Economics of Planning 28: 93-118, 1995.
© 1995 Kluwer Academic Publishers. Printed in the Netherlands.
93
Interaction of Human and Physical Capital in a
Model of Endogenous Growth
JOB GRACA, SAQIB JAFAREY "~ and APOSTOLIS PHIL1PPOPOULOS
Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex, C04 3SQ,
U.K.
Abstract. A model of developmentis presented where growth is initially driven by physical capital
accumulation, as in the neoclassical model. After a critical level of physical capital is reached, the
economy 'takes off' and enters a stage of sustained growthdriven by human capital accumulation. The
link between these two stages is providedby the assumptionthat private incentivesfor human capital
accumulation increase with the average levels of human and physical capital. At the early stages of
development, these incentives are low so the level of human capital stays stagnant until sufficient
physical capital is accumulated. Other results are that some economies may reach a steady state of
physical capital before a 'take-off' is possible.This is especiallylikely for economiesin which agents
have low savings propensities. Such economies remain stuck in a no-growth equilibrium forever.
Economies that do grow may experience endogenous cycles if the return to investment in human
capital is sufficientlyincreasing in the level of physical capital.
1. Introduction
This paper presents a model of endogenous growth where human and physical
capital interact, not just in the production technology, but in the technology for
augmenting human capital as well. The return on time spent in augmenting human
capital is assumed to be an increasing function of the economy-wide levels of
both physical and human capital. This formulation generates dynamic equilibria
which are consistent with both 'poverty traps' and 'self-sustaining' growth. It also
generates outcomes in which economies grow out of poverty traps due to the initial
accumulation of physical capital and then embark on the path of self-sustaining
growth with increases in both human and physical capital.
The 'new' growth theory, by emphasizing the role of human capital, has yielded
considerable insight into the possibility of self-sustaining growth. This outcome
sat uncomfortably with the predictions of the neoclassical theory of the 1960s. The
main reason for this was theoretical: the neoclassical theory had ignored human
capital, focusing solely on physical capital, and could thus reconcile the principle
of diminishing returns to individual factors with the possibility of sustained growth
only by appeal to an exogenous process of technical change. 1
By contrast, the new theory assumes that in addition to physical capital, improvements in the quality of labour, i.e. human capital, can be endogenously produced.
Further, it assumes that marginal improvements in human capital are not subject
To whom correspondence should be addressed
94
JOB GRACA ET AL.
to decreasing returns in the level of human capital, while the production of output exhibits constant returns to physical and human capital taken together. In the
long run, these assumptions make possible a balanced growth equilibrium, where
endogenous human capital accumulation, rather than exogenous technical change,
drive the process of growth.2
Thus, the new growth theory has drawn attention to human capital as an important factor of production and has attributed long-run growth to the accumulation of
this factor. At the same time, a branch of this theory has shown that, under plausible
modifications, models of human capital can generate 'poverty trap' equilibria in
which no accumulation of human capital and no growth takes place.
Examples of poverty-trap equilibria are in papers by Becker etal. (1990), Ehrlich
and Lui (1991), Galor and Zeira (1993) and Azariadis and Drazen (1990). The first
two papers study the interaction between numbers of children and schooling per
child in the context of endogenous fertility models. The latter two papers examine
the effects of nonconvexities in the technology for augmenting human capital.
While the frameworks are different, a common thread runs through these papers economies that start off with a low level of human capital stagnate while economies
that start off with sufficiently high levels of human capital grow. With the exception
of Becker et al., none of these papers considers the implications of initial levels of
physical capital for growth. 3
The available evidence, however, suggests that both human and physical capital
are positive determinants of growth in developing economies.4 Two contributions
are especially relevant. In his analysis of the sources of growth in the U.S. economy
during 1909 to 1957, Denison found a reversal in the relative importance of human
to physical capital. Over the period 1909-1929, improvements in the educational
composition of the workforce contributed 12% while increases in physical capital
contributed 26% towards accounting for the growth in real national income over
this period. Over the period 1929-1957, 23% of the growth in real national income
was contributed by education and only 15% by physical capital. 5 Thus, there was
a reversal in the relative magnitudes of these contributions. 6
A recent empirical study by De Long and Summers (1991) also argues for
the importance of physical capital. Distinguishing between equipment and other
types of capital (e.g. structures), the authors find a significant positive relationship
between cross-country growth rates and equipment investment over the period
1960-1985. Indeed, their results indicate a far greater importance of this type of
capital for growth than has been previously recognised, either in the theoretical
or the empirical literature. They reconcile their results with previous findings by
proposing that equipment, as opposed to other forms of capital, can potentially
generate externalities which promote growth, something which cannot be captured
by studies based on aggregate concepts of capital.
Our model incorporates the insights discussed above, albeit in a stylized fashion.
An externality from physical capital to the educational sector is crucial for the
results. This externality allows for equilibria in which physical capital formation
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
95
drives the initial phase of growth, which we term as neoclassical growth (because,
by itself, this phase cannot go on forever), while human capital begins to accumulate
at a later stage, and drives the economy into sustained growth with increases in
both factors.
Other results from our model are: First, self-sustaining growth can arise with
sufficiently high levels of either human or physical capital (thus, a high level
of human c~tpital is not a prerequisite for a 'take-off' although human capital
accumulation drives growth in the long run). Second, savings propensities matter
for whether an economy takes off or not. Third, if the externality from physical
capital to the human capital accumulation technology is strong enough, endogenous
cycles can occur along the balanced growth path.
Section 2 presents the model. Section 3 characterizes three types of equilibria
within the framework of a specific example. Section 4 solves for these equilibria. Section 5 discusses some comparative statics regarding capital mobility and
savings. Finally, Section 6 contains concluding remarks.
2. Model
2.1. ENVIRONMENT
Time is discrete and denoted by t -- 0, 1, 2, . . . . The economy consists of a series
of two-period lived, overlapping generations. A member of generation t is born in
period t and lives till period t + 1. The population is constant at 2N, so each member
of generation t is associated with one successor, born in period t + 1. Individuals
of a given generation are identical in all respects. The representative individual
of generation t derives utility only from her own consumption and not from the
consumption or utility of her successor; thus, there is no bequest motive.
