remains constant).
The key concepts
of strongly labor (or
more generally
factor) saving technology
labor complementary technology are introduced in the next definition.
—
vector x
X
on
xn )
i, ...,
and only
if
(x, t)
/
(x
if
defined on
Kn
in
d2 f
X
(x)
T
x
/dx
in x.
The concept
be nonincreasing.
2
d f
for
(x, t)
each
/dxidt
of
is
If
>
/
differentiable
is
for
each
Bl and Bz
is
and
for all
and
t,
f
(x, t")
^
i'.
In addition, a function
has increasing differences (strict
)
—f
i
supermodular
is
(x, t) is
nondecreasing (increasing)
—f
defined similarly and requires / (x, t")
TcK,
and decreasing
then increasing differences
differences
is
equivalent to
2
d f
is
that
of
We
is
is
strongly labor saving at (L,Z)
G (L, Z, 8)
Theorem
equivalent to
(x, t)
<
/dxjdt
that
say that technology
supermodular
in 8.
there exist neighborhoods
strongly labor complementary at (L,Z)
L and Z such
3 Consider
if
exhibits decreasing differences in (L, 9)
G (L, Z, 9)
is
Economy
M
if
on
Br,
x
=
1,...,K,
technology
x
all
L,
Z
and
advances, in the sense that d9*k (L,Z)
if it
9 £ 0.
and suppose that Assumption
is
©
Bl x
strongly labor saving (labor complementary) globally
3 holds
and G(L,Z,9)
Let the equilibrium technology be denoted by 9* (L,Z).
if
Bz
there exist neighborhoods
is
Then labor
scarcity will induce technological advances (increase 9), in the sense that d9*k (L,Z)
each k
Bl and
exhibits increasing differences in (L, 9) on
strongly labor saving (labor complementary) for
for
to
(x, t)
i.
L and Z such
x G.
i
X
C R n and T C K m
X
of decreasing differences
t), if
x G
for all
>
Conversely, technology
Bz
>
dxi>
for all t"
Definition 4 Technology
Bz
t
(where
increasing differences) in (x,
Recall that given a
a twice continuously differentiable function / (x)
,
and strongly
jdL <
strongly labor saving, and will discourage technological
/dL >
for
labor complementary.
17
each k
=
1,...,K,
if
technology
is
strongly
.
Proof. From Assumption
3,
C (9)
is
This together with equation
strictly increasing in 9.
from
(18) implies that technological advances correspond to a change in technology
The
is
strictly
theorem.
(k
G is supermodular in
function
=
Therefore, a small change in
and comparative
1,...,A'),
strongly labor saving,
each k
=
1,...,K.
8.
will lead to a small
9 in the
This yields the
neighborhood of L, Z, and
L and
first
JdL >
for
This gives the second part of the desired
Though
each k
(14)
(L,Z)
exhibits decreasing or
When technology is
9* (L,Z).
and thus d9*k (L,Z) jdL <
when
=
1,...,K,
G exhibits
and labor scarcity
result.
simple, this theorem provides a fairly complete characterization of the conditions
under which labor scarcity and wage push
(in
competitive factor markets) will lead to tech-
nological advances (technology adoption or progress due to innovation).
are not covered by the theorem are those where
G
in each of 9*k
change
G
3,
9'
the implicit function
part of the desired result. Conversely,
d9*k (L,Z)
9,
L by
determined by whether
statics are
G exhibits decreasing differences in L and 9,
increasing differences in
reduces
L and
L
different! able in
is
>
to 9"
by assumption. Moreover, again from Assumption
concave and the solution 8* (L,Z)
increasing differences in
for
8
9'
G
is
positive, but
L and
of labor scarcity on each technology
''direct effect"
cases that
not supermodular in 9 and those where
exhibits neither increasing differences nor decreasing differences in
permodularity, the
The only
9.
Without
su-
component would be
because of lack of supermodularity, the advance in one component
may then
induce an even larger deterioration in some other component, thus a precise result becomes
impossible.
labor supply
When G
L
exhibits neither increasing or decreasing differences, then a change in
components
will affect different
of technology in different directions
cannot reach an unambiguous conclusion about the overall
parametric assumptions).
tion
is
Clearly,
automatically satisfied and
the neighborhood of L,
Z and
when
G
makes the
are multiple
is
either increasing or decreasing differences in
9* [L,Z).
this analysis
is
results not readily generalizable to a
ways of extending
will continue to
(without making further
single dimensional, the supermodularity condi-
must exhibit
Another potential shortcoming of
this
9
effect
and we
this
that the environment
dynamic framework,
is
static.
it is
Although
clear that there
framework to a dynamic environment and the main forces
apply in this case (see subsection
5.1 for
an extension to a growth model). The advantage of the
an illustration of this point using
static
environment
is
that
it
enables
us to develop these results at a fairly high level of generality, without being forced to
r-
make
some other notion of a
functional form assumptions in order to ensure balanced growth or
well-defined
dynamic equilibrium.
