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WHY DO NEW TECHNOLOGIES COMPLEMENT SKILLS?
DIRECTED TECHNICAL CHANGE AND WAGE INEQUALITY
Daron Acemoglu
No.
97-15
August 1997
,
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
? -Y
I:
WORKING PAPER
DEPARTMENT
OF ECONOMICS
WHY DO NEW TECHNOLOGIES COMPLEMENT SKILLS?
DIRECTED TECHNICAL CHANGE AND WAGE INEQUALITY
Daron Acemoglu
No.
August, 1997
97-15
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02139
1
(
MASSACHUSETTS
I^^STITUTE
UBRAHlES
J.
I
.>.
iOLOGY
JAN 1 9 1998
UBWAFHES
This version: August 1997
Why Do New
Complement Skills?
Directed Technical Change and Wage Inequality*
Technologies
Daron Acemoglu^
Department of Economics, E52-371
Massachusetts Institute of Technology
Cambridge,
02139
MA
have benefited from many insightful conversations with Jaimie Ventxira during the gestation
and from many of his comments on earher versions of this paper. I also thank Ohvier
Blanchard, Ricardo Caballero, Francesco Caselli, Michael Kremer, Dani Rodrik, Fabrizio Zilibotti,
seminar participants at LSE, NBER and Rochester for comments and suggestions. Financial support
from the National Science Foundation Grant SBR-9602116 is gratefully acknowledged.
tTel: 617 253 1927. Fax 617 253 1330. E-mail: daronOmit edu
*I
of this project
.
This Version: August 1997
Why Do New
Skills?
Technologies
Complement
Directed Technical Change and
Wage
Inequality
Abstract
consider an
I
gies.
economy where
The rate of improvement
skilled
and unskilled workers use
of each technology
different technolo-
determined by a profit-maximizing
is
R&D sector. When there is a high proportion of skilled workers in the labor force,
market
for skill-complementary technologies is larger
upgrading the productivity of
when the
skilled workers.
An
and more
effort will
be spent
implication of this theory
relative supply of skilled workers increases exogenously, the skill
decreases in the short-run, but then increases, possibly even above
the
is
in
that
premium
its initial
value,
because the larger market for skill-complementary technologies has changed the
di-
rection of technical change. This suggests that the rapid increase in the proportion
of college graduates in the U.S. labor force
in the college
ual
premium during the 1970s and the
The paper
the 1980s.
may have been
causal in both the decline
large increase in inequality during
also derives implications of directed technical
wage inequality and shows that
change
for resid-
calculations of the impact of international trade
on inequality that ignore the change
in the direction of technical progress
may be
misleading.
Keywords: Endogenous
Technical Change, Relative Supply of
Skill,
Returns
to Education, Skill-Biased Technological Change, Skill- Technology Complementarity,
Wage
Inequality.
JEL
Classification: 014, 033, J31.
Introduction
I.
Between 1811 and 1816
that they would
make
in Britain, the Luddites destroyed
machines beUeving
their skills obsolete. In 1826 in Lancashire,
hand-loom weavers
attacked weaving machines.
In 1830 during the Captain Swing riots, agricultural
But technical progress could not be halted
workers destroyed threshing machines.
and these
skilled
workers were quite soon replaced by the machines they tried to fight
Mokyr, 1990). In contrast to the
(see
nineteenth century,
new
examples
like
replacing technological advances of the
technologies today appear to be complementary to skilled
and educated workers rather than the
historical
skill
rmskilled.^
Why
is
this?
As suggested by the
the spiiming jenny and the assembly line which increased
the productivity of unskilled labor, the answer that
very nature "skill-biased"
is
new
technologies are by their
Even computers which are seen
not satisfactory.
as
the prototype example of "skill-biased" innovations can be and are sometimes used
as
complementary to unskilled
and most employees
labor.
For example, most workers making deliveries
at fast food restaurants
and supermarkets use computers and
scanners. Motivated by this reasoning, this paper starts from the premise that
new
technologies are not complementary to skilled labor by nature, but by design. This
premise then raises the question of why over this century, and especially over the past
two decades, the productivity of
than that of the unskilled.^ This
Most
many
skilled
is
and educated labor has been upgraded more
the question addressed in this paper.
technologies, once invented, are largely nonrival goods: they can
firms
and workers
at low marginal cost.
When
the market for skill-complementary technologies
able to obtain higher returns.
high,
more
effort
As a
result,
when
wiU be devoted to the invention
is
there are
larger
more
be used by
skilled workers,
and the inventor
will
the proportion of skilled workers
be
is
of skill-complementary technologies
rather than those complementary to unskilled labor. This implies that the impact of
an increase
in the supply of skilled workers
on the
skill
premium will be determined by
^See Grilliches (1956) and Bartel and Lichtenberg (1987) for micro estimates. See, among others,
Berman, Boimd and Grilliches, (1994), Goldin and Katz (1996), Autor, Katz and Krueger (1997)
for studies that document the relation between relative wages and technology.
^In practice, skill is multi-dimensional. The skills replaced by nineteenth century inventions are
different than those complemented by computers today. For most of the paper I focus on education
which is the most easily observable component of skills. I will then return to other dimensions of
skill in an attempt to explain the rise in residual (within group) wage inequahty.
two competing
forces: the first is the conventional substitution effect
economy move along a downward
directed technology effect
demand curve
for skills.
sloping relative
demand
curve.
which makes the
The second
which can be thought of as shifting the "short-run"
This
effect is
is
the
relative
driven by the fact that an increase in the
supply of skilled workers increases the profitability of technologies complementarity
to skilled labor.
Two
effect is sufficiently
skilled workers
implications of this approach are: (1)
effect,
accumulates more
the directed technology
pronounced, in the long-run an exogenous increase in the supply of
can increase the
the substitution
if
skill
premium;
(2) irrespective of
whether
it
dominates
the directed technology effect implies that as an economy
skills,
the fraction of
workers should increase, which
is
new
technologies complementary to skilled
consistent with the fact that technological change
appears to have become more skill-complementary over time.
This theory suggests an alternative explanation for the changes in the structure
The conventional wisdom
of wages in the U.S. over the past two decades.
is
these changes are due to exogenous technological developments. In particular,
believe that there
was an exogenous
The most
starting in the 1970s. ^
of any reason
why exogenous
that
many
acceleration in the skill-bias of technical change
serious
problem
for the conventional
skill-biased technical
view
is
the lack
change should have accelerated
in the 1970s, coinciding with the unprecedented increase in the relative supply of
skilled workers.
"for the
The
puzzle
is
in fact
more
striking.
In Katz
and Murphy's words;
1963-87 period as a whole and most strongly for the 1980s, the groups
with the largest increases in relative supplies tended to have the largest increases
in relative wages" (1992, p. 52). In contrast to the conventional approach, suppose
we
are in a world of directed technical change and consider a large and exogenous
increase in the supply of college graduates (as in the U.S. diiring the late 1960s
1970s, see the discussion in the text).
graduates in the labor force will
technology) relative
first
demand curve
Then, the induced change
This increase in the proportion of college
move the economy along a short-run
as in Figure
Suppose
effect.
The improvement
first
1,
(constant
reducing the college premium.
in the direction of technical progress will increase the
marginal product of college graduates and
right.
and
shift
the relative
demand curve
to the
that the substitution effect dominates the directed technology
in skill-complementary technologies will then increase the
^Observe following Autor, Katz and Krueger (1997) that from 1940 to 1970 relative supply of
by 7.7 percentage points and the college premium fell shghtly. In contrast,
from 1970 to 1990, the supply increased by 11.6 percentage points and the skill premium increased.
A constant rate of skill-biased technical change cannot account for this pattern.
college graduates increased
Relative
Short-run
Wage
Relative
)emand
Long-run Relative
Long-run
Rel
Demand For
Skills
Wage
Initial
Rel
Wage
Demand
Short-run
Shift in Rel.
Response
due to Technical
Change
H/L
Exogenous
Shift in
Relative Supply
Figure
1:
DjTiamics of the
nates the substitution
college premiimi
from
it
its
college
is
the college
its level
complete explanation
before the supply shock. In contrast,
increase. This
for the
its initial level.
premium increased during the
sufficiently strong, the
premium should
five years. I will also
short-term low but never above
why
increased above
technology effect
directed technology effect domi-
effect.
the model can explain
why
premium when the
skill
is
model predicts that
In this case,
1980s, but not
if
the directed
in the long-run, the
the case drawn in Figure
1
and
offers
a more
changes in the U.S. college premiimi over the past twenty
show how this mechanism may account
wage inequality during the 1970s while the
college
for the increase in residual
premium was
falling.
Furthermore,
the theoretical results also apply to any group using technologies different from the
rest of the labor force
is
and experiencing an increase
in relative supply.
So the model
consistent with a positive association between changes in the relative supplies of
more disaggregated groups, and
their relative
wages over the medium run as found by
Katz and Murphy (1992) and Bound and Johnson
this
(1992). Therefore, the analysis in
paper suggests that the rmprecedented increase in the supply of college graduates
during the 1970s
may have been
causal both for the technological developments and
the changes in the structure of wages during the 198Gs as well as the decline of the
college
premium during the
1970s.
There are other episodes
in
which a large increase in the supply of
to have affected the direction of technical change.
skills
appears
The enrollment and graduation
rates for high school doubled in the 1910s, mostly due to changes in the location
curricula of schools
and the decline
high school premiiim
fell
in transport costs (Goldin
and Katz, 1995). The
sharply in the 1910s. Yet, despite the even faster increase
in the supply of high school skills during the 1920s, the relative
graduates levelled
that the
demand
off
and started a mild
increase. Goldin
for high school graduates
the 1920s, presumably due to changes in the
from new industries such as
An
this theory.
the direction of technical change.
LDC
If
unskilled labor will increases and
by only a small amount.
protection,
it is
not possible to
sell
induce further
effort in
and increased demand
on wage inequality provides
is
that trade will affect
the U.S. starts trading with the
market
in
and chemicals.^
observation
for technologies
wage inequality
However,
increase the relative price of the
will
office technologies
The key
firms, the size of the
skill
and Katz (1995) conclude
electrical machinery, transport
an interesting application of
wages of high
must have expanded sharply starting
analysis of the implications of international trade
technologies to
and
LDCs and
sells
complementary to
will decline, or at
most increase
due to lack of international property rights
if
new
technologies to
skill
intensive good.
LDC
firms, trade will
simply
This relative price change
upgrading skill-complementary technologies.
I
show that
in this case conventional calculations underestimate the impact of trade
on wage
inequality because they ignore the change in the direction of technical change.
