Tempo effect in first marriage table: Japan and China. Kiyosi Hirosima (Shimane University, Japan)

Transcription

1 Temo effect in first marriage table: Jaan and China Kiyosi Hirosima (Shimane University, Jaan) Introduction Recently many East Asian countries and regions are undergoing the raid increase in age at marriage. The ostonement of or retreat from marriage has been the redominant factor of fertility decline in these regions. Researchers there are increasingly aware of the need of most recise measurement of marriage decline. It is well-known that for the measurement of most recent situation of marriage, the roortion according to the first marriage table is more referable to the total first marriage rate since the former is based on the intensity of first marriage at the time of observation without the influence of ast marriages. However, as Bongaarts and Feeney (1998) argued, the quantum of vital events calculated by life table at certain time is also influenced by temo change of the occurrence of the events of cohorts. This aer rovides, first, a theoretical exlanation of the temo effect in the quantum by the life table in a clearer way than has been reviously rovided, and an examination of the way to adjust this influence. Second, it rovides an alication of these theoretical examinations to the first marriage ta of Jaan and China (Taiwan) and examines the effectiveness and the racticability of the theory. Theory of temo effect in the life table Let denote the roortion of ersons who were born at time T t x and are surviving without exeriencing an vital event, say death, at age x at time t, d( denote the density of the event of this cohort at age x at time t, and μ( denote the corresonding intensity, or force of mortality for death. It should be noted that excet μ(, these functions are different from those for the synthetic cohort hyothesized in the conventional eriod life table. I call these functions as two-dimensional cohort life table function. These notations are exactly the same as aeared in Bongaarts and Feeney (23). I will first clarify the relationshis among these functions and define the quantum and the temo of vital events; and second, define the eriod life table functions using these cohort life table functions; and third, rove the existence of the temo effect in the eriod quantum of the events obtained by the eriod life table. l At time t that is after x from time T (t-x), can be exressed by txt ( t x x) and ( also by as follows. 1

2 ( Noting of - ( - ( t x x) log txt 1 d t x x) d log t x x) dx dx l ( t x a) x txt txt, integrating the both sides of (1.1), we obtain x ( t x a), x (1.1) ex ( t x a), (1.11) Using the relation of d( l ( (, we get the following exression x d ( ( ex ( t x a), (1.13) Using this exression, after multilying to the both sides of equation (1.1) and integrating them from age to age we obtain x d( t x a) ( t x a) x 1- l ( (1.14) l Hence, Q c (T) the mean exerience frequency er erson is given as follows for the cohort born at time T t x at age x at time t t T. Q c (T) d( T x) dx 1 l (, T ) (1.15) Meanwhile, the eriod life table based on ( is exressed by the eriod life table functions as follows. Note that the integration and the differentiation in these functions are not in the direction of the cohort life course line but in the direction of age of the hyothetical cohort at time t. l ( x l (, ex (, d ( (2.1) x x Also, ( d l ( ( d ( ( ex x ( (2.2) From the occurrence density d (, the quantumq,and mean of events occurrence M can be defined as follows, ostulating l (, 1. Q ( d ( dx l ( ( dx dx 1 l (,, (2.3) x 2

3 M ( xd ( dx l ( dx l (, (2.4) d ( dx 1 l (, I call Q ( as the eriod life table quantum, and ( as the eriod life table mean age at event. M Temo change in cohorts and the influence on eriod quantum McKendrick equation for a cohort with size can be exressed as follows; 1 1 ( t x (1.19) 1 If we exress here the growth rate of at age x at time t as r (, t then we obtain where 1( r( 1( ( (3.1) d1( 1 and d 1( x l ( x This relation is well-known for age secific oulation(bennett and Horiuchi, 1981; Preston and Coale, 1982; Arthur and Vauel,1984). The Integration of the first equation of (2.5) yields l ( ex x 1( (2.7) Using this equation, another eriod quantum can be defined as follows. Q 1( d 1( dx 1( x, dx dx x (2.5) 1,, (2.8) Substituting 1( ( r( in equation (2.7) and using equation (2.1), it reduces to l ( ex x { ( r( } l ( ex x r( (3.3) Temo effect in life table functions Let us examine the temo effect in the quantum in the eriod life table Q ( defined by equation (2.3). Let xω in equation (3.3). Then we obtain 3

4 ω, l (, ex r( (3.8) Substituting this in equation (2.8), it reduces to Q 1( 1-l (, ex r( (3.9) According to equation (2.3), l (, 1- (, equation (3.9) reduces to Q Q 1( 1 1 Q ( ex r( (3.1) This equation reresents the relation between a eriod quantum Q1( table Q (. and the quantum by the life Now we ostulate that the roortion of eole who have never exerienced an event at maximum age ω, ω, be constant. That is the ostulation that the quantum of events of the cohorts Q c (T) is constant even if the temo of the event M c (T) changes with regard to cohorts. According to this ostulation, equation (1.15) reduces to Q c (T)1- ω,. This is the same equation with (2.8), i.e. Q c (T) Q 1( 1- ω,. Hence, under this ostulation, there is a relation between Q c (T) and Q ( shown below. 1 1 Q ( ex r( (3.11) Q c (T) l (, 1 Q (, 1 Q ( T) c ex r ( (3.12) Hence, if r( >, then aears the following relation. l (, > ω,, Q ( < Q c (T) (3.13) The reverse relation emerges in the reverse case. These relations show the existence of the distortion in the quantum obtained through eriod life tables ( Q ( ) in that it will differ from cohort quantum (Q c (T)) when the temo of cohort life table changes, i.e. r (, growth rate of changes so that r(. Adjustment of quantum of eriod life table 4