Further Results
3.2
The
Theorem
results of
Economy
3 apply to
E and
generalized to Economies
and
M and under Assumption 3.
also hold
without Assumption
3.
These
Here
I
results
can be
show how
this
can be done in the simplest possible way by focusing on global results (while simultaneously
relaxing
Assumption
The next theorem
3).
a direct analog of Theorem
is
equilibrium technology corresponds to the Nash equilibrium of the
3,
except that
game among
now
oligopolist
technology suppliers. As a consequence, multiple equilibria (multiple equilibrium technologies)
As
are possible.
is
well
known
(e.g.,
Milgrom and Roberts, 1994, Topkis,
are multiple equilibria,
we can
"extremal equilibria".
These extremal equilibria
greatest
refers to
Theorem
C (9)
is
9,
an increase
we must have
in the greatest
4 Consider Economy
strictly increasing in 9.
O
8'
>
>
8
if
9'
In this light, a technological advance
and the smallest equilibrium technologies.
technology
and 8" both
is
G (L, Z, 9)
supermodular
is
increase. If technology
and the smallest equilibrium technologies
In
Economy O,
Nash equilibrium
of a
8'
the equilibrium
game among
the
S
is
advances
and the smallest
in the sense that
the greatest
and 6" both decrease.
is
given by Proposition 4 and corresponds to a
oligopolist technology suppliers.
supermodular game. In addition, when
G
G
is
Inspection of the
supermodular
exhibits increasing differences in
globally, the payoff of each oligopolist exhibits' increasing differences in its
L.
and that
strongly labor complementary
corresponding profit functions of each oligopolist shows that when
this is a
in 9
strongly labor saving globally, then labor
globally, then labor scarcity will discourage technological
Proof.
statics for
there exists another
scarcity will induce technological advances in the sense that the greatest
equilibrium technologies
there
context correspond to the
9" (meaning that
9").
and suppose that
If
unambiguous comparative
in the present
& and
and smallest equilibrium technologies,
equilibrium technology,
now
typically only provide
when
1998),
own
in 8,
L and
strategies
9
and
Then, Theorem 4.2.2 from Topkis (1998) implies that the greatest and smallest equilibria of
when L
this
game
first
part follows with the
will increase
increases. This establishes the second part of the theorem.
same argument, using — 9 instead
labor saving globally.
19
of
8,
when technology
is
The
strongly
This theorem therefore establishes the global counterparts of the results in
Economy O, which
for
features further game-theoretic interactions
two
The
corollaries.
the oligopolists.
and Economy E, and are stated
next
in the
proofs of both corollaries are simple applications (or extensions) of the
M and suppose that G (L, Z, 9)
Corollary 1 Consider Economy
is
3
Theorem 4 and thus omitted.
proof of
C (9)
M
Economy
Similar global results also hold for
among
Theorem
strictly increasing in 9.
globally,
If
technology
is
and that
in 9
strongly labor saving [labor complementary]
then labor scarcity will induce [discourage] technological advances.
Corollary 2 Consider Economy E and suppose that
and that
supermodular
is
C (9)
is
strictly increasing in 6. If
G (L, Z, 9
technology
is
l
,
9) is
supermodular
in (9
l
,
9),
strongly labor saving [labor comple-
mentary] globally, then labor scarcity will induce [discourage] technological advances.
Nevertheless,
The main reason
hood
it is
important to emphasize that these results do not apply to Economy D.
for this
of an equilibrium in
advance".
Any
that, as already discussed in the previous section, in the neighbor-
is
Economy D,
there
is
no meaningful notion of "induced technological
small change in 9 will have second-order effects on net output and any non-
small change in 9 will reduce net output at the starting factor proportions (at
9* (Z,
Z) already maximizes output
3.3
Implications of
L and Z)
since
at these factor proportions.
Wage Push
Let us consider the same environments as in the previous subsections, and also suppose that
vF(L,6)(Lfi)
2,
is
negative semi-definite at the factor supplies (L, Z), so that, from
the (endogenous-technology) relationship between labor supply and wage
decreasing function w*L (L). This implies that
we can
is
Theorem
given by a
equivalent ly talk of a decrease in labor
supply (corresponding to labor becoming more "scarce") or a "wage push," where a wage
above the market clearing
employment
as
L e = min
level is
<
imposed. In this
(w*L )~
light,
(w Le ),L>, where
ui
e
L
we can
is
generally think of equilibrium
the equilibrium wage rate, either
determined in competitive labor markets or imposed by regulation. Under these assumptions,
all
of the results presented in this section continue to hold. This
Corollary 3 Suppose that
sumptions as
in
Theorems
3
V 2 F^ Lt g^ L ^
and
is
is
stated in the next corollary.
negative semi-definite.
