A Work in Progress：Comments are welcome.

Productive Consumption and Population Dynamics
in an Endogenous Growth Model:
Why Are Population Growth Rates Declining in Poor Countries?†
Ichiroh Daitoh∗

Tohoku University

December 10, 2006
This paper shows that by endogenizing population growth rate an endogenous growthmodel under productive consumption hypothesis (PCH) can be more tractable than wehave considered so far, and finds that PCH may have important implications for
population dynamics. In contrast to Steger (2000), we focus on a BGP with a constantlevel of per-capita income. We find that the model may have a unique or multiple
saddle-point stable BGPs in both no- and positive-saving phases. In the no-saving phase
more relevant to poor economies, population growth rate may rise or decline
monotonically along a transition path. The theoretical results are realistically relevant;the recent trend of declining population growth rates in modern developing countries
could be explained. Furthermore, we find that “human development” aid promoting theaccumulation of knowledge about nutrition, health and/or education may reduce
per-capita GDP and does not always improve welfare.
JEL Classification Number: J11,O11,O41
Keywords: endogenous growth, population dynamics, productive consumption† This paper was presented at PET (Public Economic Theory) 2006 Hanoi Conference. I appreciate Professor Kazuo

Mino and Koichi Futagami for useful comments at a seminar in Osaka University. I appreciate Professor Michihiro
Ohyama (Toyo University), Ryuhei Wakasugi (Keio University), Masao Satake (Tohoku University) and Ken-ichi
Akao (Waseda University) for useful comments at a seminar of the Japan Society of International Economics. I thankProfessor Jungsoo Park (Sogang University) for useful discussion at the Winter Meeting of the Korea International
Economic Association in Hanyang University in Seoul.
∗

Department of International Economic Relations, Graduate School of International Cultural Studies,

Tohoku University; 41 Kawauchi, Aoba-ku, Sendai-city, Miyagi, 980-8576, Japan;+81-22-795-7595

Email: idaito@intcul.tohoku.ac.jp

1

TEL&FAX:1.Introduction
For the last decade “human development” has received considerable attention as animportant concept and strategy for poverty reduction on a world-wide basis. The UNDP
(United Nations Development Programme), for example, has published Human
Development Report every year since 1990. In the Millennium Development Goals, six
out of eight goals are concerned with the concept of “human development”, especiallywith an improvement in nutrition, health or education. The roles of international aid
including Official Development Assistance from developed to developing countries arehighlighted to attain these objectives.
In the traditional studies of development economics, on the other hand, therelationship between economic growth and human development (nutrition, health and/or
education) has been regarded as important for growth of developing economies.
Prominent economists have studied this idea as “productive- consumption” hypothesis(PCH): consumption improves productive potential of labor or enhances human capital(e.g.,Bliss and Stern (1978a,b), Gersovitz(1983), Dasgupta and Ray(1986), Ray andStreufert (1993)). Most of the previous papers have studied it in the form of static anddynamic efficiency wage hypothesis. An interesting exception must be Ray andStreufert (1993). They presented a dynamic analysis of the relation between landownership and involuntary unemployment by focusing on a nutritional effect on rural
workers’ labor productivity. However, their model was of high originality and thus did
not seem very tractable for growth economists. It is only recently that PCH wasintroduced in standard models of economic growth.1
PCH has been introduced into the neoclassical growth model with exogenous
1

Recently growth economics and health economics have begun to collaborate. See, e.g., Lopez-Casasnovas, Rivera
and Currais (2005).

2

population growth rate by Steger (2000, 2002) and Gupta (2003)). Steger (2002)
distinguishes two possibilities for modeling PCH. First, current consumption raises
labor productivity of workers. 2 Second, it enhances the stock of human capital(disembodied knowledge) that improves nutrition, health, public sanitation or education.
Using the latter setting, Steger (2000) shows, as a distinguished feature derived from
PCH, that the model has a zero-saving phase as well as a positive-saving phase. He
considers only a balanced growth path (BGP) with a constant growth rate of per-capita
income as the steady-state equilibrium. However, taking into account that PCH is more
relevant to developing economies, it should be no less important to examine propertiesof a BGP with a constant level of per-capita income. Gupta (2003) has derived this typeof BGP in his endogenous growth model in which productive consumption improves
labor productivity.3 At the same time, he has shown that this BGP is unstable. In a
simple AK model by Steger (2000), on the other hand, while a BGP with a constant
growth rate of per-capita income exists in the no-saving phase, it does not exist in thepositive-saving phase and, instead, only an asymptotic BGP exist. These results so far
do not fully make clear dynamic implications of PCH, e.g., how many BGPs may exist,
whether a BGP can be (saddle-point) stable, what properties a transition path may havein a phase diagram, etc.. The above properties of the model may also seem troublesome.
Growth and development economists thus might possibly have an impression that
introducing PCH into the standard growth model does not work very well and thus may
not be a productive task.

2

This kind of endogenous growth model has been analyzed by Steger (2002) and Gupta(2003).
In Gupta’s formulation, productive consumption ((1-λ)c) only improves labor productivity but does not affectutility function. This fails to capture an importance aspect of PCH: consumption improves both productivity andutility at the same time.