This paper
is
related to
some
on induced innovations, including
older literature
the theoretical work by Fellner (1961) and Kennedy (1964), the empirical studies
by Schmookler (1966) and Hayami and Ruttan
Habakkuk
(1962).
(1970),
and the
historical
work by
These studies (with the exception of Schmookler) discuss the
impact of factor prices on induced innovations.
I
treat factor prices as endogenous,
point out the importance of the fixed costs of innovation and of market
size.
These
features yield the crucial result that a larger relative supply of a factor can lead to
faster
upgrading of technologies complementary to this
factor,
which
is
the opposite
of the results that these papers discuss but in line with the evidence provided
by
Schmookler. This paper also builds on and extends the work of Aghion and Howitt
(1992) and
Grossman and Helpman
"^The high school
(1991)
premium never reached
its
increase in the supply of high school graduates.
by allowing technological change to be
1900
level,
partly because of the continued rapid
In any case, note that the theory offered in this
paper does not predict that every increase in the relative supply of skills should be associated with
increased skill premium, especially when the system is away from the balanced growth path as U.S.
is likely to have been in the 1900s. The model however predicts that every increase in supply should
have an impact on the direction of technical change.
directed towards different groups.
that changes in the supply of
made
is first
in
skills
who
are skilled, firms find
targeted for this group.
The
increase in skilled wages.
Krugman
varieties
common
result
A
features.
a
is
fall
differing
with regards to the
inequality.
an economy with
The plan
of the paper
skills
is
in the level of unskilled
skills
which
is
wages and an
model
in the proportion of skilled workers
of their high school graduates.
may
and have a high
as follows.
Section
skill
II
can
He shows
that
choose a technology which
premium.
analyzes the basic model and
contains the most important results of the paper. Section
supply of
a sufficient
profitable to create jobs specifically
high school graduates
use of high school
is
(1997) compares the technology choice of economies
skill level
less skilled
it
This point
for skills.
by Kiley (1997) considers an expanding
recent paper
Walde
wage
little
demand
the
(1997) has recently constructed a signalling
model and shows that an increase
increase
makes
may change
Acemoglu (1996) using a search model; when there
fraction of workers
with some
a number of recent papers also suggest
Finally,
III
treated as exogenous in Section
endogenizes the relative
II.
Section
IV
discusses
the impact of international trade on the direction of technical change, and on wage
inequality through this channel. Section
V concludes.
II.
The Basic Model
A.
Technology and Preferences
Consider the following continuous time
infinite
preferences over the unique consumption good, y.
exp(-r(r
Ckt
There
is
a
H of skilled workers and a continuiim L of imskilled workers with identical
continuum
where
horizon economy.
is
conditional
utility, it will also
be the
set at
time t,r
utility of
agent k at time
- t))ckTdT
the consiunption of agent k at time
upon the information
The
is
t,
Et
is
t is:
(1)
the expectations operator
the discount rate and due to linear
interest rate.^
The imique consumption good is produced from two complementary intermediate
goods, or production processes, one using skilled and the other unskilled labor.
market
^An
for intermediate
goods
equivalent formulation
is
to
is
competitive.
I
denote the total output of these
assume a general instantaneous
and borrowing at the rate r.
perfect international market for lending
The
utility function u{ckr)
and a
processes (intermediate goods) by Yi{t) and Yh{t), and the aggregate production of
the consumption good at time
t is:
=
yit)
<
where p
1.
normaUze the
I
biYr{t)+^,Yat)]'^^
price of the final
good
(2)
in each period to
1,
and denote
the prices of the two intermediate goods by pi{t) and ph{t) which, due to competitive
pricing, impUes:
demand
a
(3) is
equation, hnking the consumption of the intermediate goods to their
relative price.
There are mi and ruh firms
rest of the analysis
I
in the
will normalize,
mi
two intermediate goods sectors, and in the
=
m,h
=
The production
1.
of Y^, the skill-
intensive good, requires skilled labor while the production of the labor intensive good,
yj,
requires unskilled labor. Namely, firm
ysii.t)
where
=
s
Z, /i,
<
and
ns{i,t)
and
is
=
i
in sector s has production function:
As{i,t)[ns{i,t)f
(4)
the number of workers employed by firm
i
in sector s
at time
i, /?
sector
only employ unskilled workers, and those in sector h only hire skilled workers,
/
1
As{i,t)
the productivity of labor in this firm. Since firms in
is
feasibility requires that Jni(i,t)di
The
is
J nh{i,t)di
firm level productivity parameter, As{i,t),
and quality
There
< L and
of nonlabor inputs (machines) that firm
a continuum
js
£
[0, 1]
i
<
is
H
for all
t.
determined by the quantity
in sector s ptirchases at time
of sector-specific machines for each sector.
that each sector uses different machines
is
The
t.
fact
the sense in which skilled and unskilled
workers use different technologies in this model. The quantity of machine j that firm
i
in sector s uses at time t
after use.
for j
G
I
is
denoted by Xs{i,j,t). These machines depreciate
denote the highest qualities of these machines available at time
[0, 1]
and
s
= l,h.
In principle,
it is
t
by
fully
qs{j, t)
possible that a firm purchases inputs that
are not of the highest available quality, but this will not
happen
in equilibrium (see
below). Hence, incorporating the fact that outdated machines will not be used, the
productivity parameter of firm
i
takes the form:
rl
1
Mht) = -J
<
where a
1
— p.
and will make
if q;
= I — P,
If
what
B.
—
1
/3,
production in both sectors
follows,
same
mi and ruh are
Whether a
results.
Maximization
Profit
fijxed).
In contrast,
subject to constant returns, and free-entry
is
do not take a position on
I
(5)
intermediate good producers have decreasing returns
positive profits (hence their numbers,
will give exactly the
in
a <
Qs{j,t) [xs{i,j,t)fdj
= 1 — P does not
and
affect the results,
this.
By Firms and Labor Market
Equi-
librium
Firms (producing
yi
and
Denoting the price of machine
profits.
unskilled workers
by
inax
and
Wh,{t)
qs{j, t)
wi{t), firm
by
is
strictly concave,
sector hire the
=
it
same amount
of labor
and the aggregate demands
are Xs{j,s)
=
problem
and
skilled
at
time
has a imique solution, implying that
H
and
and the wages of
t:
Xs{j,t)xs{i,j,t)dj-Ws{t)ns{i,t)
/
=
i
t)
solves the following
normalization, m;
for all
Xs{j,
Jo
Since this problem
same
i
-
ps{t)As{i,t)[ns{i,t)f
ns{t,t),Xs{i,j,t)
in the
machines and labor to maximize static
y^) purchase
t,
rrih
1,
all
for
Given
=
Ni
=L
machine j
and
nh{i,t)
in sector s at
N^,
=
time
t,
=
this
unique solution to the profit
firms in sector s will have the
same productivity parameter
(ps(i)9s(j, 0-^f/Xs(j)^))
maximization problem,
and inputs (machines). With the
this implies that ni{i,t)
"
firms
all
'
(technology), As{t). This enables us to determine wages in terms of the final good:
Ws{t)
=
Pps{t)As{t)N-^^-l'^ for s
The
premium
skill
for this paper.
The
This
is
skill
Using
(skilled
=
wages relative to unskilled wage)
(3), this skill
premium
l,h.
increases
premirmi, u,
when
skilled
is
the main interest
is:^
workers become more scarce, jjM;
<
0.
the usual substitution effect and shows that, for given technology, the relative
^In this section, for some parameter values, skilled workers
unskilled,
i.e.
the supply of
uj
<
1.
One may want
to
impose jh/li
endogenized in Section
parameter restriction is not necessary.
skills is
III,
> {H/Ly
the
skill
may
~^'''^
premium
have lower wages than the
-p)-pp ^.q avoid this. When
is
always positive, and this
demand curve
when p G
for labor
(0,1],
is
nyA
^^
(t)
downward
-^
sloping with elasticity (1
The
conventional
wisdom
is
become more productive, not
reveal
1
an
The converse
> OJ The
is
Moreover,
when p <
skilled
workers
In support of this, most estimates
less productive.
between
obtained
when
that the skill-premium increases
elasticity of substitution
which implies p
/3p).
'^^^ implies that when the skill-complementary
^-
technology improves, the skill-premium increases.
0.
—
skilled
and unskilled workers greater than
high degree of substitution between skilled and unskilled
workers suggested by the increased share of college educated workers within almost
narrowly defined industries also corroborates this view
all
Krueger, 1997). Hence, in the remainder of the paper
p G
(0, 1)
Autor, Katz and
(e.g.
limit attention to the case
I
(though the formal analysis does not depend on this parameter restriction).
Technological Advances
C.
Technological advances take place as in Aghion and Howitt (1992) and Grossman
and Helpman
(1991). Namely,
-
f
.\
_
j ^ X Q^U,
\
for all j
A >
1
which
G
[0,1]
and
s
=
I,
t)
if
no irmovation
so that innovations improve productivity.
is
of
R&D
I
just before time
assume that X > a
R&D
t.
i-",
terms of the
t)
final
where
carried out by research firms using only final
is
free-entry into the
R&D
activity in technology j'for sector s at time
probability of innovation
Bqs{j,
Also,
i
a simplifying assumption to be discussed in the next subsection.
output as factor of production. There
amount
t
at time
h where q~{j,t) denotes the quality
Innovations are the result of
is
innovation at time
if
Q7{j)t)
is
Zs{j,t)(p{zs{j,t)).
The marginal
good) for inventing a machine of vintage
5 is a positive constant.^
decreasing and Z(p{z)
is
I
nondecr easing, that
there are decreasing returns to
R&D
assume that
is (f){z)
effort for
effort increases the probability of discovery.
+
sector. If the total
t is
Zs{j,t),
then the
R&D
effort (in
cost of
qs{j, t) in sector s at
0(.) is
z(f)'{z)
>
time
t
everywhere smoothly
0.
This implies that
any particular machine, but more
Also,
I
impose limz_,o 0(2)
=
cx3
and
between
which implies l/cr = 1 — j3p < 1, and
therefore /o > 0. Since a large part of the substitution between skilled and imskiUed workers is within
industries, p should not be interpreted as the elasticity of substitution between different goods.
^Alternatively, the cost of improving vintage qj{j,t) is BXq~{j,t).