5 The distorted quantum obtained through the eriod life table is to be adjusted so that it reresents the average level of the quantum of cohorts that constitute the eriod quantum. For adjusting the eriod quantum derived from the intensity of the vital events, some authors roosed the shift model of intensity (Bongaart and Feeney, 26; Kohler and Orteg 22). This model is not alicable, however, if the intensity at the ages higher than the average age hardly changes in site of the shift of the average age of the intensity. In fact for first marriage, it often is the case. (I will show this for first marriage in Jaan and Taiwan in the next section.) If we take the density of the events by the life table, however, the shifting of the age attern is more salient. Therefore, it is aroriate to aly the shifting model to adjust the distortion in the eriod quantum of the events Q ( that can be assumed to be derived from eriod density ( d ((2.2)) of the events. The adjustment rocedure that is considered to be aroriate is the one roosed by Bongaart and Feeney, 1998 for fertility, i.e. Q (, dm ( 1 dt using the time derivative of the average age of the event M (. It should be noted, however, that the density of the event is for the eriod or the synthetic cohort and not for real cohort. Therefore, the consistency among densities by age for a cohort is not strictly maintained. For examle, if the event is death, the sum of the densities by age for a cohort does not necessarily coincide with unity. It may be one of the sources of the error of this adjustment. In this sense, the adjustment is an aroximation. Alication to first marriage in China (Taiwan) and Jaan First marriage table was calculated by the vital statistics and the oulation census for every five years from 198 to 2 for Jaan and for 2 and 27 for Taiwan*1. Figure 1 shows that the delay of first marriage deicted by first marriage table is not caused by the simle shift of the age attern of intensity of first marriage. The intensity at the ages higher than the average age hardly changes while that at lower ages is declining. Figure 2 shows, however, that the delay of first marriage is caused by the shift of the density of the first marriage ( d although the shae is also changing in a way that the to of the curve is going down. It is lausible to assume that the change in the density of first marriage can be the shift at least for a short eriod of time. Therefore, I adjusted the eriod quantum via the first marriage table by the formula above-mentioned. 5

6 The results are resented in Table 1 for Jaan and in Table 2 for Taiwan. In both regions, the roortion by first marriage table has been getting smaller for both sexes. The adjusted roortion becomes higher by about 5 ercent than the counterart. values surassing unity for females in Jaan may be caused by the adjusting formula I adoted and by ta roblem in Taiwan cases. The roortion of among Jaanese females in 2 by the first marriage table is.83 and the adjusted value is.89. These values could be the maximum level of marriage when we roject the future trend in women s marriage in Jaan. The converging value of the roortion of women is set as.764 in the official oulation rojection conducted by the Jaanese government in 26, which is very low, comared with these values. Note 1) First marriage tables were rovided by Motomi Beu. I areciate his cooeration. 2) References Arthur, W. Brian and James W. Vauel (1984), Some General Relationshis in Poulation Dynamics, Poulation Inde 5(2): Bennett, Neil G. and Shiro Horiuchi (1981), Estimating the comleteness of death registration in a closed oulation, Poulation Inde 47(2): Bongaarts, John and Griffith Feeney (1998), On the quantum and temo of fertility, Poulation and Develoment Review, 24(2): Bongaarts, John and Griffith Feeney (23), Estimating mean lifetime, Proceedings of the National Academy of Sciences of The United States of Americ 1(23): Bongaarts, John and Griffith Feeney (26), The Temo and Quantum of Life Cycle Events, Vienna Yearbook of Poulation Research,Volume 26, Horiuchi, Shiro and Samuel H. Preston (1988), Age-secific growth rates: the legacy of ast oulation dynamics. Demograhy, 25(3): Hans-Peter Kohler, José Antonio Ortega (22), Temo- Period Parity Progression Measures, Fertility Postonement and Comleted Cohort Fertility, Demograhic Research, 6(6), Schoen, R. and Canus-Romo V. (25), Timing effects on first marriage: Twentieth-century exerience in England and Wales and the USA, Poulation Studies 59-2,

7 .25 Figure1 qx:first marriage, female, Jaan qx 198 qx 1985 qx 199 qx 1995 qx Figure 2 dx: first marriage, female, Jaan dx 198 dx 1985 dx 199 dx 1995 dx

8 Table 1 roortion and mean age at first marriage by first marriage table: Jaan Male Female Mean Mean age at age at first first roortion marriage roortion roortion marriage roortion Year (yrs ) (yrs ) Change rate of age at marriage for 2 is based on the rate of Table 2 roortion and mean age at first marriage by first marriage table: Taiwan Male Female roortion Mean age at first marriage (yrs ) roortion roortion Mean age at first marriage (yrs ) roortion Change rate of age at marriage is rovided based on the rate of

9 Figure 3 roortion and mean age at first marriage by first marriage table: Jaan roortion roortion roortion Year roortion Figure 4 Mean age at first marriage by first marriage table:jaan 3 29 Male Female 28 Ageatmarriage(yearsold)

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