Production is carried out by firms, using capital and labour. Firms acquire capital
by borrowing the savings of private agents, and labour by hiring workers in return
for a wage. Labour services consist of effective hours of labour which depend not
only upon labour-time, but also the skill level, or human capital, of the worker.
Thus, the hourly wage depends upon the skill level of the worker.
Members of each generation start life with a certain level of human capital
which they costlessly inherit from their respective predecessors (who are old by
then). They are also endowed with a fixed amount of non-leisure time (normalized
to unity) in both periods of their lives. When young, they divide this time between
employment and education; the latter activity allows them to augment their inherited level of human capital. They also divide their employment income between
consumption and savings to be carried over to old age. Upon becoming old, they
work full-time for a wage, as there are no further incentives to augment human
capital.
Young individuals augment their human capital through a technology which
generates increments to human capital as functions of the fraction of time spent in
education, the individual's initial human capital, the average level of human capital
96
JOB GRACAETAL.
in that period and a measure of the average level of physical capital (to be defined
below). The augmented level of human capital becomes available to individuals
when they become old.
Note that since individuals are identical, average quantities equal individual
quantities. However, the notion that returns to investment in human capital depend
on the average levels of human and physical capital captures the external nature
of these effects. Each individual takes the average levels of these variables as
given and does not take into account the effect of their own actions upon these
averages.
The following elaborates on the model discussed above.
2.2. PRODUCTION
Each period, a single good is produced by a constant returns to scale technology
using two factors: capital and labour. Capital depreciates fully each period.
The production function is assumed to be Cobb-Douglas:
(1)
where Y~ is the level of output at time t, K t is the capital stock at time t and Lt is
the effective labour supply at time t. The latter is given by
Lt = n t x t
(2)
where nt is total hours of labour supplied at time t and xt denotes the common
skill level of individuals working at time t. Note that the skill level xt is the same
for young and old people alive at the same time t.
Each old person supplies one unit of labour and each young person supplies
1 - rt units, where rt is the amount of time spent in education during period t.
Since there is an equal number, N, of young and old people at any time t, nt can
be written as
nt = ( 2 - r t ) N .
Therefore, equation (2) can be expressed as
Lt = (2 - T t ) x t N .
(3)
Output can be written in intensive form as
Yt = kilt
(4)
where yt = Y t / L t and kt = K t / L t .
Since private returns to scale are constant, firms earn zero profits in competitive
equilibrium. Therefore, factor prices are given by
rt = /3kt~-1
(5a)
wt = (1 - ~)kt~
(5b)
INTERACTIONOF HUMANANDPHYSICALCAPITAL
97
where rt is the rate of return on capital and wt is the wage rate at time t. Note
that the gross rate of return on capital equals the net rate of return because of the
assumption of full depreciation.
Since only young agents save and accumulate capital, we shall define per-capita
quantities in terms of the population of young agents only. Let yt and kt denote
output per young-saver and capital per young-saver respectively. In any period,
these are given by Y t / N and K t / N respectively. The relationship between per
young-saver and intensive quantities is given by
/It = (2 - "ct)xtYt
/¢t = ( 2 - Tt)xtkt.
(6a)
(6b)
2.3. HUMAN CAPITALACCUMULATION
The skill level of the ith individual worker at time t is denoted by x~, while
the average skill is denoted by xt. Although these quantities will be equal in
equilibrium, individuals take the average skill level as given in private decision
making.
Each young agent faces the following technology in deciding the amount of
time to devote to education (denoted by -r{)
Xt+xi ~. X{(1 --}-")/(xt, kt )T{)
(7)
where "/(xt, kt) represents the marginal efficiency of time spent in augmenting
human capital. 7(.) depends upon the average levels of both human and physical
capital. 7 Specifically, we assume that 7(.) is: (i) increasing in each argument; (ii)
concave; (iii) bounded above by a maximum value 7"-8
The assumption that 3'(.) is an increasing function of the average level of human
capital has been used previously by Azariadis and Drazen (1990). Its analytical
purpose is to allow for the existence of multiple steady state equilibria. This
assumption may be justified by appeal to 'peer group' effects.
The assumption that 7(.) is also increasing in the average level of physical
capital requires somewhat greater comment. While it may be more plausible to have
3'(.) depend upon private purchases of physical capital for educational purposes,
it is the presence of the external effect in our model that generates 'take-off'
equilibria. 9 Therefore, we neglect the role of private physical capital in education
in favour of examining the purely external effect of social physical capital.
In order to justify this external effect, it is helpful to appeal to more precise
concepts of both physical and human capital. First of all, if capital may be distinguished by vintage, then a higher level of physical capital may be interpreted
as embodying more modern techniques. Similarly, if we interpret investment in
human capital as the attainment of higher education which develops engineering
and management skills, then clearly physical capital, as laboratories, computes,
etc., plays an important role in such education. 1°
98
JOB GRACA ET AL.
To a great extent, this role may be manifested through private purchases of
capital inputs by educational institutions, something which we admittedly are not
considering. However, externalities may arise through two sources: (i) geographical proximity to modern technology may allow for 'hands-on' experience (e.g.
internships) which cannot be duplicated in the classroom; (ii) access to and interaction with a modem industrial sector may facilitate the adoption of training programmes (e.g. in computer-assisted design) that complement modem techniques
of production. 11
2.4.
UTILITY
MAXIMIZATION
A typical individual born in period t has a time-additive utility function
Ut
=
(8)
Oq ln(c~) + ce2 ln(C~+l)
where c~ is the consumption of generation t when young and ct°+l is the consumption of generation t when old.