'''See
Freeman
(1986). Practically, all estimates of the short-run elasticity of substitution
high and low education workers are between a
=
I
and
2,
limz^oo
<l){z)
=
These Inada type
0.
and smooth djmamics. In the Appendix,
A
it
holds a perfectly enforced patent), so
sell
many
as
Equilibrium
R&D
The aggregate demands
profit
maximizing price
input
is
follows
—
qs{j,t) is qs{j,t)
ensure an interior solution
where
it
profit
wishes.
maximizing price and
The marginal
linear in quality.
technology characterized above are isoelastic, so the
for
for vintage qs{j,
t) is Xs{j-, t)
buy
=
t qs{j,t)
qs{j,t) sold at the
q;~"/(i~°)
>
= ~^-
Hence, the price of each
which ensures that even
Xs{hj,i)
=
sector s at time
recall that
=
Xs{j,t)
t,
monopoly
(aps{t)N^]
H and Ni = L.
Nh =
These expressions make
in sector s at time
The
t is
equilibrium
it
Also
I
I
demands
wiU
in the relevant sector will buy:
as:
As{t)
=
^Qs{t) {<^Ps{t)N^]
have defined Qs{t)
=
sector s at time
R&D
technology in sector
s.
machines and the monopoly pricing policy im-
=
7r,(;,t)
-
Zs{j,t)cj){zs{j,t))V,{j,t)
R&D
effort to
is:
TTs{j,t)
=
+
Vs{j,t)
improve machine j in sector
(7)
s at
time
have made use of the fact that the leading monopolist will not perform
(Aghion and Howitt, 1992).
(7) is
rate Zs{j,t)(f){zs{j,t)), the firm loses
assumption were not
satisfied,
its
a standard value equation; at the flow
monopoly position because there
is
a
new
then the leading vintage of each technique would not be
priced at the monopoly price, but at a sufficiently low price to exclude the next best technology
a Umit
h.
/,
t is:
Zs{j,t) is the aggregate
^If this
=
Hence, the value of owning the leading vintage of machine j of
rVs{j,t)
I
where
measure of technological know-how
ply that the profit level of a leading innovator with vintage qs{j,t)
^^^^Xs{j,t)qs{j,t).
,
/J qs{j, t)dj for s
refer to this as the level of
for
would
price.^
clear that the relevant
Qs{t).
the next
Therefore, the equilibrium productivity in
.
can be written
(5),
if
qs{j,t)/^ were sold at marginal cost, firms
Given the monopoly pricing policy every firm
and
cost of
Effort
from the assiunption that A
prefer to
t,
1.
a constant markup over the marginal cost of production. This pricing policy
best technology at time
where
=
(j){z)
right over that particxilar vintage (e.g.
can charge a
it
it is
i.e.
</>(.)
also discuss the case
units of the newly discovered input as
producing input
D.
I
monopoly
firm that innovates has a
on
restrictions
price).
The
rest of the results
(1995) for a discussion of this case.
would
stiU
(i.e.
remain unchanged, see Barro and Sala-i-Martin
innovation, and the time derivative of
the fact that Zs{j,t)
may be
Finally, since there
on the right-hand
R&D
machine
=
R&D
effort for
Bqs{j,t)
(8)
the marginal return to higher
is
j in sector s (either
the right-hand side
An
side
activities, additional
profitable, thus:
4>{z,{j,t))Vs{j,t)
where the left-hand
side of (7) takes care of
time varying.
free-entry into
is
any machine must not be
V
by an existing research firm
or
R&D
effort directed at
by a new entrant), and
the marginal cost.^°
is
equilibrium in this economy requires that product, intermediate good and
labor markets clear, firms buy the profit maximizing amount of the latest vintage
of each technology, innovators follow the profit maximizing pricing policy,
is
no opportunity
Equations
for
(3), (6), (7)
equilibrium.
any research firm to enter
and
(8)
(or exit)
and increase
and there
its profits.
ensure these conditions and characterize the dynamic
The balanced growth path (BGP)
then an equilibrium such that
is
all
variables are time-invariant.
Characterization of the EquiUbrium
E.
Let us start with the balanced growth path and thus drop
Using
V = 0, equation (7) implies:
Then, the free-entry condition
(8)
Vs{j)
=
-
((1
for all j
(9):
G
[0, 1]
and
s
=
l,h.
A
number
= r + zs{j)cl>{zs{j))
increasing in Ps and Ns- Therefore, there
certain product
(9)
of insights can be obtained from equation
s, Zs(j),
flow profit of the leading iimovator for this machine
The
+ Zsij)(t){zsij)))).
BGP:
research effort to improve machine j in sector
on the incentives to innovate.
time dependence.
a) Xs{j)qs{j)) /{a{r
implies that in
^</'(^.(i)) {psN^y^^''"'^
all
is
a price
price effect
becomes more expensive, more
is
will
high.
is
(ps)
be higher when the
This flow of profit
and a quantity
is
(A^f ) effect
perhaps more familiar: when a
effort will
be spent to invent the next
vintage. This price effect also suggests the reverse intuition to the one discussed in
the introduction:
when
there are
more
skilled workers, the price of the
^°This ignores the constraint that total expenditure on
which
will
not apply
sufficiently decreasing
if
R&D
should not exceed ciurrent output,
the economy can borrow from abroad at the rate
would
also ensure that the expenditure
10
on
goods they
R&D
r.
Assuming that
is
never exceeds total output.
produce
Counteracting this
to unskilled labor.
skilled workers, the size of the
For p G
R&D
be low, so there should be more
will
market
the quantity effect
(0, 1],
makes more sense
is
the quantity
is
more powerful, and
model. ^^
premium. This
Returning to equation
s
=
l,h. In other
parameter
will require further
words, the
(9), it
rate of
=
Qh
is
p
=
to
This equation
—
is
BGR
Combining
a crucial result.
=
this
Zh.
with
Qh/Qi
this case
=
Zg for all j
is
and
(labor intensive)
a continuum of skill-intensive
is
\s
the growth
the growth rate of Qi. For
l)zi(f){zi) is
zi
larger.
workers will increase the
BGP levels of effort for all skill-intensive
Similarly (A
is
more
restrictions.
Equation
we
(3),
(9)
BGP,
then implies that
obtain:
the equilibrium technology of the
skill-intensive sector relative to the labor intensive sector,
of the
why
argued above
immediately follows that Zs{j)
be constant, therefore
{H/L)~^ along the
abundance
there are
exactly the rate of improvements. Hence {X—l)zh(j){zh)
Jo qs{j)dj.
we need Qh/Qi
I
skilled
technologies are the same. Next, note that since there
inputs, Zh(j){zh)
when
Also, note that the quantity effect
dominating does not imply that a higher supply of
skill
effect:
skill-complementary technologies
for
in the context of the
complementary
for technologies
and depends on the
two types of labor. The greater the fraction of
skilled
relative
workers in
the economy, the greater their relative productivity as compared to rmskilled workers
(this is for
>
p
equation (10)
equalized.
0;
is
as noted above, the reverse obtains
that equilibrium returns to
R&D
when p <
effort in
0).
The
intuition for
the two sectors have to be
Since profits to innovation are proportional to the market
effectively proportional to the nimiber of workers using the technology.
when
H
increases relative to L, innovation
become more
profitable.
and
R&D
Therefore,
in the skill-intensive sector
This translates into a higher Qh/Qi in the BGP.
this as the directed technology effect:
when the
they are
size,
I
refer to
relative supply of skill changes, the
direction of technical progress changes, leading to different equilibrium technologies
for skilled
result:
and unskilled
as the
make new
labor.
Equation (10) immediately implies an important
economy accumulates more
technologies
skills,
more complementary
^^For some other situations, the price effect
may
technical change will respond to
to skilled labor. Therefore, the fact that
dominate.
For example, Hayami and Ruttan
(1970) discusses the different paths of agricultural development in the U.S. and Japan.
of land in
Japan
relative to the U.S. appears to
The
scarcity
have induced a much faster rate of innovation and
adoption of fertihzers, increasing output per acre.
11
new
become more skill-complementary over time
technologies have
is
consistent with
the approach in this paper.
BGP R&D effort level can now
imposing z* = zi = Zh, which gives:
The
+ Z*(f>{z*) _ 1-Q UnMil-p)
r
we have
Finally, using (6),
Proposition
total output
Qh/Qi
is
There
1
be determined from
is
rj
=
given by (10) and the
—
l)z*(j){z*)
skill
with z* given by (11).
premium
— (1 — /5p),
BGP
(recall
equation
sloping relative
the
The
which
skill
is
(..^
and
both sectors
Along the BGP,
is:
Vh
there
is
(12)
a 1-to-l relation between the relative supply of
workers and their relative wage.
or decreasing.
by
(10)
— /3p)
(1
In the unique
skilled
and
tI^)
(BGP) where
a unique balanced growth path
grow at the rate (A
f^
i-p
(9)
(proof in the Appendix):
-= where
./3p/(l-p)]
,
(3),
But, this relation can be either increasing
make
force that tends to
it
the elasticity of the relative
This
(6)).
is
premium would be determined
is
If
is
the second term in
demand curve for
the usual substitution
demand curve for skills.
coimteracting force
decreasing
effect,
skills for
rj,
given A^/Ai
leading to a downward-
technology were exogenous in this economy,
solely
by
this effect as in equation (6).^^
The
the directed technology effect which works through changes in
relative technologies (Qh/Qi)-
An
increase in
H/L
leads to
more
R&D
activity in the
skill-complementary technologies and therefore increases Qh/Qi, shifting the "short-
run relative demand curve" to the right as in Figure
strong, that
is if
and the long-run
increase in the
f^
is
demand curve
for skills is
upward
of skilled workers relative to the
will lead to a higher relative price of skill in the long-run.
likely to
happen when p
If this effect is sufficiently
large enough, the directed technology effect can dominate,
relative
number
1.
is
sloping. In this case,
number
of unskilled workers
This "perverse" case
close to 1 so that the skill-intensive
an
is
more
and labor-intensive
^^A hybrid case is where technology is exogenous, thus there is no R&D, but As{t) is still determined by purchases of machines from a monopolist. In this case, the skill premium is again a decreasing function of relative supply of skills, H/L, albeit with the smaller elasticity, —{l—/3p—ap)/{l—ap),
which is the short-run elasticity in the full model as shown in Proposition 3.
12
production processes (intermediate goods) are close substitutes, and when
to
1
so that there are only limited decreasing returns to labor within each sector.
A
different intuition for the possibly
that there
an important nonconvexity
is
cost of discovering
new
of skilled workers
and
it
new
lim^-^oo z(f){z)
rate
equal to (A
is
=
—
(po
<
elasticity
Next,
it
—(1
the essence of the directed technology
B =
— (3p —
effect,
There
To make the economy
Then, we have
oo.
is
ap)/[l
now
—
is
an instructive
z^
well
—
>
behaved in
oo for s
=
/, /i,
and the growth
size for technologies is
decreasing in the relative supply of
ap), which
can be established that the
and proof
Proposition 2
so that if
Q<
is
no longer
2.