4 or in Corollaries
20
1
and
2,
a
Then under the same
minimum wage above
as-
the market
clearing
wage
advances when technology
level induces technological
discourages technological advances
Proof. This
result follows
when technology
is
is
strongly labor saving
and
strongly labor complementary.
immediately from the observation that, when
negative semi-definite, a wage above the market clearing level
is
V 2 F(^ m^m
is
equivalent to a decline in
employment.
The
close association
V
assumption that
demand
curves are
push can have
between labor scarcity and wage push
F(l,6){l,6)
downward
richer effects
negative semi-definite, so that the endogenous-technology
1s
sloping (recall
and
this
is
Theorem
When
2).
discussed in Section
this is not the case,
in
employment.
this
Nevertheless,
pronounced, overall output
is
the case, net output
when the
may
may
should
it
decline because of the reduction
on technology
effect of labor scarcity
employment
increase even though
declines.
sufficiently
is
Consider the
following example, which illustrates both this possibility and also gives a simple instance
technology
Example
Let us focus on
Economy
M
G (L, Z, 0) =
and the
factor to
will
where
strongly labor saving.
is
1
wage
5.
While Corollary 3 shows that wage push can induce technological advances,
be noted that even when
on the
in this result relies
cost of technology creation
Z—
1
is
and suppose that the
39Z 1/3 +
C (9) =
36>
2
-9) L 1/3
/2.
Let us normalize the supply of the
,
to start with that equilibrium
be given by marginal product. Equilibrium technology
is
function takes the form
3 {I
and denote labor supply by Z. Suppose
Equilibrium wage
G
is
then given by 9* (Z)
given by the marginal product of labor at labor supply
=
1
Z
wages
— Z1
'
3
.
Z and technology
9:
w(L,9) = (\-9)L~ 2 l i
To obtain the endogenous-technology
.
relationship between labor supply
and wages, we
substi-
tute for 6* (Z) into this wage expression and obtain
w(L,e*(L)) = Z" 1/3
This shows that there
is
a decreasing relationship between labor supply and wages.
Suppose that labor supply Z
be
4.
.
is
equal to 1/64.
Next consider a minimum wage
at
w =
21
5.
In that case, the equilibrium wage will
Since final good producers take prices
demands;
as given, they have to be along their (endogenous-technology) labor
that employment will
fall
3/4, whereas after the
L
to
=
e
minimum
this implies
Without "wage push," technology was
1/125.
wage, we have 8* (L e )
=
4/5,
which
8* (Z)
illustrates the
=
induced
technology adoption/innovation effects of wage push.
Does wage push increase
(2
—
a) /
(1
—
a)
overall output? Recall that net output
G (L, Z,9) — C (9).
It
can be verified that
for
a
is
equal to
close to 0,
Y (L, Z, 9) —
wage push reduces
net output; however for sufficiently high a, net output increases despite the decline in employ-
ment. Generalizing this example,
output
F
2.
\BF
(Z, Z, 8) /8L\
a
sufficiently small.
(Z, Z, 8) (or
is
G
When
In this section,
nology
0cl,
IL, Z, 9)) exhibits decreasing differences in
2
\d
These conditions can be
4
can be verified that when
wage push
will increase
the following conditions are satisfied:
if
1.
3.
it
is
I
F
(Z. Z, 9)
economies
in
2
<
\
\d
2
F
(L, Z, 9)
easily generalized to cases in
/dL89\
which 8
is
6;
OF
{L, Z, 9) /88;
multidimensional.
Technology Strongly Labor Saving?
investigate the conditions under which, in a range of standard models, tech-
strongly labor saving.
is
/38
L and
which
The
this is the case,
results
though
show that
in
possible to construct a rich set of
it is
most models commonly used
and economic growth, technology turns out not
in
macroeconomics
to be strongly labor saving.
Throughout,
I
provide examples in which technology can be represented by a single-dimensional variable (thus
focus on
Economy M). This
is
to simplify the expressions
and communicate the basic ideas
in
the most transparent manner. As the analysis in the previous sections illustrated, none of the
results require technology to
be single dimensional.
Cobb-Douglas Production Functions with Harrod Neutral Technology
4.1
As a
first
example, suppose that the function G, and thus the aggregate production function
of the economy, takes a "Cobb-Douglas form" with Harrod-neutral technology. In particular,
let
us write this function as
G(L,Z,9) = H(Z){8L) a
22
,
where
H M^
->
:
where a £
R+, and thus aggregate net output
(0, 1) is
G
with respect to
L and
9 in this case
It is
Therefore, technology
wage push
It is
is
H
that
Gie
=
(2
-
a)
H (Z) (GL) a / (1 - a),
-1
>
0.
discourage technological advances.
La
~l
>
same conclusion holds
the production function
if
In this case, the assumptions
.
Supposing that
strictly increasing in 9.