33

This paper shows that an endogenous growth model under PCH can be more tractableby endogenizing population growth rate and further explores dynamic implications of
PCH. In contrast to Steger (2000), we focus on a BGP with constant level of per-capita
income. We find that the model may have a unique or multiple BGPs that is
saddle-point stable in both no- and positive-saving phases. In the no-saving phase that ismore relevant to poor economies, population growth rate may rise or decline
monotonically along a transition path. Next, we explain based on data from Worlddevelopment Indicators 2004 that the theoretical results are realistically relevant; inparticular, the recent trend of declining population growth rates in modern developing
countries could be explained. Furthermore, we explore a role of foreign aid to adeveloping country from some developed countries or international institutions. We findthat “human development” aid promoting the accumulation of knowledge about
nutrition, health and/or education may reduce per-capita GDP and does not always
improve welfare.
In section 2 we present a basic model. Growth process in the positive-saving range is
examined in section 3 and section 4 investigates income growth and population
dynamics in the no-saving range. In section 5, we show the theoretical results are
consistent with data. Section 6 shows how “human development” aid may affect percapita income, population dynamics and welfare. Section 7 gives concluding remarks.

2. The Model
The aggregate production function Y = F ( K , L) exhibits constant returns to scale incapital K and labor L and satisfies the standard properties. Capital is composed ofphysical capital Kp and human capital Kh. In this paper Kh is intangible and disembodied4to labor: it can be interpreted as knowledge capital. Following Steger (2000), we assume
that physical and human capitals are perfect substitutes, that is, K=Kp+Kh.4 Labor input
is equal to population, which grows at a rate n(t) at a point in time t,
L (t ) / L(t ) = n(t )(1)

We rewrite the production function into an intensive form y = f (k ) with f ' (k ) > 0

and f " (k ) < 0 , where y=Y/L and k=K/L. The physical capital is accumulated by saving
part of income. Then kP =KP/L evolves according tokP (t ) = f (k (t )) − n(t )k P (t ) − c(t )(2)where c(t) is per capita consumption. On the other hand, under PCH, the human capital
Kh is accumulated by consumption activities. We suppose that per capita consumption
leads

to

human

capitalaccumulation

throughexchanging

andcreating

consumption-based information and new knowledge. For example, if people finddrinking water with salt and sugar stops dehydration of their children, they will
exchange this information in the society and/or create new knowledge about, e.g., howmuch salt and sugar should be put into water or how often they should give it to theirchildren. This leads to an increase in human capital at social level. By dividing thisincrement K h (t ) by the number of people L(t ) , we will get the human-capitalenhancement function φ (c) by Steger (2000). Therefore, K h (t ) = φ[c(t )]L(t ) holds.
Functionφ (c)is increasing and concave, i.e.,

lim c→0 φ ' (c) = ∞ and lim c→∞ φ ' (c) = 0 .

φ ' (c ) > 0 , φ ' ' ( c ) < 0

with

The per capita human capital kh =Kh/L

evolves according to

4

In Steger’s (2000) analysis the dynamic properties would remain unchanged if one treated these two capitals
separately.

5kh (t ) = φ[c(t )] − n(t )kh (t )(3)Therefore, the economy’s capital per capita evolves according tok(t ) = f (k (t )) − n(t )k (t ) − ψ (c(t ))(4)where ψ (c(t )) = c(t ) − φ (c(t )) is the net cost of consumption (NCC). Per capita
consumption cannot exceed per capita income:0 ≤ c(t ) ≤ f (k (t ))(5)

The representative agent maximizes the intertemporal utility functionn(t )1−ε − 1
∫0 ln c(t ) + 1 − ε  exp(− ρt )dt∞(6)

where ρ > 0 is a constant time preference rate and ε > 0 ( ε ≠ 1 ) represents theintertemporal elasticity of substitution.5 Following the standard practice of the literatureon endogenous fertility, we assume that the instantaneous utility depends on the
population growth rate n , as in Yip and Zhang (1997). One may claim that it should
depend on the number of children. However, this formulation could be justified asfollows. In Razin and Ben-Zion (1975) model in which each generation lives for one
period, the gross population growth rate ( Lt +1 / Lt ) may also be regarded as the percapita number of children of generation t. Thus in our model 1 + n could be interpreted
as the number of children. Furthermore, if we replace the gross population growth ratewith the net population growth rate n , the dynamic properties of the equilibrium willremain unchanged. Therefore we make use of this instantaneous utility function.
We assume away the cost for increasing population growth rate to reveal basic
properties of the system. In the literature on endogenous growth with endogenous
5

The representative agent here is an individual who is atomistic in the whole economy. Thus we do not incorporate
the population growth rate (n) in the exponential term.

6fertility, it is often assumed that the cost for increasing a fertility rate involves
child-rearing cost (using time or goods in the present period). Introduction ofchild-rearing cost induces a possibility of rising fertility rate along transition path (Barroand Sala-i-Martin (1995)). We will show that this model under PCH may induce a
possibility of rising population growth rate even if child-rearing cost is ruled out.
The representative individual chooses time paths of per capita consumption c(t) andpopulation growth rate n(t) to maximizes (6) subject to (4) and the inequality constraint(5).6 Following Leonard and Long (1992), the present-value Hamiltonian is defined asn1− ε − 1H (c, n, k , π , t ) = ln c +exp(− ρt ) + π [ f (k ) − nk −ψ (c)]1 − ε 
(7)

The Lagrangean function is L(c, n, k , t , π , λ ) = H (c, n, k , π , t ) + λ[ f (k ) − c] . The
first-order conditions (FOC) for this problem with inequality constraint are(i) c*(t) maximizes H (c, n, k , π , t )subject to f (k ) − c ≥ 0 .