Q^
unambiguously negative.
BGP characterized in Proposition
1 is
(saddle
Locally, there exists a unique transition path converging to
Q* then z^ and
,
Q* then
Zh
Suppose e^{z)
is
>
Q* then
,
The system
is
Zh
zi
z'
Then
>
to
zi
jump and
Q
BGP,
monotonically adjusts to Q*
.
If
along the transition path and vice versa.
nonincreasing in
unique saddle path
Q<
with
skills,
in the Appendix):
1.
,
limit
this case, also
path) stable. For this proposition, define e^ as the elasticity of the function 0.
(analysis
many
so that there are no upfront fixed costs of
Because the market
l)0o-
important, the skill-premium
an
is
their long-run relative wage.
technologies.
impose
a fixed upfront
ensures an increasing relation between the relative supply
case where this effect disappears:
discovering
is
is
This nonconvexity (nonrivalry of technology use)
This
clientele.
powerful enough,
if
economy. There
in this
demand curve
relative
R&D firms are more willing to improve the technologies
implies that profit-maximizing
designed for a larger
upward sloping
technologies/^ and once discovered, they can be sold to
firms at constant marginal cost.
and
close
/? is
BGP
Then, for
z.
along which
Q
all
Q
j^
Q*
,
there is a globally
monotonically converges
to
Q*.
If
along the transition path and vice versa.
always locally stable, and under a fairly weak assrmiption,
it is
also
globally well-behaved. Figure 2 illustrates the local dynamics with a phase diagram.
F.
The Dynamic Response
The
to a Relative Supply Shock
following result summarizes the
dynamic response
of the
economy
z*Bq/{r
+ z*(p{z*)).
unanticipated relative supply shock (proof in the Appendix):
^^The expected discounted cost of discovering vintage q along
13
BGP
is
to
an
^=Zh-
Z]
Q=Qh/Q,
Figure
Transition Dynamics
2:
Around the Balanced Growth Path.
Proposition 3 Consider an unanticipated
from a
BGP with skill premium u = u.
falls to
ill
new
— log to = —9 log
where log lj
BGP
premium u
skill
is
increase
^
from
to 5j^ at
Immediately after the
(5,
and
such that logo;
=
9
—
"^~/l^~"^
logo)
=
77
,
Zhtj zu
— ^log^.^^ The
words, Qh/Qi
short-rrm response
/9
>
0),
Figure
and the
1,
is
skill
premium
adjusts to
its
up.
The
skill
premium
falls
know-how; in other
many
is less
its
^^Recall that 8
— (ip)
to a change in
>
1
demand
than the long-run
analysis in this paper formalizes this claim.
more
in
short-term low. In terms of
to the right. Therefore,
for skills.
premimn
This result
is
not
believe that the short-run elasticity of substitution between
and unskilled workers
predictions are
1
increases (as long as
in the relative supply of skills will create a period of rising skill
very surprising:
simply —(1
from
demand curve shifts
in response to the induced shift in the relative
skilled
jumps
In terms of Figure
new BGP, Qh/Qi
starts increasing
the constant technology relative
an increase
premium
a move along the short-run (constant technology) relative
As the economy
curve.
for given technological
a stock variable and only changes slowly.
is
the introduction, this
demand
is
starting
log ^.
Therefore, immediately after the relative supply shock, the
by
t
the skill
shift,
and
time
surprising
and —6
is
and
Qh/Qu
In contrast,
original.
negative because
because for given
/3
elasticity. So,
<
when
14
>
0,
jy
1
—
<
0,
the
the model's
Because with the increase in
H/L
a. Also observe that the elasticity
the equilibrium productivities
H/L.
77
when
is
the
not
Ah/Ai change in response
size of the
market
for skill-complementary technologies increases, there is
to upgrade these technologies
Proposition
3,
and the
skill
especially in the case
for the behavior of the U.S.
premirmi increases above
where
77
>
0, offers
To put
its initial
R&D
value.
an alternative explanation
economy during the past twenty
large increase in the supply of skills in the 1970s.
more
it
five years.
There was a
into perspective, observe
that the employment share of college graduates increased by 1.7 percentage points
from 1940 to 1950, by
from 1950 to 1960, and by 2.8 percentage
2.8 percentage points
points from 1960 to 1970. In contrast to this relatively slow increase before 1970,
it
increased by 6.6 percentage points from 1970 to 1980. Also during this decade, the
share of workers with some college but no college degree increased by 7.2 percentage
points. ^^
These are very large changes
in the relative supply of skills preceding the
the college premium.
rise in
Furthermore, these large supply changes were at least partly exogenous rather
than a simple response to anticipated higher returns to education in the future. Enrollment rates had been increasing since the mid 1950s and this trend continued in the
Two
1960s.
other factors contributed to the sharper increase during the 1960s: (1)
until almost the
college
end of the war, the Vietnam era draft laws exempted males enrolled in
from military
service.
This induced
many more young males
to stay in college
during the late 1960s in order to avoid the draft (see Baskir and Strauss, 1978);
government financial aid
for college increased
programs of guaranteed student
by a large amount during
loans, supplemental loans for students
this era.
(2)
The
and parental
loans for undergraduate students began in the 1960s and provided credit to college
students,
1973,
Pell
Grant program which subsidizes college tuition costs began in
and created yet another
cially for
1991,
and the
rise in college
enrollments from 1974 onwards, espe-
the group of students with low income parents (see McPherson and Schapiro,
and Kane,
1994).
For example, the total federal aid to college students that
stood at approximately 2 million dollars in 1963 increased to $14 million in 1970-71
and then
to $24 million in 1975-76
(all
numbers
in 1989-90 dollars).
McPherson and
Schapiro (1991) estimate that without government support, college enrollments would
have been 20% lower during
this period (see also Leslie
though 20% may be an overestimate of the
is
probably in the right range when the
and Brinkman, 1987). Even
direct effect of
effects of
government support,
Vietnam era
it
draft are taken in to
^^See Autor, Katz and Krueger (1997). These figures refer to the shares of fuU-time equivalent
all workers. The changes
(hours worked) of college graduates divided by the full-time equivalent of
in the share of these workers in the labor force are very similar.
15
account. Using this number, a back-of-the-envelope calculation suggests that without
these exogenous factors, the share of college graduates which increased from 13.8%
to 20.4% from 1970 to 1980 would have only increased to 17%. Therefore, in terms
of the
model
The theory
should
of this section, there
was a substantial and exogenous increase
predicts that in response to this large increase in
and then
fall first,
H/L, the
skill
in
premium
due to the directed technical change
start increasing
H/L.
effect.
This pattern matches the broad behavior of the U.S. college premium from 1970 to
the present.
If
/;
<
0,
the increase in the
not compensate for the
skill
premium
In contrast, with
initial fall.
the technical change should take the
skill
rj
supply shock would
after the
>
0,
premium above
the model predicts that
its initial level
as
was the
case in the U.S.
It is also interesting
for reasonable
to do
some back-of-the-envelope
parameter values, the model makes
portant parameters are
/3,
a and
p.
Recall that p
realistic predictions.
and a
/? is
First,
skill
the return to machines in the
is
as a usually accepted value of the returns to machines (capital)
/3.
three im-
a measure of differential
production process. These parameter are not easily pinned down.
pieces of information to determine
The
the degree of substitution between
is
the skilled and unskilled production processes, and
intensity of the two production processes^^,
calculations to see whether
I
.
choose
I
a
=
0.3
used two different
Dunne, Doms and Torske (1997) report
the share of workers with college degrees across plants using different technologies.
Approximately 33% of employees at establishments using the most advanced technologies are college graduates, whereas the
same
ratio
the least amount of advanced techniques. Second,
industries at the 90th
and 10th percentiles
I
is less
than 10%
for those using
calculated the difference between
in terms of the
wage
bill
share of workers
with at least some college education among the 142 industries used in Autor, Katz
and Krueger (1997)
differential is
number
wage
in the census years 1970, 1980
all
three years, this
approximately 0.4 (0.40 in 1990, 0.41 in 1980 and 0.38 in 1970).
of (full-time equivalent) workers with
bill
and 1990.^^ In
some
college
is
I
^^In the model,
p
choose values for
is
/5
between 0.3 and
the
used instead of their
share, the 90-10 differentials across industries are quite similar.
these niimbers,
If
Based on
0.45. Finally, there is
also related to the share of labor in total output, but this
is
a range
due to the simple
A more reaUstic setup would involve both production processes using skilled
for example, Yh — Ah (Hh)
In this case, /? corresponds to (3i — 02{Lh)
structure of the model.
and unskilled
labor,
Unfortunately, this more general model does not yield analytical solutions.
^^I
thank David Autor
for providing
me
with these industry data calculated from the censuses by
Autor, Katz and Krueger (1997).
16
of estimates for the short-run elasticity of substitution between college graduate
non-college workers.
a
=
1.41.
it
as
cr
=
Katz and Murphy (1992) estimate
Bound and Johnson
1.70.
concludes that
this short-run elasticity as
(1992) using a slightly different technique, estimate
Freeman (1986) surveys a number of estimates of
it
should be in the range between
U.S. data put in between 1.2 and
a situation where Ah/Ai
is
1.8.
1
and
In the model,
we would have
constant,
if
1
we
—
/5p
=
[1
— f5p —
ap)/{l
—
=
ap)
consider the short-rim as
=
1/cr.
a and
1/a. Therefore, for given
both of these numbers between 0.55 and
0.85,
and
most studies using
In fact,
2.
this elasticity
Or
/3, I
if
we
consider
we would have
the short-run to involve Qh/Qi constant but Ah/Ai variable, then
9
and
choose p to place
which implies a betweenl.2 and
1.8,
approximately equal to the estimates of Katz and Murphy and Bound and Johnson.
take the relative supply shock to be the increase in the ratio of college graduates
I
to high school graduates from 1971 to 79, because 1971
large change in the skill composition of the labor
shock,
A\og{H/L)
=
log 5, as 0.4 (Katz
force. ^^
was the starting year
for the
This gives the relative supply
and Murphy, 1992, Table
8).
In addition to
the implied (inverse) elasticities of substitution between skilled and iinskilled workers
and the implied value
of
7y,
I
The
report three nrmibers.
the impact
first is
effect,
—9 log 6, which is the immediate effect of the increase in supply as given by Proposition
3.