0,
so that the
in Substitution
Patterns
ccHq (Z, 9)
Changes
=
always strongly labor complementary in this case and labor scarcity
G (L, Z,9) = H (Z,9) L a
must be
(L, Z, 9)
straightforward to verify that the cross-partial
a 2 H(Z)(9L) a
also straightforward to verify that the
(L, Z, 9)
4.2
is
will necessarily
modified to
Y
is
G L9 (L,Z,9) =
or
given .by
the parameter of the production function in (10), measuring the elasticity of
aggregate output to the subcomponent G.
of
is
it
is
same conclusion
imposed
so far
also differentiable,
is
imply
we have
reached.
Technological change that alters the substitution patterns across factors often turns out to be
strongly labor saving.
The
analysis in subsection 4.4 will illustrate this in detail, but a simple
example based on Cobb-Douglas production functions
G
the function
illustrates the intuition.
Suppose that
takes the form
G(L,Z,9) = A(9)H{Z)l}-\
where
A (9)
technology
is
a strictly increasing and differentiable function. From Proposition
Therefore, whether
we have
Glb{L,Z,6*) =
C (9*)
equilibrium).
which
will
-
'
- A{9*)
ta
[(1
-9*) A'
made
using (19),
be negative
C (9)
and the
positive or negative.
we can
C
'9"
\nl =
C {9*)
is
level of labor
.
(19)
supply and the economy,
In particular, suppose that in equilib-
chosen appropriately for this to be the case in
write
= (l-e*)e-A (9*) H (Z) L~ e
G LB
{L,Z,6*)
for
small enough.
e
1
-(1 -9*)A(9*)lnl-A{9*)]H(Z)L' B \
[9*)
eL (where the function
Then
H (Z) l
strongly labor saving technology depends on the sign of
that by modifying the function
L, this expression can be
rium
equilibrium
satisfies
A'(9*)H (Z)l 1
It is clear
3,
23
'
,
Factor-Augmenting Technological Change
4.3
Let us next turn to constant elasticity of substitution (CES) production functions with factor-
augmenting technology, which are
will see
also
commonly used
below that the important restriction here
simplify the discussion, let us continue to focus
by a single-dimensional
and
variable, 9,
we have
9
to distinguish between
two
on cases
Z
cases,
one
in
e
(0, 1),
and once
is
factor augmenting.
is
which
G
"augments"
9
To
represented
only one other factor of
This means
also single dimensional.
is
We
literature.
Z
and one
in
which
function can then be written as
[(l-^^ZJ^+r/L 2^
G{L,Z,9) =
r]
is
which technology
in
suppose that there
also
"augments" labor. Let us start with the former. The
for
macroeconomics
that technology
is
production, for example land or capital, and thus
that
in the
again, net output
equal to the same expression multiplied by
is
(2-a)/(l-a).
Straightforward differentiation then gives
G Le (L, Z, 9) =
7<r
+
1
—a
-!
777(1
a
-
,- g-i
Z~ y L)s
.
r,)
-,,-,
1
(i
-
g-l
v (9Z)^~
)
This expression shows that technology will be strongly labor complementary
either of the following two conditions are satisfied:
a <
1
(1)
Q--1
a -I
(i.e.,
G^e >
+ VL
L
7
=
1
if
(constant returns to scale); (2)
(gross complements).
Therefore, in this case for technology to be strongly labor saving
and a >
1
we would need both 7 <
(and in fact both of them sufficiently so) so that the following condition
is
(20)
a
This result can be generalized as to any
it is
also
homothetic
G(9Z,L), where
sufficient for
G
is
in
Z
and
G
featuring Z-augmenting technology (provided
L). In particular, for
homothetic.
It
1
satisfied.
l-7>-.
that
0)
any such G, we can write
can then be verified that (20)
is
G (L, Z,9) =
again necessary and
technology to be strongly labor saving, with 7 corresponding to the local degree
of homogeneity of
G
and a corresponding
qualifiers are added, since these
to the local elasticity of substitution (both "local"
need not be constant).
This result shows that with Z-augmenting technology, constant returns to scale
to rule out strongly labor saving technological progress. In addition, in this case
24
is
we
sufficient
also
need
—
Since 9
a high elasticity of substitution.
augmenting the other
is
factor, Z, a high elasticity
of substitution corresponds to technology "substituting" for tasks performed by labor.
somewhat
intuition will exhibit itself
when we turn
differently next,
to the
CES
This
production
function with labor-augmenting technology.
With labor-augmenting
G
technology, the
function takes the form
tr-l
G{L,Z,9)= (l-rj)Z—+ri(6L)-z
Straightforward differentiation
G L e{L,Z,e) =
Now
1
+
l 1 r {6L)°^
}
now
gives
a
^^{l-T )Z -^1 \x 1 ri{6Lr°
,,
I
1
,..<//.
.
defining the relative labor share as
the condition that
G^g <
is
—>
—
r](9L) _ wL L
~ =
=
w zZ
[\-r])Z
SL
;
Q>
equivalent to
sl<
As with condition
likely to
(20), (21) is
more
likely to
be strongly labor saving, when 7
returns. However,
21
is
now technology cannot be
be
satisfied,
and technological change
is
also intuitive.