∂L ∂H
1
=
− λ = e − ρt − πψ ' (c) − λ = 0
∂c ∂c
c

(8-1)

with λ ≥ 0 , f (k ) − c ≥ 0 , λ[ f (k ) − c] = 0(ii) n*(t) maximizes H (c, n, k , π , t )∂L= e− ρt n −ε − πk = 0∂n(iii) π (t ) = −

(8-2)

∂L
= − [π (t ) + λ (t )] f ' (k (t )) + π (t )n(t )∂k∂L＝ f (k (t )) − n(t )k (t ) − ψ (c(t ))(iv) k(t ) =
∂π

(8-3)
(8-4)

As in Steger (2000), this model has two phases: no-saving ( c = f (k ) ) and6 One could distinguish the equilibrium and optimal growth paths by assuming that an individual does not take into
account the effect of φ ( c ) . However, since the no-saving phase would not occur on the equilibrium path, we will
focus on the optimal path along which each individual recognize this effect correctly.

7

positive-saving ( c < f (k ) ) phases.7 If the marginal NCC (ψ ' (c) = 1 − φ ' (c) ) is positive,
renunciation of current consumption will promote capital accumulation. Thus saving
will be positive. Conversely, if it is negative, an increase in current consumption will
promote capital accumulation. Thus there is no incentive for saving. In the no-savingphase where c = f (k ) lies in [0, cz ] , the economy moves along the production functionf (k ) . When the value of k exceeds the critical value kZ with φ'( f (k z )) = 1 , theeconomy switches into the positive-saving phase, in which per capita consumption
diverges from f (k ) . We will first consider the positive-saving phase and then proceed
to the no-saving phase.

Figure 1. Human-capital Enhancement Function

φ
φ (c)

Slope=1０ccz

3. Growth Process with Physical Capital Accumulation
In this section we consider the positive-saving phase (λ(t)＝０) in which physicalcapital accumulation takes place. Per capita consumption and population growth rate
7

Steger (2000) discusses transition and asymptotic ranges separately for the positive-saving phase.

8evolve according toc(t ) =c(t )[ f ' (k (t )) − n(t ) − ρ ]1 + η (c(t ))(9)n(t ) f (k (t )) ψ (c(t )) f ' (k (t )) − ρ −+ε k (t )k (t ) (10)n (t ) =where η (c) = cψ " (c) /ψ ' (c) . Since population growth rate is endogenous, (9) is a little
different from the Modified Keynes-Ramsey Rule shown by Steger (2000).
The dynamic system is given by (4), (9) and (10). The steady-state equilibrium(c * , k * , n* ) is defined as the balanced growth path (BGP) in which c = k = n =0 holds.
As shown in Appendix, there may exist at most two BGPs. We can examine thestability of BGP and transition dynamics: how population growth rate may be related toper capita income along the transition path. Although the transition path is in the(c, n, k ) space, we will examine the correlation between population growth rate and per
capita income by taking a projection onto the (n, k ) plain. Setting c =0, we obtainn (t ) 1 f (k (t )) ψ (c(t )) = n(t ) −+n(t ) ε k (t )k (t ) (11)

Thus we can immediately obtain n / n = −(1 / ε )(k / k ) . From an arbitrary initial valuek (0) < k * , population growth rate will decline as per capita income grows along the
projection of the transition path on the (n,k) plain.

Proposition 1 (BGPs in Positive-saving Phase)In the positive-saving phase, (i) there may exist at most two BGPs. (ii) A BGP can beeither saddle-point stable or unstable. (iii) Along the transition path, population growthrate will decline as per capita income increases.

9Let us mention that the average saving rate rise during this growth process. Steger(2000) claims that the average saving rate needs to rise as a least requisite for thegrowth model, showing it by a simulation of AK model. If we assume the AK- typeproduction function y = Ak , we can show it analytically.
In Steger (2000), population growth rate is given, Setting

f ' (k ) − ρ = [ f (k ) −ψ (k )] / k . Using it, we getn = 0yieldsc / c = [1 /{1 + η (c)}](k / k ) . A change inaverage propensity to consume isd (c / y ) / dt − η (c)  k  =(c / y )1 + η (c)  k 

(12)

When k / k > 0 , the average saving rate (c / y ) declines as per capital income
increases. Therefore the average saving rate ( 1 − (c / y ) ) rises as per capital income
increases.

Result 1 (Saving Rate along Transition Path in Positive-saving Phase)

Under the AK production function, the average saving rate ( 1 − (c / y ) ) rises as per
capital income increases along the transition path.

4. Population Dynamics with Human Capital Accumulation
4.1 Equilibrium Conditions
Now let us move on to the no-saving phase ( λ (t ) > 0 ) in which only human capital isaccumulated through productive consumption. The FOCs are
1
∂L ∂H
=
− λ (t ) = e− ρt − πψ ' (c) − λ = 0
∂c
∂c
c

10

(13-1)

 1  − ρt
 ε e = π kn (13-2)π (t ) = − [π (t ) + λ (t )] f ' (k (t )) + π (t )n(t )(13-3)k(t ) ＝ φ ( f (k (t ))) − n(t )k (t )Since

∂H / ∂c = λ (t ) > 0

leads to(14)
c = f (k ) , (8-4) takes the form of (14).

