In: Economics
Explain the simple version of the endogenous (neo-classical) growth model. Contrast this model with the standard Solow model of growth. In particular, explain why the endogenous growth model does not necessarily lead to interregional convergence
1. BASIC NEOCLASSICAL SETUP
Consider an economy populated by a large (but constant) number of
separate
households, each of which seeks at an arbitrary time denoted t = 1
to maximize
u(c1) + βu(c2) + β
2
u(c3) + . . . , (1)
where ct
is the per capita consumption of a typical household member
during
period t and where β = 1/(1 + ρ) with ρ > 0 the rate of time
preference. The
instantaneous utility function u is assumed to be well behaved,
i.e., to have the
properties u
0 > 0, u
00 < 0, u
0
(0) = ∞, u
0
(∞) = 0. The analysis would not be
appreciably altered if leisure time were included as a second
argument, but to
keep matters simple, leisure will not be recognized in what
follows. Instead, it
will be presumed that each household member inelastically supplies
one unit
of labor each period.
It is assumed that the number of individuals in each household
grows at the
rate ν; thus each period the number of members is 1+ν times the
number of the
previous period. In light of this population growth, some analysts
postulate a
household utility function that weights each period’s u(ct) value
by the number
of household members, a specification that is effected by setting ψ
= 1 in the
following more general expression:
u(c1) + (1 + ν)ψ βu(c2) + (1 + ν)2ψ β
2
u(c3) + . . . . (10)
With ψ = 0, expression (10
) reduces to (1) whereas ψ values between 0 and 1
provide intermediate assumptions about this aspect of the setup.
Most of what
follows will presume ψ = 0, but the more general formulation (10)
will be
referred to occasionally.
Each household operates a production facility with input-output
possibil-
ities described by a production function Yt = F(Kt
, Nt), where Nt and Kt are
the household’s quantities of labor and capital inputs with Yt
denoting output
during t. The function F is presumed to be homogeneous of degree
one so, by
letting yt and kt denote per capita values of Yt and Kt
, we can write
yt = f(kt), (2)
where f(kt) ≡ F(kt
, 1). It is assumed that f is well behaved (as defined above).
Letting vt denote the per capita value of (lump-sum) government
transfers
(so −vt = net taxes), the household’s budget constraint for period
t can be
written in per capita terms as
f(kt) + vt = ct + (1 + ν)kt+1 − (1 − δ)kt
. (3)
Here δ is the rate of depreciation of capital. As of time 1, then,
the household
chooses values of c1, c2, . . . and k2, k3, . . . to maximize (1)
subject to (3)
and the given value of k1. The first-order condition necessary for
optimality
can easily be shown to be
(1 + ν)u
0(ct) = βu
0(ct+1)[ f
0(kt+1) + 1 − δ], (4)
and the relevant transversality condition is
limt→∞
kt+1β
t−1
u
0(ct) = 0. (5)
The latter provides the additional side condition needed, since
only one initial
condition is present, for (3) and (4) to determine a unique time
path for ct and
kt+1. Satisfaction of conditions (3), (4), and (5) is necessary and
sufficient for
household optimality.
To describe this economy’s competitive equilibrium, we assume that
all
households are alike so that the behavior of each is given by (3),
(4), and
(5). The government consumes output during t in the amount gt (per
person),
the value of which is determined exogenously. For some purposes one
might
want to permit government borrowing, but here we assume a balanced
budget.
Expressing that condition in per capita terms, we have
gt + vt = 0. (6)
For general competitive equilibrium (CE), then, the time paths of
ct
, kt
, and vt
are given by (3), (4), and (6), plus the transversality condition
(5). In most of
what follows, it will be assumed that gt = vt = 0, in which case
the CE values
of ct and kt are given by (4) and
f(kt) = ct + (1 + ν)kt+1 − (1 − δ)kt
, (7)
provided that they satisfy (5).
Much interest centers on CE paths that are steady states, i.e.,
paths along
which every variable grows at some constant rate.
