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DEv'siE)
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Labor-
and Capital-Augmenting
Teclnnical Ctiange
Daron Acemoglu
Working Paper 03-02
December 2002
Room
E52-251
50 Memorial Drive
Cambridge,
This
MA
02142
paper can be downloaded without charge from the
Network Paper Collection at
Social Science Research
http://papers.ssm. com/abstract_id=372820
December 2002
Labor- and Capital- Augmenting Technical Change^
Daron Acenioglu'
Abstract
^s-
I
analyze an economy in which firms can imdertake both labor- and capital-augmenting
economy resembles the stfrndard grow'th
model with purely labor-augmenting technical change, and the share of labor in GDP
is constant. Along the transition path, however, there is capital-augmenting technical
change and factor shares change. Tax policy and changes in labor supply or savings typically change factor shares in the short nm. but have no or little effect on the long-nm
technological improvements. In the long rim, the
factor distribution of income.
KeyM'ords: Economic Growth, Endogenous Growth, Factor Shares, Technical Change.
JEL
'I
Classification: 033, 014, 031, E25.
thank Ruben Segura-Cayuela
for excellent research assistance,
and Manuel Amador, Abhijit Ba-
Chang Hsieh, Steven Lin, Robert Lucas, Torsten Persson, Mathias Thoenig,
Jaume Ventura, two anonymous referees, and seminar participants at the Canadian Institute for Advanced
Research, Harvard, MIT, the NBER Economic Fluctuations meeting, and Rochester for useful comments.
I also gratefully acknowledge financial help from The Canadian Institute for Advanced Research and the
narjee, Olivier Blanchard,
National Science Foundation Grant SES-009.52-53.
t
Massachusetts Institute of Technology-, Department of Economics, E52-.371, Cambridge,
for Advanced Research; e-mail: daronQmit.edu
and the Canadian Institute
MA
02319,
Introduction
I.
Figures
and
1
show the shares
2
of
GDP
accrmng
The
over the past 80 years (with the remainder accrmng to capital).'
well
known, pattern
significant capital
is
is
years. For example, in
Almost
II.
and exogenous
movements
all
both countries, there
run (despite
all
a large increase in the share of labor after
is
models of growth and capital accumulation, of both endogenous
types, explain the stability of factor shares using
technical change
With an
in the long
but
in the share of labor over periods as long as 10 or 20
either the elasticity of substitution
or
show no trend
first striking,
deepening during the same period). The second important observation
that there are large
World War
that these factor shares
and France
to labor in the U.S.
is
between capital and labor
one of two assumptions:
is
taken to be equal to
assumed to be labor augmenting (Harrod
elasticity of substitution
between capital and labor
1,
neutral).''
ecjual to 1,
i.e.,
with a
Cobb-Douglas production fvmction, the shares of capital and labor are pinned down by
technology alone (as long as firms are along their factor
suppose that the aggregate production function
and L
is
labor.
Then, the share of labor
will
is
y =
demand
curves).
AL'^K^~°' where
reality,
K
is
capital,
always be ecjual to a. There are reasons to
be skeptical that the Cobb-Douglas production function provides an
approximation to
For example,
entirely satisfactory
however. First, most estimates suggest that the aggregate elas-
ticity of substitution is significantly less
than
1.^
Second, a production fimction with an
does not provide a framework for analyzing fluctuations in
elasticity of substitution of
1
factor shares, such as those
shown
in Figures 1
and
2.
'The French data are from Piketty (2001), and the U.S. data are from Piketty and Saez (2001), who
Income and Product Accounts data.
-A third possibihty is that the aggregate production function is Y' = F{K,H) where H is human
capital, accumulating at the same rate as A', so that there is no "capital deepening". However, the rate
of accumulation of human capital appears to be substantially less than that of physical capital. For
example, in the U.S., average schooling of the workforce increased by about one year in every decade in
in turn use the National
the postwar era, which translates roughly to a 6 percent increase in the
human
capital of the workforce
Acemoglu and Angrist, 2000), compared to an approximately 4 percent a year growth in the capital
stock between 1959 and 1998 (see The Economic Report of the President, 1999).
^For example, Nadiri (1970), Nerlove (1967) and Hamermesh (199.3) survey a range of early estimates of
the elasticity of substitution, which are generally between 0.3 and 0.7. David and Van de Klundert (196.5),
(e.g.,
similarly estimate this elasticity to be in the neighborhood of 0.3. Using the translog production function.
and Gregory (1976) estimate elasticities of substitution for nine OECD economies between 0.06 and
See also Eisner and Nadiri (1968) and Lucas (1969). Berndt (1976), on the other hand, estimates an
elasticity of substitution equal to 1. but does not control for a time trend, creating a strong bias towards
Griffin
0.52.
Using more recent data, and various different specifications, Krusell, Ohanian, Rios-Rull, and Violante
1. Estimates implied
by the response of in\'estment to the user cost of capital also typically yield an elcisticity of substitution
between capital and labor significantly less than 1 (see, e.g., Chirinko, 1993, Chirinko, Fazzari and Mayer,
1999, and 2001. or Mairesse, Hall and Mulkay, 1999).
1.
(2000) and Antras (2001) also find estimates of the elasticity significantly less than
The
sical
patterns depicted in Figures
1
production function, but require
and 2 are consistent with a more general neoclas-
all
technical change to
be labor augmenting and to
More
take place exactly at the same rate as the rate of capital deepening.
consider an aggregate production function of the form
Y = F{MK, NL).
specifically,
The assump-
tion of labor-augmenting technical change implies that technical progress only increases
A'',
and does not
ital axis.
A
affect
M—or in other words,
it
rotates the isoquants around the cap-
neoclassical production function with (purely) labor-augmenting technical
change provides an attractive framework
for
macroeconomic
analysis, since
tent not only with the long-run stability of factor shares, but also with
why do
It is in fact
Barro and Sala-i-Martin, 1995).
(e.g.,
However, this framework raises another important question: why
labor-augmenting? Or equivalently,
consis-
medium-term
swings in response to changes in capital stock, labor supply or technology.
the starting point of graduate textbooks on growth
it is
technical change
is all
profit-maximizing firms choose innovations
that only increase A^? Although starting with Romer's (1986, 1990) and Lucas' (1988)
contributions a large literature has investigated the determinants of technological progress
and growth,
the direction of technical change
of increases in
A'^
—has received
In this paper
technical change.
I
I
little
—the reason why
all
progress takes the form
attention.
investigate the forces that
push the economy towards labor-augmenting
analyze an otherwise standard endogenous growth model where profit-
maximizing firms can invest to increase both
M and
Y = F{MK,NL).
that capital,
The only asymmetry
labor, L, cannot.^
I
show that
in this
is
economy
A'^
in
all
terms of the production function
K, can be
accimiulated, while
technical progress will
be labor-
augmenting along the balanced growth path. Hence, given the standard assimiptions
endogenous
gi-owt-h,
the result that long-run technical change must be labor-augmenting
follows fi'om profit-maximizing incentives.
capital
and the
for
interest rate
remain
to labor- augmenting technical change
Conseciuently, in the long run, the share of
stable, while the
and
wage rate
capital deepening.
In
increases steadily due
some
therefore provides a microfoundation for the basic neoclassical growth
sense, this
paper
model with labor-
augmenting technical change.
Notably, however, while the balanced growth path of this economy resembles the stan-
dard neoclassical model, along the transition path there
technical change.
That
is,
pvirely
is
typically capital-augmenting
labor-augmenting technical change
is
only a long-nm
phenomenon.
I
also
show that
as long as capital
and labor are gross complements,
i.e.,
as long as the
"•The important assumption is that (efficiency units of) labor cannot be accumulated asymptotically,
which appears reasonable with finite lives, since individuals will have on a limited time to invest in human
capital. See Jones (2002) on this, and also footnote 2.
between these two factors
elasticity of substitution
with
labor-augmenting technical change
pvirely
and
eciuilibrivim,
stable.
it is
The
capital-augmenting techniques
is
complement or use
than
1
,
is
increjising in the share of capital in
capital increase the
capital. Consequently,
demand
when the share
less
than
new
for
new
GDP —both a higher
new
of capital in
be further capital-augmenting technical change. With the
these
the profitability of
intuitive:
will
1,
the balanced growth path
the unique asymptotic (non-cycling)
is
stability property
and a larger supply of
interest rate
is less
technologies that
GDP is large,
there
elasticity of substitution
technologies will reduce the share of capital, pushing the
economy
towards the BGP.
In addition to providing an explanation
why long-run
ft)r
technical change
augmenting, the framework presented here also suggests a reason
for the long-rim stabil-
The
shares despite major changes in taxes and labor market institutions.
ity of factor
neoclassical
labor
is
growth model with labor-augmenting technical change predicts a constant
long-run share of labor, but this share should respond to policies which affect the capitallabor ratio. In contrast,
show that
I
framework here with both capital-augmenting
in the
and labor-augmenting technical change, a range of
second-order effects) on long-nm factor shares:
will also
have an offsetting
effect
suggest that the framework here
growth model, but
tive statics that
policies will
they
have no
effect (or
will affect capital-deepening,
on capital-augmenting technical change. These
is
only
but
results
not only useful as a microfoundation for the standard
for policy analysis as well:
it
points out that a
number
of compara-
assume capital-augmenting technical change away may give misleading
answers.
It is
useful to briefly outline the intuition for
long-run technical change
is,
the invention of
new
labor-intensive goods. ^ Similarly, capital-
augmenting progress corresponds to the invention of new capital-intensive goods. In
economy, new goods
sale.
When
there are
intensive
good
workers,
and
wage
rate, w.
is
will
be introduced because of future expected
n labor-augmenting goods, the
proportional to
its profits
profitability of
profits
are given by a
markup over the marginal
Similarly, w^hen there are
m
this
from their
an additional labor-
— because each intermediate good producer
will hire
cost of production
^
— the
capital-intensive goods, profits from further
capital-augmenting progress are proportional to
When
will
Suppose labor-augmenting progress takes the form of "labor-
be labor augmenting.
using" progress, that
why
—
,
where r
is
the rental rate of capital.
technical progress relies on the use of scarce factors such as labor, long-run growth
^In principle, there are two ways to model labor-augmenting technical progress: as the introduction
of
new production methods
that directly increase the productivity of labor, or as the introduction of new-
goods and tasks that use labor. Here, I discuss the first formulation. Later, I will show that the same
results appl\' when labor-augmenting progress takes the form of "lalsor-enhancing" progress.
requires that further innovations build "upon the shoulders of giants"
in
n and
m
have to be proportional to their existing
fLirther resoui'ces to
labor-augmenting innovation
the return to capital-augmenting innovation
be balanced
will
is
is
levels.'^
,
that
The return
is,
to allocating
therefore proportional to
proportional to
m
•
^.
increases
n
•
—
These two returns
for a specific factor distribution of income.
Furthermore, when the elasticity of substitution between capital and labor
than
1
,
n
a high level of
m
relative to
implies that the share of capital
to the share of labor. This will encourage
increase
while
,
m. The converse applies when
m
more capital-augmenting
is
is
is less
high compared
technical progress
and
too high. Eciuilibrium technical progress
will
therefore stabilize factor shares.
Finally, capital
accumulation along the balanced growth path implies that technical
n more than m.'
progress will increase
Intmtively, there are
two ways to increase the
production of capital-intensive goods, via capital-augmenting technical change and capiaccumulation, and only one way to increase the production of labor-intensive goods,
tal
through labor-augmenting technical change. Capital accumulation, therefore, implies that
technical change has to be,
on average, more labor-augmenting than
capital- augmenting.