The ith young agent at time t supplies (1 - r~) labour hours (which implies
an effective labour supply of (1 - r~)x{ for employment, and r~ labour hours for
education). At time t + 1, her human capital is increased to xt+
l i and this is devoted
full-time to income-earning activity. Thus, her budget constraint in each period is
given by
(1 - r~)wtx~ = cYt + st
i
(9a)
0
(9b)
Wt+lXt+ 1 + rt+lSt = ct+ 1
where st is savings carried over by a young person at time t.
Capital per young s a v e r kt+l equals st. The intensive measure of capital is
related to st by
st = (2
-
(10)
~+l)Xt+lkt+l.
The maximization problem facing a young agent can be expressed as
max o/1 ln((1 - r~)wtx{ - st) + o~2in(Wt+l (1 --}-"y(xt, kt)r~)x~ + rt+lat)
7"~~8t
subject to r~ _> 0 . 1 2 In stating the maximization problem, (7) has been substituted
into (9b), and (9a) and (9b) into (8).
The first-order conditions with respect to st and r~ are, respectively,
o
°/1Ct+l
a2ctY
_
rt+l
¢-'~2
"
i
c~+"17(xt,
kt)Wt+lX,
- °e---L1
g y W t Xit < O
(lla)
(= 0 i f r ~ > 0).
(1 lb)
99
INTERACTIONOF HUMAN AND PHYSICALCAPITAL
(The superscripts denoting individual agents and the generations to which they
belong were relevant for keeping track of private decision variables while taking
the first-order conditions. These shall henceforth be dropped on the grounds that
individual and average quantities will be equal in equilibrium.)
Combining (1 la) and (1 lb) gives the following inequality
(12)
")'(.)Wt+l ~ rt+l"
wt
Condition (12) states that the yield on investment in human capital must equal
that on physical capital in any interior maximum, i.e. for any % > 0. If (12) holds as
a strict inequality, that implies that no training takes place in period t, i.e. Tt = O.
Equilibrium paths are characterized by (1 la) and (12) (which determine the
optimal choices of st and "rt), along with (5a) and (5b) (which determine wt, wt+l
and rt), (9a) and (9b) (which determine cty and ct°+l). Taking into account (5) and
(9), and using (10) to eliminate st, the optimality conditions may be written as
/3k
kt+l
> 7(xt, kt)
/3kG1 = ¢ [(2 =
(13a)
+
,t)22
[(1 -/3)(1 - "rt)k~ - ( 2 -
Tt+l)(1 + "ytTt)kt+l]
(13a)
where ¢ = OZl/O~ 2 and 3't = 7(xt, kt).
Equations (13a) and (13b) constitute a set of two equations and two unknowns
(Tt and kt+l). If (13a) holds as an inequality, then Tt = 0 and (13b) alone determines
kt+l as a function of kt and the parameters/3 and ¢. Equation (7) completes the
system by describing the evolution of the stock of human capital.
Equation (13b) may be solved for kt+l as a function of kt, Tt and Tt+l. Let
¢ denote this function; thus, kt+l = ¢(kt, Tt, Tt+l). 13 It can be shown that ¢ is
increasing in kt and "rt+l, and decreasing in Tt. Thus (13b) represents a downward
sloping locus in ('rt, kt+l) space, i.e. for fixed values of kt and "rt+l. Equation (13a)
does not depend upon Tt, therefore it is a horizontal line in (Tt, kt+l) space. An
interior equilibrium occurs at time t if (13a) and (13b) intersect at a positive value
of Tt (see Figure la). However, if they intersect at a negative value of %, then
Tt = 0 (see Figure lb).
3. Equilibrium paths
Depending on initial values of human and physical capital, three of the possible
types of equilibria are (i) underdevelopment equilibria in which the level ofhuman
capital does not change over time while the level of physical capital reaches a
steady state value; (ii) sustained growth equilibria in which the level of human
capital grows steadily; (iii) take-off equilibria in which the level of human capital
stays constant over some initial period of time but then begins to grow as in the
sustained growth equilibria. 14
100
JOB GRACA ET AL.
kt+l
kt+l
(13~)
¢ ( to,,r,, r~+~)
1
I
1
I
"rt
Fig. la.
Equilibrium with r~ > 0.
kL+t
(13~)
k~+l
¢(k,,T,,r,+l)
Fig. lb.
Equilibrium with ~-t = 0.
3.1. UNDERDEVELOPMENTEQUILIBRIA
An underdevelopment equilibrium (UDE) is characterized by "rt = 0 for all t _> 0.
In this equilibrium (13a) holds as an inequality for all t >_ 0. The paths of physical
and human capital accumulation are given by:
kt+l
fl(1 - fl)
= ¢+(~+2)fl
• kt~
(14a)
INTERACTIONOF HUMANAND PHYSICALCAPITAL
101
kt+t
-I
kO
Fig. 2.
k~
Dynamics of the capital stock in the UDE.
x t = xo.
(14b)
Equation (14a) is derived from (13b) after setting rt = rt+l = 0. It defines an
upward sloping, concave locus in (kt, k t + l ) space. The unique fixed point of this
locus is denoted by k which represents the steady state level of physical capital in
a UDE (see Figure 2).
The analytic expression for k is
=
/3(1 - / 3 )
2/3 + ¢(1 -+-/3)'
(15)
From any given value of k0, there is monotonic convergence to k. The level of
per-capita income may be higher or lower at this steady state, depending upon the
starting value of human capital, but there is no growth in per-capita income once
is reached.
The defining feature of the UDE is that (13a) holds as an inequality at all points
of time. In the steady state of the UDE, (13a) reduces to
/3
>
~l--fl
--
which holds for any xo _< g, where g is a critical value such that the above barely
holds as an equality. Thus, as we shall prove below, any economy that starts off
with xo _< g and k0 _< k remains in the UDE.
102
JOB GRACA ET AL.
k*
(16a)
L
I
l
l
(]~b)
T*
Fig. 3.
The existence of a balanced growth path.
3.2. SUSTAINED GROWTH EQUILIBRIA
A sustained growth equilibrium (SGE) is characterized by (13a) holding as an
equality and by the fact that 7-t > 0 for all t. Equations (13a), (13b) and (7) jointly
determine the evolution of the variables r~, kt+l and Xt+l.