Differentiating (13-2) with respect to time yields ε (n / n) + k / k + π / π + ρ = 0 . We
transform (13-3) by eliminating λ (t ) using (13-1) and (13-2)


π
nε
= n − f ' (k ) 1 +−ψ ' ( f (k ))π { f (k ) / k}

(15)

Substituting and rearranging the terms, we obtainn (t ) = [n(t ) / ε ]Γ(k (t ), n(t ))

(16)

where

 φ ( f (k ))nεΓ(k , n) = f ' (k ) φ ' ( f (k )) +−−ρ
{ f (k ) / k} kThe dynamic system for this phase is given by (14) and (16).

The steady-state

equilibrium (k * , n * ) is defined as the BGP on which k(t ) ＝ n (t ) = 0 holds. It ischaracterized by

φ[ f (k * )] = n*k *

(17)


n*ε 
*
= b( k * ) + ρf '(k ) φ '[ f (k )] +* a(k ) 
(18)*

where a(k ) = f (k ) / k and b(k ) = φ[ f (k )] / k .
Let us call a locus of (k,n) on which k = 0 holds “kk curve”. First, taking into
account that (17) leads to φ ( f (k )) / k = n and that φ ( f (k )) / k is decreasing in k , the11slope of kk curve is always negative:dn φ ' ( f ) f ' (k ) − n
<0
=
dk
k

(19)

Second, a locus of (k,n) on which n = 0 holds is called “nn curve”. The slopeΓ ( k , n)
dn
=− k
dk
Γn (k , n)(20)

can be either positive or negative, whereΓn (k , n) =kf ' (k ) ε −1
εn >0
f (k )(21)nε a ' ( k )
nε 
2
−
f
'
(
k
)
− b' (k ) (22)Γk (k , n) = f ' ' (k ) φ '+
+
φ
'
'
(
f
(
k
))[
f
'
(
k
)]

a(k ) a(k ) a(k ) 

4.2 Balanced-growth Equilibrium and Stability: Case of Increasing nn Curve
We will first consider the case where nn curve is increasing. This case will happen when
the production function f (k ) and the human-capital enhancement function φ (c) arestrongly concave. To see this, we should look at (22). While the third and fourth termson the right-hand side of (22) are positive, Γ (k , n) < 0 holds when the sum of the firstkand second terms (negative) is dominant, that is,

f ' ' (k ) and φ ' ' (c) are large

enough.

Proposition 3 (A Unique Saddle-point Stable BGP in No-saving Phase)In the no-saving phase, there exists a unique BGP (k * , n* ) that is saddle-point stable if
and only if nn curve is increasing ( Γ (k , n) < 0 ).
k

(Proof) When nn curve is increasing, it intersects with kk curve at one point. Thus a

12

BGP uniquely exists. If the slope of kk curve is smaller than the slope of nn curve,

φ ' ( f ) f ' (k ) − nk<−Γk (k , n)holds. This is equivalent to Det J*< 0, where J* is theΓn (k , n)coefficient matrix of the linearlized system of (14) and (16), evaluated at a BGP. Det J*<
0 means that only one of the two eigen values of J* is negative. Thus the BGP is
saddle-point stable. (Q.E.D.)

Figure 2 shows transition dynamics for this case: population growth rate n is positively
related to per capita capital k . From a low initial level of k 0 , the population growthrate rises as per capita income y = f (k ) increases along the transition path (Anintuitive explanation will be given later).

Figure 2: Rising Population Growth Rate

n

k
n

E

n(0)

k
n

k

O

k0

13

4.3 Case of Decreasing nn Curve
Next we consider the case where nn curve is decreasing ( Γ (k , n) > 0 ). From whatkIhave just explained above, this case will happen when the production function and thehuman-capital enhancement function are weakly concave: f ' ' (k ) and φ ' ' (c) are
relatively small.
Since the slope of nn curve may be either larger or smaller than that of kk curve, they
may intersect with each other at more than one point. Thus there may exist multiple
BGPs. If nn curve is flatter than kk curve,

φ ' ( f ) f ' (k ) − nk<−Γk (k , n)holds. This isΓn (k , n)equivalent to DetJ*<0, implying that the BGP is saddle-point stable. Figure 3 shows thecase of a unique saddle-point stable BGP. Along a transition path, population growthrate will decline as per capita income grows.8

8

The BGP is unstable when Γk(k , n) > 0 and nn curve is steeper than kk curve. This case will happen whenf ' ' (k ) and φ ' ' (c) are very small.

14

Figure 3: Declining Population Growth Rate

n
n

k

E

n(0)

n
k

k

O

k0

Intuitively, the change in population growth rate along the transition path is linked to
the growth of per capita income by (8-2): The discounted present value of the marginalutility from an increase in population growth rate (1 / nε )e − ρt is equal to the imputed
value of capital πk . When the imputed price πe ρt declines at a rate higher than thegrowth rate of k , the population growth rate n will rise, and vice versa.
Let us now consider the case of multiple BGPs. First, consider the case where there
exist only two BGPs. Since they are saddle-point stable or unstable, only the stable one
is economically meaningful. Next, if there are more than two BGPs that are saddle-point
stable (Figure 4), optimal path will be indeterminate. This is because both BGPs satisfy
sufficient conditions for the intertemporal utility maximization and thus the maximized
utilities should be equal among them. In this case, since kk curve is downward-sloping,one BGP involves a high population growth rate and low per capita income, while the15other a low population growth rate and high per capita income. Expectations will matterin deciding which BGP will be realized. The transitional dynamics is the same
qualitatively both when BGP is unique and when it is multiple. From a low initial levelof k 0 , population growth rate n will decline as per capita income increases.