In fact, the model implies a stronger result: with an elasticity of substitution between capital
and labor
less
than
1,
run there
in the long
change, rn will remain constant, and
The
all
will
be no net
capital- augmenting technical
technical change will be labor augmenting.
ideas in this paper are closely related to the induced innovation literatm-e of the
1960s and to Hicks' discussion of the determinants of eciuilibrium bias of technical change
in
The Theory of Wages (1932). Hicks wrote: "A change in the
of production
is itself
a
spvir to invention,
and to invention of a
to economizing the use of a factor which has
124-5). Fellner (1961)
expanded on
relative prices of the factors
become
this argimient
partici.ilar
kind
relatively expensive."
—directed
(1932, pp.
and suggested that technical progress
was more labor-augmenting because wages were growing, and were expected to grow, so
technical change woi_ild try to save
on
this factor that
was becoming more expensive. Li
an important contribution, Kennedy (1964) argued that innovations should occur so as to
keep the share of
GDP
®Rivera-Batiz and
work
accruing to capital and labor constant. Samuelson (1965), inspired
Romer
(1991) refer to this case as the knowledge-based specification.
in this area supports the notion of substantial spillovers from past research, e.g.
and Henderson (1993).
what n and m, correspond to in
Empirical
Caballero and
Jaffee (1993), or Jaffee, Trajtenberg
^An important question
is
this question precisely within the context of a stylized
model,
Although it is difficult to answer
seems plausible to think of many of the
practice.
it
major inventions of the 20th century, including electricity, new chemicals and plastics, entertainment, and
computers, as expanding the set of tasks that labor can perform and the types of goods that labor can
produce. In contrast, some of the early important technological improvements, such as the introduction
of coke, the hot blast, the Bessemer process, can be viewed as capital-augmenting advances, since they
reduced the costs of capital and other nonlabor inputs, see Habakkuk (1962, pp. 1-57-159).
by the contributions of Kennedy and
firms chocjse
M and
in
A'^
reduced form model where
Fellner, constructed a
terms of the production function above in order to maximize the
He showed
instantaneous rate of cost reduction.
that under certain conditions, this would
imply equalization of factor shares. Samuelson also noted that with capital
tecfmical change would tend to be labor-augmenting.
(1973), criticized this whole literature, however, because
was not
clear
My
who undertook
paper
change, as
in,
RfcD
among
others,
Romer
for the analysis here,
I
allow for both
is
lacked microfoundations:
it
priced.
Anant and Dinopoulos
(1990),
and Aghion and Howitt (1992, 1998), where innova-
(1991a,b),
paper
it
and how they were financed and
(1990), Segerstrom,
by profit-maximizing
tions are carried out
rest of the
activities,
Others, for example Nordhaus
but starts from a microeconomic model of technical
revisits this territory,
Grossman and Helpman
The
the
accrrnivilation,
firms. In contrast to these papers,
la]:)or-
and
crucial
and capital-augmenting innovations.
organized as follows.
The next
section outlines the basic
environment, and characterizes the asymptotic equilibria and the balanced growth path.
Section
III
analyzes transitional dynamics, and shows that with an elasticity of substitu-
tion less than
the
1,
economy tends
to a balanced growth path with stable factor shares
and labor-augmenting technical progress. Section IV analyzes the consequences of a range
of policies on the factor distribution of income.
Section
V
investigates the implications
of alternative formulations of the "irmovation possibilities frontier," extends the model to
allow for the production and
same
R&D
sectors to
compete
for labor,
and
all
Modeling The Direction of Technical Change
A.
The Environment
consider an
sector,
and S
scientists
for workers.
I
economy consisting of L
"scientists"
is
VI concludes,
the proofs.
II.
and
shows that the
results obtain with different formulations of technical change. Section
while the Appendix contains
I
also
unskilled workers
who perform RfcD. The
who work
distinction
in the
production
between unskilled workers
adopted to ensure that the production and RfcD sectors do not compete
This
is
only to simplify the exposition, and will be relaxed in Section V.
assume that the economy admits a representative consumer with the usual constant
(CRRA)
relative risk aversion
preferences:
oo
c{t)'-
/
where
C (t)
When
^
neutral.
when
is
-"'dt
1-e
consumption at the time
>
and ^
t
(i;
is
the elasticity of marginal
utility.
= 0, the utility function in (1) is linear, and the representative agent is risk
When 9-^1, the utility function becomes logarithmic. I drop time arguments
this causes
The budget
no confusion
(I
use the time arguments in the proofs in the Appendix).
constraint of the representative consumer recjuires that consumption and
investment expenditm'es are
less
than total income:
C + 1 <ivL^ rK + uJsS + H,
where / denotes investment,
the capital stock, uis
is
w
is
the wage rate of labor, r
the wage rate for scientists, and
resource constraint of the
Y
is
the interest rate,
11 is total profit
The
(3)
an output aggregate produced from a labor-intensive and a capital-intensive
For simplicity,
I
assume that there
is
no depreciation of
f,
K=
labor-intensive
where
is
<
oo.
change
in the
given by
I.
(4)
(CES) production functions of labor-intensive and
intensive intermediates, with elasticity
z^
=
1/(1
—
yi
{l)^di
capital-
/3):
1//3
1//3
y(/)'s
£
and capital-intensive goods are produced competitively from con-
stant elasticity of substitution
Yr
<
capital, so the
capital stock (and in the representative consLmaer's asset level)
where
income.
7^^^ +(l-7)i^K
good, respectively Yl and Yk, with elasticity of substitution
The
K denotes
economy implies that
wL + rK + LosS + n = y
where
is
(2)
and Yk
denote the intermediate goods and
intermediate goods are gross substitutes.^
Vkiifdi
/?
€
(0, 1),
so that v
(5)
>
\
and
different
This formulation implies that there are two
*The presence of two types of agents, scientists and workers, causes no problem for tlie representative
consumer assumption since with CRRA utility functions these preferences can be aggregated into a CRRA
representative consumer. See, for example, Caselli and Ventura (2000).
® Alternatively, preferences could be directly defined over the different varieties of ,y(i), with identical
results.
different sets of intermediate goods,
of those that are produced with labor,
7i
An
that are produced using only capital.
labor-intensive intermediates
— corresponds
m corresponds to
an increase in
n
increase in
— an
and
7n
expansion in the set of
to labor- augmenting technical change, while
capital- augmenting technical change.
Intermediate goods are supplied by monopolists
who
hold the relevant patent, and
k{il
(6)
are produced linearly from their respective factors:
=
rnii)
where
l{i)
for labor
and
and
=
and y,CO
and capital used
k{i) are labor
good
in the production of
/.
Mai'ket clearing
capital then requires:
I
/
^0
To
l{i)
close the model,
di
[t)
=L
and
k
/
(?:)
di
= K.
need to specify the innovation
I
(7)
Jo
possibilities frontier
new
the technological possibilities for transforming resources into Ijlueprints for
of capital-intensive
created by the
There
it
is
and labor-intensive intermediates.
R&D
free-entry into the
R&D
R&D
n
bi,
b^
and
varieties
employed by
firm invents a
new
R&D
firms.
intermediate,
and becomes the perpetual monopolist of that
firms have access to the following technologies for invention:
- =
where
R&D
is,
assume that these blueprints are
are, in turn,
Once an
sector.
receives a perfectly enforced patent
intermediate.
who
efforts of scientists,
I
— that
bi(f)
{Si
)
5,
-
6
and
- = b,0
6',-
(5',)
-6,
(8)
777
5 are strictly positive constants
and decreasing fimction such that
d)
(s) s is
and
((>{) is
a continuously differentiable
always increasing, and 0(0)
denote, respectively, the nimiber of scientists working to discover
new
<
oo.
Si
and
labor-intensive
5';-
and
capital-intensive intermediates, with the meirket clearing condition
Si
I
also
assume that the economy
starts at
+
i
S'k
=
=
S.
with
(9)
77
>
and
777
>
0.
Eciuation (8) implies a nrrmber of important features:
1.
Technical change
faster
is
directed, in the sense that the society (researchers)
improvements
in
one
tj-pe of
can generate
intermediates than the other. This featiu-e
-will
enable the analysis of whether equilibrium technical change will be labor- or capital-
augmenting.
The
2.
means that there are intra-temporal decreasing
fact that 0(-) is decreasing
R&D effort;
returns to
when more
scientists are allocated to the invention of labor-
This might be, for ex-
intensive intermediates, the productivity of each declines.
ample, because scientists crowd each other out in competing for the invention of
similar intermediates.
This decreasing returns assumption
— when
the analysis of transitional dynamics
Sk
is
constant, the behavior of 5/ and
discontinuous.
Research
3.
4>{-) is
adopted to simplify
is
devoted to the invention of labor-intensive intermediates, 4>{Si)Si,
effort
leads to a proportional increase in the supply of these intermediates at the rate
while the
same
effort
devoted to the discovery of capital-using intermediates leads
to a proportional increase at the rate
differ since
more
The parameters
b^..
new
the discovery of one type of
difficult
intermediate
bi
and
b^ potentially
may be
"technically"
than discovering the other type (the standard model with only labor-
augmenting technical change can be thought as the special case with
assume that the crowding
by individual
one more
bi,
R&D
effect
captured by the function
firms, so each
scientist to
R&D
The
is
=
0). I also
not internalized
firm takes the productivity of allocating
each of the two sectors, bi4>{Si) or
deciding which sector to enter.
</)(•)
6^:
results are identical
as given
when
R&D firms act
"non-
6^-0
when
(S'fc),
competitively" and form global research consortiums, internalizing these crowding-
out
effects.
Each intermediate disappears
4.
devoted to a particular type of intermediates,
6
=
0,
its
when
no research
effort
stock declines exponentially.
With
at the rate 6, so that
there
is
the results are similar, but there will exist multiple balanced growth paths
(see Proposition 4 below).
Notice that in
knowledge
(8) scientists
spillovers
are "standing on the shoulders of giants"
from past research. This type of knowledge
growth when technical change uses scarce
—benefiting from
spillover is necessary for
factors, such as labor or scientists (see, e.g.,
Romer, 1990, Rivera-Batiz and Homer, 1991, or Barro and Sala-i-Martin,
equation
(8) is
a direct generalization of the accumulation equation in the standard en-
dogenous gi'owth model where we would have 7n
(e.g.,
equation
1995). In fact,
(3) of
Romer,
1990).
An
=
by assumption, and h/n
=
bi4>
additional assi_miption implicit in (8)
a higher stock of knowledge accumulated in one sector benefits only that sector
higher
n
increases the productivity of scientists working in the n-sector).
discussion of this assumption later.
I
is
(S)
S
that
(i.e.,
a
retiu-n to a
Finally, define S^
and S^ as the number of
technology in each sector constant,
Assumption
1:
5;*+ SI
which implies that there
in
both
is
<
i.e., bicp
(Sf)
scientists required to
S'l
=
and
6
6j.0 (5^)
S^
keep the state of
=
6.1 impose:
S,
enough
scientists in the society to enable technological progress
sectors.
Consumer and Firm Decisions
B.