When
and labor requires a high
9
augments
a <
9
more
smaller and thus there are strong decreasing
strongly labor saving
when a •>
opposite of the restriction on the elasticity of substitution in the case
This
is
when
1,
9
which
is
the
augments Z.
augments Z, a high degree of substitution between technology
elasticity of substitution, in particular,
labor, a high degree of substitution
a >
1.
In contrast,
when
between technology and labor corresponds to
1.
This result can again be extended to labor-augmenting technology in general.
again that
G (L, Z,9) =
G(Z,9L), with
G
homothetic.
Then
Suppose
(21) characterizes strongly
labor saving technology with 7 corresponding to the local degree of homogeneity of
G
and a
corresponding to the local elasticity of substitution.
This discussion highlights that with the most
common
production functions in macroeco-
nomics, Cobb-Douglas and production functions with a factor of minting technological change,
technology tends to be strongly labor complementary rather than strongly labor saving. Nevertheless, the latter possibility
is
not ruled out, at least
25
when
technological change affects the
patterns of substitution.
next turn to a setup where technological change more explicitly
I
replaces labor and thus affects the patterns of substitution between labor and other factors.
Machines Replacing Labor
4.4
Models
in
which technological change
labor have been proposed by
is
caused or accompanied by machines replacing
Champernowne
(1963), Zeira (1998, 2006)
and Hellwig and Irmen
and generalizing the paper by Zeira (1998).
(2001). Let us consider a setup building on
economy and explain why we need
describe a fully competitive
human
I
first
to depart from this towards
an
environment such as Economy M.
Aggregate output
given by
is
" ]
£=1
y(v)
where y
*
"
dv
denotes intermediate good of type v produced as
(u)
fc(")
jtjt
if
v uses new "technology"
it
v uses old
technology
I
where both
that T](v)
r\
is
[y]
and
/3
{v) are
assumed
to
a strictly increasing function,
,
be continuous functions and goods are ordered such
and
/3(f)
capital used in the production of intermediate u.
I
decreasing. In addition, k(y) denotes
is
use capital as the other factor of production
here to maximize similarity with Zeira (1998).
Firms are competitive and can choose which product to produce with the new technology
and which one with the old technology. Total labor supply
that capital
is
using the
L. For now, let us also suppose
supplied inelastically, with total supply given by K. Let the price of the final
good be normalized
is
is
new
to
1
and that of each intermediate good be p
technology. Clearly,
n
[y)
=
RT]{y)
where
w
is
which
is
strictly increasing
the wage rate and
be decreasing or
(5
R
is
1
We
write
n
(v)
—
1 if
v
whenever
<
w/3{i/)
,
the endogenously determined rate of return on capital. Let
by assumption. In
[v) could
[y).
fact, this is all that is required, so
be increasing as long as 7
26
(u)
is
strictly increasing.
rj
{u) could
we
In the competitive equilibrium,
=
n(v)
Since 7
increasing, its inverse
is
ages higher levels of 9*
now see
Let us
.
This
that this
is
have 9* such that
will
1
for all
1/
<
9*
wage to
and exploited
effect is highlighted
]
7
(w/R), so that
6>*.
also increasing, so a higher
is
=
rental rate ratio encour-
in Zeira (1998).
indeed related to decreasing differences.
With
the
same reasoning,
suppose that
=
n{v)
for
some
€
6
(0, 1) (since, clearly, in
1 for
any equilibrium or optimal allocation,
crossing" must hold). Then, prices of intermediates
P{
<9
a\\u
must
this type of "single
satisfy
(VMR Xv<0
\ 0(v)
'
w
v>9
if
'
Therefore, the profit maximization problem of final good producers
is
E-l
max
[y(
t,
dv
y{v)
)L S [o,i]
rj{v)y(v)dv-w
ill
j
Uo
0(v)y(v)dv,
which gives the following simple solution
y[
Now market
K, and
'
\{f3{v)w)-
clearing for capital implies
similarly,
market clearing
{u)
for labor gives
\l-e
A (9)
JQ k
n\y)
e
du and
Y \iv>9
dv
Jd
= fQ
fi
(v)
r/
~E
(u) y (u)
=f
dv
w~ E Y dv =
B {9)
L
/3{ls)
-E
77
(v)
~'~
R~ e Ydu =
L. Let us define
dls
(22)
IQ
Then the market
clearing conditions can be expressed as
l-e
R
Using (22) and
(23),
l-e
[|a OT
we can
,.„i
It
jB{9)
(23)
A(6)'K^ +B{9)c irr
Equation (24) gives a simple expression
9.
(
write aggregate output (and aggregate net output) as
Y
sector
Y
u,i-e =
can be verified that
Y
for
(24)
aggregate output as a function of the threshold
exhibits decreasing differences in
L and
9 in the competitive
equilibrium. In particular, equilibrium technology in this case will satisfy
dY
_
1
T}\
A(9*)~
K— -P{9*)
27
1
~e
B{9*)~ L
—
Y? =0.
