Chinese Studies
2013. Vol.2, No.2, 77-83
Published Online May 2013 in SciRes (http://www.scirp.org/journal/chnstd) http://dx.doi.org/10.4236/chnstd.2013.22011
Age Distribution of a Zero-Growth Population:
Implications for China
Song Jian1,2
1Chinese Academy of Engineering, Beijing, China
2Chinese Academy of Sciences, Beijing, China
Received December 2nd, 2012; revised February 17th, 2013; accepted March 7th, 2013
Copyright © 2013 Song Jian. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
It is generally accepted that zero-growth population would be the long-term destiny of any population.
China’s population is expected to reach 1.4 billion with zero-growth around 2030, if the low fertility pol-
icy continues up to then. Demographic dynamics indicate that the age composition of a steady zero-
growth society would asymptotically approach the population mix of today’s many developed countries.
Here we present a brief analysis and some insights into the age composition of a zero-growth society and
the connectedness between total fertility rate, net reproduction rate and replacement level of fertility.
Other formulas useful for demographic studies are also provided to further the analysis. Our results reveal
that the age composition of China’s population in 2050 would be similar to those of some developed
countries today. We argue that the misgivings about “population aging” or the fear of a “winter of human-
ity” in China stem from rather oversimplified estimations.
Keywords: Age Composition; Zero-Growth Population; Total Fertility; Replacement Level; Population
Aging; China
Introduction
It is self-evident that, given the size of the Earth’s surface
and its limited resources, the number of human population that
the Earth or a country can support cannot be infinite. In 2011,
the world population reached 7.0 billion, with an annual growth
rate of 1.2% since 2000. It is projected that the world’s total
population is likely to exceed 9.3 billion by the middle of the
century (United Nations Population Division, 2011). Mainland
China (hereafter China) has now 15.2 million newborns every
year, a net growth of 5.03 million and a natural growth rate of
0.445 percent in 2011. Under the medium scenario (i.e., total
fertility rate ranges from 1.5 - 1.8 in the projection period),
China’s population would reach as high as 1.48 billion by the
middle of the century. In the context of scarcity in land and
water, even one per thousand annual growth is likely to be un-
sustainable in the long run.
There is a consensus that China has witnessed the demo-
graphic transition and achieved a noticeable economic growth
within three decades (Hussain, 2002). The success of family
planning program has contributed the lion’s share to this de-
velopment (Li, 2009). According to the United Nations Popula-
tion Division (UNPD) (2011), China’s total fertility rate has
dropped from 5.0 in 1970 to around 1.5 - 1.7 at present, which
is below the replacement level of fertility for 15 consecutive
years. It is projected that China’s population will be stabilized
at 1.4 billion in the years of 2025-2030 if the current fertility
rate follows its predicted trajectory from a Bayesian Hierarchi-
cal Model that captures the world’s fertility evolution trajectory
and China’s own historical trend (Raftery et al., 2012; UNPD,
2011). From a long-term perspective, most countries in the
world will have to achieve zero population growth eventually.
However, lowering total fertility rate, extending life expec-
tancy at birth, and heightening the median age of a population
would all imply inevitably the increase of the percentage of the
elderly population. This is what some Chinese scholars have
been concerned about recently. They began talking about
“population aging”, “senior boom” and even fears of a “winter
of humanity” (e.g., Wang, 2005; Zou & Yang, 2009).
The aims of this paper are: 1) to derive the demographic in-
dicators, including age distribution, of a zero-growth population
from population dynamics; and 2) to compare the age distribu-
tion of China’s population in 2050 with that of a zero-growth
population given the similar life expectancy.
Demographic Dynamics
Stationary population, aka zero growth population, has long
been extensively discussed in research literature and many
textbooks (e.g., Keyfitz, 1973; Preston, Heuveline, & Guillot,
2001; Ryder, 1975). Because a life table or model life table
population is a stationary population, previous studies, in most
cases, rarely linked it with a real population when they were
addressing the age structure of a stationary population, for no
real population was (or close to) a stationary population thirty
or forty years ago. Instead, they used the distribution in a life
table or model life table at a given mortality level. This is justi-
fiable and acceptable, although age distribution of a real popu-