An
good
equilibrium in this economy
prices,
w,
saving decisions,
scientists
r, a;^,
[pi{i)]"^o,
between the two
maximize the
[Pk{'i)]iLo^
[^{^)\T=o^
[/(?')ir=0'
given by time paths of factor, intermediate and
is
Pl
^"^d
p^, emplo}Tnent, consumption and
^
[y/lOliLc {yk{^)]iLo^
sectors,
5';
and 5^, such that
utility of the representative
consumer given
f^d
7,
and the allocation of
[y((?)]"=o'
[yA-(0]ilo'
factor, intermediate
^
'^^^ ^
and good
and [/(OliLc [^(0]™ni [P;('0]r=o ^"^ LPfc(0]i^o maximize profits of intermediate
imply zero-profits for all R&rD firms, and all markets clear.
goods monopolists, 5/ and
prices;
S'/,.
I
start with the optimal
satisfies
consumption path of the representative consumer, which
the familiar Euler equation:
''^
%-\[r-p),
where
recall that r
is
(10)
The consumption sequence
the rate of interest.
[C'(i)]^ also satisfies
the lifetime budget constraint of the representative agent (the no Ponzi
lim
(—»oo
Consumer maximization
K
r{r)dv
exp
[t)
game
constraint):
(11)
0.
'0
gives the relative price of the capital-intensive
good
as:
= i^(^)",
P-^
\iLj
PL
7
where
I
p/^- is
the price of Y}^ and pi
is
the price of Yi.
choose the price of the consumption aggregate,
[i^pY^
+
(1
-
iYp^k']'^"
PK =
[7
V
=
+
- 7)1
,
To determine the
in
level of prices,
each period as mmieraire,
i.e.,
impfies that:
1, '^vhich
(1
Y
(12)
'
'
and PL
= [Y +
(1
- ifP 1-.17^
^"Equation (10) implies that when consumption grows at a constant rate, the interest rate will be conwhich is a well-known feature of CRRA preferences. It may therefore appear that these preferences
ensure a stable interest rate in the long run. This is not the case, since there may not exist an ecjuilibrium
with a constant growth rate of consumption. Conversely, if preferences were not CRRA, there could never
stant,
exist
an equilibrium with a constant growth rate of consmnption and constant interest
9
rate.
Next, consumer maximization and the
lastic
demand
fyi
PL
will
be
fimctions in (5) yield the following isoe-
cvirves for intermediates:
Pi {i)
Given these
CES
isoelastic
{i)
Pk
and
Pk
Yr.
demands,
(14)
maximization by the monopolists implies that prices
profit
markup over marginal
set as a constant
fVk («)
Yu
{i)
cost (which
is lu
for the labor-intensive
intermediates and r for the capital-intensive intermediates):
1
1
Pi{i)
Since,
from
=
^«
and
-^
=
pk{i)
f
1
(15), all labor-intensive intermediates sell at
=
implies that yi{i)
the same price, y^
for all
yi,
(i)
=
k
and
i,
for all
Then from the market
T.
=
=-
l{i)
=
^
(15)
the same price, equation (14)
we obtain
yi{i)
^
j
since all capital-intensive intermediates also
as well.
i
--
and
yk{i)
n
=
k{i)
=
sell at
clearing equation (7),
—K
(16)
m
Substituting (16) into (5) and integrating gives the total supply of labor- and capitalintensive goods as:
Yl
These eciuations
reiterate that
technologies. Greater
= n~T~L and Yk = mT'K.
n and
in correspond to labor-
n enables the production of a
(17)
and capital-augmenting
greater level of
Yj, for
a given quantity
of labor, and similarly an increase in vi raises the productivity of capital.
Equations
(14), (15), (16)
and
(17) give the
wage rate and the
rental rate of capital
as:
w = [3n
Finally, using (12)
and
=
value of a monopolist
r{t)
is
pl and r
Pl£
1-7
PL
7
who
= Pm
1-3
i^
px-
the interest rate at date
t,
is
good
is
1-3
m \ -0h
Ln
new
(19)
L
J
(r(u;)
6
(18)
T^-l
/
invents a
exp
Vf{t)
where
"
(17), the relative price of the capital intensive
P
The
1-3
/-intermediate, for
+
6)duj
Kf{v)dv,
/ =
/
or
At,
is:
(20)
the depreciation (obsolescence) rate of existing
intermediates, and
1-pivL
TTZ
(3
n
,
and
TTfc
1-prK
m
(3
10
(21)
and
are the flow profits from the sale of labor-
wage ujs^
Scientists are paid a
ensure that this wage
find
ecjual to the
is
competition between the two sectors and free-entry
maximum of their contribution
RA'D
the two sectors. Recall that
olists in
so the marginal value of allocating one
intermediates
where
and
bi(f){Si)nVu
is
and 14 are given by
Vi
capital-intensive intermediate goods.
firms
more
do not
internalize the crowding effects,
scientist to the invention of labor-intensive
for capital-intensive intermediates,
it is
h);4>{Sk)mVk,
(20). Therefore, free-entry reciuires:
= max {6;0 {Si) nV], bkcf) {Sk) rnVk}
ujs
to the value of monop-
Equation (22) implies zero expected
(22)
.
profits for all firms at all point in time, so 11
=
in
(2).
An
equilibrium in this
and
satisfy (18)
(22),
economy
good
prices,
therefore a set of factor prices, w, r
is
\pi{i)]^^Q,
and
u>s that
that satisfy (15), intermediate
b^k{'i)]^Q,
production levels given by (16), output levels given by (17), sequences of aggregate consvimption and investment levels that satisfy (10) and (11), and sequences of Si and
6'^
that satisfy (22).
Asymptotic and Balanced Growth Paths
C.
I
to as
define an asymptotic path
—
i
hTiit^oo
GO,
>
(-
[t)
/
or \\int-,ooC
possibly
and does not include
C (t)
{t)
=
/C {t)
oo,
=
i.e.,
at the
same
is
we can have
either
consiimption grows more than exponentially (explodes),
Qc, i.e.,
the rate of consimiption growth tends to a constant,
defined as an
finite
In an AP,
limit cycles."
(including the case where limj^ooC'(i)
growth path (BGP)
grow
(AP) as an equilibrium path that the economy tends
AP
constant rate,
=
as a special case).
A
balanced
where output, consumption and the capital stock
i.e.,
Vmit^^C {t) /C (t) = Yimt^^Y
{t)
/Y {t) —
\imi^^k{t)/K{t)=g:^
This subsection
is
will
show that with £ <
1
,
only
BGPs can be an AP,
so
if
the economy
going to tend to a non-cycling path, this has to be a BGP. In contrast, with ^
exists asymptotic paths
different rate
Tq
than
>
1,
there
where consmnption grows more than exponentially or grows at a
capital.
facilitate the analysis, it is useful at this point to define
A =n
"Unfortunately,
I
am
1-3
>^
and
M=m
1-3
^^
,
unable to rule out limit cycles, except in the case with
risk neutrality.
See Section
III.
^"This definition
is
con\'enient for the purposes here.
consumption and capital grow at
different rates as
BGP.
11
Some authors
also refer to
growth paths where
which
in a
with
simplifies the notation below, and, together
(17), allows
me
to write output
more compact way:
y = 7(iVL)^ +
In addition,
(23)
define a normalized capital stock,
I
,
k
which
(1-7)(MA')"
^
MK
^,
(24)
a direct generalization of the normalized capital stock defined in the standard
is
growth models as capital stock divided by the
effective units of labor.
Here the numerator
contains the "effective imits of capital" as well, since there can be capital-augmenting
technical change. Then, using (13), (18), (19)
and
(24),
we can write the
= R{M, k)=p{l-j) M jk-^ +
r
(1
-
interest rate as:
'^'
7)
(25)
•
Also, define the "relative share of capital", a^-, as^^
= —r=pk =
(TK
^-k—
luL
The
relationship between the relative share of capital
depends on
s,
which
is
if
£
>
1
and labor
,
and
Now we
will
is
also the elasticity of substitution
in this economy. In response to
decrease
1:
With
^
technical change,
is
capital stock
if
<
£
an increase
in k,
gk
between
will also increase
1
can state (proof in the Appendix):
Proposition
This
and the normalized
the elasticity of substitution between capital-intensive and labor-
intensive goods. Equation (26) shows that s
capital
26
7
the
first
<
i.e.,
1,
all
APs
are
BGPs and
they have limt_,oo
M
[t)
important result of the paper.
feature purely labor-augmenting
/M [t) =
It
0.
demonstrates that with s
<
1, i.e.,
with labor and capital as gross complements, the only asjmiptotic (non-cycling) paths
feature purely labor-augmenting technical change. There will
to the invention of capital-intensive intermediates, but this
technology in that sector at a constant
is
be research
'
^
Note that
2:
•
With
£
>
1
,
devoted
only to keep the state of
level.
For completeness, the next proposition covers the cases with £
Proposition
effort
will
>
1
and s
=
1.
there are three APs:
this "relative share of capital" leaves out the
nominator.
12
income accruing
to scientists
from the de-
1.
limt^oc,
C (0 /C (t) =
lim.^oo
M
/M (t) =
[t)
3.
=
S;
K (0 /!< {t) = liint-,o. Y {t) /Y (t)
limt-^,^
k
{t)
/
K
=
[t)
l\mt^,^Y
<
g
and
cxi
{t)
/Y
=
[t]
oo and
and
lim(_,oo
Sk
lim,^oo
^ (0 /'^ (0 = gc<oo. Ihnt^.y,. K (f) /A' (0 = g^ <
(t)
=
0;
=
\imt-,ooC{t)/C{t)
2.
limt-,.x>
g,,
and lim,^oo S^
(i)
0.
With
•
With the
tion to the
5
=
1,
there
is
a Linique
elasticity of substitution
BGP
AP
which
is
a BGP.
between capital and labor greater than
with purely labor- augmenting technical change, there
is
1
and another equilibrium path where consumption grows
stant finite rate greater than the rate of growth of the capital stock,
change
stable,
is
We
labor augmenting.
and the economy
will
below that the
will in fact see
but since ^
tV,
purely
at the con-
all
technical
in this case
not
is
=
1,
Cobb-Douglcis, the type of technical change does
is
not matter, and the only possible asymptotic equilibrium path
M and
and
is
tend to one of the two other APs. In the case with £
the aggregate production function
growth of both
BGP
in addi-
an equilibrium
path where consumption grows faster than exponentially, and technical change
capital augmenting,
,
=
1,
is
a
BGP
both of these are neutral,
(which features
i.e.,
neither capital
nor labor augmenting).
D.
Growth Path
Characterization of Balanced
We
saw above that with
change can be an AP.
5
<
1,
Now I show
BGP
BGP as long as
(5
>
0,
this eqmlibriuni path.
from the Euler equation,
Moreover, since from Proposition
constant.
with pvirely lal^or-augmenting technical
that there in fact exists a imicjue
and characterize the properties of
First note that
only a
(10), the
1
M /M
BGP
=
rate of interest has to be
0, ecjuation (25)
immediately
implies that the price index for capital-intensive goods, pi^, and therefore, the relative
price of capital-intensive goods, p,
BGP,
In addition, in
gi'ow at a
common
output,
rate, g.
—
(3)
(or
N
Y
,
the wage rate, w, and the capital stock. A', will
rate. Therefore,
has to grow at the rate
with
g).
We
allowing for the depreciation of technologies at the rate
M constant, n has to grow at the
can then integrate equation
6,
and the growth of w,
to obtain the values of inventing labor- and capital-intensive goods
T.