Figure 4. Multiple Saddle-point Stable BGPs

n

n
k
Ｅ１

Ｅ２
E3

n
O

k

k0

k

Proposition 4 (Multiple Saddle-point Stable BGPs in No-saving Range)

Suppose that nn curve is decreasing in the no-saving phase. Then (i) there may exist
multiple BGPs (k * , n * ) . (ii) A BGP is saddle-point stable if and only if the slope of kk
curve is smaller than the slope of nn curve. (iii) When there are more than one
saddle-point stable BGPs, the optimal path is indeterminate. (iv) Along a transition path,
population growth rate declines as per capita income grows.

16

4.4 Switching from No-saving to Positive-saving Phase
Finally, let us elucidate how the no-saving phase may switch to the positive-savingphase. When per capita income exceeds the critical value f (k z ) in the no-saving phase,the economy moves onto the saddle-point path toward a BGP for the positive-savingphase. In the case of increasing nn curve, population growth rate will rise at first (in the
no-saving phase) and, after the phase switches, will decline as per capita income
increases. In the case of decreasing nn curve, population growth rate will keep declining
as per capita income increases, regardless of the phase switching.

5. Relation to Empirical Evidence
We have shown how transition dynamics could explain the relationship between percapita income and population growth rate in poor developing economies. In general,
we cannot support the theory if its results contradict the facts observed in the realsociety. The consistency with empirical evidence does not mean that the theory actually
explains the facts. However, we need to examine, as a minimum requirement, whether
these theoretical results does not contradict empirical evidence. In this section, let us
check it using data from World Development Indicators (2004).
Before proceeding, the recent casual observations and empirical studies have often
shown that population growth rates have been declining not only in the world as awhole but also in developing countries (see e.g. Table 6.1 on p.105 in Tietenberg(2006)). A decline in population growth rates in some developing countries could be
explained by the theory of demographic transition: as nations develop, they eventuallyreach a point where birth rates fall. However, one should note, this applies to economies
that have succeeded in income growth in the last several decades (e.g., Mexico, Brazil,

17

Indonesia etc.).9 For relatively poor developing countries in South-East Asia, Latin
America or Africa, the income growth has not been so smooth that the theory of
demographic transition can apply. Thus it will be important to explore a possibility of adifferent explanation for the declining population growth rates that are widely observed
mainly in these areas. We can do it by focusing on the no-saving phase of the present
model.
First, we will look at the rather exceptional case for positive correlation between
per capita income and population growth rate (Figure 2). This positive correlation isconsistent with the data of Nepal on Table 1. Second, a negative correlation between
population growth rate and per capita GDP (Figure 3 and 4) is consistent with the dataof India and Columbia on Table 2 and 3. The negative correlation has recently been
observed very frequently in data of modern developing economies. However, looking
more carefully into the data of WDI, one can also find the data from African countries
such as Ghana and Sudan (on Table 4 and 5) that exhibit more complicated, or scattered,relations. These could also be consistent with our model, as explained in Appendix 3.

6. Human Development Aid
We will examine effects of “human development” aid by focusing on the role of thehuman-capital-enhancement function. Let us replace φ (c) with θφ (c) , where θ > 0 is
an exogenous parameter. Since we assume away the cost for a rise in θ , we could
interpret it as representing the effect of an introduction of aid from foreign countries orinternational institutions. Then the definition of BGP changes into

9

In our model phase switching could explain a growth driven by physical capital accumulation of economies suchas Mexico, Brazil or Indonesia, etc..

18θφ[ f (k * )] = n*k *(23)
n*ε  θφ ( f (k * ))*+ρf '(k ) θφ '[ f (k )] +=a (k * ) 
k*


(24)

*

6.1 Welfare on Balanced-growth Path
As a preparation, we will examine the welfare on BGPs, though welfare along transition
path can hardly be evaluated. Since c* = f (k * ) holds in the BGP, (6) leads to
U* =1(n* )1−ε − 1 * ln f (k ) +ρ
1− ε 

(25)

Welfare on BGP is higher when k * and n* are larger. When a new BGP is locatednortheast of the initial BGP, welfare on the new BGP is higher than that in the initial
BGP.

6.2 Comparative Statics and Dynamics
A rise in θ shifts up both kk and nn curves. A new BGP can be located either
northeast or northwest of the initial BGP. To see this, let us check how much a rise in θ
will shift kk and nn curves upward (how much n needs to rise with k fixed)
respectively, by using (23) and (24). From (23), one unit increase in θ raisesn by

φ ( f (k * )) / k * . From (24), it raises ( f '( k * ) / a (k * ))nε by less than φ ( f ( k * )) / k * . Sincef '(k * ) / a(k * ) is smaller than unity, n may have to rise by either more or less thanφ ( f (k * )) / k * . Therefore, a shift of kk curve, in general, may be either larger or smaller

than the shift of nn curve. I will discuss separately the implications of these two cases,by focusing on the more frequently observed case when population growth rate declines
as per capita income increases ( nn curve is downward sloping).