SONG J.
lation differs greatly from the life table or model life table.
With life expectancy at birth in the more developed world
reaching or exceeding age 80, populations in many developed
countries have now demonstrated some similar characteristics
of a stationary population. This development improves our
knowledge and enables us to better understand the age distribu-
tion of a currently non-stationary population in the future when
it approaches a stationary population. Below I will provide
some basic mathematical formulas that are important in deter-
mining the age structure of a stationary population.
Let N(t) be the population size in the census year t and Ni(t)
be the population aged i but less than I + 1 (i can be a sin-
gle-year age group or a five-year age group), I = 0, 1,···, z, with
z being the highest age to which one can live; M(t) and Mi(t) be
the number of annual total deaths at age i in year t; m(t) =


M
t
N
t and mi(t) =



i
i
M
t
N
t be crude mortality rate and age-
specific mortality rate in year t; B be the number of total live
births in year t; and b(t) =


B
t
N
t be the crude birth rate in
year t.
Assume that the total newborns in year t is B(t), and imagine
that they go through their lives at the same age-specific mortal-
ity {mi(t), I = 0, 1,···z} of the year. There are m0(t)N0(t) =


0
0
M
t
N
t N
0(t) = M0(t) infants who would die in the first year,
(1 − m
0(t))N0(t) survive the first year and enter age one. Pro-
ceeding in this way, mi(t)Ni(t) would be the number of deaths at
age i, and (1 − m
i(t))Ni(t) would be number of survivors who
enter the next age I + 1 a year later. Assuming that Qi(t) is the
net migrants of entering and leaving age i, we get a system of
discrete equations of population dynamics from year t to year t
+ 1:
    
    
1
1000
11 ,
11 ,
1,2,; 0,1,
iiii
Nt mtNtQt
,
N
tmtNtNtB
izt
 
 

t
(1)
Now assume Ei(t) is the remaining life expectancy at age i
for all the members of Ni(t) in year t. By definition, the sum of
total years survived by all members of Ni(t) is Ei(t)Ni(t). If mi-
gration is not accounted for, from Equation (1) we get
















  
    
    




  
11
11
1
1
1
2
11
1.
11
1111
111 2
111
iii iii
zz
ii i
ii z
ii
zi
EtNtmt Ntmt Nt
mtNt
mt Ntmt
mtNtm t
mt zmt z
mt Ntz






 

 
 
 
 


(2)
Now we assume that in a steady zero-growth society mi(t) 
mi(t + 1) and B(t)  B(t + 1)  M(t)  M(t + 1), Equation (1)
gives a unique stationary solution Ni(t)  Ni(t + 1) and N(t) 
N(t + 1) (Song, 1982; Song et al., 1982). As we further drop the
time dimenson, Equation (2) can be simplified and re-arranged
as






 
1
11
111
11 1
iiii i
ii z
ENmm m
mm mN


 

.
i

 



(3)
That is,






 
1
11
111
11 1
ii ii
ii z
Em mm
mm m


.

 
 

 (4)
Set i = 0, we get the life expectancy at birth E0 of the infant
group B,



 

001 1
12 1
11 11
11 1
z
Emm mm
mm m

  
2
.

 


(5)
From here, one can see that infant mortality m0 and un-
der-five mortality m0-4 have the biggest impact on E0. When m0
changes by 0 m

, we can make an approximate estimation:

0
000
0
1
1
E
EmmE
m

00
m


 (5)
For example, when E0 is around 75, a decrease of m0 (or m0-4)
by 10 per thousand would raise the life expectancy at birth E0
by 0.78 years if m0 is around 40 per thousand. Today, low in-
fant and under-five mortality is the main reason for high E0 in
all developed countries.
Now denote the population density of age i by Pi = i
N
N
. We
here also drop the time dimension by assuming no changes in
the total population and age-specific mortality rates over time.
Obviously,
0
z
i
i
P


= 1. Since b = B/N and m are crude birth
and death rates, from Equations (3) and (5) we can write