1
-P
all
Furthermore, for p to remain constant, (12) implies that Yi
and Yk should grow at the same
rate (5g/ (1
must remain constant.
wL/n
as:
1-p rK/m
K
(20),
and
n,
Notice that these vakies also grow at a constant rate along the
The denominator
n are growing.
growth rate
the balanced growth path, while
BGP, p and
Recall that in
remaining scientists
is
are constant, so there
=
S/bk,
i.e.,
BGP
grows along
Sk
is
no net capital-augmenting
=
SI as defined above.
The
(10)
p
>
0.
then gives the
when the growth
BGP interest rate as r* = p+9g*. The interest
rate
is
higher in order to convince consumers to
delay consumption, and the elasticity of marginal
utility, 6,
determines
how
strong this
needs to be.
Let k
clear
M
tt;,
its
and
remains constant.
71
ensures that g*
rate has to be higher
is
because
denominator of
in the
is
Vfc
K
therefore
The Euler equation
effect
from that of
because w,
work on labor-augmenting technical change. The growth rate of
will
P
1
m
m
This implies (f){Sk)Sk
technical change.
Assumption
different
is
lower than that of 14: n, which
is
the economy
for V/
BGP
and
= G{M)
such that
from (25) that G'
This
k.
is
so capital has to
>
M and k are consistent with BGP
r* = R{M,
—that there a
increasing relationship between
k)).
(i.e.,
strictly
is
is,
It
because a greater k implies a lower price of capital-intensive goods,
become more productive,
i.e.,
M has to increase in order to keep the
interest rate at r*.
Next,
be the
let k*
stock and at
M/M
technical change,
=
i.e.,
level of
normalized capital such that at this normalized capital
R&D firms are indifferent between capital- and labor-augmenting
0,
bi4>{S
—
Sl,)7iVi
=
bk4>{Sl)7nVk, or from equation (27),
bi^{S-Sl)xoL
r*
This implies that, at k
^
with g* given by
+6-{l-2p)g*/{l-p)
=
M*
technology that
+
r*
k*, the relative share of capital,
5
(29)
- g*
ax, must
satisfy:
k4>{S-Sl){l-p){p + 6 + {e-l)g*)
-bk(t>{si){{i-p){p + 6) + i{i-p){e-i) + p)g*y
_.._
(28). In
= k*^
be such that k*
is
^
'
other words, using ecjuation (26), we have:
k
Finally, let
bk4>{Sl)r*K
=
'
(i?^)
G{M*),
i.e.,
^=^
M*
''^'
is
=
^*-
the level of capital-augmenting
consistent with the equilibrium interest rate taking
14
^^^^
its
BGP
value
when k =
r*
and the
when
result,
=
A:
MM
=
0,
N /N
while
M~
and
k*
relative share of capital will
BGP,
In
gies,
As a
k*.
be
h*
>
0.
—
<
have to make equal
which
0.
new
we need conditions
profits, so
M = M*
can therefore state (proof
Proposition
=
k
3:
labor-intensive
^
by
M
(31),
=
so that r
(29)
and
This proposition
is
(30) to hold,
and
I
>
6
0.
= M* =
Then
there exists a unique
=
G''^ {k*), r
r*
=
p
+
Oy*,
A;
BGP
there
the second main result of the paper.
It
In this
BGP, most
is
just
is
=
equal to aj^
RfeD
b*,
firms
ai'e
Intuitively,
re-
enough
— that
result, despite gi^owth
capital deepening, factor shares remain constant in the long run.
relative share of capital
where
characterizes the unique
features purely labor-augmenting technical change.
no net capital-augmenting technical change. As a
is
A;*,
and output,
capital-augmenting technical change to keep the productivity of capital constant
is,
=
(28).
devoted to the invention of labor-intensive intermediates. There
is
i.e..
r*.
consumption and wages grow at the rate g* given by
search
we
in the text):
Suppose that £
k* as given
BGP, which
and capital-intensive
This implies that firms working to invent both types of goods
in turn requires that
We
the interest rate will be equal to
there were no research directed at capital-intensive intermediates,
if
M/M
would have
,
Because of the depreciation of technolo-
there must be both research to invent
intermediates
M*
and
when the
just indifferent between in-
venting capital-intensive and labor-intensive intermediates; so in equilibrium they allocate
their effort
We
between the two sectors precisely to keep the relative share of capital
have already seen that when
change
is
5
<
different initial conditions, the
CRRA
Given the
and growth
—
change
a
BGP
BGP
with purely labor-augmenting technical
the only possible asymptotic equilibriimr path.
below that, i_mder certain conditions,
est rate
the
1.
is
this
economy
BGP
will
is
In addition,
we need
not surprising. Wliat
is
M
tend towards
= M* — i.e.,
we
will see also
dynamically stable, so starting from
this
preferences, the conclusion that for a
rate,
at b*.
growth path.
BGP
with constant inter-
no net capital-augmenting technical
important (perhaps
siu-prising),
however,
is
that such
exists despite the possibility of capital-augmenting technical change.^''
The results
but there are
g* (given
We saw
by
are similar in spirit
now many
there
is
balcinced growth paths.
(28) evaluated at
(5
=
in Proposition 2 that with £
We
when
0),
>
1,
no technological depreciation,
i.e.,
5
These paths have the same growth
=
0,
rate,
but different factor distributions of income. This
there are other equihbrium paths with capital-augmenting
and V that the equilibrium path with purely laboraugmenting technical change is unstable and that for other formulations of the innovation possibilities
frontier such a balanced growth path typically fails to exist.
technical change.
will also will see in Sections III
15
reflects that the equilibrium
in 6 at 6
=
correspondence
Summarizing (proof
0.
Proposition
M > M* = G-i
(k*),
lower-hemi continuous, but not continuous,
in the text):
^
Suppose that £
4:
is
1
where k*
and
is
6
=
BGP
Then, there exists a
0.
given by (30) and (31) with
<5
=
0.
In
for
all
each
BGPs,
output, consumption, wages, and the capital stock grow at the same rate g* given
by
(28) with 6
=
0,
and the share of labor
normalized capital stock, k
The
is
= G (M),
intuition for the multiplicity of
reqmred
BGP
for a
is
and a
BGPs
is
constant.
Each
BGP
has a different
different relative share of capital, aj^.
is
simple:
without depreciation,
that
all
that labor-augmenting improvements should be more profitable
than capital-augmenting improvements,
T4
i.e.
^
^^nd this
^'^h
can happen
range of
for a
capital (labor) shares.
Dynamics
Transitonal
III.
The previous
(when
6
>
section established the existence of a unique balanced growth path
with a constant interest rate, stable factor shares and purely labor-
0)
augmenting technical change, very much resembling the textbook growth model. Nevertheless,
balanced growt.h would be of limited interest
no other APs are
in Proposition 1 that
cient to establish stability, since there
dynamics in
I
to this
by
possible, but this,
can also be limit
cycles.
I
now
BGP.
itself, is
I
will establish
analyze local
stability,
already
not
suffi-
discuss transitional
economy. Unforti_mately, the transitional dynamics are rather
this
to analyze. So
starting from an arbitrary capital
economy did not tend
stock and factor distribution of income, the
showed
if,
and then prove global
difficult
stability in a
special case.
The key
is less
than
unique
result
1, i.e.,
BGP
when the
that
when
s
<
1,
elasticity of substitution
dynamics
transitional
will
between capital and labor
take the economy towards the
with purely labor- augmenting technical change. Along the transition path,
however, there
change.
is
will also
In contrast,
—that M
be net capital-augmenting technical change
when
£
>
1,
the economy will tend to an
is,
AP
that
is
will also
not a
(explosive growth or different asymptotic growth rates of consumption and capital)
A.
BGP
.
Local Stability
The key
Proposition
stable
result in this section
5:
when
Suppose 6 >
£
<
1,
0.
is
that:
Then the BGP
and vmstable when
s
16
>
characterized above
1.
is
locally saddle-path
This proposition
proved in the Appendix. The argument
is
BGP, the equihbriuni behavior
k
= MK/NL,
latter
and
SV,
two are control
c
approximated by four hnear
is
= C/K. The
variables.
show
I
is
standard: around the
differential
two of those are state
first
Appendix
in the
that, with r
equations in
variables, while the
<
1,
the set of linear
has two positive and two negative eigenvalues, and
differential equations
M,
is
thus locally
saddle-path stable.
The
recjuires
(Ta'
>
b*
intuition for local stability can
cr^-
=
then
,
<
6^-0 {SD mVk >
With
b*.
£
the relative share of capital, aj^,
I,
{S
bi(f)
technical change than along the
(Ta' is
be obtained from equations
—
S'l)
and there
nVi,
BGP. This
when
£
>
arbitrarily close to
Finally,
when
1
k.
If
be more capital-augmenting
will
BGP.
BGP
(31).
decreasing in
Clearly, this
But because
argument applies
and the economy moves away from the BGP, even when
,
it
starts
it.
£
=
1,
the economy
is
identical to a standard
with a CobbDouglas production fimction, jind the
BGP
endogenous growth model
locally (and globally) stable.
is
Global Stability with Risk Neutrality
B.
I
next characterize the global stability properties in the special case where ^
where the representative agent
is
and
implies that k will increase.'^
decreasing in k, the economy will approach the
in reverse
is
(30)
is
risk neutral.
I
also
=
0, i.e.,
assume that negative consimrption
allowed. This immediately implies that the interest rate,
r,
always has to be equal to
the discoimt rate p, and removes the Euler ecjuation of the representative consumer, (10),
and the
that at
becomes a control
capital stock (and therefore k) also
all
points in time p
= R {k, M) where R {k, M)
is
variable.
This ensures
given by (25). In other words,
the relationship k = G [M) has to hold at all points in time, and as before, G is strictly
increasing in M, with k* = G (A/*). These properties imply (proof in the Appendix);
LemiTia
1:
With
6*
=
0,
the transitional d3Tiamics of the
^=
where
V [M*) =
i'{M) I
This
lemma
<
1, i'
{M) =
(32)
for all
M
= M* and when
5
>
1,
implies that transitional dynamics can be represented by Figures 3 and
'^Unfortunately, this
0,
and when s
(A/)
M = M*.
for all
4 for the cases with £
m/m >
0.
i'
economy are given by
<
is
1
and
£
>
I,
respectively. Inspection of these figures immediately
not true globally, since, in general, k
because we can have
K /K < 0.
may
Hence, the argument here
17
not increase despite the fact that
is
a local one.
implies that the
BGP
The intmtion
the same as in the last subsection:
is
is
when
globally stable
e
<
and globally
1,
with s
<
^instable
when
1,
when
M
s
>
1.
and k are
BGP levels, there will be capital-augmenting technical change, reducing both
towards their BGP levels. Because the dynamics of k are pinned down by the behavior
of M via the equation k = G (M), we can also rule out limit cycles, and the result is one
of global stability. In contrast, with £ > 1, levels of M and k greater than M* and k*
above their
lead to further increases, taking the
economy towards the asymptotic path with
augmenting technical change and explosive growth, while
capital-
M < M* leads to the AP with
purely labor-augmenting technical change, and consumption growing faster than capital.
Finally, as
noted above, the economy with s
The next
Proposition
IV.
With 9 =
when
s
>
0,
BGP
the
always converges to the imique BGP.