The long-run behaviour of SGE is characterized by a balanced growth path
associated with steady-state values of kt and 7-t (denoted by k* and 7-* respectively)
and a constant growth rate for xt (denoted by v).
Along the balanced growth path the equations characterizing equilibrium reduce
to
9
(16a)
(1¢*) 1-/3 -- ~*
(k*) 1-~
3
1 [
(1 - 3)(1 - 7-*)
[
~b [(2 - 7-*)3 + (1 - 3)](1 + 7*r*)
(2-7-*)/~+(1-/3)J
x +l = (1 + 7*7-*)xt
(16b)
(16c)
where 7" is the upper bound on 7.
Equations (16a) and (16b) jointly determine k* and 7-*. Equation (16a) determines k* independently of 7-*. Equation (16b) describes a locus in (7-*, k*) space.
The LHS of (16b) is increasing in k* while the RHS is decreasing in both 7-* and
k*. Thus, (16b) describes a downward-sloping locus in (7-*, k*) space. Figure 3
illustrates the balanced growth equilibrium, where both (16a) and (16b) hold.
Figure 3 shows that k* < k, i.e. capital intensity is lower in the SGE than in
the UDE. This is because the solution for k* from (16b), at the point where 7-*
equals zero, must equal the solution for k from (14a). Thus, for the case 7-* > 0, the
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
103
equals zero, must equal the solution for k from (14a). Thus, for the case r* > 0, the
solution k* must be smaller than k. This of course does not imply that per-capita
stocks of physical capital are lower in the SGE than in the UDE. The reason that
the ratio of per-capita physical capital to per-capita human capital is lower in the
SGE has to do with the fact that human capital is much higher, not that physical
capital is lower, in the SGE than in the UDE.
The above analysis of long-run equilibria in a SGE is identical to that in Azariadis and Drazen (1990). However, the dynamics of SGE in our model may differ
from those in the latter paper. Azariadis and Drazen show that the balanced growth
path is locally saddle-point stable. This need not be the case in our model, where
3`(.) is an increasing function of kt. If the elasticity of 3`(.) to changes in kt is large
enough, the balanced growth path in our model may display oscillations (this is
analysed in the next section).
3.3. TAKE-OFFEQUILIBRIA
An equilibrium in which an economy experiences a transition from the growth path
associated with an underdevelopment equilibrium (UDE) to a sustained growth
equilibrium (SGE) is defined to be a take-off equilibrium (TOE).
A TOE is characterized by (14a), (14b) for t E [0, T - 1]; and by (13a) holding as
an equality, along with (13b) and (7) for t E [T, oo), Thus, rt = 0 for t E [0, T - 1]
and rt > 0 at time T. The jump in the value o f t takes place because 3'(.) increases
sufficiently by t = T to induce young members of generation t to start investing
in education. In the following section, we shall characterize the set of equilibria,
and show how they arise from the initial combinations of physical and human
capital.
4. Characterization of equilibrium
We shall now prove that for the economy described in Section 2 there is a threshold
3`t (denoted by "~) such that if 3`t < "Yfor all t, then the economy remains in a UDE
for all t. We shall also prove that if 3`t exceeds -~ at some time t, then this economy
must be either in a TOE or a SGE; If it is in the TOE, it may begin to accumulate
human capital before % crosses the threshold "~.
LEMMA 1. For any t with kt+l given by q3(kt, O, O) and for any kt, (13b) implies
that
/3kt~ -- A( a constant) = ~b(1 + [3) +2/3
kt+l
1Proof. with "rt = "rt+l = 0, (13b) becomes
¢~kt~+l1 = V5
(1 -
(1 +/3)kt~+l
e - 2kt+l"
104
JOB GRACA ET AL.
Multiplying both sides of the above equation by ]¢~t/k~tq_l , setting ~]¢~t/kt+l = A
and inverting both sides gives the following equation
.
A
1 +/3)¢
(1)
(1 + / 3 ) ¢
which after straightforward manipulation leads to
A = ¢(1 + f l ) + 2 f l
1-¢~
[]
PROPOSITION 1. Let ~/ = A. Then 7t <_ @for all t is a necessary and sufficient
condition for a UDE.
Proof. (Sufficiency) Let 7t <_ "Y - A for all t. Assume initially that ~-t = 0 for
all t so that xt = xo. Then, from Lemma 1, the LHS of (13a) is equal to A along the
entire path followed by the variable kt as it converges to k. If A _> 7(x0, kt) over
the entire path for kt, then our initial assumption that rt = 0 for all t is consistent
with the equilibrium.
(Necessity). Suppose that ~/t > @ at some time t. Then, Tt = 0 cannot be an
equilibrium irrespective of future values of T. To see this, first consider the case
where Tt+l = 0. If "rt = 0 as well, then, from Lemma 1, the LHS of (13a) equals
"~. But the RHS equals 7t > @by assumption. Equation (13a) is violated, implying
that rt = 0 cannot be an equilibrium.
Now consider the case ~'t+l > 0. IfT-t = 0 as well, then since kt+l is an increasing function of Tt+l, the LHS of (13a) will be even lower than in the case where
rt+l = 0. Thus, the LHS of (13a) is less than @ while the RHS equals 7t >_ 7,
implying that (13a) will be violated even more strongly. Therefore, regardless of
the value of "rt+l, ~'t = 0 cannot be an equilibrium when 7t > @.
[]
The intuition behind the necessity proof is as follows. When 7t > @, the lefthand side of (13a) is too small when evaluated at a point where "rt = 0. Equality in
(13a) is achieved by increasing Tt, which given (13b) implies a fall in kt+l. This
causes an increase in the left-hand side of (13a) leading to the required equality.
We next look at the conditions under which an economy that starts off with no
growth in human capital takes off at some future date. Proposition 1 has already
established that a precondition for take-off is that 7t > "~ at some date t. As argued
above, such a situation requires that "rt > 0 in order for (13a) to be-satisfied.