19

First, consider the case in which E’ lies northeast of E (Figure 5). The economy
moves from the initial point to E. When the aid comes in, it jumps up to F and moves
along a transition path to E’. In this process, per capita income keeps rising andpopulation growth rate continues to decline ( n jumps up to F at the time when θrises). Since the BGP values of k * and n* on E’ are both higher that those on E, thehuman development aid will improve welfare on the BGP. Not only that, if the economy
gets into the positive-saving phase, the physical capital accumulation will set in. In thissense, human development aid may help the economy escape from underdevelopment
trap.

Figure 5. Human Development Aid (Declining Population Growth Rate)

n
ｎ’
k’
ｎ

k
F

E

n(0)

ｎ’
k’

Ｅ

ｎ
k

k

O

k0
Next, consider the more noticeable case in which E’ lies northwest of E (Figure 6). Until
the economy reaches to E, population growth rate will decline as per capita income

20increases. When the aid comes in, the economy jumps up to F and then per capita
income will begin to decrease and population growth rate begins to rise along thetransition path toward E’. Since E’ has a lower value of k * , one cannot say anything
definite about whether human development aid will improve welfare. Furthermore,since per capita income decrease, the economy cannot get into the positive-saving phase.
In this sense, human development aid will not be useful for escaping fromunderdevelopment trap.

Figure 6. Human Development Aid (Per-capital income rises and then declines)

n
ｎ’

k’
E
F
ｎ

k
k’
ｎ’

n(0)
Ｅ
k

O

ｎ

k

k0
7. Concluding Remarks
We have shown that an endogenous growth model under PCH can be more tractable
than we have considered so far by endogenizing population growth rate and further
investigated dynamic implications of PCH. In contrast to Steger (2000), we focus on a21

BGP with a constant level of per capita income. We have found that the model may havea unique or multiple saddle-point stable BGPs in both no- and positive-saving phases. In
the positive-saving phase there may be one saddle-point stable BGPs. Along atransitional path, population growth rate declines as per capita income increases. In the
no-saving phase more relevant to poor economies, population growth rate may rise or
decline monotonically along a transition path. The theoretical results turn out to berealistically relevant in reference to data from World Development Indicators (2004): inparticular, the recent trend of declining population growth rates in modern developing
countries could be explained, and exogenous changes in time preference rate could
explain complicated relations between population growth rate and per capita GDP insome African countries. Furthermore, we find that “human development” aid enhancing
human capital accumulation may reduce per-capita GDP and does not always improvewelfare.
Let us elucidate qualifications of this paper. First, we have assumed awaychild-rearing cost. It is important to examine how the qualitative results or properties of
BGPs will change if this cost is explicitly incorporated. Second, the present model is ofa one-sector closed economy. Extension to an open economy may be useful for
obtaining further implications of PCH. The present paper will only be a starting point
toward future research.

AppendicesA.1 Proof of Proposition1:

Defining µ (t ) = π (t )e ρt , FOCs (8) leads to22

1= µψ ' (c)c(A1-1)n(t ) ε µ (t )k (t ) = 1(A1-2)µ (t ) = ρµ (t ) − µ (t )[ f ' (k (t )) − n(t )](A1-3)and limt →0 π (t )k (t ) exp(− ρt ) = 0 . First, differentiating (A1-1) and eliminating µ / µ
by

using

(A1-3),

we

get

(9).

Next,

differentiating(A1-2)

yieldsε (n / n) + µ / µ + k / k = 0 . Using (4) and (A1-2), (A1-3) leads to (10). The dynamic
system for positive-saving phase isc(t ) =c(t )[ f ' (k (t )) − n(t ) − ρ ]1 + η (c(t ))n (t ) =n(t ) f (k (t )) ψ (c(t )) f ' (k (t )) − ρ −+ε k (t )k (t ) k(t ) ＝ f (k (t )) − n(t )k (t ) −ψ (c(t ))First, we will examine the existence of BGP. Eliminating µ using (A1-1) and (A1-2),
we get kn ε = cψ ' (c) . From c =0, we get f ' (k ) = n + ρ holds. Combining them leads
tok[ f ' (k ) − ρ ]ε = cψ ' (c)(A1-4)The slope of this curve is
dc [ f ' (k ) − ρ ]ε −1{ f ' (k ) − ρ + kεf " (k )}=dkψ ' (c) + cψ " (c)(A1-5)

where ψ ' (c) + cψ " (c) = [1 − φ ' (c)] − cφ " (c) > 0 . The locus of (c,k) that satisfies (A1-4)
takes an inverse U-shape. Next, using n =0, we getf (k ) − kf ' (k ) + ρk −ψ (c) = 0(A1-6)The slope of this curve is
dc ρ − kf " (k )=>0ψ ' (c )dk

23In addition, the locus of (c,k) that satisfies (A1-6) starts from a positive value c0 on thevertical axis.10 Therefore the two curves typically intersect at point E1 and E2.

