Mankiw, Romer, Weil, QJE 1992

Transcripción

Does the Solow Model Explain the
International Variation in the Standard of Living?
(Mankiw, Romer, Weil, QJE 1992)
Solow Model with
Technological Progress
• Treat productivity (A term in our model) as
due to level of technology.
• Assume that technology is labor-enhancing.
• So, can view labor input in terms of
“effective units of labor.”
• Assume growth rate of technology is
exogenous and equal to “g.”
Solow Model with
• Similar to way we modeled human capital,
except now we assume this “quality”
feature is due to technology.
• Later we will enter human capital into the
model as a separate input.
• Specification allows production function to
be expressed in terms of output per
effective labor units and capital per effective
labor units.
data. growth,
focus
on the model
model'stakes
implications
the ratesforofcross-country
saving, population
Solow's
and
technological
progress as exogenous. There are two inputs,
A. The
Model
capital and labor, which are paid their marginal products. We
of saving,
model
takes the rates
population growth,
ECONOMICS
OF so
JOURNAL
QUARTERLY
410Solow's
a Cobb-Douglas
production at time
function,
assume
production
and technological progress as exogenous. There are two inputs,
t is given by
We s, is
andmodel
which arethat
products.
paida their
marginal
labor, assumes
capitalThe
ECONOMICS
of output,
OF
fraction
QUARTERLY JOURNAL
constant
OF
0 < so
a.production
<ECONOMICS
1.
410 a Cobb-Douglas
Y(t) = QUARTERLY
K(t)a(A(t)L(t))l(1)
at time
function,
assume
productionJOURNAL
invested. Defining k as the stock of capital per effective unit of
t is given
by is standard: Y is output, K capital, L labor, and A the
The
notation
per effective unit of
the level of
as output,
and yof
k = KIAL,
labor,
is
s,0tooutput
fraction
a(1)
constant
odel assumes that level
fraction of
a
constant
that
assumes
model
The
L
A
at output, s, is
exogenously
of
and
are
assumed
grow
technology.
=
<
a.
<
1.
Y(t)
K(t)a(A(t)L(t))llabor, y = Y/AL, the evolution of k is governed by
ECONOMICS
OF
JOURNAL
QUARTERLY
410
of
unit
per
effective
of
capital
stock
as
the
Defining k
n and g:
rates
unit of
perAeffective
of capital
the stock
k Yas
invested.
K+
L labor,
and
the
output,
capital,
The
notation Defining
is standard:
issy(t)
=
+
8)k
(t)
g
(n
k(t)
(4)
per
effective
output
level
KIAL, and y as the (2)
unit of
per effective
of
output
level
the
y are
= KIAL,
and
=as
kmodel
L
labor,
()entunit
L
A
toof
exogenously
ofof
and
assumed
grow
(t)
technology.
level
s, atis
of output,
fraction
a Lconstant
that
assumes
The
=
+
+
8)k(t),
g
(n
sk(t)0
rates
k isn and
governed
Y/AL, the evolution of
k is governed
=g:Y/AL,
the
y Defining
labor,
per effectivebyunit of
of of
capital
= A (r)ent.
as the
kby
invested.
Aevolution
(3)
(t)stock
unit of that k
per effective
asLthe
k =8 KIAL,
labor,
=level
(4) implies
Equation
depreciation.
ratey of
is the and
where
L of output
()ent
(2) number of effective units(t)
A- (t)L
at rate
of labor,
+ grows
(t) n + g.
g + 8)k
(n(t),
k(t) =
(4)
+to 8)k
(n y+ =gY/AL,
k(t) = sy(t) -The
by
ofsy(t)
k isk*
governed
evolution
the(t)
labor,
sk
*a = (n + g + 8)k *,
by
defined
value
a
steady-state
converges
=
A (t) A (r)ent.
(3)
or(n + g + 8)k(t),
= sk(t)0 (4)
8)k+
(t)g + 8)k(t),
(n + g -+ (n
k(t) = sy(t)= - sk(t)0
-The
number of effective units of labor, A (t)L (t), grows at rate n + g.
=
sk(t)0 - (n + g + 8)k(t),
k* =of [s/(n
+ g + 5)]1I(1-a)
(5)
depreciation.
