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Demographic transition pattern in a small country
a , b
*
Akira Momota
, Koichi Futagami
a
Faculty of Economics, Tezukayama University, 7-1-1 Tezukayama, Nara 631-8501, Japan
b
Faculty of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka 560-0043, Japan Received 9 June 1999; accepted 2 November 1999
Abstract
We investigate how the population growth rate changes along the transition path when the mortality rate decreases. It is shown that the existence of an altruistic household and child rearing costs is one of the factors which can explain the stylized fact called the ‘demographic transition pattern.’ 2000 Elsevier Science S.A. All rights reserved.
Keywords: Altruistic household; Child rearing cost; Demographic transition; Small country
JEL classification: O41
1. Introduction
1
There are several approaches concerning endogenous fertility theory. Among these approaches, we consider in this paper the one that parents care about their children and determine the fertility rate endogenously. We investigate how birth rate and population growth rate change not only at the steady state, but also in a transition dynamics, when a permanent decline of the mortality rate occurs unexpectedly, for example foreign aid to improve hygiene. It is shown that the result obtained from the analysis is consistent with the stylized fact called the ‘demographic transition pattern.’
2. Model
We consider a small open economy and thus world interest rate r is given as exogenous. Households live for only an instant and they work, consume and raise their children during the instant.
*Corresponding author. Tel.: 181-742-48-9536; fax:181-742-46-4994.
E-mail address: [email protected] (A. Momota)
1
See Razin and Sadka (1995) on this subject.
Each household at period t bears n children instantaneously, but d children die as soon as they aret
born. d represents the (instantaneous) mortality rate, the level of which is assumed to be exogenous. Thus, the following equation holds concerning the total population size, denoted by N :t
~
Nt5(nt2d )N .t (1)
We postulate that the economy starts at time 0 and N051 without loss of generality.
Households derive utility from their own (instantaneous) consumption c and the number oft children they have. Furthermore, we assume that the households are altruistic in the sense that they care about the utility of their children. In this case, as pointed out by Becker and Barro (1988) and Jones (1997), the households as of time 0 maximize the following dynastic utility function:
`
2rt
U05
E
u(c ,N )et t dt, (2)0
where r is the time preference rate that applies across generations. In other words, it represents the
a e
degree of ‘selfishness.’ We shall specify the functional form of u(c ,N ) as c N , where 0t t t t ,a,1 and 0,e,1.
Each household is endowed with one unit of labor. When the households raise nt2d children,
b(n ,d ) units of labor are necessary. Hence, the households supply 1t 2b(n ,d ) units of labor to thet where a denotes the asset level of the household. w represents the wage rate, and it is constantt because of a small open economy, as will be shown later.
The behavior of the household can be derived as maximizing (2) subject to (1) and (3). To do this, we shall set up a current value Hamiltonian:
a e
H5c Nt t1lt[(r2nt1d )at1(12b(nt2ud ))w2c ]t 1m[(nt2d )N ].t (4) The optimal conditions for the household can be obtained from the first-order conditions of (4). Performing some manipulations, we obtain
where nt;mt/N . Thus, the optimal behavior of the household is described as (3) and (5)–(7). At
transversality condition with respect to N is expressed ast
e 2rT
lim NT Tn e 50. (8)
T→1`
Firms are identical and act competitively, and the aggregate production function is represented as Yt5F(L ,K ), where Y , L and K are output, labor input and capital stock, respectively. Thet t t t t
production technology is assumed to be constant returns to scale. Thus, it can be expressed as
˜ ˜
yt5f(k ), where yt t;Y /L and kt t t;K /L . The optimal conditions of a representative firm aret t
˜ ˜ ˜ ˜ ˜ ˜
expressed as r5f9(k ) and wt 5f(k )t 2f9(k )k . As the level of r is constant, so are k (t t t ;k ) and thus w.
Market clearing conditions of asset and labor markets are represented as at5kt1b and Lt t5(12
b(nt2ud ))N , respectively, where kt t;K /N and b is defined as per capita foreign asset level. It ist t t ˜
useful to remember that k(12b(nt2ud ))5k holds. As a result, the dynamic systems of thet economy are expressed as (5), (7) and the following two equations:
21 12a
˜
(12b(nt2ud ))k1bt1b9(nt2ud )w5a nct , (9)
~ ~
kt1bt5(r2nt1d )(kt1b )t 1(12b(nt2ud ))w2c .t (10) Using (5), (7) and (9), (10) can be rearranged as
c jumpable variables, but these variables must satisfy (9). The degree of freedom of the variables is two in this case. Thus, the steady state is saddle-point stable when the characteristic equation of the system has one stable root and two unstable roots. We shall check in Appendix A that the condition is satisfied when a,e.
The non-trivial (i.e. c ±0) solution is expressed as follows:
a
More accurately, we can draw the loci in (n,c,n) (i.e.R ) space, instead of the (n,c) plane. However, note that both
1
~ ~
Fig. 1. Projection of the dynamic system on the (n,c) plane.
4
*
In the steady state, n ,r1d must hold in order to satisfy (8). (See also Fig. 1.) We shall confine
*
our analysis to the natural case where c .0. The curve QAR in Fig. 1 represents the projection of
5
*
the saddle path on the (n,c) plane. The level of nt in the steady state, which is denoted by n , is obtained from (7):
The surface representing nt50 can be drawn in (n,c,n) space. It can be immediately checked that
~
nt.0 holds above the surface, and vice versa. Contours of the surface are drawn in Fig. 1. As the
6
level of n becomes larger, the contour exists farther from the origin. Now let us section the (n,c,n) space along curve QAR, and draw the sectional plan in Fig. 2. The saddle path of the economy is
* *
~
drawn above nt50 when (n ,c )t t ,(n ,c ), and vice versa.