However, a take-off can occur even before 7t crosses the critical value @.
To see this, consider an economy that is growing in physical capital but not in
human capital. Along this growth path, 7t rises due to physical capital accumulation.
Suppose that if the economy remains on this path, 7t crosses the critical value for
the first time at some date s', i.e. %, > @. Then, "rs, > 0. However, this will raise
savings at date s' - 1, and therefore ks, will rise (as implied by the ¢(.) function
above). This will imply that the LHS of (13a) will be less than A, its value in a
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
105
UDE. In that case, if %,_ 1 was very close to ~, then the inequality required by
(13a) may be reversed, requiring T~,-1 > 0.
Thus, human capital may begin to accumulate even before "/t crosses ~. The
following Lemma allows us to establish another critical value of "7 below which
an anticipated future increase in T does not induce current investment in human
capital.
LEMMA 2. For any t, with kt+l given by ¢(kt, O, 1) and for any kt, (13b) implies
that
t3k3t -- A' (a constant) =_ ¢ +13
kt+a
1 - fl
Proof. (Follows the steps outlined in the proof of Lemma 1, with rt = O, "rt+1
1).
=
[]
Note that A' < A for any positive/3. Let '} - A'; therefore "~ < @. While
7t _< ") implies that rt = 0 if rt+l = 0, 7t <_ £/implies that ~-t = 0 even if'rt+l is
positive; in particular, even if Tt+l = 1.
At one extreme, consider an economy in which 7t _< £/at date t. The discussion
following Lemma 2 implies that no growth in human capital will take place at
date t, regardless of the value of ~-t+l. At the other extreme, if 7t > 5', then as
argued in the proof of Proposition 1, *t > 0, again regardless of "rt+l. However,
if ~ _> 7t > ~, then the value of 7t depends upon the value of Tt+I. Two cases
may be distinguished. If-yt _< @ over all future periods, then current ~-t will remain
zero since all future values of r will also be zero. However, if at some future date
s', %, > ~, then future values of Tt will not remain zero. This may feed back to
cause Tt+l and consequently rt to turn positive. This result is summarized in the
following Proposition.
PROPOSITION 2. Given s' > s > O, if vt <_ 5"/for t 6 [0, s), £¢ < Vt <_ ~ for
t E [s, s') and ~/t > ~/at t = s', then there is some date T E [s, s'] such that "it = 0
for t < T, and 7T > O.
[]
4.1. INITIAL CONDITIONS AND EQUILIBRIUM PATHS
We now turn to characterizing UDE and TOE on the basis of initial values of the
state variables z0 and k0.
Define ~?t - x (-~, kt) as the level of xt that satisfies the equation -y (x (~/, kt ), kt) =
@. The properties of 7(.) imply that, for given @, x(.) is a downward sloping and
convex function of kt.
Lemmas 3 and 4 establish regions for (x0, k0) that lead to UDE, while Lemmas
5 and 6 establish regions for (x0, k0) that lead to TOE.
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JOB GRACA ET AL.
LEMMA 3. Let ~ = x('~, k) where ~/ is the critical value and-k is the steady state
level o f physical capital in the UDE. Then, for any ko <_ -k, xo < ~ implies that the
stock of physical capital increases towards -k and the economy converges to (xo, k)
in a UDE.
Proof Since 7(~, k) = "~ and xo _< ~, k0 < k, 7(xo, ko) < "~. Thus, the economy grows towards the steady state level k of physical capital with no growth in
human capital. As kt rises do does 7, but since x0 < 5, 7(xo, k) < "~. ?(.) never
crosses the threshold level required for growth in human capital to take place, and
the economy remains in the UDE.
[]
LEMMA 4. For any ko > -k, there exists a critical value :~o - z('~, ko), such that
xo < xo implies that the stock of physical capital falls towards k and the economy
converges to (x0, k) in a UDE.
Proof Since xo _< xo, 7(xo, ko) _< ~. a s kt falls with no change in xt, 7t also
falls. Thus, the inequality is strengthened and ~/t _< "~ for all t along this path. []
LEMMA 5. For any ko < k, if xo lies in the interval (~, :~o], then 7(.) will cross
the critical value zy at some finite date t.
Proof Note that since x(~, ko) is a downward sloping function of ko, ko <
implies that ko >_ ~. Since x0 < 5:o, 7(x0, ko) < "~. However, a s kt grows towards
k, 7t will rise. Even with no change in xt along the way, xo > ~ would imply that
7(xo, k) > "Y. If xt begins to grow along the way, the above inequality would be
even stronger. Thus, at some point, 7t has to cross the critical value -~.
[]
For the following lemma, define ~:t - x(,~, kt), where "~ is as defined in the
previous section.
LEMMA 6. For any ko < -k, if ~:o > ~ and xo lies in the interval (~, 5:o], then: (i)
7(.) will cross the critical value ;y at some future date t, and (ii) the economy is in
a TOE, i.e. shows zero growth in human capital initially, butpositive growth at a
future date.
Proof (i) Since "~ > ~, then k0 > x0 and the fact that xo lies in the interval
(~, 5:o] implies that it also lies in the interval (~, ko]. Hence, from Lemma 5, 7t
will cross ~ at some future date. (ii) xo < xo implies that 7(xo, ko) < "~. Thus,
TO = 0, and there is no growth in xt initially. The growth path from (x0, ko) satisfies the conditions for Proposition 2. Thus the economy experiences a 'take-off'. []
Finally, note that if x0 > ~0 the economy will experience growth in human
capital from the very start. This is similar to the threshold property required for
growth in Azariadis and Drazen (1990). However, in this case the threshold level
is a decreasing function of the initial capital stock.
Figure 4 shows how the space for initial conditions (xo, ko) may be divided up
into regions for UDE, TOE and SGE respectively. 15
107
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
x
SGE
\
Fig. 4. Initial conditions for the three types of equilibria.
7
Fig, 5a. Dynamics.