Since the term in square brackets must be equal to zero,
82 Y
1
-B{e*)
390 L
-£
l-e
there
is
<0
an intimate connection between machines replacing
labor and technology being strongly labor saving. However,
economy because we
1
1
B(9*)-r L—'Y-<
e
why
This argument suggests
L
we must have
Theorem
3 does not apply to this
and thus dY/89
are in a fully competitive environment,
=
in
equilibrium
(and hence induced changes in technology do not correspond to "technological advances").
Motivated by
us consider a version of the current environment corresponding to
this, let
Economy M. To do
this, set
Z=
K and suppose that
G (L, K, 9) =
with cost C(9), e >
1,
and
economy ensures that an
we only have
fore,
that
Gl6
is
and
K^ +B{0)' L^
B (9)
whether technology
L and
9.
is
a >
in this
strongly labor saving, or whether
G
exhibits
proportional to
l-e
=
fact that
Straightforward differentiation and some manipulation imply
E-l
-0 (6*y- e B(6*)— L—G(L,
with Sl
The
defined as in (22).
increase in 9 indeed corresponds to a technological advance. There-
to check
decreasing differences in
A (9)
\A (9)°
u,'iL/[(2
be negative when
—
a)
C (9*)
further functional forms,
technology that
is
G (L, K, 9) / (1
is
small or
we cannot
—
when
A',
0)*
+ (2 -
a)
C {9*) SL
a)] as the labor share of income.
the share of labor
is
small.
,
This expression will
But without specifying
give primitive conditions for this to be the case. Instead,
strongly labor saving obtains easily
if
we
consider a slight variation on this
baseline model, where the constant returns to scale assumption has been relaxed, so that the
G
function takes the form
K^ +B{9) L^
1
G{L,K,9) = \a(6)*
Then
it
*
can be verified that
G Le =
so that technology
is
-^!3 (9*)
l
-£
B (9*)^
L~i <
0,
always strongly labor saving and a decrease in
advances.
28
L
will
induce technological
The
analysis in this subsection therefore shows that models
takes the form of machines replacing
technology. This
to
labor create a tendency for strongly labor saving
intuitive, since the process of
is
new technology
human
where technological progress
machines replacing labor
is
connected
closely
substituting for and saving on labor.
Extensions and Discussion
5
In this section,
discuss a
I
number
of issues raised by the analysis so far.
First,
show that
I
technology being strongly labor saving does not contradict the positive impact of secular technological changes
on wages. Second,
into this framework. Finally,
labor scarcity
show how endogenous
factor supplies can be incorporated
how wage push can
lead to very different results than
discuss
Technological Change and
5.1
One
demand curve
the endogenous-technology
with the conditions provided in Theorem
line
is
when
I
I
for labor
Wage
in the
upward sloping
(in
2).
Increases
objection to the plausibility of strongly labor saving technology
accompanied by a steady increase
is
wage
is
that the growth process
rate, while strongly labor saving
technology
implies that further technological advances will tend to reduce the marginal product of labor.
In this subsection,
I
increase wages while
My
reason,
purpose here
I
show that a simple dynamic extension allows technological change to
still
is
maintaining technology as strongly labor saving.
not to develop a
only use a slight variant of
communicate the main
ideas.
full
dynamic general equilibrium model.
Economy E and a simple demographic
The exact form
of the production function
is
For this
structure to
motivated by
the models in which machines replace labor such as those discussed in subsection 4.4, though
various different alternative formulations could also have been used to obtain similar results.
The economy
is
in discrete
time and runs to infinite horizon.
It
is
inhabited by one-
period lived individuals, each operating a firm. Therefore, each firm maximizes static profits.
The
total
measures of individuals and firms are normalized to
production,
L and
Z, both inelastically supplied in each period with supplies equal to
Z. Past technology choices create an externality similar to that in
suppose that
all
There are two factors of
1.
Economy
E. In particular,
firms are competitive and the production function of each at time
yi(Ll,Zi,elA t )=At
1+7
$)
(^ +
29
(i-0j)
1+7
(i4)
L and
t is
firm
(25)
where 7 <
Z and
and we assume that
£
is
sufficiently close to
1,
so that (25)
is
jointly concave in L,
This production function implies that higher 9 will correspond to substituting factor
6.
Z, which
may be
capital or other
human
or
nonhuman
factors, for tasks
performed by labor.
Suppose that
At =(l + g(6t-i))A -u
(26)
t
where g
at
time
(1986).
is
an increasing function and 9
t
= fie ~6ldi
the average technology choice of firms
is
t.
This form of intertemporal technological externalities
It
may
result, for
is
similar to that in
Romer
example, from the fact that past efforts to substitute machines or
capital for labor advance (and also build upon) the knowledge stock of the economy.