8 is the rate
where
that k Equation (4) implies that k
(4)
implies
Equation
the rate of depreciation.
k +rate
implies
=to (n
g + 8)k *,
by sk *athat
k*
of depreciation.
rate
defined
value
8 is the
a steady-state
toby
converges
the
related
is
ratio
The
steady-state
*, (4) positively
+Equation
skcapital-labor
*a = (n + g
8)k
k* defined
o a steady-statevaluewhere
+ g + 8)k *,
by sk *a = (n growth.
k*rate
defined
a steady-state
converges
or
of population
to the
negativelyvalue
of savingtoand
or
The central predictions of the Solow model concern the impact
k* = +[s/(n
+ 5)]1I(1-a)
+ greal
(5)
k* = [s/(n growth
income. Substituting (5)
on
and population
g + 5)]1I(1-a)
ofg saving
(5)
k* = [s/(n +
+ 5)]1I(1-a)
we find
logs,
takingispositively
and
function
the production
intosteady-state
to the rate
positively
related
capital-labor
The
steady-state
ratesteadyto
thethat
related
is ratio
ratio
capital-labor
The
is
income
capita
per
state
rate
to
the
positively
related
is
state capital-labor ratio
growth.
of population
ratethe
to the to
andand
negatively
ofofsaving
growth.
of population
rate
negatively
saving
OF
ECONOMIC
GROWTH
THE EMPIRICS
impact411the411
theconcern
concern
the Solow
ofECONOMIC
predictions
central
growth.
of
population
nd negatively to the rateThe
OF
GROWTH
THE
EMPIRICS
impact
model
Solow
of themodel
predictions
(6)tl
[Ot]ln()gt
central
The
(5)8).
income.
Substituting
on 1realIn(s)
growth
and population
of(6)
saving In
+
+
1
+
+
ln(n
=
A
In
(0)
g
gt
impact
the
concern growth
model
Solow
ntral predictions of the
income.
Substituting
on
real
andendowments,
population
ofbut
saving
so
andsteadyon; it may (5)
institutions,
that
we find
taking logs,
function andclimate,
production
theresource
into
so
it
may
and
on;
climate,
institutions,
endowments,
but
resource
that steadytaking
(5) arelogs,
function
production
income.
the
into
on
real
nd population growth
that
We
assume
across
countries.
therefore
differ
is Substituting
income
capita
per
state
marginal
theirfind
paid we
thatand
factors
assumes
model
the
Because
assume that
across
countries.
therefore
differ
isWe
income
capita
per
state
that
we
findonly
taking
and
oduction function
the steadysigns
not
predicts
itlogs,
products,
= a +but
(6)tl
[Ot]ln()gt
E, also the magnitudes of
lnA(O)
+ g + 8).Specifically,
1
+
1
+
ln(n
=
A
In
In
In(s)
(0)
(6)
gt = and
growth.
a + population
E,
saving
the coefficients onlnA(O)
e per capita is
(6)tl
[Ot]ln()gt
E is a country-specific
log
shock.
Thus,
where acapital's
and
is a constant
the
oneln(n
third,
in +income
is roughly
share
(a) In(s)
because
- 1 marginal
+
+
+
1
8).
=
A
In
In
(0)
g
(6)
gt
E
log
shock.
Thus,
is
a
country-specific
where
a
constant
and
a
is
their
paid
are capita
that
factors per
model
theper
Because
incomeimplies
at a given
capita
with respect to the
income
Ot]ln()gt
elasticity
of time-time
an assumes
model
0 for simplicity-is
also the magnitudes of
the
notln(n
only
it predicts
for
-ofa1approximately
+ g signs
+ 8).
1 capita
= In A (0) +income
In(s)
time-time
per
given
gt + products,
0but
to
0.5
and
ansimplicity-is
elasticity with respect marginal
rate at
saving
paid their
that factors
assumes
model
the on
Because
Specifically,
growth.are
and population
saving
coefficients
the
-0.5.
+ g + In
n(7)
8 of approximately
a+ in income
In(s)-(a)
In(n+g+8)+E.
third,
roughly
share
because
capital's
magnitudes of
the the
butonealso
theis signs
not
only
predicts
it
products,
In
a+
In(s)(7)
In(n+g+8)+E.
marginal
their
paid
are
factors
e model assumes that
the
to
respect
with
income
capita
per
elasticity
of
an
model
implies
Specifically,
and population ingrowth.
onbasic
saving
coefficients
the
this section.
empirical
is our
Equation
to
with respect
0.5 and an specification
elasticity
of(7)
rate but
approximately
saving
of
magnitudes
the
also
B.signs
Specification
t predicts not only thebecause
third,
inspecification
roughly
income
share
capital's
inisthis
of saving(a)
and
are the
section.one
the
rates
empirical
population
growth
is
our
basicthat
assume
Equation
-0.5.
n + g(7)+We
8 of
approximately
Specifically,
growth.
population
support to the
ients on saving and
dataproduction
the the
is whether
consider
natural
The
respect
with
capita
pershifting
elasticity
of income
implies
factors
ofanquestion
country-specific
independent
of to
and
are
Wemodel
that the
rates
population
growth
assume
saving
of
the
determinants
concerning
model's
Solow
predictions
the
the
one
third,
is
roughly
income
(a)
pital's share in
to
with
0.5
and
an
elasticity
of approximately
saving
n are the
of e.respect
that s and
This
independent
function.