4.2. DYNAMICS
Figure 5a shows the dynamics of these equilibria, while Figure 5b compares equilibria in our model with those in Azariadis and Drazen (1990).
Any path that starts off in a UDE converges monotonically along a horizontal
ray towards k. A path that starts off in the TOE region grows horizontally until it
crosses ~; after this, it may enter the transition to a SGE at any point until it hits
108
JOB GRACA ET AL.
I
I
I
Fig. 5b.
Dynamics in the Azariadis-Drazen model.
5~. A path that starts off in the SGE follows an upward pointing arrow, indicating
growth in human capital, and converges towards the steady state level k* of physical
capital.
This convergence will be locally monotonic if e'~k' i.e. the elasticity of 7t with
respect to kt in a neighbourhood of 7", is small. However, if this is not the case,
oscillations may result. This is captured by Proposition 3 below (the proof of which
is in the Appendix).
Let "Yk = 07t/Okt and ¢7k =- ")'k • kt/7t. Let 7~ --- 7k evaluated at 7" and
k, = k* respectively and ¢7k = 7~(k*/7").
PROPOSITION 3. If e~k <-/3 (resp. >/3), SGE growth paths display monotonic
convergence (resp. display oscillations) in the neighbourhood of the steady state
Note that/3 represents capital's share in total output. Empirical estimates of this
parameter tend to be around 0.2 to 0.4. Thus, locally monotonic convergence of
SGE growth paths requires that the proportional response of 7(.) to changes in
physical capital be less than about 20%. Offhand, this appears to be a plausible
restriction.
One possibility that our analysis has not accounted for is that growth paths in
the SGE region may hit the boundary of the UDE region and lead to stagnation
in human capital. This is especially true for paths that begin to the right of "~ but
below ~. Such paths lead to declining levels of physical capital, evan as the level
of human capital rises. The effect of this combination on 7(.) is ambiguous. It is
possible that 7(.) may fall below ~ along such paths. The assumption that 7~, the
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
109
3;
SGE
5: ~
~
slope= -w/v
I
"-L"<.
Fig. 6. The state space for the Section 4.3 example.
sensitivity of 7(.) to kt, is small may rule this out. Note that this is similar to the
condition required for locally monotonic convergence of SGE growth paths.
4.3. AN EXAMPLE OF 7 ( x t , kt)
In order to illustrate the results of the above sections, we shall assume that 7(x, k)
follows
7(x, k) = 7* - b. exp -(vx+~°k)
where b, v and w are all non-negative constants. It is easy to verify that 7 :
R+ x R+ ~ [7* - b, 7*] is everywhere continuous and differentiable, increasing
at a decreasing rate in each argument and that it is weakly concave throughout.
Note that b = 0 implies no externality from human or physical capital to 7(.), while
v = 0 (resp. w = 0) implies that the externality depends upon physical capital
(resp. human capital) alone.
For a fixed value of7t = 7, the equation for x ( 7 , kt) becomes
xt =
ln(b) - In(-/* - 7)
72
w
- - - " kt
1)
(17)
which defines level curves in (x, k) space for given values of 7(.)- Figure 6
illustrates these curves and shows how the regions for UDE, TOE and SGE may
be determined for this example.
The level curves associated with the 7(.) function are straight lines. These level
curves get flatter as w / v , which measures the external effect from physical capital
110
JOB GRACAET AL.
SGE
/
slo,. . . . . .
/v
TOE
Fig. 7.
The effect of steep level curves upon the state space.
relative to that from human capital, gets smaller. As w approaches zero, the level
curves become horizontal as in Azariadis and Drazen (1991].
At the other end, as w approaches infinity, the level curves become vertical.
Given k, the steady state level of physical capital in a UDE, there is a critical value
of w such that UDE get eliminated altogether. 16 This means that starting from
a n y positive x0, an economy will eventually increase 7(.) sufficiently to begin
accumulating human capital. This critical value is derived from (17) by equating
the RHS to zero at O't = "~ and kt = -k:
ln(b)
W c
=
-
-
k
If w _> we, the level curve associated with "~ intersects the horizontal axis to the
left of k (see Figure 7). Thus, all growth paths hit 5' before they hit k.
5. Comparative statics
5.1.
INTERNATIONAL CAPITAL MOBILITY
The preceding analysis was confined to closed economies. However, a simple
extension to the open economy framework may proceed as follows. Consider a
small economy, hitherto closed to trade with the world, that is growing along a
horizontal line in the TOE region of Figure 4a. Opening this economy up to trade in
a world dominated by 'developed' countries, i.e. countries that are growing along
a SGE with physical capital equal to k* will prevent this economy from increasing
its capital beyond k* (note that the return on capital is greater at k* than at k).
I N T E R A C T I O N OF H U M A N A N D PHYSICAL CAPITAL
111
X
X*
TOE
to UDE~
I
I
I
k*
Fig. 8.
The effect of capital mobility upon the state space.
If this economy cannot attain the critical value ~ before it hits K*, it will remain
in a UDE forever. In other words, the scope for an economy remaining in a UDE
will increase at the expense of the scope for an economy to experience a take-off.
This is because, given two steady states k and k*, with k > k*, the boundary
between the UDE and TOE regions shifts up from ~ to x* -- x(5/, k*), so that any
economy starting off with x0 < x* and k0 < k* will never raise "Yt sufficiently to
induce self-sustaining growth.
This is shown in Figure 8. The shaded region in that diagram represents a set of
initial conditions that belong to the TOE region in the case of a closed economy,
but switch to the UDE region in the presence of capital mobility.
The international mobility of physical capital does not imply the equalization
of growth rates of per-capita output in our model, unlike in Rebelo (1992). Returns
to capital do equalize, but not per-capita incomes, either in levels or in growth
rates. 17
The reason is that 'residents' of SGE economies experience growth in per-capita
savings, and therefore per-capita stocks of physical capital, at a rate equal to the
growth rate of their individual human capital. 'Residents' of UDE economies end
up with stagnant levels of individual human capital; thus, their per-capita savings
and stocks of physical capital also stop growing. Although interest rates equalize
along with the intensive measure of physical capital, per-capita quantities remain
divergent.