It can be shown that in
the present setup, with no technical progress, any steady state is
characterized
by stationary (i.e., constant) values of ct and kt
.
(These constant values imply
growth of economy-wide aggregates at the rate ν, of course.) Thus
from (4) we
see that the CE steady state is characterized by f
0(k) + 1 − δ = (1 + ν)(1 + ρ)
or
f
0
(k) − δ = ν + ρ + νρ. (8)
This says that the net marginal product of capital is approximately
(i.e., ne-
glecting the interaction term νρ) equal to ν +ρ, a condition that
should be kept
in mind. If the more general utility function (10) is adopted, the
corresponding
result is f
0(k) + 1 − δ = (1 + ρ)(1 + ν)1−ψ . Thus with ψ = 1, i.e.,
when
household utility is u(c) times household size, we have f
0(k) − δ = ρ.
It can be shown that, in the model at hand, the CE path approaches
the CE
steady state as time passes. Given an arbitrary k1, in other words,
kt approaches
the value k∗
that satisfies (8) as t→∞. This result can be clearly and
easily
illustrated in the special case in which u(ct) = log ct
, f(kt) = Akα
t
, and δ = 1
(Below we shall refer to these as the “LCD assumptions,” L standing
for log
and CD standing for both Cobb-Douglas and complete depreciation.)
In this
case, equations (4) and (7) become
(1 + ν)
ct
=
βαAkα−1
t+1
ct+1
(9)
and
Akα
t = ct + (1 + ν)kt+1. (10)
Since the value of kα
t summarizes the state of the economy at time t, it is a
reasonable conjecture that kt+1 and ct will each be proportional to
kα
t
. Substi-
tution into (9) and (10) shows that this guess is correct and that
the constants
of proportionality are such that kt+1 = αβ(1+ν)−1Akα
t and ct = (1−αβ)Akα
t
.
These solutions in fact satisfy the transversality condition (TC)
given by (5),
so they define the CE path. The kt solution can then be expressed
in terms of
the first-order linear difference equation
log kt+1 = log[αβA/(1 + ν)] + α log kt
, (11)
which can be seen to be dynamically stable since | α |< 1. Thus
log kt con-
verges to (1 − α)−1
log[αβA/(1 + ν)]. For reference below, we note that sub-
traction of log kt from each side of (11) yields
log kt+1 − log kt = (1 − α)[log k∗ − log kt], (12)
where k∗ = [αβA/(1 + ν)]1/(1−α)
, so 1 − α is in this special case a measure of
the speed of convergence of kt
to k∗
.
It might be thought that the complete-depreciation assumption δ = 1
ren-
ders this special case unusable for practical or empirical
analysis. But such a
conclusion is not inevitable. What is needed for useful
application, evidently,
is to interpret the model’s time periods as pertaining to a span of
calendar
time long enough to make δ = 1 a plausible specification—say, 25 or
30 years.
Then the parameters A, β, and ν must be interpreted in a
corresponding manner.
Suppose, for example, that the model’s time period is 30 years in
length. Thus
if a value of 0.98 was believed to be appropriate for the discount
factor with
a period length of one year, the appropriate value for β with
30-year periods
would be β = (0.98)30 = 0.545. Similarly, if the population growth
parameter
is believed to be about one percent on an annual basis, then we
would have
1+ν = (1.01)30 = 1.348. Also, a realistic value for A would be
about 10k
(1−α)
,
since it makes k/y = 3/30 = 0.1. So the LCD assumptions could
apparently
be considered for realistic analysis, provided that one’s interest
is in long-term
rather than cyclical issues.
WEAKNESSES OF THE NEOCLASSICAL MODEL
The evident trouble with the neoclassical growth model outlined
above is that
it fails to explain even the most basic facts of actual growth
behavior. To a
large extent, this failure stems directly from the model’s
prediction that output
per person approaches a steady-state path along which it grows at a
rate γ
that is given exogenously. For this means that the rate of growth
is deter-
mined outside the model and is independent of preferences, most
aspects of
the production function, and policy behavior. As a consequence, the
model
itself suggests either the same growth rate for all economies or,
depending on
one’s interpretation, different values about which it has nothing
to say. But
in reality different nations have maintained different per capita
growth rates
over long periods of time—and these rates seem to be systematically
relatedto various national features, e.g., to be higher in
economies that devote large
shares of their output to investment. These and other failings were
stressed by
Romer (1986, 1987, 1989) and Lucas (1988).