That
is, we assumefactors
production
shifting
ofrate
country-specific
independent
Solow Model with
• Assume g and δ (depreciation rate) are
constant across countries.
• A(0) term reflects not only technology but
resource endowments, climate, institutions,
etc., so it may differ across countries.
• Identifying assumption is that the country-
specific shock is independent of s and n, so
can estimate using OLS.
Solow Model with
• Coefficients on ln(s) and ln(n + g + δ)
should be equal in magnitude and opposite
in sign.
• Using factor share of 1/3 for capital and 2/3
for labor, the coefficients should equal 0.5
and -0.5.
• Assume value for g + δ equal to 0.05 and
use data on population growth rates to
compute ln(n + g + δ).
Estimation Results forTextbook
Solow Model
• Coefficients on investment rate and
population growth have the predicted signs
and are significant in two of the three
samples.
• Equality restriction can’t be rejected.
• Regressions explain large fraction of the
cross-country variation in income per
capita.
414
QUARTERLY JOURNAL OF ECONOMICS
TABLE I
ESTIMATION
OF THETEXTBOOK
SOLOWMODEL
Dependent variable: log GDP per working-age person in 1985
Sample:
Observations:
CONSTANT
ln(I/GDP)
ln(n + g + 8)
H2
s.e.e.
Restricted regression:
CONSTANT
ln(I/GDP) - ln(n + g + 8)
1?2
s.e.e.
Test of restriction:
p-value
Implied a
Non-oil
98
5.48
(1.59)
1.42
(0.14)
-1.97
(0.56)
0.59
0.69
Intermediate
75
5.36
(1.55)
1.31
(0.17)
-2.01
(0.53)
0.59
0.61
OECD
22
7.97
(2.48)
0.50
(0.43)
-0.76
(0.84)
0.01
0.38
6.87
(0.12)
1.48
(0.12)
0.59
0.69
7.10
(0.15)
1.43
(0.14)
0.59
0.61
8.62
(0.53)
0.56
(0.36)
0.06
0.37
0.38
0.60
(0.02)
0.26
0.59
(0.02)
0.79
0.36
(0.15)
Note. Standard errors are in parentheses. The investment and population growth rates are averages for the
period 1960-1985. (g + 8) is assumed to be 0.05.
Three aspects of the results support the Solow model. First,
the coefficients on saving and population growth have the predicted
Estimation Results forTextbook
Solow Model
• Impacts of saving and labor force growth
are much larger than model predicts since
value of α implied by model for
intermediate sample is 0.59 (SE = 0.02),
much larger than the capital’s income share
of 1/3.
• Growth (development?) Accounting shows
adjusted R-squared of only 0.28 in
intermediate sample.
Solow Model with Human
Capital Accumulation
• Add human capital to production function.
• Include additional transition equation for
adjustment of the stock of human capital.
• Solve for steady-state values of k and h.
analogous to those in Table I to see whether proxies for human
capital
can resolve the anomalies found in the first section.7
A.
The Model
Let the production function be
A. The Model
(8)
Y(t) = K(t)H(t)P(A(t)L(t))1-a-,
Let the production
function be
where H is the stock of human capital, and all other variables are
(8)
Y(t) =
defined as before. Let SkK(t)H(t)P(A(t)L(t))1-a-,
be the fraction of income invested in
physical
and
Sh of
thehuman
fraction
invested
in all
human
where Hcapital
is theTHE
stock
and
capital,
othercapital.
variables
EMPIRICS
OF ECONOMIC
GROWTH
417Theare
evolution
of the
economy
is determined
by GROWTH
defined as
before.