Therefore, the international mobility of capital can have adverse implications
for the distribution of incomes across countries. Of course, if both labour and
112
JOB GRACA ET AL.
1: TOE to UDE
III: SGE to TOE
/¢¢
Fig. 9. The effectof savingsbehaviouruponthe state space.
physical capital were internationally mobile, human capital growth rates could
equalize, allowing convergence of other per capita variables as well. 18
5.2.
SAVINGS BEHAVIOUR
Returning to the closed economy framework, consider the effect of a decrease in
savings propensities. In the context of the example of Section 4, this can be achieved
through assuming that the relative preference for young-age consumption, given
by ¢ ~ oq/o~2 increases.
The steady state level of capital in the UDE, given by (15), is repeated below.
Clearly, an increase in ¢ causes k to fall.
=
/3(1 - / 3 )
2/3 + ¢(1 +/3)"
Further, given the proof of Lernma 1 and Proposition 1, the critical value "~
depends positively upon ¢:
~ = ¢(1 +/3) + 2/3
1-/3
An increase in the value of ¢ causes the set of initial conditions that lead to UDE
to expand at the expense of both TOE and SGE, while initial conditions that lead
to TOE expand at the expense of SGE. Thus, the likelihood of UDE increases, of
SGE decreases while that of TOE may increase or decrease. All in all, the change
reduces the set of growth equilibria.
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
113
These shifts are depicted in Figure 9. k~ and "~' denote the new values of
and ~ respectively, and ~ denotes the new boundary between the TOE and UDE
regions.
The implication is that two economies that differ in savings propensities can
grow differently even if they start with the same initial conditions.
6. Conclusions
The model presented in this paper shows the implications of allowing the technology for human capital accumulation to depend upon the level of physical capital.
Long-run growth takes place through constant returns to factors that can be accumulated. These factors are human and physical capital, and the production technology
displays constant returns jointly to these two factors.
The engine of growth in our model is human capital accumulation. Allowing
an interaction of human and physical capital at the level of the human capital
technology has allowed us to characterize a richer set of equilibria than in previous
papers. Thus, not only do we have the two long-run equilibria of 'poverty trap'
models, but we also have the possibility of some economies escaping what appears
to be a poverty trap through an endogenous mechanism.
The dependence of the human capital technology on physical capital also reemphasizes the role of physical capital, at least for driving the early stages of
growth. In our model, unlike other 'poverty trap' models, an economy with sufficiently high physical capital may escape the poverty trap, even if its initial human
capital is low.
This has both positive and normative implications: the positive implications
are reflected in the take-off equilibria, where a period of simple physical capital
accumulation precedes a take off into sustained growth. The normative implications
are that exogenous increases in the physical capital stock (perhaps through foreign
aid) may have the indirect effect of raising private incentives for human capital
accumulation, thus allowing an otherwise stagnating economy to cross the threshold
for sustained growth.
These considerations may provide s o m e justification for capital controls in
developing countries. Because of the external nature of the effect that physical
capital has on human capital accumulation in our model, private decision makers
will underinvest in physical capital. In the presence of capital mobility, this underinvestment may result in underdevelopment equilibria in countries that may have
grown if physical capital were immobile across countries.
Acknowledgements
The authors would like to thank S. Lahiri, seminar participants at the Federal
Reserve Bank of Cleveland and participants at the ESRC Development Study
JOB GRACAET AL.
114
Group Conference (University of Leicester, March 1994), especially J. Knight and
P. Levine, for comments. All errors are our own.
Appendix: Proof of Proposition 3
In this Appendix, we shall present the analysis of local dynamics around the SGE
steady state (k*, 7.*).
The first-order conditions (13a) and (13b) are totally differentiated, with all
relationships evaluated at the steady state values, k*, 7"* and 7*. This yields the
following system:
]
[dkt
Idkt+l
A1
dT.t+l ] = [ B1 0B2 j LdT.t j
]
(A.1)
where dkt, dT.t, etc. represent small deviations from the steady state values. At the
steady state, (13a) and (13b) imply:
/3(k*) ¢~-1 = 7*
(a.2a)
(
1
2-7.*+¢,1+~-7.*.
]
=
(1 -/3)(1 - 7.*)(k*)~-I
1+7"7.*
(A.2b)
Taking account of (A.2a) and (A.2b), the coefficients A1 and B2 may be written
as
A1 ---- / 3 7 " - ' 7 1 " k *
[
127'
]
B2 = (1 -/3)(k*) ~-l (1 + ¢ ) ( 1 +7*7.*)23 .
The coefficient B1 is not reported because it does not affect the local dynamics
of the system. Because the system (A. 1) is block-diagonal, it has two, distinct real
roots, given by
/~1 = A1
A2 = B2.
Monotonicity requires both roots to be positive. If this fails, then oscillations
occur, although the system could remain saddlepoint stable, i.e. converge to the
balanced growth equilibrium along a unique path. Since we have one predetermined
variable (k) and one jump variable (7.), saddlepoint stability requires the magnitude
of one root to be larger and the other to be smaller than unity. The unstable root
implies (given k0) a unique choice of 7.o such that explosive paths are ruled out.
LEMMA A. 1: .~2 ---~B2 _~ 1.
Proof. t32 may be multiplied and divided by (1 - T*). Therefore,
B2 = (1 - 7 . * ) ( 1 - / 3 ) ( k * )
~-1
( 1 - + ¢ ) ( 1 + 7"7.*)2(1 - 7.*)
'
115
INTERACTIONOF HUMANAND PHYSICALCAPITAL
B2 may be rewritten as:
B2
(1 - 7-*)(1 -/3)(k*) fl-1
(1 + 7*7-*)
•
1 +7"
(1 + %b)(1 + 7*r*)(1 - 7-*)"
Using (A.2b), and rearranging, B2 becomes
2 + ~b(1 + / 3 - 1 )
B2 =
_ (1 q- ~/3)7-*
(a +
1 +7"
1+'7"7-*
1
1-7-*
The above expression consists of three terms, each of which may be shown to be
no less than unity. The second and third terms obviously satisfy this, given that
0 < ~-* < 1. The first term may be broken into
2 + ~(1 +/3-1) _ 7-*.