Of course, the neoclassical model does imply that transitional
growth rates
will differ across economies, being faster in those that have
existing capital-
to-effective-labor ratios relatively far below their CE
steady-state values. This
observation is what prevented fundamental dissatisfaction from
being openly
expressed before the appearance of the Romer and Lucas papers and
is one of
the two lines of defense recently mentioned in a lively discussion
by Mankiw
(1995, p. 281). But transitional phenomena cannot provide a
quantitative
explanation of the magnitude of long-lasting growth rate
differences under the
standard neoclassical presumption that the production function is
reasonably
close to the Cobb-Douglas form with a capital elasticity of
approximately one-
third (roughly capital’s share of national income). One way to
describe the
problem is to consider a comparison in which one economy’s per
capita output
increases by a factor of 2.9 relative to another’s over a period of
30 years, which
is the factor that would be relevant if the first economy’s average
growth rate
exceeded the second’s by about 3.6 percent per year. (This last
figure is twice the
standard deviation of per capita growth rates among 114 nations
over the years
1960 to 1990, as reported by Barro and Sala-i-Martin [1995], p. 3,
so a sizable
fraction of all nation pairs have had differences exceeding that
value.) Then,
with a capital elasticity of one-third, the capital stock per
capita would have
to increase by a factor of 2.93 = 24.4 relative to the second
economy, if their
rates of technical progress were the same. Thus the real rate of
interest—i.e.,
the marginal product of capital—in the first economy would fall by
a relative
factor of 24.42/3 = 8.4. So if the two economies had similar real
interest rates
at the end of the 30-year period, the first economy’s rate would
have been 8.4
times as high as the second’s at the start of the period! But of
course we do
not observe in actual data changes in capital/labor ratios or real
interest rates
that are anywhere near as large as those magnitudes, even though we
observe
many output growth differentials of 3.6 percent and more. Some
evidence
that this argument is robust to production function assumptions,
and a dramatic
comparison involving Japan and the United States, is provided by
King and
Rebelo (1993).
The same general type of calculation is also relevant for
cross-country
comparisons. The level of per capita incomes in the industrial
nations of the
world are easily 10 times as high as in many developing nations.
With a pro-
duction function of the type under discussion, this differential
implies a capital
per capita ratio of 103 = 1000, and therefore a ratio of marginal
products of
capital of 1000−2/3 = 1/100 = 0.01. In other words, the real rate
of return to
capital is predicted to be about 100 times as high in the
developing nations
as in those that are industrialized. But surely a differential of
this magnitude
would induce enormous capital flows from rich to poor countries,
flows entirely
unlike anything that is observed in actuality.
Another perspective on the neoclassical vs. endogenous growth issue
in-
volves the question of “convergence,” which has been much discussed
in the
literature. From equations (14) and (23) above we see that if all
nations had the
same taste and technology parameters, and the same population
growth rate,
then they should, according to the neoclassical model, have the
same steady-
state level of per capita income. Thus as time passes, per capita
income levels
in different countries should converge to a common value, with low
income
countries growing more rapidly than those in which beginning per
capita in-
come levels are high. Empirically, however, it is the case that
growth rates over
periods such as 1960 to 1985 are virtually uncorrelated with
initial-year income
levels. In fact, there is a small, positive coefficient in the
Mankiw-Romer-Weil
(1992) sample of 98 “non-oil” countries; their cross-section
regression is
log y1985 − log y1960 = −0.27 + 0.094 log y1960
(0.38) (0.050)
R
2 = 0.03 SE = 0.44.