Let Sk
be
the fraction
of income invested
THE
EMPIRICS
OF ECONOMIC
417 in
there is nocapital
steady and
stateSh
thisfraction
model. We
discussinthis
possibility
in The
physical
the
invested
human
capital.
sky(t) - (n + g + 8)k(t),
(9a)
k(t) =for
Section
Equations
and
(9b)We
imply
economy in
evolution
of the
economy
is determined
by thatthisthepossibility
there
is noIII.)
steady
state for(9a)
this
model.
discuss
= Shy(t) - (n + g + 8)h(t),
h(t)
(9b)
converges
to a steady state(9a)
defined by(9b) imply that the economy
Section
- (n + g + 8)k(t),
(9a) III.) Equations
k(t) = sky(t)and
= K/AL,
where y =to Y/AL,
k state
converges
a steady
andbyh = H/AL are quantities per
defined
= Shy(t) - (n + g + 8)h(t),
h(t) We
(9b)
effective
unit of labor.
are assuming that the same production
+g +
\n
function
to human
applies
capital,
physical
capital,
consump-per
=
=
y
where
k
h = H/AL
Y/AL,
K/AL,
and
areand
quantities
(10)
+
+
tion.
In other
oneWe
unit
of
consumption
transformed
g assuming
\n are
effective
unit words,
of labor.
that can
the be
same
production
a
1/(1-a-)
-S a
k
(
*
(10)
into
costlessly
either
one
unit
of
or
one
physical
capital
unit of
418
QUARTERLY
JOURNAL
OFphysical
ECONOMICS
function
to human
applies
capital,
capital, and consump,
n
a
+g+
a /assuming
human
In addition,
1/(1-a-)
we
are
that
capital
k-S of
(* one unit
tion. Incapital.
other words,
canhuman
consumption
be transformed
at(10)
depreciates
the
same
as Ifphysical
capital.
Lucas
[1988]
= 1/3,
value than into
a =Pfunction
the
coefficient
onrate
ln(sk).
for
the
example,
Substituting
into
and
taking
logs
, the
nproduction
costlessly
either
one
unit
or one
capital
unit of
+g+ of /physical
coefficient
on
+ g income
+
ln(n
models
function
production
for
8) would
behuman
-2. similar
Incapital
as
fundamenthis
model
high
gives
anthe
equation
per
capita
to
equation
(6)
human
Infor
capital.
addition,
we are assuming that human capital
Substituting
into
the
(10)
production
function
and
taking
logs
population
growth
lowers
income
pergoods.
capita
because
the that,
amounts
different
from
that
for
tally
other
We
believe
at least
above:
at
depreciates
the
same
rate
as
physical
capital.
Lucas
[1988]
of an
both
physical
and
must be
spread
more
gives
aninitial
equation
forhuman
income
pernatural
capita
similar
to
equation
(6)
for
itcapital
is
to
examination,
assume
thatthinly
the two
the
models
production function for
human capital as fundamen+
over
the
population.
above:
types of
functions
similar.
-are
Inproduction
[L(t)|
In
(I ln(n
(1 1) There
A(O)
different
from
that+ for
tally
other
We
gt way
goods.
g + aof human
that, at least
is
an
alternative
to
the+ believe
role
Y
(t)]_
We assume that a + ,B < 1, express
which implies
that there are
+
for
an initial
it is natural
to assume(11)that
examination,
in determining
capital
in
model.
withthe two
- this (If
decreasing
returns
to income
all
a(I + Combining
capital.
there
,B= +1,g then
are
In
[L(t)|
In
+
+
a
(1
1)
A(O)
ln(n
gt
the equation
for the functions
steady-stateare
level
of human capital given in
of production
types
similar.
Y
(t)]_
constant
to scale in+ the reproducible
returns
case,
+factors. In this
In(Sh)
(10)We
yields
an equation
aIln(Sk)
as which
function
of thethat
rate
of
assume
that afor+ income
there
are
,B < 1,
implies
investment
in
physical
capital,
the
rate
of
population
growth,
and
decreasing
returns
to all
+ ,B= 1,
(If a depends
then
there are
+capital.
This
equation
shows
how
income
per capita
on of
population
+
7.level
Previous
authors
have
provided
evidence
ofIln(Sk)
the importance
human
capital
In(Sh)
the
of
human
capital:
constant
to
in
returns
scale
the
In
reproducible
factors.
this
for
growth
in
income.
Azariadis
and
Drazen
[1990]
find
that
no
country
able case,
was
to
growth and accumulation of physical and human capital.
grow quickly during the postwar period without a highly literate labor force. They
Like
the
Solow
model,
model
augmented
predictswith
This
shows
how
capita
depends
onassociated
population
interpret
this
astextbook
evidence
that
there
a the
threshold
externality
[L(t)
(12)equation
In
= InA(O)
+income
+ 1is_per
gt
ln(sk)
human
7.
capital
accumulation.