In this section,
when
£
<
1,
Dynamics
analyze the effect of policy on the factor distribution of income
I
income
The main
II.
is
result of this analysis is that the long-run
independent of
fiscal
policy
and labor market
approximately independent of the discount rate (the savings
with the implications of the standard
gi'owt.h
rate).
policy,
countries, tax
and
This result contrasts
model with only labor- augmenting technical
change, where such policies would affect the long-run factor distribution of income.
many OECD
and
1.
model of Section
factor distribution of
globally (saddle-path) stable
is
Policy and Comparative
in the basic
1
proposition simimarizes these results (proof in the text):
6:
unstable
=
and labor market
policies
In
have changed substantially over the
past 100 years (see Persson and Tabellini, 2003, on tax policies, and Saint-Paul, 2000,
on labor market
policies).
Figures
distribution of income, but
is
difficult to reconcile
change, but
is
in line
Changes
and
2
show large medium-run changes
no long-run changes. The long-rim
in the factor
stability of factor shares
with the standard model with only labor-augmenting technical
with the predictions of the framework outlined
the discussion, in this section,
A.
1
I
focus on the case with
in Capital
e:
<
here.^*"'
To
simplify
1.
Income Taxation and Discount Rates
First consider taxation of capital
income
at
some
rate
t.
Assume that the proceeds
from capital income taxation are distributed lumpsum to consumers. This implies that
^^
Obviously, long-run stability of factor shares in response to policy changes
is
consistent with the
Cobb-Douglas production function, but with such a production function, we caiinot explain/analyze
short-run and medium-run swings in factor shares as those shown in Figures 1 and 2.
18
the budget constraint of the representative agent changes to
C + I < wL +
where r
is
{1
-
that pre-tax interest rate and
t)
T
rK + usS + H + T,
is
the hrmpsiim redistribution to consLuners
from the proceeds of taxation. The government budget constraint implies that
Clearly, the resource constraint of the
the representative consumer
after-tax one, (1
-
which
at g,
M
=
will also
equation (10), the
and
be the
BGP
-
r) r
-
is
of
the
p)/e.
consider the case of exogenous labor-augmenting technical
first
0,
((1
rrK.
The Euler equation
identical to (3).
is
similar to (10), except that the relevant interest rate
C/C =
r) r, thus
For comparison,
change, where
is
economy
T=
N =
BGP
e^'.
The
BGP
growth rate
now exogenously
is
given
growth rate of consumption. Therefore, from the Euler
after-tax interest rate
has to satisfy r*
still
=
{p
+ 6g) / (1 —
r).
Since the pre-tax interest rate must equal the marginal product of capital, this also implies:
/3(l-7)A/k-^ + (l-7)l^ =
J
L
where k
is
the
BGP
^,
—
1
value of the normalized capital stock in this case.
immediately implies a decreasing relationship between r and k
there
is
only labor-augmenting technical change, so
elasticity of substitution, c, is less
reduces the
BGP
=
than
1,
an increase
M
is
This equation
(recall that,
by assumption,
As long
constant).
as the
income taxation
in the rate of capital
value of the normalized capital, eind through this channel, increases the
share of capital income in
case where £
r
1, i.e.,
GDP
>
(the case with s
\
when the production function
would give the
reverse).
Only
in the
takes the Cobb-Douglas form,
is
the
long-run factor distribution of income independent of the rate of capital income taxation.
Now
cal
consider the case where both capital-augmenting and labor-augmenting techni-
change are allowed and endogenous. Equation (10)
of consumption, but for balanced growth
we need
still
determines the rate of growth
to have both
M
=
and
N
>
0,
which implies that there should be both capital-augmenting and labor-augTnenting innovations (otherwise
we would have
interest rate, equilibrium
factor distribution of
BGP
still
income
M/M
<
requires a^is
0).
=
Since firm profits depend on the pre-tax
b*
<==^ k
=
k*
Therefore, the long-nm
.
unaffected by capital income taxation.
In addition, in
consiimption must grow at the rate g* as given by (28), and so the Euler equation
(10) implies that the pre-tax interest rate has to satisfy r
=
(p
+ 6g*)
/ (1
therefore an increasing function of the rate of capital income taxation.
to ensure both capitalinterest rate,
K/NL,
also
and labor-augmenting research,
and since k
falls.
= MK/NL,
is
M
—
t),
Since k
and
=
is
A;*,
has to increase to raise the
also implies that capital to effective labor ratio,
Therefore, with endogenoios capital- and labor-augmenting technical
19
change, capital income taxation reduces the capital-labor ratio, but creates an exactly
and
offsetting capital- augmenting technical change,
leaves the long-rim factor distribution
of income vmchanged.
Next, consider a change in the discount rate
The
p.
analysis
is
standard model with only labor-augmenting technical progress, this
ings rate, the capital-labor ratio
R&D
will
requires
still
gk =
towards both types of technologies.
b*.
so that
on the factor distribution of income because the change
BGP
interest rate faced
and through
r,
in
p
BGP
growth rate
zero.
is
when
Similarly,
Therefore changes in the discount rate
on the
g*
is
this channel,
it
will
be an
change
b*
the
and
when the
be second
order.
have small or second-order
effects
small, this effect will
will generally
will
will also influence
Inspection of ecjuation (30) immediately shows that this effect disappears
k*.
in the
remains profitable
However, now there
effect
by consumers,
it
In the
change the sav-
and the factor distribution of income. In contrast,
framework here, long-run equilibrium
to imdertake
analogous.
factor distribution of income.
Labor Market Policy
B.
Next to analyze labor market policy
imposes a (binding)
level of
in a simple way,
minimum wage w, and
suppose that the government
moreover, indexes this
minimum wage
to the
income. In particular, assume that
w=
x~^ {fK
+ wL)
,
> ^^^ L is the level of employment, which is now determined endogenously.^^
the minimum wage is binding, the eciuilibrium wage rate has to be lu = w at all
where x
Since
points in time. Multiplying both sides of this equation by L, and rearranging
we obtain
the quasi-labor supply curve, relating employment, L, to the relative share of capital, ax-
L = X"'
I
refer to (33) as the quasi-labor
(1
+
(33)
'JK-r'
supply cm-ve of the economy, since any equilibrium has
to be along this curve.
As
before,
BGP
requires
long-run share of capital in
of employment.
An
M/M
GDP
increase in
will
\-
=
0,
thus
a^ =
be imchanged
will
via this channel raise k (recall that k
b*
and k =
k*.
Therefore, the
—irrespective of the equilibrium
level
immediately reduce employment, however, and
= MK/NL).
In the case where the elasticity of
^
This expression makes the minimum wage proportional to the sum of capital and labor income rather
than total income, which also includes scientists' earnings. This is only to simplify the expressions, without
any substantive implications.
'
20
substitution between capital and labor,
BGP
Subsequently, the economy adjusts back to
process, the share of labor in
relates
this
GDP falls,
employment to the share
than
£, is less
1,
the labor share will also increase.
starting with k
and returns to
k*
Thi-oughout this
.
its initial level in
GDP, employment
of labor in
>
BGP.
Since (33)
also falls steadily during
adjustment process.
This result
is
interesting in light of the developments in
many European
kets over the past several decades. For example, Blanchard (1997)
unemployment and the labor share
in a
number
labor mar-
documents that both
of continental European economies rose
Hammour
sharply starting in the late 1960s. Both Blanchard and Caballero and
(1998)
interpret this as the response of these economies to a wage-push; the militancy and/or
the bargaining power of workers increased because of changes in labor market regulations
taking place over this time period, or because of the ideological effects of 1968.
This
wage-push translated into higher wages and lower employment. During the 1980s, we see
a different pattern: unemployment in these countries continues to increase, but the labor
share
falls sharply.
Blanchard documents that the decline
explained by capital-labor substitution, and conjectures that
ased technical change''.
The framework presented here
wage-push shock,
in response to a
and unemployment
and employment
increase.
Then
The
falls further.
i.e.,
is
it
may have been due
to "bi-
consistent with these patterns:
GDP
an increase in Xi both the share of labor in
as technology adjusts,
fall
cannot be
in the labor share
in k
is
k returns to
accompanied by an
which corresponds to capital-biased technical change.
BGP
value,
A;*,
offsetting decline in
M,
its
^^
Discussion and Extensions
V.
The analysis
so far has established that in a natural
capital-augmenting technical change, there
is
model with potentially labor- and
a unique balanced growth path equilibrium
with no net capital-augmenting technical change, stable factor shares and a constant
long-rim interest rate. Moreover, as long as capital and labor are gross complements
the elasticity of substitution
is less
than
the economy converges to this
1),
(i.e.,
BGP. This
analysis relied
on a number of assumptions. For example, technical change took the form
of invention of
new
R&D
goods;
sectors did not
RfcD
compete
w£is carried
for labor;
on the number of existing goods, with
out by scientists so that the production and
and productivity
spillovers
in the
from past research.
of these assumptions are importeint for the substantive results.
'"Because the elasticity of substitution,
technical change.
and
See Acenioglu (2002)
s, is less
for
than
1.
R&:D
a decline in
We
I
sector
now
will see
depended
clarify
which
that the only
M corresponds to "capital-biased"
a discussion of the relationship between factor-augmenting
factor-bicised technical change.
21
important assumption
(or the
form of
spillovers
Possibilities Frontier
There are two important assumptions embedded in
tier (8):
first,
Second, there
R&D
possibilities frontier
from past research).
The Innovation
A.
form of the innovation
for the results is the
this innovation possibilities fron-
uses a scarce factor (scientists, or labor as in the next subsection).
a specific form of spillovers from past research; an increase in n raises
is
the productivity of
R&D in the n-sector,
but not in the 7n-sector.
dependence, since the relative productivities of
R&D
in the
I
refer to this as state-
two sectors depend on the
state of the system, (n, m).
Let us
now
relax each of these two assumptions on the form of the innovation possi-
The
bilities frontier.
an
alternative to
R&D
sector using scarce labor
what Romer and
is
Rivera-Batiz (1991) refer to as the lab-eqmpment model where the final good (or capital)
is
used for
R&D.
For example, we coLild have (implicitly setting
(/){)
=
1 in
terms of
(8)
to simplify the notation)
n
where Xi and Xk are the
R&D
m = bkX^ — 5m,
— 6n and
biXi
two sectors
expenditvires in the
good, and the resource constraint needs to be modified to
important point
is
that long-rim growtrh
from past research, because
Consecjuently, there
in the
two sectors
is
is
also
biVi
or
6fc
1
and bkVk
=
1,
is
BGP
rK
=
is
we need
m=
0,
condition (35) implies that
good.
R&D
R&D now requires
used to invent
6/
labor-intensive
recjuires:
wL
bi
m.
(35)
.
n
and w, n and
K
final
remains unchanged.
not consistent with balanced gi'owth, however.
to remain constant,
the
BGP
final
Y. The
relative productivity of
rest of the setup
since one unit of final output
h
This condition
—only the
so far applies, but the free-entry condition into
capital-intensive goods. Therefore,
terms of the
possible without knowledge spillovers
no state-dependence, since the
The
in
C + / + A'; + Xk =
does not use any scarce factors
always constant. ^^
Much of the analysis
=
R&D
now
is
(34)
and 7n
will
K
For the interest rate
to grow at the
same
rate.
But
grow together. Therefore, with the
innovation possibilities frontier given as in the lab-equipment specification, there exists
no
BGP
(though there exist other
'^Equation (34)
is
APs
with constant gi'owth of consumption).
equivalent to the formulation of the innovation possibility frontier
I
used in Acemoglu
(1998) in the context of technical change directed at skilled and unskilled labor. See Acemoglu (2002) for
a more detailed discussion of the implications of different forms of the innovation possibilities
22
frontier.