0
0
000
11
zzz
ii iii
iii
EB Ebm PPmPm
N
,

 
 (7)
or abbreviated this as
0
1. m
Eb

 (8)
This implies that two of the three indicators (m, b and E0) are
interdependent.
Let ki(t)Ni(t) be the number of females reaching age i in year
t, and k(t)N(t) be the total female population in year t. Suppose
all women of age i have borne Fi(t) children in year t, so fi(t) =


 
i
ii
Ft
ktNt is the age-specific fertility rate per woman in
year t. The sum of age-specific fertility rates, denoted by T(t), is
defined as the total fertility rate (TFR) in year t,
 
2
1
,
a
i
ia
Ttf t

 (9)
where and are the lower and upper bounds of the
1
a2 a
childbearing age for females. Let hi(t) =


i
f
t
Tt , = 1,

2
1
a
i
ia
ht


denote the women’s reproductive age distribution. Census data
from different countries have shown that hi(t) fits the gamma
distribution in statistics (Song & Yu, 1985). Thus the total
number of infants borne by all women in the census year t must
78

SONG J.
be
 
2
1
a
ii i
ia
FtTth tktNt

. (10)
Now suppose that all infants born this year, as denoted by
B(t), go through their lives at the same age-specific mortality
rates mi(t) and give births to their own children according to the
reproductive age distribution at the census year (i.e., no change
in age-specific mortality rate over time), then i year later, there
are Ni(t + i) among B(t) would be able to survive and become a
member of Ni(t + i).










011
11 1
i. i
N
timmm Bt

  (11)
By putting Ni(t + i)  Ni(t) (or B(t)  B(t + 1)) and substitut-
ing the right-side value of Ni into (10) after dropping the time
dimension, we get
 
2
1
011
11 1
a
ii i
ia
FThkmmm B



. (12)
This means that, if B infants go through their lives as all
people do in the census year, they will have given births to F
children of their own. In this context, F defined by (12) can be
understood as the fertility level of the whole population in the
census year. If F = B, the total number of B’s children would be
exactly the same as B and just enough to replace B in the future.
In this special case, the total fertility rate (TFR), as denoted by
Tr, is called replacement fertility (RLF). Hence, from (12), we
have,
 
2
1
1
01 1
11 1
a
rii i
ia
Thkmm m







.
(13)
It is evident that if total fertility rate T is bigger than Tr, then
F > B, meaning that the population is growing, and vice versa.
This implies that if a real population keeps T = Tr for a long
period of time and age-specific mortality rate stays unchanged
or changes only slightly, then the population growth equation
will approach a unique stationary solution of (1):




 
00
1, 0,1,
1.
ii ,
N
tNti
NtNtconst
 
 
z
(14)
This also means that all of the population densities and the
total population size are remaining constant over time, so long
as the natural growth rate “g” is zero, i.e., crude mortality rate
m and birth rate b are equal. One can infer from (8) that in a
zero-growth population the birth rate is just about inversely
proportional to life expectancy E0 at birth,
00
1, .
11
N
bBNb
EE


(15)
In other words, given the appropriate size of population N, in
order to maintain a stationary population, the annual total births
B should be N/(1 + E0).
Females are always in the focus of demography, since they
take on the primary mission of reproduction. The family lineage
of their children makes census data mostly reliable. The aver-
age number of daughters born per woman who survive beyond
childbearing age is defined as net reproduction rate (NRR),
denoted by Rn.
In a stationary population, let i
N
be the number of females
in Ni, and i F be the total number of girls born in i
N
that
year, i m be female mortality in i
N
, and i h be the daugh-
ter-bearing age distribution of women in i
N
. We then have

2
1
2
1
011
11 1
1
,.
a
nf ii
ia
a
ii
fi
ia iif
R
Tmmm
FF
Th
NNT



h











(15)
When Rn = 1, every woman would have only one daughter
able to survive the childbearing age in average to replace her
mother. This total girl fertility rate, denoted by Tf, could be
called net replacement reproduction rate (NRRR), denoted by
Tfr.