The neoclassical model does not actually require, however, that
population
growth values are equal in various countries and does not imply
that taste
and technology parameters must be the same. So convergence in the
“uncondi-
tional” sense of the foregoing discussion is not, it can be argued,
relevant to the
performance of the neoclassical model. What that model does imply,
is a concept that has been termed “conditional convergence.”
It should be noted that the foregoing discussion does not imply
that the
neoclassical analysis was unproductive. On the contrary, it played
a major and
essential role in the development of dynamic general equilibrium
analysis, the
basis for much of today’s economic theory. It is only as a theory
of growth
that it is here being criticized.
6. ENDOGENOUS GROWTH MECHANISMS
In response to the various failures of the neoclassical model,
Romer, Lucas,
King and Rebelo, and other scholars have developed models in which
steady
growth can be generated endogenously—i.e., can occur without any
exogenous
technical progress—at rates that may depend upon taste and
technology pa-
rameters and also tax policy. There are numerous variants of such
models, but
several important points can be developed by focusing on three
basic mecha-
nisms. Two of these, one involving a capital accumulation
externality and the
second relying upon the accumulation of human capctal.
Let us consider first the externality model. For its presentation
we will mod-
ify the setup of Section 1, in which there is no exogenous
technical progress.
There is, however, an externality in production so that the typical
household’s
per capita production function is
yt = f(kt
, ¯kt) f2 > 01, f22 < 0, (28)
where ¯kt
is the economy-wide average capital stock per person. Quoting
Romer
(1989, p. 90), the “rationale for this formulation is based on the
public good
character of knowledge. Suppose that new physical capital and new
knowledge
or inventions are produced in fixed proportions so that [¯kt] is an
index not only
of the aggregate stock of physical capital but also of the
aggregate stock of
public knowledge that any firm can copy and take advantage of.” But
each firm
or household is small, so it views ¯kt as given when making its
choice of kt+1
and other decision variables.
So as to highlight the effect of the resulting externality, suppose
that the
production function is Cobb-Douglas,
yt = Akα
t
¯k
η
t
, (280
)
and that the other LCD assumptions hold as well (i.e., u(ct) = log
ct and δ =
1). Also, let gt = vt = 0. Then the household’s budget constraint
is
Akα
t
¯k
η
t = ct + (1 + ν)kt+1 (29)
and its first-order optimality condition is
(1 + ν)
ct
=
βαAkα−1
t+1
¯k
η
t+1
ct+1(30)
.
In addition, for general competitive equilibrium the following
condition must
be satisfied, since households are alike:
kt = ¯kt (31)
. Given these relations, it is a reasonable conjecture that in a CE
both kt+1
and ct will be proportional to kα+η
t
. Substitution into (29) and (30), using (31),
shows this guess to be correct and that the resulting expression
for kt+1 is
kt+1 = αβ(1 + ν)−1Akα+η
t
. (32)
There are two interesting points relating to this solution. First,
with η > 0
the CE path is not socially optimal. For social optimality, the
problem is to
maximize (1) subject not to (29), but to (29) with (31) imposed. In
that case,
the equation comparable to (32) that results is
kt+1 = (α + η)β(1 + ν)−1 Akα+η
t
. (320)
Clearly, if α + η < 1, then (32) implies that kt approaches a
constant value,
but it is one that is smaller than the steady-state value implied
by (320)—an
outcome that reflects the failure of individuals to take account of
their own
actions’ effect on the economy-wide state of knowledge. Second, if
by chance
it happened that α+η = 1, then kt would grow forever at a constant
rate equal
to αβ(1+ν)−1A−1. Thus, it is possible, within this framework that
excludes
exogenous technical progress, for steady-state growth to be
generated, in which
case its rate will be dependent upon α, β, ν, and A. Admittedly,
the case with
α + η = 1 exactly might be regarded as rather unlikely to prevail.
That issue
will be taken up below.
Now let us consider the second of the two basic endogenous growth
mech-
anisms, this one involving the accumulation of human capital—in the
sense of
labor-force skills that can be enhanced by the application of
valuable resources.