A
slightly
modified version of Proposition 2 applies in this environment and implies that
equilibrium technology 9* (Z, Z)
given by the solution to
is
dYJ(Z,Z,9*{L,Z),A
t
)
do
6* (L,
This implies that
2)
uniquely determined and independent of
is
A
t
,
in particular,
given
by
6*(Z,Z) =
1
Since 7
<
and e
>
1,
6* (L,
2)
is
+
(Z/L)
^£(0,1).
y
decreasing in Z, so labor scarcity increases 9* (Z, Z). Then,
(26) implies that a higher equilibrium level of 0* (Z,
and wages. This
is
Z
will lead to faster
growth of output
despite the fact that, at the margin, labor scarcity increases 9* (Z, Z)
substitutes for tasks previously performed by labor.
in
Z)
will also increase 9* (Z,
Z)
.
It
highlights that in a
can be easily verified that an increase
The immediate impact
of wages, but this change will also increase the rate of
and
of this will be to reduce the level
wage and output growth. This
result
dynamic framework with strongly labor saving technology, the short-run
and long-run impacts
of technological advances
on wages
will typically differ.
This analysis thus shows that in a dynamic economy, there
nological changes leading to a secular increase in wages
is
no tension between tech-
and technology being strongly labor
saving.
5.2
To
Endogenous Factor Supplies
highlight the
new
treated the supply of
results of the
all
framework presented
in this paper, the analysis so far
has
factors as exogenous. This ignores both the response of labor supply
30
to changes in
wages and also the adjustment of other
factor supplies or labor market regulations that create
is
factors,
such as capital, to changes in
wage push. Endogenous labor supply
further discussed in the next subsection.
Here
us focus on endogenous supply of other factors.
let
situation in which one of the other factors of production
is
we can imagine
For example,
capital that
is
a
elastically supplied. In
will affect
both technology and the supply of capital so that
the rental rate of capital remains constant.
Consequently, the overall impact on technology
change in labor supply
this case, a
will
be a combination of the direct
effect of labor
supply and an indirect effect working through
the induced changes in the capital stock of the economy. Although the details of the analysis
are
somewhat
in Section 3
we can note
different in this case,
that the essence of the
main
remain unchanged. In particular, those results were stated
in
results presented
terms of strongly
labor saving [complementary] technology or decreasing [increasing] differences of the function
G holding the supply of other factors fixed.
One can
alternatively define notions of labor saving
[complementary] technology (or decreasing [increasing] differences) holding the price of capital
constant.
It is
then straightforward that Theorems 3 and 4 would apply with these modified
definitions.
Wage Push
5.3
Let us
now suppose
function
L
s
(wl).
Labor Scarcity
vs.
that the supply of labor
From
results will be affected
ticular, in this case,
if
endogenous, given by a standard labor supply
the analysis leading to Corollary
(u>l), or
then clear that none of the
3, it is
vF^L,e)(L,e) i s negative semi-definite
we can study the impact
where L s {wi) < L s
is
and L s
(u>l) is increasing. In par-
of a shift in labor supply
the impact of a binding
minimum
from L s (wl) to L s (wl),
wage.
Since
V
F(Le)(L6)
1S
negative semi-definite, the endogenous-technology relationship between employment and wages
is
decreasing. Therefore, a leftwards shift of the labor supply schedule from
will
reduce employment and increase wages. The implications
by whether technology
Theorems
3
and
4;
is
for
Ls
(wl) to
L s (wl)
technology are determined
strongly labor saving or strongly labor complementary (according to
see Corollary 3).
However, the close connection between wage push and labor scarcity highlighted in Corollary 3
is
broken when
technology
vF^ LS Lg
demand curve
^,
is
\
is
not negative semi-definite. In this case, the endogenous-
upward sloping and thus a decrease
31
in labor supply reduces wages,
.
Figure
whereas an increase
when
labor supply
is
in labor
The next example
Example
by
that
V
F(L,e)(L,e) 1S
from Theorem
2,
in the previous subsection. In this case, multi-
and wages, become
different levels of labor supply, technology
illustrates this possibility using a simple extension of
2 Suppose that the
L 2 / 3 /2, and
8)
This becomes particularly interesting
supply increases wages.
G
C (9) —
is
no longer negative semi-definite (where as
we expect the endogenous- technology
supply) and wage to be increasing.
We
will
now
see
2
3# /4.
was
it
in
this interacts
1
except
2' 3
/2
+
can now be verified
It
Example
relationship between
how
1,
G(L,Z,8) = 3#Z
In particular, suppose that
the cost of technology creation
Example
Example
function takes a form similar to that in
for a slight variation in exponents.
3(1 —
Multiple Equilibria
endogenous as discussed
ple equilibria, characterized
possible.
1:
1).
Therefore,
employment (labor
with endogenous labor
supply.