2
1
1
01 1
01 1
11 1
111.
a
f
ri
ia
a
Tmmm
mm m



i
h


 




 


(17)
In order to see the relationship between common indicators
Rn, Tr and T, assume coarsely ki = 1
2 in (10)-(12), mi = i
m, hi
= i
h, Tf = 1
2T. Then one gets an approximate relation from
(15),






1
1
01 1
11
0
11 1
1.
ri
a
in
i
Tmm mR
mRT




 





n T
(18)
In the life table of the UNPD Database, only Rn and T are
available. The value of replacement level of fertility (RLF) can
be derived from (17) when needed. Table 1 shows some values
of T, Rn and Tr of China and other selected countries in
1970s-1980s for comparison (Song & Yu, 1981; UNPD, 2011).
Age Distribution of a Zero-Growth Population
In order to obtain the age composition of a zero-growth
population with E0 = 80, we assume its age-specific mortality
rates to be the same as shown in Table 2, which is the esti-
mated average for countries and areas like Japan, Iceland,
Switzerland and Northern Europe with E0 around 80 in
2005-2010 (UNPD, 2011). According to historical data of de-
veloped countries, when the life expectancy at birth, E0,
reached 75 - 80 years, the infant and under-five mortality was
reduced to 10 per thousand and lower.
Starting from (15) and (14), and taking mi from Table 2, we
have, as shown in Column 1 of Table 3, worked out the com-
position of a zero-growth population known as “standard dis-
tribution”. Note that the “standard” age distribution in a sta-
tionary population is irrelevant to a population size.
The results show that in a sophisticated society with zero
population growth, 11 percent of the population are youths aged
15 - 24, 59 percent are working-age population aged 15 - 54
who support 15 percent of children, and 25 percent are seniors
aged 55 or older, which is similar to the state of affairs in most
developed countries today. Also listed here for comparison are
age distributions in China, developed countries, and the world

SONG J.
Table 1.
Comparison of the replacement levels of fertility (RLF) in China and
other selected countries.
Year TFR (T) NRR (Rn) RLF (Tr)
China 1970-1975 4.8 2.04 2.32
1975-1980 2.9 1.28 2.27
1980-1985 2.5 1.15 2.24
1995-2000 1.8 0.78 2.24
2000-2005 1.7 0.73 2.28
2005-2010 1.6 0.71 2.29
USA 1970-1975 2.0 0.95 2.13
1975-1980 1.8 0.85 2.12
2000-2005 2.0 0.98 2.09
2005-2010 2.1 1.00 2.09
Japan 1970-1975 2.1 1.01 2.10
1975-1980 1.8 0.88 2.09
2000-2005 1.3 0.53 2.08
2005-2010 1.3 0.54 2.08
UK 1970-1975 2.0 0.95 2.10
1975-1980 1.7 0.83 2.09
2000-2005 1.7 0.80 2.09
2005-2010 1.8 0.88 2.08
France 1970-1975 2.3 1.09 2.11
1975-1980 1.9 0.89 2.09
2000-2005 1.9 0.91 2.09
2005-2010 2.0 0.95 2.08
India 1970-1975 5.3 1.87 2.78
1975-1980 4.9 1.83 2.54
2000-2005 3.0 1.25 2.55
2005-2010 2.7 1.17 2.50
Bangladesh 1970-1975 5.9 1.97 3.41
1975-1980 5.5 2.35 2.79
2000-2005 2.9 1.25 2.57
2005-2010 2.4 1.07 2.55
Pakistan 1970-1975 5.5 2.43 2.58
1975-1980 5.5 2.53 2.59
2000-2005 4.0 1.59 2.53
2005-2010 3.7 1.57 2.48
Nigeria 1970-1975 5.5 2.08 3.12
1975-1980 5.8 2.18 3.05
2000-2005 5.8 1.98 3.01
2005-2010 5.5 2.00 3.00
Note: author’s calculation from Formula (13) using data from UNPD (2011).
Table 2.
Age-specific mortality rates in current developed countries with life
expectancy at birth 80 - 85 and Replacement Level of Fertility (RLF)
around 2.1.
Age mx (‰)Lower 95% CI Bound Upper 95% CI Bound
0 4.48 3.04 5.92
1 - 9 0.17 0.13 0.20
10 - 190.24 0.20 0.29
20 - 290.52 0.43 0.52
30 - 390.57 0.54 0.80
40 - 491.53 1.25 1.80
50 - 593.90 3.45 4.34
50 - 599.85 9.39 10.34
70 - 7928.52 27.15 29.89
80 - 8454.91 51.85 57.97
85+ 151.59148.18 157.47
Note: CI: Confidence Interval. Source: author’s calculation using data from
UNPD (2011).
in 2010 and 2050 as projected by the UNPD (2011). It is as-
sumed that China will stick to its low fertility policy (TFR = 1.5
- 1.6) till 2020 and gradually raise TFR to 1.8 in 2050. As a
consequence, China’s total population will peak out at 1.4 bil-
lion by 2030. Table 3 shows that, with a couple of exceptions,
the age composition of China’s population in the middle of the
21st century will be more or less like that of some developed
countries today and that of the zero-growth population with a
similar life expectancy at birth. For example, according to the
medium scenario of the world population prospects (UNPD,
2011), the proportion of senior citizens aged 55 or older (25.7%)
in China in 2050 will be slightly higher than that of Japan
(22.7%) today, but much lower than Japan (35.5%) and South
Korea (32.8%) in 2050. The total working-age population in
China in 2050 is projected to be around 790 million. The pro-
portion of working-age population in 2050 in China will be
around 51%, close to that of today’s Japan (54%), but higher
than Japan (51%) and South Korea (54%) in 2050. The parent
support ratio is an important indicator for measuring the burden
of care that the middle-aged or young elders have to shoulder in
taking care of their oldest-old parents, the ones among senior
citizens who need care the most (Poston, 2008). The ratio is
formulated with population aged 80 or older over those aged 50
- 54 in 2050. This ratio in China is also close to both that of
Japan today and the standard population. Overall, the age
structure of Chinese population is more or less similar to those
of Western and Northern European countries in 2050. Figure 1
further reaches a similar conclusion in both low and high sce-
narios.
Concluding Remarks
We have presented a brief analysis of, and offered some in-
sights into, the age composition of a zero-growth society with
the connectedness between total fertility rate, net reproduction
rate, and replacement level of fertility. As an illustration, a
comparison is conducted between China’s population distribu-
ion by age in the middle of this century with a stationary t
80