Next,
us retmii to the formulation with scientists undertaking
let
from past research, but modify
In (36), there are
But there
case.
is
remove state-dependence (and again
(8) to
n =
hirfm}'''*'Si
still
spillovers
— 6n and
7h
^
R&D
in
one of the sectors
which leads to equation (35) as a
uJs-
also, there is
=
1):
(36)
affects
in this
both sectors
This contrasts with (8) where current research for the invention of
R&D
for labor-intensive
goods in the
=
R&D
now
requires 6;V/
BGP condition.
As a
result, in this case
future, but not for capital-intensive goods. Free-entry into
bkVf;
set 0(-)
— 6m.
bf;n'^Tn^~'^Sk
labor-intensive goods increases the producti\ity of
and
spillovers
from past research to ensure long-run growth
no state-dependence:
ecjually in the future.
=
RfcD and
cvg
no BGP.
Therefore, the
BGP
with purely labor-augmenting technical change
is
only consis-
tent with an innovation possibilities frontier with a strong degree of state-dependence.
Intuitively,
venting
ing
new
balanced growth with capital accumulation requires the profitability of
new
in-
capital-intensive goods not to increase faster than the profitability of invent-
labor-intensive goods
—so that
in equilibrium firms are
happy
to imdertake only
labor-augmenting improvements. Since capital accumulation increases the profitability of
research towards capital-augmenting technologies, a strong form of state-dependence in
the
R&D
technology, whereby labor-augmenting technical change raises the profitability
of further research towards labor-augmenting technologies,
necessary to balance this
is
eS-ect.20
Competition For Labor Between Production and
B.
In the baseline model, there are
two
tj^pes of workers, unskilled labor
R&D
and
scientists,
with scientists specializing in R&:D and thus no feedback from the relative price of labor
to growth. This assumption was
made
an innovation possibilities frontier
'-'Is
Unfortunately,
analyzed
(1992)
by,
I
am
among
in
-Jaffe
(1993),
the model to
(8),
plausible?
The data on patent
citations
and Henderson (1993), Trajtenberg, Henderson and
may be
A
now modify
with a strong degree of state-dependence, like
others, Jaffe, Trajtenberg
citations of patents by other innovations.
rele\-ant in this context. Tliese
citation of a previous patent
is
.JafFe
papers study subsequent
interpreted as evidence that a
by the previous invention. This corresponds to some
from past research. One can therefore use patent citations data to in\'estigate whether
is
degree of spillo\er
I
not aware of anj- direct im'estigation of this issue.
and Caballero and
current in\-ention
only for simplicity, and
e.xploiting information generated
state-dependence at the industry
level. Industrj' level state-dependence corresponds to patents
which they originated. Results reported in Table 1 in Trajtenberg,
Henderson and .Jaffe (1992) suggest that there is some amount of industry- state-dependence. For example,
patents are likely to be cited in the same three-digit industry from which they originated. Nevertheless,
it is currently impossible to investigate state-dependence at the factor level. This is because, although we
have infoniiation about the industry' for which the patent was developed, we do not know which factor
the inno\'ation was directed at.
there
is
being cited
in the
same
industrs- in
23
R&D
allow the production and
let
sectors to
us again focus on the case with
ix
- =
where L;
employed
in production.
condition
is
employed
in the
L+
R&D
R&D
of the analysis
eciuation (22)
is
1.
from Section
replaced by
w=
—
2P) g
=
bik,
R&D
1,
is
compete
sectors
condition binVi
where f and g are the
=w
wage
workers
for workers.
R&D
now
for scientists),
M needs to remain
new
to inventing
binVi
is
by workers
are invented
Taax{binVi,bkmVk}. In BGP,
we need
L^
the labor market clearing
(rather than the
some research devoted
The
bkTnVk given by (27) above.
and L
but the free-entry condition into
wage rate
intermediates to balance depreciation. Thus,
(1
for labor-intensive goods,
new goods
production and
to the
(37)
for capital-intensive goods,
II applies,
new innovation
constant, so there has to be
f + (5—
R&D
In this framework,
sector, so the
simplify the discussion,
6,
m
Normalizing total labor supply to
+ L^ =
L;
relates the value of a
and
in
—
= hkLk -
and
6
To
for labor.
Ecjuation (8) then changes to
1.
the number of workers employed in
is
the number of workers employed in
Most
-
hiLi
n
=
()
compete
= bkmVk =
capital-intensive
lu,
with binVi and
immediately implies that in BGP:
BGP interest
and growt.h
Now using
rates.
the Euler equation for consumption, (10), we have
{2p
Furthermore, in
pg/
{1
labor,
—
P),
and
BGP
.,
=
^
(1
+ pg/ {1 —
- P)Pb,k -
rest of the analysis is
and stable factor
6/bi
(38), the long-rim
^
The
m/m =
we again have
also L;
and equation
+ 6-iyg +
shares.
6
+p=
0,
bi^.
(38)
which implies Lk
P)bi. Using the
6/bk,
is
=
and h/n
market clearing condition
growth rate of the economy
for
therefore given by:
+ - (1 - P)bi6
{i-P)pb,e + p6-{i-P)P{i-2p)bk
(1
- P)Pbk
=
(p
rmchanged. In particular,
As
(5)
BGP
before, along the balanced
reqi.iires
m = 0, hence ax = b*
growth path, technical change
is
pm-ely labor-augmenting, with research towards capital-augmenting goods only to keep
the net productivity of capital constant.
complicated, however, because both the
In this case, transitional dynamics are more
number
of production workers
and the speed of
technical progress change along the transition path.
C.
Different
Forms of Technical Progress
Labor- augmenting technical change has so far been interpreted as "labor-using"
change, that
is,
the introduction of
new goods and
24
tasks that use labor.
I
now show
that the results of the above analysis generalize to different formulations of the techno-
change process, including a model of technological progress with new varieties of
logical
machines, and one where technical change takes the form of
Grossman and Helpman (1991a,b) and Aghion and Howitt
run technical change
cjuality
improvements as
in
In both cases, long-
(1992).
be labor-augmenting, with no net capital-augmenting technical
will
change.
To
new
discuss the consequences of technical change resulting from the invention of
machines,
let
me
modify the basic framework such that the two goods have the following
production functions:
where
z;
(j) this
quantity of the
the quantity of the j-th machine complementing labor, and Zk [j)
j-ih.
machine complementing
capital.
This
is
two factors are not accumulable, and more
solution can be found there.
Notice that n and
types of machines complementing these two factors.
monopolists, while producers of Y^ and
depreciate at the rate ^
1
in
terms of the
from
and
profit
[j)
X';.
final
zi {j)
=
for these
Xk U)
=
(^
+
+
5,
^) / (1
~
is
r
z^.
.,
/?)-^'
(j)
{pk/v)
.
=
(1
is
K, where
a constant
n and
Pr
r
=
^
Xi (j)
depend on the
we can
_„ (1^)""^
n
\r +
interest rate. Next,
see that
The monopolist
BGP
will originally
markup
over
Therefore,
p,'-
m.
profits of technolog}' monopolists are:
and
...
Ml -« (i-l)'"^.
m
\r +
assuming the innovation
recjuires a^-
produce
2;
(39)
J
profits are identical to those in (21), except for the constant
depreciated.
normalized to
straightforward to derive
is
—
is
Next, h'om market clearing, factor prices are
J
(8),
assume that machines
the interest rate plus the depreciation rate.
These equations imply that the
by
machines
L and
{pl/Xi U))
I
new machine
profit-maximizing monopoly price of machines
w=
These
These machines are supplied by
cost of producing a
The demand
on the
denote the user cost of machines. Since the demand curves for machines are
marginal cost, which
=
and the
0,
good.
maximization:
isoelastic, the
Xi [j)
>
details
the numbers of different
competitive.
cire
V/^-
now
are
the
the model used in Acemoglu
(2002), for the case where the
m
is
=
b*
,
and the
fact that they
possibilities frontier is given
and we obtain exactly the same
(or Zk) units,
results
and then replace the machines that have
and
as in Sections II
the introduction of
This demonstrates that whether technical change
III.
new
labor-intensive
and
capital-intensive goods or as the invention of
labor-enhancing and capital-enhancing machines
The second
up the
possibility
now produced
=
yL
^
where zl and zk are
Ql and Qk
-|-
(1
{qW-')
L' and Yk =
j^ {qW-') K^
machines that complement labor and
ciuantities of
capital,
and
> 0, or a new vintage of capital-complement the machines,
+ X)Qk- This R&D firm would be the monopoly supplier of
and
at the flow rate
Q'j^
it
=
would dominate the market
scientist
6;
(1
who works
until a
to discover a
(or bk). Notice that this
new
new, and better, vintage
Ql
vintage of
BGP
the resulting improvement in the "level" of productivity
loss of generality,
and normalize the marginal
I
Qk)
before: research
leads to proportionately better machines, so the greater
Without
{or
arrives.
successful
is
assumption already builds in knowledge- based
were required for the existence of a
spillovers that
is
(40)
\)Ql, where A
assume that a
Ql
competitively with the production functions
vintage of labor-complementary machines, with productivity
this vintage,
quality
replace old ones. Suppose
new
with productivity
I
new goods/machines
two formulations
denote the qualities of these machines. Technical progress results when an
firm discovers a
=
immaterial.
one where technical progress takes the form of firms moving
features creative destruction:
it
the two goods are
Q'l^
is
quality ladder (vertical innovations), which differs from the other
because
R&D
is
modeled as
is
is
on a vintage of
Qi, the greater
XQl)-
(i.e.
assume here that machines depreciate
cost of producing z to 1/(1
is
+
A).
I
fully after use,
assume that A
also
small enough that the leading monopolist will set a limit price to ensvire that the
next best vintage breaks even
(see, for
the marginal cost of production
Zl
=
is
1/(1
QlL and zk = p^ QkK-
Pl
example, Grossman and Helpman, 1991b). Since
-f-
A),
the limit price
/3(1
—
BGP values of inventing new
V,
=
^"^
.
(l
and
(5;,-
Vi
>
P{1
14
is
+
A)(r
+
.,
—
P)~^Pjl-
bi
=
(l
<5,)
fej.^;
and
5^
=
ij-^fc.
+
A)(r
+
introduced technological obsolescence
i.e..
By
From
the above as-
Similar reasoning to before implies that
Si
=S
(in addition to
26
(39).
and
5fc)'
consistent with stable factor shares. Therefore, along the
only be labor-augmenting technical change,
Q
^'^
and V,"
are the endogenous rates of creative destruction.
sumptions, we have
only
=
straightforward
(higher) quality intermediate goods
are
bi
r
it is
Hence,
^-
P)~^Pl Ql, and profits are given by an ecjuation similar to (21) or
standard arguments, the
where
Xk — Xl ~
Substituting these into (40),
to verify that the eciuilibriimi interest and wage rates are:
w=
is
and Sk
=
0.
BGP,
Because
there will
I
have not
the endogenous creative destruction
already present in these models),
now
BGP
requires V;
a range of labor shares consistent with
V^ rather than
V]
=
so there
Vj.,
is
BGP.
Conclusion
VI.
Almost
all
existing models of economic growth rely
the production function
labor-augmenting.
Much
on one of two assumptions: either
supposed to be Cobb-Douglas (an
is
between capital and labor exactly equal to
1.