This has no coimterpart in
data,
I
reality.
The second
the "short-run" response. In the
is
take this to be the proportional change in the
skill
premium
as
measured by
the average weekly wages of college graduates divided by the average weekly wages
of high school graduates from 1971 to 1979, which
Murphy. This number cannot be
directly
is
reported as —0.10 by Katz and
compared to the impact
effect of
one time
shock given in Proposition 3 because in reality the supply shock took place over a
ntimber of years, so technology must have adjusted during this period. Therefore,
compare
this
number
to
'°6^+'°g"
_
logo),
which
is
the simple average of the change
immediately after the shock and the long-rim response. ^^ Finally,
response to be the change in the
0.024 by Katz and
Murphy
skill
premium from 1971
(1992, Table 8),
and compare
in the
CPS
I
take the long-run
to 1987 which
this to
which is the long-rim response implied by the model. Table
^^Those with exactly 12 years of schooling
I
1 (at
77
is
reported as
log 5 (log a;
— log a;)
the back) summarizes
are considered high school graduates
those with at least 16 years of schooling are college graduates.
The
and
other categories are ignored, but
them and constructing other measures of supply shifts gives very similar results.
'^More appropriately, one should use a weighted average depending on the speed of convergence
reported in the Appendix. This depends on the exact time path of changes in relative supply and
on the function (p. However, I see no simple way of mapping the function (p to data at this point.
including
17
these data and the predictions of the model.
The
results in Table 1 suggest that the model's quantitative predictions are not
out of line with the data. For a range of parameter choices, the model gives numbers
very close to the data. In particular, for
/5
=
gives short-rrm elasticities in the right range
short-term
fall
in the skill
premium
3%
college
P=
above
its initial value,
premium. ^° Similar
0.45
and p
=
=
and p
0.3
rj
=
=
which
is
model
0.75, the
This predicts a
0.05.
and then a very
to the one in the data
after this initial fall that takes this
premium
quite close to the actual behavior of the
results are obtained
0.7. Naturally,
a.
and implies
premium comparable
large increase in the college
about
0.35,
when
f3
=
0.4
and p
=
0.73, or
when
these numbers should not be overinterpreted because
very abstract, and
many
other changes took place during this time period
(including further increases in
H/L
during the 1980s).
the model
is
0.3 or increased above
patterns.
/?
=
0.5,
if /? is
reduced below
the results are no longer in line with the observed
A more careful calibration exercise, taking into account the slow adjustment
of technology, the gradual nature of the supply shock
changing returns
G.
Also,
is left
for future work.
Wage
Technical Change and Residual
The paper
and education responses to the
Inequality
so far only analyzed the evolution of the skill
to changes in relative supplies.
Based on these
workers as those with a college degree,
directed technical change can
I
results
premium
in response
and interpreting the
skilled
suggested that a model incorporating the
match the evolution
of the college
premium
in the U.S.
However, there are other aspects of the changes in the structure of wages.
First,
male-female wage differentials have narrowed, and the returns to experience for low
education workers increased.
residual (within group)
wage
I
will discuss these in the
concluding section. Second,
inequality began increasing during the 1970s while the
a way of obtaining an alternative estimate of r]: use equation (12) and regress logw
sufficiently long time series, the slope coefficient gives t]. To do this, I took
the 25 years of data used by Katz and Murphy and added the data reported by Autor, Katz and
Krueger (1997) from the censuses of 1940, 1950, 1960, 1990 and CPS 1995. To obtain 1/cr, Katz
and Murphy regress log w on log {H/L) and a Unear time trend. Repeating their exercise with these
•^•^There is
on log (H/L). With a
data gives a similar (but somewhat larger) elasticity a — 1.66. I then regressed logw on log (H/L),
without a linear time trend, using robust regression {rreg in STATA). This gave rj = 0.045 with a
t-statistic of 2.3. Since there are only 30 data points, this result should be interpreted with caution.
Nevertheless, it is quite encouraging for the theory that this regression gives a value of rj very close
to the one obtained from the simple calibration exercise, which
the behavior of the
skill
premium from 1970
to the present.
18
is
also the value necessary to
match
college
premium
fell
(but see also DiNardo, Fortin and Lemieux, 1996).
consistent with the approach in this paper?
In this subsection,
extension of the model which suggests that the answer
Suppose that
A
better term).
skills
remainder have low
>
[Xh
and
ability (for lack of a
1/2 of college graduates are high
The same
ability.
suggest a simple
I
yes.
is
are two dimensional: education
fraction
fraction
<
is /ii
pattern
Is this
ability,
and the
1/2 for non-college graduates.
For example, ability could be related to the curriculum of college, but not perfectly,
so that
some
other workers do in spite of not having attended college.
world
graduates and
L
jHh {Ah ifJ-hHfy
all
+ 7w
firms use the
{Ah {iuLf)'
+m
(Ai ((1
same technology and employ
equilibrium as shown above.
is
H
college
(dropping time arguments):
is
which basically combines the equivalent equations to
that
There are
low education workers. Suppose also that the aggregate production
economy
fiinction of the
Y=
Therefore, ability in this
not innate, but acquired partly through education.
is
some
of the college graduates do not acquire the necessary ability, while
(2)
all
+ -fm
/x^L)^)'
and
and
(4),
(Ai ((1
also
- iinW^
imposes
workers, which will be true in
The important assumption embodied
in this expression
that technologies are not complementary to education but to ability (see Bartel
and Sicherman, 1997,
college graduates
for
some evidence
in support of this).
Therefore, both able
An
and able high school graduates use the same technologies.
analysis similar to the one used previously implies that in
BGP, the
state of relative
technology has to be (see the Appendix for details):
Pp" a ^m
Oh
IHH {^^HHr''
+ 111
I
,..t\Pp^
(i-p)i/
if^lL)
(13)
Qi
where u
=
{1
—
ap)"^.
ratio of technologies
When
the number of high ability workers
complementary to
consider an exogenous increase in
H/L.
results.
Because
jih
increases the proportion of high ability workers in the labor force
in
is
>
1/2
H
workers, and then increases with
>
/x^,
and induces a
Qh/Qi- The college premium behaves as in the previous section:
because there are more
BGP
has to be larger.
ability to other technologies
This has exactly the same intuition as the previous
Now
larger, the
is
first it
this
rise
declines
Qh/Qi because there
a large fraction of high ability workers among the college graduates. In contrast
to the college
premium which
falls first,
immediately after the shock. To see
residual
this, consider
19
wage inequality
starts increasing
the two measures of residual wage
p-\ilp
.
—
inequality in this model; uj^
Whh/whi and
uJ-
=
wih/wu, that
is
of high to low ability workers within their education groups.
the ratio of the wages
Similar arguments to
those developed before imply that (see the Appendix):
=
where k
Also,
p{l
Q and
+ Ppa -pa-
ap)/{l
-
ap)
>
Ch are suitably defined constants.
0,
and recaU that 9
if
technologies are
>
^""ff~°P
more complementary
college graduates
to ability rather than to
schooling, the approach developed in this paper predicts that in response to
ogenous increase in the relative supply of educated workers, there should
an exbe a
first
drop in the college premium followed by a subsequent increase and throughout
process, residual
H.
wage inequality should
0.
Since after the relative supply shock
Qh/Qi begins to increase, residual wage inequality both among the
and among the non-college workers increases immediately.
Therefore,
=
this
increase.^^
Discussion
There are a number of
1
results.
issues I
wish to discuss informally.
The above model can be extended
For example, there could be
case, the
model would
predict, as
S
to
many groups without
groups, each with their
affecting the basic
own technology.
In this
Katz and Murphy (1992) and Boiuid and Johnson
(1992) found in the data, that the groups experiencing increasing relative supply
have higher wages. Whether
this
is
may
a plausible explanation will depend on whether
these groups appear to use different technologies in practice. College graduates clearly
use different technologies than high school graduates. For other groups, however, this
may be
2.
For simplicity,
in sector
and
return to this issue in the concluding section.
less so. I will
R&D
for sectors
h leads to a discovery
to the discovery of sector
I
and h were completely separated.
of better sector
/
h machines with probability
machines with probability
1
remain unchanged, only the speed of convergence would be
results of this
paper to apply,
innovations. Clearly
it is
—
If
ip
R&D
>
\/2
0, our results would
affected.
Also, for the
not necessary that economic motives determine
some innovations
are exogenous
and stem from the advances
all
in
^^The only other paper that I am aware of which accounts for the increase in residual wage
same time as the return to education was faUing is Galor and Tsiddon (1997),
which is again based on exogenous (skill-biased) technical change.
inequality at the
20
and we may
basic science,
is
these "macroinventions" following
call
Mokyr
ventions are developed. For example, the discovery of the microchip
exogenous, but using this chip to develop personal computers and
final output). If
purposefully
the
R&D
made R&D
sector
is
more
neutral in
specific"
and
suppose Yh
=
Ai are
but
I
its
demand
for skills
(i.e.
it
uses
to increase after a relative supply
Yi
=
is
distinction
between these two
BiFi{AhHi,AiLi) where Fh
all
ii
Bh
=
Bi
=B
and
if
Ah =
Ai
=
R&D
A, and
more
Af,,
R&D
and
firms
What
the results would carry over.
that
is
and
sector-specific technologies
straightforward to see that
conjecture to be true,
no difference between "sector-
is
To draw a
Now, B^ and Bi are
perform research to improve Ah and Ai,
less clear,
skill
economy there
BhFh{AhHh,AiLh) and
skill-specific. It is
Windows 95 would
goes up dviring periods of transition.
"skill-specific" technologies.
intensive than Fi.
skill
effort
Also, observe that in this
4.
cases,
R&D
largely
intensive than the rest of the economy,
skill
there will be another reason for the returns to
shock because total
may be
by the use of these products.
profit opportunities offered
The model
3.
It
economic motives determine the direction in which these macroin-
sufficient that
be due to the
(1990).
is
is
directed
towards either Bh or Bi, the same results would also obtain. Unfortunately, such a
model
is
difficult to solve.
In this model, an increase in total population leads to a higher growth rate as
5.
in the
very
models of Aghion and Howitt (1992) and Grossman and Helpman (1991). This
is
not important for the focus of the paper. All the results of interest continue to hold
if
we impose
from
Zh
(11) that
or decrease the
+
=
zi
z,
in
H/L
growth
rate.
an increase
BGP
but this scale
effect is
removed. Also,
may increase
economy reaches maximum
leaving total population iinchanged
This
is
because the
growth when the two sectors are balanced. The restriction Zh
remove
6.
this
interestingly, the fact that
economy
(^(.)
is
zi
=
z would also
decreasing implies that the growth
declines during transition to a
adjustment, Zh increases and
logical
+
dependence of the growth rate on H/L.