(1 +%b)
Since 0 <_ 7-* _< 1, the above expression will be no less than unity if
2 + ~'b(1 +/3-1) > 2
(1
-
.'. 2+%b(1+/3 - 1 ) _ > 2 + 2 ~ b
.'. 1 + / 3 -1 > 2
which holds since/3 < 1.
Thus, A2 = B2 _> 1.
[]
From the derivation of A1 above, it is a straightforward matter to see that
A1 = A~ _> (resp. _<)0 as/3 > (resp. <_)'y~(k*/7*). Proposition 3 follows by
denoting e.rk - 7k (k /7 ).
Thus, the second root is positive (resp. negative) if the elasticity of 7(.) with
respect to k, evaluated at the steady state value of 7(.) is not too large (resp.
large enough). Finally, note that if A1 >_ 0, then its magnitude is no greater than
unity, since/3 < 1. This implies the saddlepoint stability of the balanced growth
equilibrium.
Notes
1 Some economists, notably Haavelmo (1954), Schultz (1960), Arrow (1962) and Uzawa (1965)
recognized the importance of human capital for economic growth. However, their insights exerted
little influence on the vast body of literature that grew around the Solow (1956) growth model.
2 Human capital as a factor of production has been used both to endogenize technical change
(Romer, 1990) and to introduce nondecreasing returns overall without imputing them to a single
reproducible factor (see Sala-i-Martin, 1990).
3 Becker et al. (1990) derive the rather paradoxical result that an economy starting off with low
human capital and low physical capital is more likely to grow than one with low human capital
and high physical capital. This is because they assume that the level of physical capital has a
116
JOB GRACAET AL.
negative impact upon the incentives to invest in human capital. Our paper presents an alternative
setup and reaches the opposite conclusion.
4 Thus, Barro (1992) finds that growth rates for the period 1960-1985 across a cross-section
of countries are positively related to 1960 measures of human capital and also to the ratio
of physical investment to GDP. A recent World Bank report [1993] also argues for positive
effects from schooling as well as physical capital formation upon the growth of the East Asian
economies.
5 See Denison (1962). Denison's method involved assuming a constant productivity premium for
educated versus uneducated workers, taking account only of changes in the level of educational
attainment by the workforce. Had the productivity premium increased between the first and
second periods, the increase in the relative contribution of educational composition would have
been even larger.
6 In a similar vein, it is claimed by some historians of the Industrial Revolution that, at least in
Great Britain, large-scale declines in literacy followed the onset of the factory system rather
than preceding it, implying that education was not an important factor in the early stages of the
'Revolution'. However, this remains a controversial matter (see West, 1985).
7 The specification for 3'(.) is the main point of departure of our model from other models of
human capital accumulation. For example, Lucas (1988) follows Uzawa (1965) in assuming that
3` is constant, while Azariadis and Drazen (1990) assume that it is an increasing function of the
average level of human capital alone. At the same time, Caballe and Santos (1993) and King
and Rebelo (1990) allow, as we do, physical capital to affect the technology for human capital.
However, they treat physical capital only as a private input into the educational sector, while we
consider only its external effects.
8 3' may be defined as
7" =
lim 7(xt,k*)
~t~oo
where k* is the steady state value of k~ in a sustained growth equilibrium.
9 Altematively, we could have assumed an externality from some other variable, such as output per
capita, to human capital accumulation. So long as this variable could grow during periods when
there is no increase in human capital, it would induce take-offs during subsequent periods.
10 Lucas (1993, p. 257) offers a similar justification for including physical capital in the human
capital equation.
11 Note the distinction between the use of computers by students in e.g. philosophy, and the
recognition of a particularly innovative application of computers that increases the productive
potential of students in, e.g. architecture. Thus, under our maintained hypothesis, educational
institutions in less advanced countries may not offer curricula at the 'frontier' of each field, not
just because of the private cost of offering such curricula, but also because they outpace the
technical level at which production is carried out.
12 In principle, ~-should also be constrained to be less than unity. However, the utility function of our
example satisfies Inada conditions for each period's consumption. Spending all the available time
in education would reduce young-age consumption to zero while leaving old-age consumption
positive. This consumption stream cannot be optimal under Inada conditions.
13 The explicit solution of (13b) is too messy and adds little insight. Hence, it is not reported
here.
14 Other possible equilibria are: (i) human capital grows over some initial interval of time but stops
growing later on as the economy enters an underdevelopment equilibrium; (ii) human capital
accumulation cycles between intervals of positive and zero accumulation. While acknowledging
such possibilities, we shall not analyse them in this paper.
15 Note that in Figure 3 the steady states k and k* could be determined as functions of the underlying
technology and taste parameters, k is derived in (15), while the solution for k* is too unwieldy
to report.
16 This outcome results from the fact that the level curves of 3`(.) touch both axes. If this were
not the case, then sufficiently low initial values of either type of capital would always lead to
UDE.
INTERACTION OF HUMAN AND PHYSICAL CAPITAL
117
17 Other models in which the openness of an economy may have adverse implications for its growth
rate are Stokey (1991a,b) and Young (1991). However, our result is more drastic in the sense
that capital mobility may affect the type of equilibrium in which an economy finds itself.
18 Alternatively, if the 'world' stock of physical capital rather than the domestic capital stock
exerted a positive externality on domestic education, growth rates might again equalize. However,
given the justifications outlined in Section 2 for the inclusion of this externality, geographical
proximity or at least unimpeded labour mobility would be a requirement for physical capital to
have beneficial effects on human capital. Thus, assuming that the externality 'stops at the border'
appears to be plausible.
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