Previous
authors
have
Similarly,
Rauch
provided
[1988]
finds
evidence
of
the
that
among
importance
of
human
countries
capital
in
coefficients
are
equation (11)ofthat
of the
functions
factor shares.
and accumulation
growth
and human
physical
capital.
had achieved
that
95 percent
adultand
in 1960,
for growth
in income.
literacy
for to
strong
Azariadis
Drazen
tendency
[1990]there
findwas
thata no
country
was able
aaL
As before,
isWi
capital's
of
share
somodel
welabor
athatThey
expect
Like
textbook
Solow
the
model,
augmented
predicts
income
per the
capita
tophysical
converge
over
the
period
1950-1985.
Romer
grow
[1989b]
quickly
finds
during
the postwar
period
without
aincome,
highly
literate
force.
1 Gauging
-aare
+a greasonable
+and
literacy
1960
helps
+the -iffactor
explain
subsequent
that,
oneln(h*).
8)
corrects
interpret
a in
evidence
that
there
value in
of this
ofas
about
isinvestment
an(n
threshold
one(11)
third.
externality
value
of P, forwith
associated
coefficients
equation
that
of
functions
shares.
measurement
error,
literacy has
no impact
on[1988]
growth
beyond
its effect
on
human
capital
accumulation.
Similarly,
Rauch
finds
that
among
countries
human
capital's
iscapital's
moreliteracy
difficult.
Inthe
the
United
States
the
investment.
There
isshare,
also older
a is physical
stressing
As
role
of so
of
human
share
capital
income,
we
ain for
expect
had achieved
thatbefore,
95
adultwork
in 1960,
percent
there
was
a strong
tendency
Equation
is
(12)
almost
identical
to
in
In
Section
I.
equation
(6)
development;
for
example,
see
Krueger
[1968]
and
Easterlin
[1981].
minimum
the
return
to
wage-roughly
labor
without
human
income
per
capita
to
converge
over theGauging
period 1950-1985.
Romer [1989b]
finds that
value
ofina oftheabout
onehuman
third.
a reasonable
value
of
P, for
that model
levelexplain
of
ispercent
a component
capital
of
the
error
literacy
1960averaged
helps
subsequent
investment
and
that,
if one
corrects
to
capital-has
about
30
50
of
the
average
wage
human
capital's
share,
is more
In the
United
States
the on
measurement
error,
literacy
has population
nodifficult.
impact growth
on
growth
beyond
its effect
Because
the
and
saving
rates
influence
interm.
manufacturing.
This
fact
suggests
that
50
to
70
percent
of
total
investment.
There
is
also
older
work
stressing
the
role
of
human
capital
minimum
the capital
returnto to
wage-roughly
labor without
human in
h *, one should
expect see
human
be Easterlin
correlated
positively
development;
example,
Krueger
and
[1981].
labor incomeforrepresents
the
return[1968]
to human
capital,
or that ,Bis
with the saving
rate and
capital-has
about
30 to 50correlated
the population
averaged
percent ofwith
negatively
average wage
Solow Model with Human
Capital Accumulation
•
Sum of coefficients on ln(sk ) and ln(sh ) in equation
(11) should be equal in magnitude and opposite
in sign to the coefficient on ln(n + g + δ).
•
Use average percentage of working-age
population in secondary school to proxy for
human capital investment rate.
•
Again, assume value for g + δ equal to 0.05. And
again use population growth rate and investment
rate for physical capital.
Estimation Results for
Augmented Solow Model
•
Coefficients on investment rates and population
growth have the predicted signs and are
significantly different from zero in two of the three
samples. For OECD only school variable is
significantly different from zero.
•
•
Equality restriction can’t be rejected in all samples.
Regressions explain large fraction (over 3/4) of the
cross-country variation in income per capita for
two of the three samples.
the percentage of the population in secondary school. The humancapital measure enters significantly in all three samples. It also
TABLE II
OF THEAUGMENTED
SOLOWMODEL
ESTIMATION
Dependent variable: log GDP per working-age person in 1985
Sample:
Observations:
CONSTANT
ln(I/GDP)
ln(n + g +5)
ln(SCHOOL)
R2
s.e.e.
Restricted regression:
CONSTANT
ln(SCHOOL) - ln(n + g + 5)
R2
s.e.e.
p-value
Implied a
Implied ,
Non-oil
98
6.89
(1.17)
0.69
(0.13)
-1.73
(0.41)
0.66
(0.07)
0.78
0.51
Intermediate
75
7.81
(1.19)
0.70
(0.15)
-1.50
(0.40)
0.73
(0.10)
0.77
0.45
OECD
22
8.63
(2.19)
0.28
(0.39)
-1.07
(0.75)
0.76
(0.29)
0.24
0.33
7.86
(0.14)
0.73
(0.12)
0.67
(0.07)
0.78
0.51
7.97
(0.15)
0.71
(0.14)
0.74
(0.09)
0.77
0.45
8.71
(0.47)
0.29
(0.33)
0.76
(0.28)
0.28
0.32
0.41
0.31
(0.04)
0.28
(0.03)
0.89
0.29
(0.05)
0.30
(0.04)
0.97
0.14
(0.15)
0.37
(0.12)
Note. Standard errors are in parentheses. The investment and population growth rates are averages for the
period 1960-1985. (g + 8) is assumed to be 0.05. SCHOOL is the average percentage of the working-age
population in secondary school for the period 1960-1985.