>
1),
or
all
evidence suggests that the
elasticity of substitution
technical change
eleisticity
augmenting technical change
more
is
factor distribution of
A
It
is less
ecjual to 1
than
does not
model with purely labor-
attractive, but poses the question of
no capital-augmenting technological improvements.
assumed to be
of substitution
Moreover, a framework with an elasticity of substitution exactly
enable an analysis of medium-run changes in factor shares.
is
why
there are
also suggests that the long-rim
income should be a fimction of tax and labor market
policies
and of
the savings rate, while in the data, the long-run factor distribution of income appears to
be stable despite changes
in these variables.
This paper studied the determinants of the direction of technical change in a model
where the invention of new production methods
is
a purposeful
activitj^
Profit-maximizing
firms can introduce capital- and/or labor-augmenting technological improvements.
major
result
is
that, with the standard
long-rrm technical change
path, the
economy looks
will
like
The
assumptions used to generate endogenous growth,
be purely labor-augmenting.
Along the balance growth
the standard model with a steadily increasing wage rate and
a constant interest rate. Therefore, the framework here offers a microfoundation for the
standard neoclassical growth model with (exogenous or endogenous) labor-augmenting
technical change. But,
typically
it
also
shows that away from the balanced growth path, there
be capital-augmenting technical change. Furthermore,
policies that affect the long-run factor distribution of
no long-term
It
effects in this
income
I
will
showed that a range of
in the
standard model have
model.
has to be noted that the results here hold i.mder a very specific form of the innova-
tion possibilities frontier,
and the discussion
in subsection
V.A
indicated that with other
forms, a balanced growth path with purely labor-augmenting technical change typically
fails
or
to exist.
why
Work on why
technical change
area for future research.
this
may be
form of the innovation
possibilities frontier
labor-augmenting with other plausible forms
is
is
plausible,
a
fruitful
Appendix: Proofs
VII.
Throughout
this appendix,
changeably. In addition,
I first
= x and x [t)
use the notation 11™^.,^^ x{t)
I
—> x inter-
some
state the following resvilt which will be useful in
of
the proofs:
Lemma
Al: Let
<
Suppose that s
1.
,._
7n{t)V,{t)
^'
n{t)mt)-
Then, Umt^oo k
=
{t)
=^^ hmt^oo
A [t) =
M
oo, limf^r»
[t)
/M {t) =
S-6) //?, and lim^^oo N {t) /N {t) = - {I - (3) 6/15.
And limt^^ k{t) = oo=^ \imt-,^ A [t) = 0, limt_oo M [t) jM it) = -(!-/?) ^//?,
and limi^oo N (t) /N [t) = {I - (3)
{S) S - b) //?.
(1
-
{h(t> [S)
P)
{bicf)
>
Suppose that e
\.
=
Then, limf^oo k[t)
=> limt_,oo A (i) =
oo
oo, limt^oo
S-6) //?, and Wmt^^ N (t) /N (i) = -(!-/?) 6/p.
And limt^oo k{t) = 0=> fimt^oo A (t) = 0, hm^^oo M (i) /M {t) =. and limt^oo N (i) /N (f) = (!-/?) (6^0 {S) S - 6) /p.
(1
-
M
{t)
/M [t)
{hcj^ {S)
P)
Lemma
Proof of
We
Al:
(1
- /3) 6/p,
have from (27) that
E-l
^'
w{v)L(v)
y,
As k{t) -^
0,
therefore Si
{t)
(1
-
all
p) {h(p {S)
k
^
{s)
—>
exp[-/;(r(w) + «)<ia,],o(tOiH
7,
and Sk
S-6)/p
>
for all s
{t)
-^ S.
and lim.^oo
When
t.
s
<
I,
this
impHes
/N (i) = -
[t)
{I
- p)
^
A (i)
M
oo,
and
{t)
/M (t) =
The other
cases fol-
limt_,oo C" (i)
/C (t) =
This immediately gives limt_,oo
N
^'
7,
6 /p.
low analogously.
Proof of Proposition
oo cannot be
be a
BGP
limi_,oo
K
First,
eciuilibria,
1:
and that any equilibrium path with
where
limt_,oo
with pm-ely labor-augmenting technical change,
/K {t) =
(t)
I
will
/M (f = 0.
limf^oo C {t) /C [t) =
^ and
prove that
a contradiction, suppose this
lim(_,oo
The proof will show that paths with
y [t) /Y {t) =
linit^oo
k
{t)
=
oo,
oo,
Y {t) =
i—»oo
lim
lil (t)
is
which
(^)
/^ (0 =
show
N{t)L 7 +
t—KX
(1
oo cannot be an equilibrium. To derive
Then from the budget
is
and note that, from
lim
lim^^oo^
)
the case.
I will
i.e.,
C [i) /C [t) = g must
-
constraint, (2),
not possible. To start with, take the case
(23),
output can be written
<r-l
j)k {t)~
=
lim
t—*oo
28
we need
A
{t)
as:
j—L
=
since,
with 5
<
1,
k (t)^ -.
7
+
/N
(1
{t)
-
<
-> oo. Therefore, hm;^,^
{t)
'^'
is
->
^ (0 /^^ (0 =
lim^^oo
Y
so limt_oo
0,
Y {t) /Y
=constant. then clearly.
[t)
where k
oo. Finally, consider the case
j)k (f)^
neither case
A:
where hnii^oo k
oo. Next, take the case
A'^ [t)
as
{t)
/Y
(f
—f
)
°o possible, so
[t)
which case,
0. in
limt_,oo
N {t) /N
^(0/^(0 =
lim(_,,3o
N {t) /N {t)
<
{t)
=
[t)
<
As a
oo.
C (t) jC (f) =
resrUt, in
oo cannot be
an equilibrium.
This implies that
Xvaxi^^.^C
limt^oo
[i)
C {t)
APs must have
all
C {t) jC (f = g. Next will show
liTat^oo M [t) /M {t) = 0, and that in this
limt_oo
jC {t) = g also necessitates
/C {t) = limt^^ Y (t) /Y [t) -
I
)
\\mt-.^
K
[t)
jK [t) =
g.
which
\^ill
that
case
complete
the proof.
consumption to grow at the constant
First, recall that for
Euler ecjuation (10), the interest rate r
(i)
= /3M
{t)
p^- {t) to
rate,
we need, from the
remain constant. Recall also
that
1
PK
it)
= j{i-jr'k{tr
E-l
+(i-7r
^
(Al)
J
which implies
j{i-jf-^k{ty
PK
it)
1
PK
[t)
^j{l-^f~'k[t)-
Constant interest
rate,
r(i)
i.e.,
=
0,
then
M{t)
To
derive a contradiction,
first
(A2)
.^(0
(1 - 7)'- ^ \^J
+(1-7)
reciuires:
PK{ty
suppose that
M
(i)
/M {t) <
0,
then pK
(i)
/px
{t)
>
0,
< 0, or k{t) —^ 0. But then from Lemma Al,
> 0, giving a contradiction with A/ (t) /M [t) < 0.
which, from (A2), implies k{t)/k{t)
M
Next suppose that M
^
k{t)
^
/M {t)
(i) /M [t) > 0, which reciuires p^ {t) /pk {t) > 0, which, from
(A2), implies k (t) /k {t) > 0, or (t) — oo. But then, from Lemma Al, we have A {t) -^ 0.
Thus, M (0 /M (i) < 0, contradicting the supposition that M (i) /M {t) > 0.
Therefore, we must have M (i) /M {t) = 0, and pK (t) /pk (0 = 0, and thus, k {t) jk [t)
k
{t)
->
implies limt_,oo
(i)
A-
0.
by
This also implies that limt^oo
(28) in the text.
limt^oo (^
{t)
{t)
K
jK
[t)
[t]
=
limj^oo
N [t) /N
Then, we immediately have that limi^oo
/C (i) =
g*.
/C (t) =
g*
{t)
Y {t)
= g* with g* as given
/Y (0 — f/*i ^^i^l thus
completing the proof.
Proof of Proposition
Wmt—ryoC"
>
2:
Th.e case
and llmt^oo
J^-f
with s
[t)
>
I.
/M [t) =
29
First,
it is
outHned
clear that the
BGP
in Proposition 1,
with
where
<
limt_^oo
k
{t)
/k
Proposition
1
—
(t)
Moreover, the same argument as in the proof of
0, is still possible.
implies that there exists
Second, consider the case where limi^oo k
£>1
Al with
and limj^oo
M
N
[t)
immediately implies that limj^oo
{t)
/N
= -
(t)
-
(1
6/0. Next
/3)
AP
no other
with limi^oo k
/k
M
[t]
>
jM (t) =
{t)
>
(23) with s
/M {t)+K (i) /K (t), and r {t) -^ r (t) = (3M (f) (1 limt^oo y (0 /Y (i) = limt^oo K {t) /K (i) = 00 is an AP.
(t)
Finally, consider the case
Al
gives
M
hm,^oo
{t)
(23), in turn, together
lim
Y [t)
t—>oo
Thus limt_oo
00
/M (t) =
with
lim
t—oo
Y (i) /Y (t) =
impossible from
is
where
^
and e
N{t)L 7 +
-
{I
A;
/k
(i)
{t)
<
-(!-/?) 6/f3 and lim^^oo
(f)
/c
lim(_,oo
P)
(bicf)
>
k
0, i.e.,
e/(e-l)
{bk(p {S)
S-
{t)
—
0.
[t)
/N
(f)
=
{I
>
6) //?,
Y {t) /Y
Thus limf_oo C
00.
k
0, i.e.,
Lemma
{t)
(i)
/C (i) =
Then Lemma
-
P)
{bi<P
(5)
S-
implies that
\,
lim
t—00
5-
0.
—> oo, then
p)
=
7)
N
=
{t)
implies that limj^oo
1
(l-7)A;(t)-
(5)
{t)
-
{1
/k
(i)
5) //?
<
00, but
Nit)^^^
then limt_.^
C (t) jC [t) -
(2).
So we must have limj^oo
C [t) /C [t] =
Qc
<
oo.
Then,
for consiimption to
have a
constant growth rate, the Euler equation (10) requires the interest rate to remain constant.
Prom (A2) with
linif^oo i^ {t)
k{t) -^
/K (t) =
{1
0, this
—
requires k
{S)
{bi4>
p)
rate of consimiption growt.h, which
S—
is,
[t)
/k
sS)
= sM
{t)
//?,
/M (i) =
(t)
-e
{1
-
P) djp, or
giving us another AP, with a constant
however, not a
BGP,
since limf_,oo
K
{t)
/K [t) <
\imt^^C{t)/C{t).
= 1. As £ — 1, the aggregate production fimction becomes CobbDouglas, Y = B {NLy {MK) ~^, and the equilibrium relative share of capital is always
ax — (1 — 7) /7- Therefore, the equilibriimi allocation of research effort is given by
The case with £
s-
1-7
kcP (5
all
S,) {p
points in time, where g^
=
—
lim.^oo
C (0 /C {t) = lim,^oo Y [t) /Y (i) = g*
By standard arguments,
Proof of Proposition
using
k
k
(2), (8),
_
~
K_
and
(17),
we
5:
5
+ eg*-
+e
gj)
-9n)
bi<j>{S-S,){S-S,)
bi4> {S - Sk) {S - Sk) -6,gn
7i/K+(l
7) g^.
+
bk(t>{s,){p^6
7
at
-
=
hk(j)
[Sk) S^
- 6, and
the unique eciuilibrium path has limj^oo
First, exploiting the definition of
K
{t)
M N
c
{bkcP{Sk)Sk-bi(P{Si)Si]
K
30
/
K
k given by (24), and
obtain:
K~^Ji~N'
j{NL)~ + {l-j){MKy
=
g*
[t)
6) /(3.