SONG J.
Table 3.
A comparison between a zero-growth population with e0 = 80 - 85 and age distributions in China, developed regions and the World in selected years.
China Developed regions*Japan Western & Northern EuropeThe world
Standard age distribution
of zero-growth population201020502010 2050 201020502010 2050 20102050
Total population (billion) N 1.341.301.23 1.31 0.1270.1090.029 0.031 5.909.31
% Children, aged 0 - 4 5.2 5.14.45.7 5.5 4.34.45.5 5.5 9.25.9
% Children, aged 5 - 14 10.7 13.49.110.8 11.1 9.18.911.0 11.0 17.513.5
% Youth, aged 15 - 24 11.0 15.89.512.8 11.0 10.09.112.5 11.1 17.513.5
% Primary school-age population,
aged 5 - 11 5.4 7.85.45.4 5.7 5.45.45.5 5.5 10.58.2
% Secondary school-age population,
aged 12 - 14 3.2 4.42.83.3 3.3 2.82.73.4 3.3 5.24.1
% High school-age population,
aged 15 - 17 3.3 4.52.83.5 3.3 2.82.73.5 3.3 5.34.1
% University-age population,
aged 18 - 24 8.5 12.25.89.3 7.7 7.25.49.0 7.8 12.39.4
% Women in reproductive ages,
aged 15 - 49 among total women 39.4 55.435.245.4 38.4 41.532.345.7 38.5 52.144.9
% Working-age population,
aged 15 - 59 52.8 58.252.551.7 51.5 55.245.250.0 51.5 52.257.7
% Working-age population,15 - 54 58.7 72.451.057.5 57.5 54.051.155.1 57.5 55.553.3
% Population aged 50 or older 31.3 12.333.921.7 31.9 30.541.523.4 31.9 11.021.8
% Population aged 55 or older 25.4 8.225.515.9 25.7 22.735.517.4 25.9 7.515.2
% Population aged 80 or older 9.5 1.47.54.3 9.4 5.314.54.8 10.3 1.54.3
Median age (years) 44.5 34.548.739.7 44.3 44.752.340.9 44.4 29.237.9
% Population aged 80 or
older/population aged 50 - 54 38.8 8.433.322.2 52.9 30.785.525.3 59.8 11.724.9
% Dependency ratio 71.0 38.254.048.0 73.4 55.495.851.4 74.4 52.458.1
Life expectancy at birth (years) 80 - 85 73.279.477.5 83.0 83.287.779.7 84.7 58.575.9
Note: Developed regions comprise Europe, Northern America, Australia/New Zealand and Japan. Source: The 2010 World Population Prospects (UNPD, 2011).
0
20
40
60
80
100
120
% Children, aged 0-4 % Children, aged 5-14 % Youth, aged 15-24 % Primary school-age population, aged 6-11 % Secondary school-age population, aged 12-14 % High school-age population, aged 15-17 % University-age population, aged 18-24 % Women in reproductive ages, aged 15-49 among total
women
% Working-age population, aged 15-59 % Working-age population,15-64 % Population aged 60 or older % Population aged 65 or older % Population aged 80 or older Median age (years) % Population aged 80 or older /population aged 50-64 Dependency ratio (%)
Standard Age Distribution
China, 2010
China, 2050, Medium Scenario
China, 2050, Low Scenario
China, 2050, High Scenario
Developed Countries, 2010
Developed Countries, 2050, Medium Scenario
Japan, 2010
Japan, 2050, Medium Scenairo
%
Figure 1.
A comparison between a zero-growth population with e0 = 80 - 85 and age distributions in China, Japan and developed regions in
selected years. Source: The 2010 World Population Prospects (UNPD, 2011). The high and low scenarios assume that total fertil-
ity rate will be, respectively, 0.5 more or 0.5 less than total fertility rate in the medium scenario. Developed regions comprise
Europe, Northern America, Australia/New Zealand and Japan.