Figure A1. BGPs in Positive-saving Phasecn = 0
E2

c0

E1

k
O

Second, let us examine the stability of BGPs. The linearlized system around the BGP is

 0

c
  
 n  =  nψ ' (c)   εkk   −ψ ' (c)



−

c
1 + η (c )0−k

f " (k ) *
1 + η (c ) c − c n[kf " (k ) − ρ ]  n − n * εk*f ' (k ) − n  k − k c(A1-7)

The characteristic equation for the coefficient matrix J* evaluated at BGP isλ3 − TraceJ *λ2 + BJ *λ − DetJ * = 0 . Three characteristic roots λ1, λ2, λ3 satisfy thefollowing relations.
10

Setting k = 0 we get c = 0 and c = co > 0 with co = φ (co ) . In the positive-saving range

only the latter is valid24

TraceJ * = λ1 + λ2 + λ3 = f ' (k ) − n = ρ > 0
BJ * = λ1λ2 + λ2 λ3 + λ3λ1 =cψ ' (c)  n
 n+ f " (k ) + [kf " (k ) − ρ ]
1 + η ( c )  εk εDetJ * = λ1λ2 λ3 = 0

In the positive-saving phase, ψ ' (c) > 0 and thus η (c) > 0 hold.
Since DetJ * =0 holds, at least one of the three eigen values is zero. However,TraceJ * > 0 means that λ1 = λ2 = λ3 = 0 is impossible.

Suppose that BJ * = 0

holds. If λ3 = 0 , then BJ * = λ1λ2 = 0 . Thus one of the other two eigen values is zero.
If λ2 = λ3 = 0 , then λ1 > 0 . Therefore, there are no negative eigen values. Supposethat BJ * > 0 holds. Clearly λ2 = λ3 = 0 is impossible. If λ3 = 0 holds, we get

λ1 > 0 and λ2 > 0 . Therefore, there are no negative eigen values.

Suppose that

BJ * < 0 holds. If λ3 = 0 holds, we get λ1 < 0 and λ2 > 0 . The number of negative
eigen values equals the number of state variables =1. Therefore, the BGP is saddle-point
stable. In all three cases above, there is no possibility of two negative eigen values. Thusa BGP cannot be perfectly stable.

(Q.E.D.)

A.2 Local Stability of BGP in the No-saving Phase

We will now examine the stability of a steady-state equilibrium. The properties oftransitional dynamics can be investigated, focusing on the relation between the slopes ofk(t ) ＝0 and n (t ) = 0 curves. The linearlized system around the steady-state equilibriumis25k   φ ' ( f ) f ' (k * ) − n k − k * − k* = *
*
*
*
*
* 
*
 n  (n / ε )Γk (k , n ) (n / ε )Γn (k , n )  n − n (A2-1)We denote the coefficient matrix by J*.
Trace J*= [φ ' ( f ) f ' (k * ) − n * ] + (n* / ε )Γn (k * , n * ) = ρ > 0(A2-2)

Det J*= [φ ' ( f ) f ' (k * ) − n * ][(n * / ε )Γn (k * , n * )] + k * [(n * / ε )Γk (k * , n * )](A2-3)

From (A2-2), the BGP cannot be perfectly stable. The BGP may be either saddle-point
stable or unstable, depending on DetJ * is negative or positive.

A.3 Evidence from African Countries

Taking into count that African countries have often experienced exogenous shocks,
we will explore a possible explanation for these data by comparative statics anddynamics.
We will suppose here, as one of the possible explanations, that the time preference
rate ρ changes exogenously. For example, when military conflicts or a domestic wars
occur, people in African countries may become more myopic. When the war ends, they
will come to think their lives on a long-run basis again. Let us present an explanation
for the case of decreasing nn curve (one can easily make a similar discussion for thecase of increasing nn curve).
In Figure A.2, the economy moves from the initial point ( k0 , n(0) ) to E (BGP).
Suppose that ρ rises exogenously. Then nn curve shifts upward while kk curve
remains unchanged. Thus the economy will jump from E to F and then moves along the
new transition path toward E’: per capita income f (k * ) is lower while population
growth rate n * is higher. When the wars end and ρ declines to the initial value, the26

economy will jump from E’ to F’ and moves toward E. If the economy experiences thiskind of movements, the data will probably be scattered. The present model does notalways contradict the data exhibiting non-monotonic relations between population
growth rate and per capita income.

Figure A.2 Change in Time Preference

n
ｎ’

k
E
F

ｎ
n(0)

ｎ’
F’

Ｅ

ｋ

O

k0

27

ｎ

k

References
Barro, Robert J. and Xavier Sala-i-Lartin, 1995, Economic Growth, McGraw-Hill, Inc..
Bliss,Christopher, and Nicholas Stern, 1978a, “Productivity, Wages and Nutrition Part
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Bliss,Christopher, and Nicholas Stern, 1978b, “Productivity, Wages and Nutrition Part
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Dasgupta, P. and D. Ray, 1986, “Inequality as a Determinant of Malnutrition andUnderemployment: Theory,” Economic Journal 96, pp.1011-1034.
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Leonard, Daniel and Ngo Van Long, 1992, Optimal Control Theory and Static
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Lopez-Casasnovas, Rivera and Currais, eds. 2005, Health and Economic Growth:Findings and Policy Implications, The MIT Press.

Ray, D. and P. Streufert, 1993, “Dynamic Equilibria with Unemployment due toUndernourishment,” Economic Theory 3, pp.61-85.
Razin, Asaf and Uri Ben-Zion, 1975, “Intergenerational Model of Population Growth,”
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Steger, Thomas M. 2000, “Productive Consumption and Growth in Developing
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Steger, Thomas M., 2002, “Productive consumption, the intertemporal consumption
trade-off and growth,” Journal of Economic Dynamics & Control 26, pp.1053-1068.