More
rate of the
can be observed
it
zi
falls,
and with
new BGP. This
(p{.)
is
because during
decreasing, the faster techno-
improvements in skill-complementary technologies do not compensate
slowdown
in the productivity
can account
change".
for the
growth of unskilled workers. Therefore,
growth slowdown during the process of
this
for the
approach
"skill-biased technical
This result has some similarity to Greenwood and Yorukoglu (1996) and
Caselli (1997)
who
also obtain slower
growth during the process of adjustment to new
technologies because of costs of adoption and learning. In contrast to these papers,
21
technological change
adjustment
endogenous here, and the slower growth during the process of
is
because the economy invests mainly in improving skill-complementary
is
technologies at the expense of technologies complementary to unskilled labor.
Endogenizing The Choice of
III.
Section
II
Skills
treated the relative supply of skills as exogenous,
The
the supply of college graduates in the data.
and mapped
it
to
increase in the supply of college
graduates during the late 1960s and 1970s was argued to be largely exogenous rather
than a simple response to anticipated higher returns
in the future.
Nevertheless,
education choices are to some degree forward-looking and respond to the returns.
Therefore,
it is
important to endogenize the choice of
There are two other
results are robust.
skills is
be
important. First, as noted above,
than
less
issues for
1,
choice of skills
that
is
skilled
is
workers
endogenized,
uj
will
paid
and ensure that the main
which endogenizing the choice of
when H/L
may be
skills
treated as exogenous,
is
less
uj
can
than the unskilled. Once the
always be greater than
Second, the above
1.
theory explained the changes in the structure of wages by using a large increase in
H/L.
It is
important to know whether for the approach to work, exogenous factors
need to be responsible
for all of the increase in relative supplies, or
a relatively small
impulse to the cost of education, coupled with the general equilibrium changes in the
skill
H/L.
premium, could lead to a large increase
in
Suppose now that there
of unskilled infinitely lived agents in the
economy
=
at date t
0.
is
a continuum
Each unskilled worker chooses whether and when to acquire
education to become a skilled worker.
skilled,
and during
1
this time,
the function V{K) which
is
For agent x
it
takes
K^
periods to
he earns no wages. The distribution of
K^
is
become
given by
the only source of heterogeneity in this economy. This can
be interpreted as due to credit market imperfections or differences
in innate ability.^^
The
assume that T[K)
rest of the setup
is
unchanged. To simplify the exposition,
has no mass points other than at ff
I
now
constant.
define a
It is
BGP
= 0.
as a situation in
which
straightforward to see that in
type property. That
another with K^i
is, if
I
H/L
BGP,
and
relative
wages remain
there will be a single-crossing
an individual with cost of education K^ chooses schooling,
< K^ must
also acquire skills. Therefore, there will exist a cutoff
^^This formulation is general enough to nest the case in which the economy starts out with a
combination of skilled and imskiUed workers, and then the unskilled decide whether to acquire skills.
This is because T{K) can have a mass point at
who wiU immediately become skilled.
22
level of talent,
K, such
that
BGP
that must hold along
K^ >
all
K do not get education.
Then the
first
equation
the relative skill condition:
is
^ = JW^
The second condition
Once
skill level.
that he
any
we can simply
Then, his return at time
skills.
look at an agent with talent
between acquiring
indifferent
is
BGP imposes that all workers choose the privately optimal
for a
again,
(14)
skills
R"'^,
t,
and
K does not
Suppose
not.
can be written
K and make sure
acquire
as:
= wu /o°° exp(— (r — g)T)dT = wit{r — g) where I have
used of the fact that along the BGP wages grow at the constant rate g = {\ — \)z*(f){z*)
R^^
=
/j°°
which
—
exp(— r(r
is itself
t))wirdT
H/L.
a function of
In contrast,
K decides to acquire education,
if
K, and
receives nothing for a segment of time of length
Therefore, the retiirn to agent
Rt{K)
as:
In
BGP,
=
BGP, Rl{K)
is
-
/j+^exp(-r(r
=
K
=
t))whrdT
i?"^ for all
exp(-(r
A BGP
is
given by the intersection of (14) and (15).
H/L
The
The
everywhere increasing as drawn in Figure
3.
be decreasing or
when
increasing.
sloping,
and multiple
when
>
7y
0,
equilibria, as
H/L increases oj,
increasing H/L further.
can think of government policy
Vietnam era
left.
In particular,
a higher
education and
We
BGP
K
such a left-ward
retiirn to education
u would
drawn
r]
(e.g.
side of (15)
skill
formation
relative skill condition, (14),
is
indifference condition, (15), can
>
0,
(15) is likely to
encouraging workers with high
shift of
is
(15)
be upward
in Figure 3, are possible. Intuitively,
K to obtain
the grant programs in the U.S. or the
T{K) would
also rise, thus raising
prediction of the model in this case
g).
and the left-hand
draft laws) as reducing the cost of education,
For given
-
equilibrium with endogenous
the
(12).
(r
= eM{r-g{H/L))K)
growth rate as a ftmction of
premium from
g)K)wht/
K:
is
skill
-
Hence, the second equation that must hold in
t.
[jJ
BGP
from then on.
receives Wht
from acquiring education, R^{K), can be written
the indifference condition for
where g the
he
and
shifting
increase
H/L.
K and H/L further.
T{K)
If
77
>
to the
0,
the
Therefore, the
that subsidies to education lead to an increased
tendency to acquire education and also to a larger education premium due to the
directed technology effect.
23
relative skill
H/L
condition
indifference
condition for
K
K
Figure
Balanced Growth Path Equihbrium With Endogenous
3:
Unfortunately,
it is
quite difficult to
technology and wages. This
agent with
Kx
may prefer
to wait;
finds
is
work out the
due to two reasons:
dynamics of education,
away from the BGP, even
(i)
beneficial to get education at date
it
full
Skills.
i,
an
if
K < Kx
some agents with
agents acquiring education will not take part in the production
(ii)
process and this will influence relative wages and education incentives. Nevertheless,
without carrying out the
full analysis,
the equilibrium with the highest
than the
BGP
equilibrium, instead
way
creating
since
even when
(b)
level;
it
H/L
more skill-complementary
by construction,
to the
new
H/L
BGP
is
>
locally stable starting
by building more
jump
(a)
smaller
to a
new
and on the
skills
technologies. For the first property, note that
can never decrease, the system cannot cycle, and retains
is
locally stable.
but adjusts slowly
only adapt slowly {Qh/Qi
H/L
with
the economy will never
0,
will travel there slowly
the same properties as before, thus
jump
rj
two key properties can be determined:
is
a stock variable).
is
The reason why
the same as in Section
When
II:
it
does not
technologies
a large fraction of agents decide
to acquire education, for a long while wages will actually be lower for the skilled
workers. Moreover, with
H/L
when
is
r?
>
0,
at its highest level. So,
all
the relative wage of unskilled workers
is
lowest
when
an ideal time to make himian capital investments
other workers have already completed their investments.
Therefore,
is
some
of the workers will have an incentive to wait a long while before starting to invest
in skills, creating a type of
war of
pattern of slowly increasing
H/L
attrition
and leading to a slow
and a gradual
24
rise in
shift in technologies
H/L. This
towards more
skill-complementarity
economy over the past
similar to the experience of the U.S.
is
centm'y.
Perhaps the most interesting exercise to perform
crease in
H/L
in the 1970s can
how much
to see
is
of the in-
be attributed to the endogenous propagation of the
government impulse. For example,
the model does not offer any propagation,
if
the change in relative supplies must be due to exogenous factors
all
of
none of these
(i.e.
agents would have gone to college without the government's subsidy and Vietnam
era draft laws). For a simple calibration,
g{H/L)
as constant
and equal to
I
0.02.
I
=
take r
0.05,
and
take the distribution of education costs,
V{K), to be triangular with one unknown parameter, which
imation to the normal distribution.
= ~1 + IK H/L = 0.16) which is
r{K)
^
ioi
K
>
In particular
a.
I
start the
and
K=
wage
14.4.
K
<
economy with r{K)
=
0.14
(approximately
its
I
—
and
16.7
value in 1990).
H/L
1980,
it
H/L
a and
=
l.M which
is
(i.e.
How much would H/L
w=
1.65, the
would have increased
to 0.23.
if
we
new
^
27
rose to 1.65
have increased without any
cutoff level
This
u
la-
the college
Solving equations (14) and (15) gives a
implied by the dynamics of the model. So,
increase in
cu
perform the following exercise. Suppose that
change in the distribution F? With
K
= ^K'^
approximately the relative share of college graduates in the
differential in 1970.
Next,
the simplest approx-
is
ior
T{K)
bor force in 1970 (Autor, Katz and Krueger, 1997), and
high-school
growth rate
treat the
a
is
would have been
44%
increase in
limit ourselves to the original
H/L
40%
between 1971 and 1979 or to the 54% increase between 1970 and
would have been
sufficient for the
of the poor (constrained) agents,
government to induce only a small fraction
who would have
otherwise remained unskilled, to
go to college. The rest of the agents would have acquired
skills in
response to the
changing returns, creating the large increase in the supply of college graduates and
the resulting changes in equilibrium technologies. However, this calculation has to be
interpreted with care.
H/L
continued to increase after 1980. For example, the frac-
tion of full time equivalents with a college degree increased from
in 1990. So,
if
we compare the
predicted increase in the supply
from 1970 to 1990, approximately half of the increase in
by the government, rather than about 10% of
it.
skills
21%
in 1980 to
skills to
must be
26%
the change
directly caused
Nonetheless, this simple exercise
suggests that small changes in education costs can cause large increases in the relative
supply of
skills
and important changes
in the structure of wages.
25
The Impact
IV.
of Trade on Technology
LDCs
Increased trade with the
(the South)
where
skilled labor
is
scarce
is
often
suggested as a potential cause of increased wage inequality and contrasted to the
explanations based on technology.
wage
in the
Since technology has been treated as exogenous
inequality literature, there has been
between these two explanations. This section
change
will
little effort in
uncovering the links
show that the
direction of technical
influenced by trade, thus modifying or qualifying
is
many
of the conclusions
reached in the previous literature regarding the impact of trade on inequality.
Suppose that an economy in BGP, the North, with the
ratio of skilled to unskilled
H^ /L^, begins trading with an economy, the South, which has a
H^ /L^ < H^ /L^ There is no endogenous skill accumulation in either
workers equal to
skill ratio
economy.