The initial point at which the economy starts can be derived from (9). The surface representing (9) can be drawn in (n,c,n) space since the initial foreign asset position, b , is given exogenously. As can0 be checked immediately from (9), the contours of (9) drawn in the (n,c) plane are downward (or upward)-sloping, and the direction of decreasing n is down and to the left (or up and to the left), if
˜ ˜
and only if b0w,(or .)b9k holds. Thus, if b0w,(or .)b9k holds, the surface representing (9) reduces to a downward (or upward)-sloping curve on the sectional plan of (n,c,n) space along the
˜
curve QAR. Therefore, we can find a unique initial point ifb0w,b9k holds, as seen in Fig. 2; on the other hand, there is a probability that multiple initial points emerge or that we can find no initial point
˜ if b0w.b9k.
˜ We shall concentrate our analysis below on the case where b0w,b9k.
4
*
It can be easily checked from (13) that r $ er is the necessary and sufficient condition for r2n 1d.0, which is assumed in this paper.
5
The saddle path of this economy exists in (n,c,n) space.
6
As can easily be checked from (14), the level ofn is indeterminate at (n,c)5[(r/e)1d,0]. However, it is useful to note
* *
that n is always less than (r/e)1d since (r/e)1d.r1d.n holds. The first inequality holds on the assumption of
Fig. 2. Saddle path.
4. The effect of a declining mortality rate on the fertility rate
Suppose that the economy is initially in the steady state and then a permanent decline of d occurs unexpectedly. We shall examine in this section how a steady state changes and how the economy moves along a transition path to reach the new steady state. As for the transition dynamic analysis, we shall focus on the movement of the fertility rate n and the population growth rate n2d.
Firstly, let us consider the change of the steady state. It can easily be checked from (5) and (11)
~
that the ct50 line represented on the (n,c) plane shifts to the left as the level of d decreases, but the
~
direction of a shift of the nt50 curve is ambiguous. Nevertheless, we can see by making a comparative static analysis that the steady state point necessarily moves downward and to the left as
*
*
the level of d decreases, that is, we can obtain from (12) and (13) that dn /dd51 and dc /dd.0
*
hold, as shown in Fig. 3, and from (14) that dn /dd.0 holds.
Secondly, we shall analyze the transition dynamics. As mentioned earlier, we shall pay attention to the change of the levels of the fertility rate n along the transition path to reach the new steady state.
Fig. 4. Transition dynamics.
The foreign asset which the economy possesses and the level ofnt in the initial steady state (i.e. A in
*
*
Fig. 3) are denoted by bA and nA, respectively. Then, using (9), the hyper-plane where the initial point must exist is expressed as follows:
nt 12a
˜
*
]bA5 ct 2[b9(nt2ud )w1(12b(nt2ud ))k ]. (15)
a
To see from what point the economy starts on the new saddle path, let us section the (n,c,n) space along curve Q9BR9 in Fig. 3 and draw the sectional plan in Fig. 4. Note that, at the new steady state,
*
B, the level ofn on the saddle path is less thannA, while that on the initial hyper-plane (15) is greater
7
*
than nA. This implies that the birth rate does not decline as much as the mortality rate as soon as a permanent decline of the level of d occurs unexpectedly, that is, dn /ddt ,1 holds in the short run. In the long run, however, dn /ddt 51 holds. In other words, the population growth rate n2d increases in the short run, and decreases gradually to reach the same level as before the level of d decreases. Such a change is consistent with the stylized fact called the ‘demographic transition pattern,’ which states that while a country experiences a decline in the mortality rate, the birth rate of the country does not decrease immediately, that is, in the short run we observe a time lag between the decline in the fertility and the mortality rates in reality, and that the gap observed in the short run is alleviated in the long run.
7
*
The former is obtained immediately from dn /dd.0, as mentioned earlier. As for the latter, by differentiating (15)
* *
totally w.r.t. nt, c , n and d, evaluating at point A, and noting dn /ddt t 51 and dc /dd.0 hold when we consider the change from point A to B, we obtain the following:
12a
*
(c )A dn* ˜ 12a 2adc*
]] ]5(12u)[b0w2b9k ]2]]n* *A(c )A ],0,
a dd a dd
* *
5. Concluding remarks
We have investigated the movement of the population growth rate in a small country using the model where households derive their utility from the number of children they have and the fertility rate is then determined endogenously. It was shown that the result obtained from the model is consistent with the stylized fact called the ‘demographic transition pattern.’ In other words, it can be stated that the existence of an altruistic household and child rearing costs is one of the factors which can explain the stylized fact called the ‘demographic transition pattern.’
Appendix A
In this appendix we show that the dynamic system of the economy is saddle-point stable when
* * *
a,g. First, linearizing (5), (7) and (11) around the steady state equilibrium (c ,n ,n ), we obtain:
*
Let us check the signs of the trace and determinant of matrixD, denoted trDand detD, respectively. In the dynamic system represented by a 333 matrix, it is well known that the steady state is saddle-point stable if trD.0 and detD,0. Using (13), trD and detD are calculated as
Now it is useful to remember that n ,r1d must hold in order to satisfy (8). Thus we can see that trD.0, and that detD,0 holds when a,e.
References
Becker, G.S., Barro, R.J., 1988. A reformulation of the economic theory of fertility. Quarterly Journal of Economics 108, 1–25.