One simple way to represent this phenomenon is to specify that
physical output
is accumulated according to
Akα
t
(htnt)1−α = ct + (1 + ν)kt+1, (33)
where nt
is the fraction of the typical household’s work time that is
allocated
to goods production and ht
is a measure of human capital—i.e., workplace
skills—of a typical household member at time t. These skills are
produced
by devoting the fraction 1−nt of working time to human capital
accumulation.
In general, physical capital would also be an important input to
this process,
but for simplicity let us initially assume that the accumulation of
productive
skills obeys the law of motion
ht+1 − ht = B(1 − nt)ht − δhht
,where the final term reflects depreciation of skills that occurs
as time passes.
In this expression, and for the rest of this example, we let ν =
0.
Maximization of (1) subject to constraints (33) and (34) gives rise
to the
following first-order conditions:
c−1
t = βc−1
t+1αAkα−1
t+1
(ht+1nt+1)1−α (35a)
c−1
t Akα
t h
1−α
t (1 − α)n−α
t = µtBht (35b)
µt = βµt+1[B(1 − nt+1) + 1 − δh] + βc−1
t+1Akα
t+1
n
1−α
t+1
(1 − α)h−α
t+1
. (35c)
Here µt
is the shadow price of human capital, i.e., the Lagrange
multiplier
attached to (34). With gt = vt = 0, the CE is given by the five
equations
(33), (34), and (35a) – (35c), which determine time paths for
ct
, kt
, ht
, nt
, and
µt
. Since (33) and (34) are the same from the private and social
perspectives,
there is no departure from social optimality implied by the
CE.
Now consider the possibility of steady-state growth in this system.
Since
nt
is limited to the interval [0,1], it must be constant in any steady
state. If its
value is n, then (34) shows that ht will grow at the steady rate
B(1 − n) − δh,
which we now denote as ξ. Then (35a) implies, since ct+1/ct must be
constant,
that kt must also grow at the rate ξ—and by (33) the same must be
true for
ct
. Finally, (35b) shows that 1/µt must grow like ct—and these
conclusions
are consistent with (35c) having each term grow at the same rate.
To find out
what this growth rate will be, we can equate µt from (35b) and
(35c), using
µt+1 = µt
/(1 + ξ) in the latter, and after some tedious simplification find
that
ρ(1 + ξ) = Bn. (36)
Since also ξ = B(1 − n) − δh, we can solve for
n =
ρ(1 + B − δh)
(1 + ρ)B
(37)
in terms of basic parameters of the problem. Then ξ is found easily
from
expression (36).
An important property of (37) to be noted is that the steady-state
value of
n increases with ρ. Thus ξ, the growth rate, decreases with ρ, the
rate of time
preference. In other words, the more impatience is exhibited by the
economy’s
individuals, the lower will be the steady-state growth rate. This
is precisely the
sort of result that some analysts have found highly plausible but
is not generated
by the neoclassical model. If ν 6= 0 is assumed, moreover, the
growth rate is
negatively related to ν.An obvious objection to the model based on
(33) and (34) is that produc-
tion of ht should be specified as dependent on the use of
capital—i.e., physical
goods—in that process. That extension has been studied by Rebelo
(1991), who
uses the following in place of (34):
ht+1 − ht = B(mtkt)
a
[ht(1 − nt)]1−a − δhht
. (38)
Here mt denotes the fraction of the capital stock that is devoted
to production
of human capital, so (1− mt)kt replaces kt
in (33) in this model. Rebelo finds
that the same conclusions involving steady growth and its
dependence upon ρ
hold with this extension. Furthermore, if production of physical
output is taxed,
say at the rate τ , then the steady-state growth rate will be
negatively related to
τ .
Of the two mechanisms considered, knowledge externalities and
human
capital, it is not obvious which is the more plausible as a source
of major
quantitative departure from the neoclassical model. But there is no
reason to
consider them on an either-or basis; both could be relevant
simultaneously.
Indeed, the Lucas (1988) model, of which our (33) and (34) are a
special
case, posits human capital accumulation as in (34) together with a
production
function for physical output in which there is an externality
involving average
economy-wide human—rather than physical—capital.