28

Tietenberg, Tom, 2006, Environmental and Natural Resource Economics, seventh
edition, Pearson Education Inc..

Yip, Chong K. and Junxi Zhang, 1997, “A simple endogenous growth model withendogenous fertility: Indeterminacy and uniqueness,” Journal of Population
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前掲書.

三野和雄、「経済成長モデルにおける不決定性」、西村・福田編

29

前掲書.

Table 1. GDP per capita and Population Growth Rate (Nepal)

3.00E+00
2.50E+00
2.00E+00
1.50E+00

系列1

1.00E+00
5.00E-01
0.00E+00
0.00E 5.00E 1.00E 1.50E 2.00E 2.50E 3.00E
+00
+01
+02
+02
+02
+02
+02
GDP per capita (constant 1995 USD)

GDP per capita (constant 1995
USD)

Nepal
3.00E+02
2.50E+02
2.00E+02
1.50E+02

系列1

1.00E+02
5.00E+01
0.00E+00
1

5

9

13 17 21 25 29 33 37 41
Years (1960-2002)

Nepal
Population Growth Rate (annual%)

Population Growth Rate (annual %)Nepal (1960-2002)

3.00E+00
2.50E+00
2.00E+00
1.50E+00

系列1

1.00E+00
5.00E-01
0.00E+00
1

5

9 13 17 21 25 29 33 37 41
Years (1960-2002)

30

Table 2. GDP per capita and Population Growth Rate (India)
India (1975-2002)

2.00E+00

1.50E+00
系列1
1.00E+00

5.00E-01
0.00E+00
0.00E+0 5.00E+0 1.00E+0 1.50E+0 2.00E+0 2.50E+0
0
2
3
3
3
3
GDP per capita (constant 1995 US $)GDP per capita (constant1995 US $)India
2.50E+03
2.00E+03
1.50E+03
系列1
1.00E+03
5.00E+02
0.00E+00
1

4

7

10 13 16 19 22 25 28
1975-2002

India
Population Growth Rate (%)

Population Growth Rate (%)

2.50E+00

2.50E+00
2.00E+00
1.50E+00
系列1
1.00E+00
5.00E-01
0.00E+00
1

4

7

10 13 16 19 22 25 28
1975-2002

31

Table 3. GDP per capita and Population Growth Rate (Columbia)

3.50E+00
3.00E+00
2.50E+00
2.00E+00

系列1

1.50E+00
1.00E+00
5.00E-01
0.00E+00
0.00E+ 5.00E+ 1.00E+ 1.50E+ 2.00E+ 2.50E+ 3.00E+
00
02
03
03
03
03
03
GDP per capita (constant 1995USD)
Columbia

GDP per capita(constant 1995USD)

3.00E+03
2.50E+03
2.00E+03
1.50E+03

系列1

1.00E+03
5.00E+02
0.00E+00
1

5

9

13 17 21 25 29 33 37 41
Years (1960-2002)

Columbia
3.50E+00
Population Growth Rate(annual %)Population Growth Rate (annual %)Columbia

3.00E+00
2.50E+00
2.00E+00
系列1
1.50E+00
1.00E+00
5.00E-01
0.00E+00
1

5

9

13 17 21 25 29 33 37 41
Years (1960-2002)

32

Table 4. GDP per capita and Population Growth Rate (Ghana)

5.00E+00
4.50E+00
4.00E+00
3.50E+00
3.00E+00
2.50E+00
2.00E+00
1.50E+00
1.00E+00
5.00E-01
0.00E+00
0.00E 1.00E 2.00E 3.00E 4.00E 5.00E 6.00E+00 +02 +02 +02 +02 +02 +02
GDP per capita (constant 1995 USD)

GDP per capita (constant 1995
USD)

Ghana
6.00E+02
5.00E+02
4.00E+02
3.00E+02

系列1

2.00E+02
1.00E+02
0.00E+00
1

5

9

13 17 21 25 29 33 37 41
Years (1960-2002)

Ghana
Population Growth Rate (annual%)

Population Growth Rate (annual %)Ghana (1960-2002)

5.00E+00
4.50E+00
4.00E+00
3.50E+00
3.00E+00
2.50E+00
2.00E+00
1.50E+00
1.00E+00
5.00E-01
0.00E+00

系列1

1

5

9 13 17 21 25 29 33 37 41
Years (1960-2002)

33

系列1

Table 5. GDP per capita and Population Growth Rate (Sudan)

3.50E+00
3.00E+00
2.50E+00
2.00E+00
系列1

1.50E+00
1.00E+00
5.00E-01
0.00E+00
0.00E+00 1.00E+02 2.00E+02 3.00E+02 4.00E+02
GDP per capita (constant 1995 USD

GDP per capita (constant 1995
USD)

Sudan
3.50E+02
3.00E+02
2.50E+02
2.00E+02

系列1

1.50E+02
1.00E+02
5.00E+01
0.00E+00
1

5

9

13 17 21 25 29 33 37 41
Years (1960-2002)

Sudan
Population Growth Rate (annual %)Population Growth Rate (annual %)Sudan (1960-2002)

3.50E+00
3.00E+00
2.50E+00
2.00E+00
系列1

1.50E+00
1.00E+00
5.00E-01
0.00E+00
1

5

9

13 17 21 25 29 33 37 41
Years (1960-2002)

34