Augmented Solow Model
•
Implied values for α and ß are close to 1/3 for the
first two samples.
•
Hence, model fits better with other evidence that
physical capital’s share is about 1/3.
•
Poor performance of OECD sample may be due
to violation of assumption that economies are in
steady state by 1985, because it is more likely that
rich countries were further from steady state due
to WWII’s effects being concentrated in those
countries.
•
So consider analysis out of steady state.
Convergence to Steady State
• Distance from steady state determines rate
of growth in output per worker.
• Countries that are further from steady state
will grow faster than those that are closer.
•
Introducing human capital accumulation
implies slower rate of convergence for a
given gap between current output per
worker and steady-state output per worker.
about the speed of convergence to steady state. Let y be the
steady-state level of income per effective worker given by equation
(11), and let y(t) be the actual value at time t. Approximating
around the steady state, the speed of convergence is given by
(13)
d ln(y(t))
dt
=
X[ln(y*) -ln(y(t))],
where
A = (n + g + a) (1-a
-
THE EMPIRICS OF ECONOMIC GROWTH
423
For example, if a = P = 1/3 and n + g + 8 = 0.06, then the
convergence rate (A) would equal 0.02. This implies that the
economy moves halfway to steady state in about 35 years. Notice
that the textbook Solow model, which excludes human capital,
implies much faster convergence. If IB= 0, then X becomes 0.04,
and the economy moves halfway to steady state in about seventeen
years.
The model suggests a natural regression to study the rate of
convergence. Equation (13) implies that
(14)
ln(y(t)) = (1 - e-At) ln(y*) + e-Atln(y(0)),
where y(O) is income per effective worker at some initial date.
Convergence to Steady State
• Equation (13) can be used to derive an
estimating equation to test for
convergence.
• This leads to an equation that expresses
growth in output per worker as a function
of the steady state and the initial level of
output per worker:
about the speed of convergence to steady state. Let y * be the
of income per effective worker given by equation
steady-state level
423423
THE
EMPIRICS
OFECONOMIC
ECONOMIC
GROWTH
OF
423
be the actual
value atGROWTH
time t. Approximating
(11), and let THE
y(t) EMPIRICS
around
the steady
state,= the
speed of convergence is given by
==
For
example,
if
a
1/3 and n + g + 8 = 0.06, then the
P
=
For
example, ifif aa = PP = 1/3
1/3 and
and nn ++ gg ++ 88 == 0.06,
0.06, then
then the
the
For
example,
convergence rate (A)
would
equal
0.02.
This
implies
that
the
ln(y(t))
convergence
rate d
(A)
would= equal
equal
0.02. This
This implies that
that the
the
convergence
rate
(A)
would
0.02.
(13)
dt
X[ln(y*)
-ln(y(t))],
in
economy moves halfway
to steady
state
about implies
35 years. Notice
economy moves
moves halfway to
to steady
steady state
state inabout
about 35years.
years.Notice
Notice
that the textbookhalfway
Solow model,
which in
excludes 35
human capital,
that
the textbook
textbook Solow model,
model, which
excludes human capital,
capital,
where
implies
much fasterSolow
convergence. which
If IB= excludes
0, then Xhuman
becomes 0.04,
implies much
much faster
faster convergence.
convergence. IfIf IB
IB
then XXbecomes
becomes0.04,
0.04,
==state
0,0,then
and the economy
moves halfway to steady
in about seventeen
(1-a state
A = halfway
(n + g +to
and the economy
economy moves
moves
halfway
toa)
steady
state
aboutseventeen
seventeen
ininabout
steady
years.
years.
The model suggests a natural regression to study the rate of
model
suggests
natural that
regression totostudy
The model
studythe
therate
rateofof
suggests
aa natural
regression
convergence.
Equation
(13)
implies
Equation
convergence.