.
What happens
under three
to
wage inequality
different scenarios: (a)
in the
North?
I
will
no directed technical change;
answer this question
(b) directed technical
change and new technologies sold to firms in the South on the same terms as firms
in the North; (c) directed technical change
and no property
rights enforcement in the
South.
The
scenario
first
between 2 and
3,
is
for
benchmark, and the truth presumably
so that there
is
some
be no endogenous
somewhere
sale of technology to the firms in the South,
the enforcement of intellectual property rights
section, there will
lies
skill
is less
than perfect. Throughout this
formation and
Also, in this model, factor price equalization
is
but
I will
simply compare BGP's.
guaranteed without further restrictions
because each sector employs only one of the non-traded factors (see Ventura, 1997,
for a similar structure).
No
A.
Technical Change
Let Ai and
mium
in the
Ah be exogenously
North before trade opening by
opening by uj^ and
,
let
A logo; =
logo;^
L^ =.L^ + L^, H^/L^ = 6H^/L^
skill
Denote the steady state (BGP)
given.
—
where
o;-^,
and the
logw'^.
skill
premium
Also define:
6<lhy the fact
skill
pre-
after trade
H^ = H^ + H^,
that the North
is
more
intensive than the South. Then, equation (6) from Section II implies:
AlogujNTC
=
-{I
26
- Pp)^og6 >
(16)
NTC denotes the fact that this expression refers to the case with
where the subscript
no technical change. This
trade with the South increases the relative price of
is less skill-intensive,
this exercise
the standard effect of increased trade: since the South
is
took Ah/Ai as given, and as discussed above, one can also take Qh/Qi
I
as given. In this case, the result
similar,
and does not
affect
any
would be AlogcuArrc
Let us
now
return to the analysis of Section
and assume that
(i)
= —OlogS >
0,
which
is
very
of the comparisons below.
Endogenous Technical Change and
B.
In
skills.
II
Property Rights
Full
where Ai and Ah are endogenous
before trade opening, there were no sales of technology to the
firms in the South (and no foreign direct investment in the South by firms in the
North); and
(ii)
and property
We
after trade opening, firms in the
rights of
R&D
producers in the North are fully enforced in the South.
can then use equation (12) from Section
and without
and
trade,
this
immediately
PR
that
reduce the
if
skill
77
>
when
it
R&D
BGP
skill
=
(17)
is
endogenous technical
firms are enforced in the South.
for exactly the
in Section 11.^^
The important
may
actually
same reasons that a higher supply of
More
skill
skilled
generally, the directed technology effect
intellectual property rights are fully enforced,
international trade will increase the
premia with
rj\og6
contrary to conventional wisdom, trade opening
premium
workers increased
implies that
0,
to obtain the
indicates that in this case there
change and property rights of
resiilt is
II
gives:
A\ogujpii
where the subscript
South and the North are symmetric
premium by a
it is
unlikely that
large amount. Nevertheless,
the assumption that intellectual property rights are fully enforced in the South
vmrealistic.
The next
subsection looks at the other extreme case where there
enforcement of property rights of Northern
R&D
A
Intellectual Prop-
crude and simple way of modelling the lack of intellectual property rights
to suppose that firms in the South can use the latest machines invented
^^However, note that the equivalent of Proposition 3 appUes, therefore trade opening
it back to a lower level.
wage inequaUty, and then reduces
27
no
firms in the South.
Endogenous Technical Change and No
erty Rights in the South
C,
is
is
by
first
is
R&D
increases
firms in the North without paying patent fees to Northern firms (but use the
impHes that the market
quantities). This specification
unchanged
However, there
after trade opening.
same
machines are
sizes for different
be an impact on the direction
will still
of technical change because the relative price of skill intensive goods will change.
BGP
In this case (9) also has to hold in
R&D
to equate the return to
complementary and labor complementary machines. This implies that
p
= (H^/L^j
,
therefore, the relative prices will
original level in order to restore equilibrium.'^^
which
The
skill
in this case implies:
relative
wage
is
now:
Q^/Qi
u^ =
premiimi after trade opening
BGP
In turn, the price of
skill
is
intensive
also given
by
1^^"'^^'"^ 6-\
{ih/lif'^^'"^ (h^ /
=
(H^/L^j
['^h/lif'
5~^.
The change
in the
therefore:
is
A\ogujNPR
where the subscript
in the
have to adjust back to their
intermediate good relative to the labor intensive intermediate good
(3),
in skill
= -log6>0
NPR indicates that there
is
(18)
endogenous technical change but no
enforcement of intellectual property rights in the South.
We have: A log u)npr > ^ log uj^tc > ^ log i-^pr ^-nd in fact if >
77
0.
That
is, if
supply of
if
0,
A log lopr <
property rights are fully enforced in the South, the decline in the relative
skills
should not lead to a large increase in the
skill
premium. In contrast,
intellectual property rights are not enforced in the South, simple calculations that
ignore the induced change in the direction of technical progress
may be
derestimating the impact of the international trade on inequaUty.
case, the
market
size of different technologies
seriously un-
Namely, in this
remains unchanged, but trade creates a
relative price effect, increasing the profitability of
R&D
for the skill intensive goods.
This magnifies the static impact of trade on factor returns and results in a large
long-run elasticity (—1).
To
get a sense of
how
conventional conclusions, consider the calculations
uses the share of trade with
LDCs
growing trade on wage inequality.
in the skill
may modify the
by Krugman (1995). Krugman
the presence of directed technical change
premium. Based on
in the
He
this,
OECD
output to calibrate the impact of
finds that this
can explain a 4.7% increase
he concludes that growing trade with
LDCs
is
^^Modest (and conflicting) effects of trade on the relative prices of skill intensive goods (e.g.
Lawrence and Slaughter, 1993, and Krueger, 1997) are in line with the prediction that the relative
price of the skiU intensive good staying close to its original level. However, note that a large part of
the substitution between skilled and unskilled workers may be taking place within industries.
28
unlikely to have been the
Krugman is
(7
=
factor causing the changing structure of wages. Since
using a model of constant technology, in the context of the current model,
this translates into log 6
between
main
skilled
1.7, this
~
0.05a where a
is
and unskilled workers. Using Bound and Johnson (1992) 's estimate
implies log^
~
premium (from
of
0.085. Therefore, in the absence of international property
rights enforcement, growing trade with the
skill
the short-term elasticity of substitution
(18)). Recall that
LDCs would
imply a 8.5% increase in the
from 1980 to 1990, the
premium increased
skill
approximately by 11% between 1980 and 1990 and by 13.5% between 1980 and 1995.
Hence, with these calculations, trade emerges as a potentially major cause. Although
this
number should be
interpreted with
ment may be an extreme assumption,
technical change
may modify some
some care
since
no property
suggests that recognizing the endogeneity of
it
of the calculations.
Concluding Comments
V.
The wages
of college graduates (and of other skilled workers) relative to unskilled
labor increased dramatically in the U.S. over the past fifteen years.
direct consequence of the complementarity
between
new
why new
technologies.
skills.
More
History
It is
is full
not however clear
of examples of
generally, inventions
its
it is
magnitude. In
its
"why do new technologies complement
direction of technical change
inventions.
complement
I
When
there are
skills is larger,
skill (in its
To many,
this
is
a
many dimensions) and
technologies should
complement
technologies designed to save on skilled labor.
and technology adoption are the outcome of a process of
different technologies. Therefore,
change as well as
new
we could have chosen
choice; as a society,
The
rights enforce-
is
more
to develop (or attempted to develop)
many
necessary to analyze the direction of technical
simplest form, this
skills?"
.
size of the
skilled workers, the
will
question:
This paper gave a preliminary answer.
determined by the
hence more
means to pose the
market
market
for different
for technologies that
be invented.
formalized this observation and discussed
its
implications.
I
showed that an
exogenous increase in the ratio of skilled workers or a reduction in the cost of acquiring
skills
could increase wage inequality. The likely path
larger increase in the skill
premium.
is
a decrease
These observations
fit
first,
and then a
the U.S. facts where
the large increase in the ratio of college graduates during the late 60s and 70s
depressed the college premium and then increased
I
it
first
to higher levels than before.
conclude with some comments about the implications of this approach and
29
potential future work:
The most important
1.
and
nical change,
the model
is
its
area for future work
is
to develop a test of directed tech-
impact on the structure of wages. The testable implication of
that after an increase in the supply of college graduates,
complementary to
at technologies
rapidly.
to determine which technologies are
Unfortunately,
complementary to
it is
directed
and these
college graduates should increase,
more
nologies should be upgraded
R&D
tech-
general
difficult in
skilled workers. Nevertheless,
most economists believe that computers are more complementary to
and ed-
skilled
For example, Autor, Katz and Krueger (1997)
ucated workers than the unskilled.
reports that in 1993 only 34.6% of high school graduates use computers in contrast
to
70.2% of college graduates. Moreover, Krueger (1993) shows that controlling
education, workers using a computer obtain a wage
are
more
R&D expenditure data reported by the NSF,
R&D expenditure for office computing was 3%
company funded
in 1960
premium which suggests that they
Prom the
skilled.
R&D
company funded
expenditure.
This ratio has increased to
gesting that during this period of rapid increase in the supply of
significantly
If
more
R&D
we
of the total
13% by
skills,
see that
1987, sug-
there has been
directed to one of the technologies that complements
other technologies and
R&D
for
skills.
expenditure can also be classified as complementary
to college graduates, the hypothesis of this paper can be tested.
2.
and
tially
is
Since 1970s, the participation of
likely to
their
wages
relative to those of
new
explanation.
male workers increased. Part of
different sectors
to which
women
and occupations, and these jobs use
male jobs
to higher wages,
(e.g.
carefully
also
may have
3.
is
wage
differentials.
relative
relative rate of technical
to
different technologies
it is
men
work
than
in
tra-
conceivable
once again not purely a response
affected the direction of technical change,
by studying the
be developed
women tend
desk jobs versus construction). Therefore,
channel, reduced male-female
and the
change
use different technologies than
probably limited. Nevertheless,
is
that the greater participation of women, which
more
this
than female workers, the approach in this paper suggests
The degree
within a plant or sector
ditionally
has increased substan-
be due to reduced discrimination. However, to the extent that male work-
ers use different technologies
a
women in employment
and via
this
This hypothesis can be investigated
growth of industries that employ more women,
change in these industries.
A
similar approach can
for the analysis of the return to experience.
Throughout the paper,
First, is it appropriate to
I
only discussed the U.S. case. This begs two questions.
view the U.S. as an economy rather than the
30
OECD?