In what follows it will be useful to have at hand the full dynamic,
period-
by-period solution for a representative endogenous growth model. It
is possible
to derive such a solution for the Lucas model, even with the human
capital
production externality included, provided that we use the LCD
version, which
in this case requires that human capital fully depreciates in one
period. Ac-
cordingly, let us now modify the model of equations (33), (34), and
(35) by
using Akα
t (htnt)1−αh¯η
t as the production function and setting δh = 1. Also, we
shall permit population growth again, which implies that ht+1 in
(34) and µt
in (35c) are multiplied by (1 + ν). Then the household’s optimality
conditions,
other than the transversality conditions, can be written as
follows:
Akα
t
(htnt)1−αh¯η
t = ct + (1 + ν)kt+1 (39a)
(1 + ν)ht+1 = B(1 − nt)ht (39b)
(1 + ν)ct+1 = ctβαAkα−1
t+1
(ht+1nt+1)1−αh¯η
t+1
(39c)
c−1
t Akα
t h
1−α
t (1 − α)n−α
t
¯h
η
t = µtBht (39d)
(1 + ν)µt = βµt+1[B − (1 − nt)] + βc−1
t+1Akα
t+1
n
1−α
t+1
(1 − α)h−α
t+1
h¯η
t+1
. (39e)
In competitive equilibrium we will also have ht = h¯
t
, so in what follows we
assume that condition to hold. To solve these equations for
ct
, kt+1, ht+1, nt
, and
µt
, we proceed by guessing—in analogy with the method of Section
2—that
those five variables are determined in response to the state
variables kt and ht
by expressions of the form
ct = φ10kφ11
t hφ12
t (40a)
kt+1 = φ20kφ21
t hφ22
t (40b)
ht+1 = φ30kφ31
t hφ32
t (40c)
nt = φ40kφ41
t hφ42
t (40d)
µt = φ50k
φ51
t h
φ52
t
. (40e)
If we can determine the implied values of the φ’s, we will have
substantiated
this guess.
We begin by substituting (40c) and (40d) into (39b),
obtaining
(1 + ν)φ30k
φ31
t h
φ32
t = Bht − Bφ40k
φ41
t h
φ42
t ht
. (41)
But then for (40) to be valid for all values of kt and ht
, it must be that φ31 =
φ41 = φ42 = 0 and φ32 = 1. Continuing in this manner of reasoning,
we end
up with various sensible-looking results such as that ht grows
steadily at the
rate [Bβ/(1+ν)]−1, the fraction of physical output saved is αβ, and
especially
that kt evolves as
kt+1 =
αB(1 − β)1−α
(1 + ν)
Akα
t h
1−α+η
t
. (42)
We shall make use of this last solution expression in Section
8.
An interesting and influential variant results when we again
suppress the
externality, by setting η = 0, but assume that human capital is
produced by a
production function of type but with a = α, i.e., with the same
parameters
as pertain to production of consumption (and physical capital)
output. With log
utility and Cobb-Douglas production functions, mt and nt will be
constant over
time; and with the production functions the same as well, the
relative price of
a unit of human capital in terms of output will be 1.0. Thus the
sum of the
two outputs is of the form (const.) kα
t h
1−α
t = (const.) kt(ht
/kt)1−α. But in thisspecial case it is also true that kt
/ht
is constant, so the foregoing expression
reduces to a constant times kt
, often written as yt = Akt
. Hence, this is one
case of the so-called “AK ” model, which from a growth perspective
is similar
to an extreme special case of the neoclassical model—one in which
the capital
elasticity parameter α equals one. In this case, kt and therefore
output per person
grows without limit at a constant rate, even with no technological
progress, as
inspection of equation (11) shows clearly. Furthermore, even if a
and α differ
so that kt
/ht varies from period to period, the model works as indicated from
a
steady-state growth perspective. Consequently, the AK model—which
may also
be rationalized in other ways—has played a prominent role in the
discussion
of endogenous growth possibilities.