Equation (13)
(13) implies
implies
that
Solving the differential
equation
(13)that
yields:
(14)
ln(y(t)) = (1 - e-At) ln(y*) + e-Atln(y(0)),
ln(y(t))
(14)
ln(y(t)) == (1
(1 -- e-At)
e-At)ln(y*)
ln(y*) ++e-At
e-Atln(y(0)),
ln(y(0)),
where y(O) is income per effective worker at some initial date.
where y(O) is
per
effective
worker at some initial date.
is income
income
perboth
effective
from
sides, worker at some initial date.
Subtracting In
(y(O))
In
both
In rewritten
from
both sides,
Subtracting
(y(O))
sides,
(y(O))from
which
can
be
as:
(15)
ln(y(t)) - ln(y(0)) = (1 - e-t) ln(y*) - (1 - e-At) ln(y(0)).
(15)
ln(y(t))
ln(y(0)).
(15)
ln(y(t)) -- ln(y(0))
ln(y(0)) == (1
(1 -- e-t)
e-t) ln(y*)
ln(y*) - - (1(1- - e-At)
e-At)
ln(y(0)).
Finally, substituting for y*:
Finally, substituting
Finally,
substituting for
fory*:
y*:
(16)
(16)
(16)
ln(y(t)) - ln(y(0)) = (1 - e-t)
ln(y(t))
ln(y(t)) -- ln(y(0))
ln(y(0)) == (1
(1 -- e-t)
e-t)
+ (1
++ (1
e-t)
(1 - - e-t)
e-t)
-
ln(sk)
− λt
ln(sk)
ln(sk)+ [1− e ]ln A(0)
ln(sh)
ln(sh)
ln(sh)
_ lIn(n + g + 8) - (1 - e-t) ln(y(0)).
__ lIn(n + g + 8) - (1 - e-t) ln(y(0)).
lIn(n + g + 8) - (1 - e-t) ln(y(0)).
Thus, in the Solow model the growth of income is a function of the
of income
of the
Thus, in the Solow
the growth
is a initial
function
of themodel
and the
determinants
ultimate
level
in the Solow
model
of income
ofofthe
Thus,
the steady
is a function
growthstate
determinants of the ultimate steady state and the initial level of
income.
of the ultimate steady state and the initial level of
determinants
-
(1 - eAxt) _
(1
(1 -- eAxt)
eAxt) __
Convergence Hypothesis
• Consider first unconditional convergence,
where the steady state y* is assumed to be
the same across all countries.
• Coefficient on initial output per worker in
1960 should be negative and significantly
different from zero.
• Results show evidence of unconditional
convergence only for the OECD sample.
425
TABLE III
TESTS FOR UNCONDITIONALCONVERGENCE
Dependent variable: log difference GDP per working-age person 1960-1985
Sample:
Observations:
CONSTANT
ln(Y60)
R2
s.e.e.
Implied X
Non-oil
98
-0.266
(0.380)
0.0943
(0.0496)
0.03
0.44
-0.00360
(0.00219)
Intermediate
75
0.587
(0.433)
-0.00423
(0.05484)
-0.01
0.41
0.00017
(0.00218)
OECD
22
3.69
(0.68)
-0.341
(0.079)
0.46
0.18
0.0167
(0.0023)
Note. Standard errors are in parentheses. Y60 is GDP per working-age person in 1960.
essentially zero. There is no tendency for poor countries to grow
faster on average than rich countries.
Table III does show, however, that there is a significant
tendency toward convergence in the OECD sample. The coefficient
•
•
Allow for different steady states across countries.
•
Consider first controlling only for different investment
rates in physical capital and in population growth.
•
Next consider controlling as well for different
investment rates in human capital (using school
variable).
Estimate growth regression controlling for these
different steady states--i.e., testing for “conditional
convergence.”
426
QUARTERLYJOURNAL OF ECONOMICS
TABLE IV
TESTS FOR CONDITIONALCONVERGENCE
Sample:
Observations:
CONSTANT
ln(Y60)
ln(I/GDP)
ln(n + g + 8)
R72
s.e.e.
Implied X
Non-oil
98
1.93
(0.83)
-0.141
(0.052)
0.647
(0.087)
-0.299
(0.304)
0.38
0.35
0.00606
(0.00182)
Intermediate
75
2.23
(0.86)
-0.228
(0.057)
0.644
(0.104)
-0.464
(0.307)
0.35
0.33
0.0104
(0.0019)
OECD
22
2.19
(1.17)
-0.351
(0.066)
0.392
(0.176)
-0.753
(0.341)
0.62
0.15
0.0173
(0.0019)
Note. Standard errors are in parentheses. Y60 is GDP per working-age person in 1960. The investment and
population growth rates are averages for the period 1960-1985. (g + 8) is assumed to be 0.05.