I
believe the answer to this question to be yes, but this only affects the calibration
exercise.
Second, and more important, can the approach in this paper account for
To answer
cross-country differences?
this question,
one has to distinguish between
economies such as the U.K., Australia, Canada and Japan where the college premium
Germany where
increased during the 1980s, and those like Prance and
change. For the
first
did not
it
group of countries, there was also a large increase in the supply of
college educated workers during the 1970s, also partly caused
by increased government
support for education. As predicted by the model in this paper, in the U.K. and Japan,
the college
premium fell
in France there
in the 1970s
was a decline
and then increased during the 1980s. In
in the college
contrast,
premium during the 1970s but no
during the 1980s (see Katz, Loveman and Blachfiower, 1995).
It
increase
has to be noted that
the increase in the relative supply of college educated workers was less pronounced in
Prance. But,
it is
likely that, as
also prevented the skill
type of labor market
imskilled labor
is
argued by
premium from
many researchers,
increasing.
The
labor market institutions
interesting point
rigidities will also affect technical change.
priced higher than
its
market clearing
affecting the direction of technical change:
(i)
level,
is
In particular,
because unskilled labor
their marginal product
is
is
as there are fewer unskilled workers
more expensive, the value
higher.
These two
can be quantified in future research. But,
labor market institutions
if
there will be two forces
employed, the market for technologies complementary to unskilled workers
(ii)
that this
effects
is
smaller;
of technologies that increase
work
in opposite directions
and
this simple reasoning already suggests that
may have an important
effect
on the direction of technical
change, which requires further study.
4.
Finally, a different application of the theoretical
framework developed here
would involve having capital-complementary and labor-complementary technologies.
Such a model would imply
depending on their
large reduction in
different rates of labor productivity
initial capital-labor ratios.
Another implication would be that a
employment should be associated with
complementary technologies.
31
growth across economies
faster
upgrading of capital-
Appendix
VI.
Proof of Proposition
(3), (9)
z*
and
(10)
because the
and imposing
LHS
of (11)
BGP
establishes that the
for
AhjAi
1:(10) uniquely defines
=
=
Zh
exists
and
that in
zi
gives equation (11),
uniquely defined.
is
BGP,
=
determine the time path of
all
out of
Zs{t)
(12) has to hold,
Zh, zi,
BGP
The
Then, using
which uniquely defines
RHS
2:
is
constant. This
(10) to substitute
it.
Equation
(9)
immedi-
Thus we only have
to
free-entry condition (8) holds at
Therefore, differentiating this condition, 0(zs(i))V's(j,
times.
zi.
which completes the proof.
as well as along
Qi and Qh-
=
Now using
Transition Dynamics and Proof of Proposition
ately implies Zs{j,t)
for Zh
increasing in z* and the
is strictly
we obtain
in (6),
z*
Qh/Qi
i)
=
Bqs{j,t), with
respect to time and using (7) gives:
-H^s{t))Vs{j,t)
-(A=-
''^'^
<l>'{zs{t))Vs{j,t)
ioi s
=
l,h
and
also normalize .B
this
with
(7)
and
for all j
=
(1
—
and using
.
,
-
''^'
and where
t
Q;)/a in this
(8),
_
~
r
we
+
e^
_
>
Vs{j,t)z,{t)
^'''>
e4zsit))Vs{j,t)
is
the elasticity of the function
(j).
I
Appendix to simplify the notation. Combining
obtain:
Zs{t)<t>{zs{t))
-
0(z.(t))
{pstN^f^''"^
^'"^
^,{zs{t))/z,{t)
Finally, noting that Qs{t)/Qs{i)
'"'^'"]f'""
=
(A
-
=
{^
—
^)4'{^s{i))zs{t),
we
also have:
DMm^dt)) - .,M.A(^.W))||
(21)
Equations (20) and (21) completely describe the djraamics of the system. To analyze
the local dynamics and stability in the neighborhood of the
equations, and let
Q=
ignoring constants,
we
Qh/Qi- Then, around the BGP,
have:
32
BGP,
zi
=
Zh
let
=
us linearize these
z*,Q
=
Q*, and
(
o-{z*){zi
-
z.
-
z*)
,^(,.)/,.
Q =
where
I
have defined
a{z*)Q*{zf,-Zi)
=
cr{z*)
(f)'{z*)z*
+ 4>{z*) >
are analogously defined, and are both positive.
Q*
and Zh
affect zi
below
pi is
BGP
its
differently
Q
and
g =
ative
=
'tp{z*,Q*)
and one
The reason why
Q >
when
that
is
to simpHfy the notation.
Q*,
deviations of
above
p/i is
its
BGP
and 02
V'l
Q
from
value and
value. This linearization enables us to reduce the three variable
system to two variables:
where
^
.
r,*\fr^
r,*\
Mz,Q)iQ-Q)
(^
= Zh —
Specifically:
zi.
a{z*)Q*C
i^i{z*,Q*)
+
>
'ip2{z*,Q*)
This linear system has one neg-
0.
and thus a unique saddle path converging to
positive eigenvalue,
BGP equilibrium, as is drawn in Figure 2. The rate of convergence to the
BGP can be calculated as J sjffv^O ^ + a{z*)ij{z*, Q*) - ^eliz-yz- Thus, when
is more steeply decreasing, convera{z*) = (j)'{z*)z* + (f>{z*) is lower, or when
gence is slower. In fact, in the extreme case of (j){z) = 1 (where cr{z*) is at its highest),
all our BGP results would be unchanged, but ignoring nonnegativity constraints on
consumption, there would be no transitory dynamics. That is, when Q < Q* we would
have zi = and Zh ^- oo for an infinitesimally short while. There would be transitory
the
(/>(.)
dynamics, however,
Next,
path
I
nonnegativity constraints on consumption are imposed.
establish that
if
e^{z)
is
nonincreasing, the system
paths that do not cycle must go to
all
Suppose
to establish that there are no cycles.
PhH^ >
Then
piL^.
fact that e^(z)
and
Zh,
zi
>
and
Q <
Zh-
is
0,
Now
also as
Q
the case that
consider case (A) where
nonincreasing, zi/zi
thus, there cannot
>
>
increases ph falls
zi/zi
<
also globally saddle
Q <
zi
>
Zh-
be any cycles and
and
Q*.
>
zi
and
all
Therefore,
Note that in
Then using
zi
we only have
will
this case
(20)
and the
remain larger than
paths go to infinity
zi/zi
pi increases, therefore
<
Zh/zh.
from
,
it
when
Now Q >
will
>
zi
and
zi/zi
>
Zh/zh-
then we converge to Zh
33
=
zi
If
as t
= z*
—
and
>
oo, Zh{t)
Q=
0,
always be
Hence, in this case too, cycles are not possible.
Zhjzh-
Zh{t)/zh{t),
infinity.
Zh/zh, therefore,
consider case (B) where Zh
consider case (C) where Zh
zi{t)/zi{t)
is
Since paths cannot cross and there are no other stationary points of
stable.
the system,
if
>
Q*, and
zi{t)
Now
and
we know
there
is
a unique saddle path locally and paths cannot cross, therefore,
on that path. Instead
if
cycles can be ruled out.
=
Zh{t)
We
zi{t) at
must have
some point where
either that Zh{t)
puts us in case (A), and cycles are not possible and
could be the case that Zh{t)
>
zi{t)
and
Q>
points
Q<
The proof
Q*.
Proof of Proposition
monopoly
pricing
and
profit
3:
substituting for Ph{t)
Q*
Take Qh{'t)/Qi{t)
and
paths go to
Q >
Q* which
infinity.
Or,
it
is
analogous.
as given.
Then, given optimal
maximization by firms, we have:
Mt)
Now
zi{t)
Q*, then once again
must be a imique saddle path from
Q>
for the case of
>
^^
Q*, which, by the analogous argument
to case (A), again rules out cycles. Thus, there
all
all
Q
we must be
Qiit)
and
pi{t)
[piit)L^)
and rearranging:
(!)
Substituting into
(6),
we
obtain:
U.
{Qiit))
where 9
=
{1
— Pp— ap) / {1 — ap)
=
{Qh/Qi), AlogLo
as defined in the text. Therefore, at given technology
-9log6.
Once, the technology adjusts to
Proposition
1,
thus
A log a; =
Details of the
mands
for sector
Model
77
log
its
BGP
new
level,
then we have the result of
5.
w^ith
Two-dimensional Heterogeneity: The
h machines now come from firms employing high
ability college
graduates and high ability high school graduates, and vice versa for sector
These demand curves are the same as
the optimal pricing policy
is
h machines, analogous to
(9),
l-CK,.
aB
.
'(Pizh)
and similarly
for
f/
.
can be written
„^N 1/(1-0)
where phh
ability college graduates in
and have the same
/
machines.
elasticity,
thus
the same. Therefore, the free-entry condition for sector
{p..{f^.Hry'
zi,
in the text,
de-
is
/
as:
^x1/(1-q)
>+{piUf^iLr)
= r + Zh(p{zh)
(22)
the price of the intermediate good produced by high
terms of the
final
graduates, etc.
34
good, pih
is
for
high ability high school
Then, from competitive
VHU
and
pih, Phi
we obtain
=
pricing:
Q-(l-p)(l-«)/(l-ap) (^^^)-/5(l-p)/(l-ap) y(l-p)/p
ipiHuf-''^^^'-'''^
and pu are
similarly defined. Substituting these into (22)
and
simplifying,
(13) in the text.
Finally, consider:
""
^'~
Whl{t)
llH
\AHl{t))
[{l-fXhWj
Substituting for Ahh{t) and Aih{t) gives the expression in the text.
similarly.
35
uj''{t)
is
derived
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Higher Educa-
pp 181-204.
^^McPherson, Michael and Morton Schapiro (1991); Keeping College Affordable: Gov-
ernment and Educational Opportunity The Brookings
Institution,
Washington
DC.
^^Mokyr, Joel (1990); The Levers of Riches: Technological Creativity and Economic
Progress Oxford University Press,
New
York.
^^Schmookler, Jacob {1966);Invention and Economic Growth, Cambridge, Harvard
University Press.
^^Ventura,
Jaume
nomics Vol
(1997); "Growth
112,
pp
and Interdependence" Quarterly Journal of Eco-
57-84.
38
MIT LIBRARIES
3 9080 01444 1395
^^Walde, Klaus (1997); "Relative Quality of Education, Induced Technological Change
and Wages" Universitat Dortmund Discussion Paper
39
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Date Due
Lib-26-67