M
=
jk-'^ + il-^)
+^
where
{b,4> (5',)
recall that c
-
5,
= C/K.
(A3)
bicp
-
{S
-
5,) [S
S,))
Linearizing around the
BGP. we have
the
first
linear differ-
k*)
(A4)
ential equation of the four-equation system:
k
- =
with aks
>
0,
aks {Sk
7
Ofr,,
using the definition c
-
+
SI)
'
(A:*)
= C/K,
akm
{M - M*) +
+
-
(1
>
^
7)
akc {c
0,
the Euler equation
-
c*)
+
= -1 <
a^.f.
and an- <
and equations
(8),
-
a^- {k
(10)
Similarly,
0.
and
(25):
K
C
(A5)
C~^K
1
1
= -(/9(l-7)M 7A;-^ + (l-7)
M
Linearizing around the
BGP
a-cm
with
a^c
ac^
where
=
=
^/5
a^.^
>
1
0,
7^-^+(l-7)
is
P
c
gives the second linear differential equation:
{M - M*) + ace (c -
c*)
+
a^-
(A;
-
(A6)
A;*)
and
(1-7) 7(^-*)"~ + (l-7)
>
+
-
defined above, and
In addition, a^h
is
>
a'
Q.m
-
^kr,
0.
yk-^ +
the derivative of
=
7 (A:*)-— + (1-7;
(1
-7)
{
where both terms have negative derivatives, hence
<
Qca-
'-'
i/?(l
-7) -7A;-^ -
(1
0.
Next, recall that (8) gives:
M
M
Linearizing this relationship,
1
- 9
M(5V)SV.-<5]
p
we obtain the
M—
M
with a^,
>
third differential equation:
0,ms {Sk
0.
31
—
SI)
,
(A7)
-7)
Finally, recall that in
BGP, we
hict)
Define
=
(pi
4>
—
{S
Sk) and
{S
have:
-
=
7nVk
bk<t> (S'fc)
{Sk) to simplify the notation,
4>
0^,
Sk) nVi
and
differentiate the
above equation to obtain:
where efc
=
ating (20),
0' {Sk)
Sk
n
Vi
Sk
Sk
n
Vi
Sk
<
Ski 4) {Sk)
and
TT/
and
jVk
-
{S
Sk)
(A8)
Vk
Sk/(f>
{S
-
Sk)
<
0.
Next
differenti-
and rVk
5Vi
-
F,
=
-
iVk
(A9)
<514,
given by (21). Therefore,
Sk
-
[Gk
+
m
m
ei\
Sk
-
[ek
+
a constant proportional to the
C
= (pilp + eg* + 6 -
-
2(3)
BGP
g*/
fom-th differential equation
(1
—
1
n
/?
(5
-
fact that
is
{I
h
4>kSk
ei]
where the second step uses the
oiu-
0'
Vk
we have
rVi-Vi = 7Vtwith
=
e;
m
m
,
<Pi
{S
-
we are
cus/wL
- p)r\
f wL
rK
\nVi
mVk
Sk)
+
in the
[b*
^-j^<Pk
- ok)
neighborhood of the BGP, and C
ratio given by,
Also recall that
[e^
+
e;]"^
<
0.
Therefore,
is:
Sk
ass {Sk
-
SI)
+
a^k {k
-
k*)
(AlO)
,
Sk
with
Uss
>
and
Usk
Expressing these
>
0.
ecjuations together,
fotu-
/ Sk/Sk \
1
M/M
Since
M and
A:
\ aks
I
are state variables
\
1 ^^ \
M
ams
r-^
k/k
ask
ass
c/c
\
we have
acm
ace
ack
akm
akc
a.kk
and Sk and
\ k 1
J
c are control variables,
requires the determinant in (All) to have two positive
a.ss
-
ams
det
A
ask
V
aks
\
-A
acm
akm
=
dec
akc
32
'^
ack
akk
saddle-path stability
and two negative eigenvalues. To
find these eigenvalues, write:
/
(All)
c
-
^ /
0,
which gives the following forth-order polynomial:
-
A^
+
[uss
—
+
ace
The
four roots, Aj, A2,
—
[amsO-kmO-sk
-]-ams
+
a^k] A'
OsA-
"
—
CLkm
-
akcO-ck
A'
aksO-sk]
=0.
+ UssO-ccClkk] ^
CLsslkcIck
[O^cc
•
-
[assUkk
^kc
^cm)
'
and A4 give the eigenvalues. By standard arguments we have
A.3
that these four roots satisfy:
Ai
Now
•
A2
•
using the fact that Occ
>
Since a^m
>
0- Qsi-
0,
and
A3
=
A4
•
=
Ums
•
{Cikm
0.sk
o^c
= — 1 and
'^l
^^2
A3
a/.,„
>
1
,
A4
•
^
ams
"
acm)
have
,
Qctti-
Ai
either foiu- positive roots, or four negative roots, or
O-kc
— akm we
cl'^^
dsk
we have that
0.
—
O-cc
=
Ccm
?
'
•
A2
A3
•
>
A4
So we must have
0.
two positive and two negative
roots.
Next also note that
Aj
A2
+
Aj
•
A3
+
Ai
•
A4
+
A2
•
A3
+
A2
•
+
A4
A3
=
A4
•
Uss
—dkc
Qkk
—CLks
O-ck
-(-)•(-)
(+)•(-)
-(+)(+)
which means that we can have neither four positive roots nor four negative roots, establishing that there
must be some
must be two positive and two negative
With
c
>
1,
we have
agk
and some negative
positive roots,
<
0,
roots, so the
thus Aj
A2
•
•
A3
system
A4
•
is
BGP
have three positive and one negative eigenvalues, and the
roots. Therefore, there
locally saddle-path stable.
= — Gms
'
'
^sA-
<
a'^^^
equilibrimn
0.
and we
not locally
is
stable.
Proof of
Lemma
1:
I
will
lemma
prove this
that if bi(p*n{t)Vi[t) < bk0lm{t)Vk{t) where
>
bi(()*in{tf)Vi{if) < bk(plm{t')Vk{t') for all
(f>*i
t'
dynamics can be represented by
(32),
where
ti'
=
t.
(A/*)
two
in
(p{S
Next
=
0,
steps.
First,
=
-
S*^)
and
I
uill
show that
and
(A/)
(pl
=
I
I
will
show
0(5^), then
transitional
for all
M =| M*
0 (M) | for all M | M* for £ > 1.
Recall that k{t) = G{M{t)) at all i, with G'(-) strictly increasing. Next recall that
bi4)ln [t) Vi [t) = bk^lm (t) Vk («), when aj^- = 6*, or when M = M* and k = k*. Also in
this case, M =
and N/N = g*. Now I will show that these properties imply that, for
£ < 1, Gji [t) > h* if and only if 6;0^n (i) V; (f) < bkCplrn {t) \4 {t). Moreover, these imply
>
{t') for all
bi(p*n {t') V; (f < bkdlm {t')
for £
<
1
and
)
V'l,
t'
t.
Clsk
<
Suppose that a^
>
biifiJi (i) Vi (i)
and thus
m
m<
bkcplm
Vk
bk<P {Sk (i))
{t)
+
bi(f>*in{t
[t)
>
bk(plin (t) \4
(i)
Vk
First
{t).
some S^
for
{t)
and to derive a contradiction, suppose
b*,
if
this is the case,
<
(t)
Now
{t) if bi(/)*in (t)
At)Vi{t
+
>
Vi (i)
fefe^^m (f) 14
=
At)
+ At)Vk{t +
=
At)
therefore,
limt^oo
k
(i)
''™ {t)
Vk (0 /n
(t) Vi
(0
—
bk<f>
(0)
m
and 6/0^n
At -^
+
6i<^;n(i)VH0
\4
{t)
(t) V] (t)
{t)),
>
(i)
V; {t)
0,
we have
bi(t,*n {t) Vi {t)
>
>
bk4>lm{t)Vk{t)
bu<t)lm{t)Vk{t)
bk<t)lm{t)Vk[t)
M
°°! giving a contradiction. Therefore,
bi(j)*n{t)Vi {t)
= G (M (t)), we have
establishing that
to
b*
t'
M
addition, recall that
<
>
n
{t))
>
A;
(i)
<
<
bkcplm
k*
and therefore
{t)
Vk
{t)
But whenever
.
M
(t)
< M*.
1,
M
aK{t)
when
{M*)
cta- (i)
< M*, we
(t)
<
M = M*
ip
b*,
.
>
=
= 0.
(i)
6*,
M\
The arguments
Finally,
ax
it
(0
=
b*,
M
(i)
< M*
^
<
b*
{t)
b*,
from
/M (t) = V' (M (t)). In
[t) = M* and k (t) = k*,
(t)
and
34
0,
Since
0,
>
And
(Ta' (i)
M[t) /M[t) >
we have M[t) /M{t) <
for a^- (f)
>
cr^' (t)
>
ax
has already been shown that in the case with
we have M[t) /M{t) >
have that
M{t) >
requires
whenever
This estabhshes that the
M
therefore M
law of motion of the economy can be represented simply by
£
S^
> bk(t>lm (t) \4 {t) implies m < 0, which in tvirn, from
implies k (t) < 0. So a^ [t) > 0. Therefore, we must have
> t, and consequently
(i) /M {t) < 0, and
bk4>lm [t') 14 {t') at all
/k (t) < 0, and hmt_>oo k (i) = 0. But Lemma Al implies that in this case
we must have
(t)
{t) Vi {t)
ax (0 >
+
-
have i;0 (5
recall that bi(t)*n [i] Vi [t)
A;
k
will
Third, note that for
[t).
bi(l)*n{t)Vi{t)
{t) = G {M (i)),
bi(p*in {t') Vi {if) >
b*,
n
(5)
bi<t>
Second, from (A9), we can only have
0.
>
bk(f>lm{t
5* (or
we
also that
0.
proving that
for the case
where £
>
?/;
1
b*
is
equivalent
similarly,
(M) =
when
for all
are analogous.
VIII.
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38
NBER
90%
oa
85%
•a
•a
o 80%
"5
>
A
TO
2 75%
'I
o
\/\
/
m
A ^/-\N
t\
V^
/
S 70%
A
65%
^^ A^
K^ vy^
/
/
"\
V
60%
3>
n
CO
r*
Figiu-e
1:
Labor share
1
a>
CO
1
CD
in total value
lO
o
added
Saez (2001). Source: National Accounts,
CD
o
CO
to
00
00
in the U.S. corporate sector
NIPA Table
39
1,16.
00
CO
a>
05
from Piketty and
90%
M
V
a>
Figure
2:
^
CM
o>
Labor share
«
n
on
r-
n
m
in total value
T—
t
o>
lO
t
a>
added
o>
Tt
o>
in the
(2001) based on French National Accounts.
40
^
(O
o>
lO
to
at
o>
U5
o>
CO
ho>
French corporate
hha>
^
CO
o>
sector.
lO
00
O)
a>
CO
o»
to
o>
a>
t~.
o>
O)
From Piketty
p {b,(/>{S)S-d^)
p
M
Figure
3:
Transitional dynamics uith risk neutrality
41
and
£
<
I.
M_
M
\-p
5
Figure
3837
4:
Transitional d}Tiainics with risk neutrality
.37
42
and
^
>
1.
Date Due
MIT LIBRARIES
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iji
nil
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