TABLE V
TESTS FOR CONDITIONAL CONVERGENCE
TABLE V
TESTS FOR CONDITIONAL CONVERGENCE
Sample:
Observations:
CONSTANT
ln(Y60)
ln(I/GDP)
ln(n + g + 8)
ln(SCHOOL)
R2
s.e.e.
Implied X
Non-oil
98
3.04
(0.83)
-0.289
(0.062)
0.524
(0.087)
-0.505
(0.288)
0.233
(0.060)
0.46
0.33
0.0137
(0.0019)
Intermediate
75
3.69
(0.91)
-0.366
(0.067)
0.538
(0.102)
-0.551
(0.288)
0.271
(0.081)
0.43
0.30
0.0182
(0.0020)
OECD
22
2.81
(1.19)
-0.398
(0.070)
0.335
(0.174)
-0.844
(0.334)
0.223
(0.144)
0.65
0.15
0.0203
(0.0020)
population growth rates are averages for the period 1960-1985. (g + 8) is assumed to be 0.05. SCHOOL is the
average percentage of the working-age population in secondary school for the period 1960-1985.
THE EMPIRICS OF ECONOMICGROWTH
A.
Unconditional
0
co 66
(0
Q) 4
--
427
~~00
2
2
00
Oo
0O
0
~~~~~0
o
0
-~0
0 0
0
R
?
0
0
0
o
0
00 Oo0
? -2
5,5
s
6,5
7,5
8,5
9.5
10.5
Log output per working age adult:1960
B.
Conditionalon saving and populationgrowth
~~
~~0 0
0
0
o0c~
O? ???0 08 8
6
060
W, 4
_CP
=
2
o
020
0
o
D
0
0
5s,5
6.5
7, 5
8.5
9.5
10,5
LO
C.Conditionalon saving, populationgrowthand humancapital
6
0~~~~~~
0
?
2 0~~~~~~
?
w
?
0
O??b 9
lb8
-2
5.5
6.5
7.5
8.5
9.5
FIGUREI
Unconditional
versusConditional
Convergence
10.5
•
Coefficient on initial output per worker is negative and significant
in all samples when controlling for differences in steady states,
both with and without human capital accumulation, indicating
substantial convergence.
•
But speed of adjustment parameter is much smaller than we’d
expect (assuming physical capital share of 1/3) when controlling
only for differences in investment rates for physical capital (and
population growth).
•
When also control for differences in human capital accumulation,
the speed of adjustment is closer to the value of 0.02 (for the
intermediate and OECD samples), which is what we’d expect if
we assume physical capital share of 1/3 and human capital share of
1/3.
429
THE EMPIRICS OF ECONOMICGROWTH
TABLEVI
TESTS FOR CONDITIONAL CONVERGENCE,
RESTRICTED REGRESSION
Dependentvariable:log differenceGDP per working-ageperson 1960-1985
Sample:
Observations:
CONSTANT
ln(Y60)
ln(SCHOOL) - ln(n + g + 5)
R2
s.e.e.
p-value
ImpliedX
Implied(x
Implied 13
Non-oil
98
2.46
(0.48)
-0.299
(0.061)
Intermediate
75
3.09
(0.53)
-0.372
(0.067)
OECD
22
3.55
(0.63)
-0.402
(0.069)
0.500
0.506
0.396
(0.082)
(0.095)
(0.152)
0.238
0.266
0.236
(0.060)
0.46
0.33
(0.080)
0.44
0.30
(0.141)
0.66
0.15
0.40
0.0142
(0.0019)
0.48
(0.07)
0.23
(0.05)
0.42
0.0186
(0.0019)
0.44
(0.07)
0.23
(0.06)
0.47
0.0206
(0.0020)
0.38
(0.13)
0.23
(0.11)
population growth rates are averages for the period 1960-1985. (g + 5) is assumed to be 0.05. SCHOOL is the
average percentage of the working-age population in secondary school for the period 1960-1985.
specifications that do not consider out-of-steady-state dynamics.
Similarly, the greater importance of departures from steady state
for the OECD would explain the finding of greater unconditional
•
Sum of coefficients on ln(sk ) and ln(sh ) in equation (16)
should be equal in magnitude and opposite in sign to the
coefficient on ln(n + g + δ).
•
•
Cannot reject this restriction.
•
These convergence regression estimates give larger weight
to physical capital and a smaller weight to human capital
compared to the estimates explaining variation in output
per worker (Table II).
Implied values for income share of physical capital of range
from 0.38 to 0.48 and the implied values for income share
of human capital is 0.23 in all three samples.

Mankiw, Romer, Weil, QJE 1992

Transcripción

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