Z
Let us again normalize the supply of the
L e Equilibrium
.
w (L e
,
9)
=
(1
—
technology then
9)
(L e )~
'
satisfies 8*
(L e )
for a given level of
response of 8 to employment
L
e
,
factor to
=
illustrates the potentially
is
also responsive to
(L e )
'
.
and denote employment by
Equilibrium wage
is
given by
we have
{L
e^l/3
e
(27)
)
upward-sloping endogenous-technology relationship between
employment and wages demonstrated more generally
supply
—
'= 1
technology 9 and once we take into account the
w(L e ,9*(L e )) =
which
1
Z
in
Theorem
wage and takes the form L s (w)
32
2.
= 6w —
2
Now
llu>
suppose that labor
+ 6. Now
combining
this supply relationship
levels of labor
with (27), we find that there are three equilibrium wages, with different
supply and technology,
and labor supply
is
w=
highest at
w=
Thus the implications
of
destroy the other equilibria. Figure
wage indeed destroys the
there
is
existed,
no employment, or
more
likely.
it
it
1
and
Moreover, technology
3.
This
is
most advanced
will typically destroy
illustrates the situation diagrammatically.
the
first
w =
1
and
w —
2.
Nevertheless,
it
two
may
The minimum
some caution
is
also
can also introduce an extreme no-activity equilibrium, where
may make
When we
2
3.
"wage push" are potentially very different when
equilibria at
necessary. In certain situations
and
3.
Next consider a minimum wage between
equilibria.
2
1,
are in
such a no-activity equilibrium, which
Economy M, such a
may have
already
no-activity equilibrium does not exist,
because the monopolist acts as a "Stackleberg leader" and chooses the technology anticipating
employment. However,
of
6
minimum wage
is
in
Economy O, such a
no-activity equilibrium
may
arise
if
a high level
imposed.
Conclusion
This paper studied the conditions under which the scarcity of a factor encourages technological
progress (innovation or adoption of technologies increasing output). Despite a large literature
on endogenous technological change and technology adoption, we do not yet have a comprehensive theoretical or empirical understanding of the determinants of innovation, technological
progress, and technology adoption.
abundance
Most importantly, how
or scarcity of labor, affect technology
is
factor proportions, for example,
poorly understood.
In standard endogenous growth models, which feature a strong scale effect, an increase in
the supply of a factor encourages technological progress.
In contrast, the famous
Habakkuk
hypothesis in economic history claims that technological progress was more rapid in 19th-
century United States than in Britain because of labor scarcity in the former country. Related
ideas are often suggested as possible reasons for
why high wages might have encouraged more
rapid adoption of certain technologies in continental Europe than in the United States over
the past several decades.
The famous Porter hypothesis has
a similar logic and suggests that
environmental regulations can be a powerful inducement to technological progress.
This paper characterizes the conditions under which factor scarcity can induce technological
advances (innovation or adoption of more productive technologies).
33
I
introduced the concepts
of strongly labor (factor) saving
particular, technology
is
and strongly labor
strongly labor saving
differences in an index of technology, 6,
complementary
main
is
and
if
(factor)
complementary technology.
the production function exhibits decreasing
Conversely, technology
labor.
is
strongly labor
the production function exhibits increasing differences in 6 and labor.
if
paper shows that labor scarcity induces technological advances
result of the
if
is
strongly labor complementary.
wage push
I
also
show
that,
The
technology
strongly labor saving. In contrast, labor scarcity discourages technological advances
nology
In
if
tech-
under some additional conditions,
— an increase in wage levels above the competitive equilibrium—has similar effects
to labor scarcity.
I
also provided
examples of environments
labor saving and showed that such a result
results clarify the conditions
is
in
which technology can be strongly
not possible in certain canonical models. These
under which labor scarcity and high wages are
likely to
encourage
innovation and adoption of more productive technologies, and the reason
why such
obtained or conjectured in certain settings but do not always apply in
many models used
results
were
in
the growth literature.
While the theoretical analysis provided here
scarcity
and wage
conditions
may
(factor price)
or
may
clarifies
push may induce innovation and technology adoption, these
not hold depending on the specific application (time period, institu-
tional framework, the industry in question, etc.).
necessary to shed light on
may encourage
effect of
when
factor scarcities
This suggests that empirical evidence
and various regulations
innovation and technology adoption.
a possibility, but
is
the conditions under which factor
not conclusive.
is
affecting factor prices
Existing evidence suggests that this
is
For example, Newell, Jaffee and Stavins (1999) show an
changes in energy prices on the direction of innovation and on the energy efficiency
of household durables
and Popp (2002) provides similar evidence using patents.
and Finkelstein (2008) show that the Prospective Payment System reform
Acemoglu
of Medicare in the
United States, which increased the labor costs of hospitals with a significant share of Medicare
patients, appears to have induced significant technology adoption in the affected hospitals.
In a different context, Lewis (2007) shows that the
skill
mix
in
US
metropolitan areas ap-
pears to have an important effect on the choice of technology of manufacturing firms. Further
research might shed
more systematic
light
on the empirical conditions under which we
may
expect greater factor prices and factor scarcity to be an inducement, rather than a deterrent,
to technology adoption
and innovation.
34
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