SONG J.
population and its corresponding demographic parameters in
some developed countries. We have found that the age compo-
sition of China’s population in 2050 will be close to that of
some developed countries today, many of developed countries
then; and it will also be close to the age composition of a sta-
tionary population if the fertility level follows its historical
trend with linkage to the world’s fertility evolution trajectory.
This finding helps us better understand how the age structure
changes from a non-stationary population today to a state close
to a zero-population growth in the future.
Our results further demonstrate that the proportion of old
adults aged 55 or older in the middle of the 21st century is close
to that of Japan today but lower than those of Japan and South
Korea in 2050. Some researchers recently argued that the ef-
fects of changing age structure on economic growth mainly
depend on support ratio, a ratio of the effective number of pro-
ducers to the effective number of consumers (Lee & Mason,
2011). Considering that 1) health condition of elderly popula-
tion is improving (Gu et al., 2009); 2) the postponement of
retirement age has recently received increasing attention from
both scholars and governments (Gu, 2000; Li, 2012; Zeng,
2007; Zhang, 2012); and 3) productivity would improve in the
future, the actual support ratio in China in 2050 would expect to
be higher than what some scholar has projected (Miller, 2011).
Thus, we argue that the worries of “aging”, a “senior boom”, or
a “winter of humanity” lack solid evidence. However, this does
not mean that China does not need to pay attention to popula-
tion aging. In the contrary, China needs to speed up her devel-
opment of policies and facilities to meet elderly care needs, not
only because of its sheer size, but also due to the rapid growth
(Gu & Vlosky, 2008; Zeng & George, 2010). The roots of the
rapid population aging in China do not simply lie in the current
fertility policy, but in a combination of a low fertility rate, ris-
ing life expectancy, and the cumulative effect of past changes
in birth and death rates (Banister, Bloom, & Rosenberg, 2010).
How large a population a country can support should be studied
comprehensively and systematically by the scientific commu-
nity and decided by legislative institutions that are well in-
formed of political, economic, resource, and environmental
backgrounds and technical factors (Cohen, 1995; Mahadevan et
al., 1994; Qian, 1982). However, it is almost certain that having
more people will put additional pressure on an already fragile
environment (Banister, 1998; Edmonds, 1994; Niu & Harris,
1995).
As Confucianism believes, health, longevity, wealth, be-
nevolence, and natural death are the five great blessings for
human being. It is the eternal pursuit of human societies to
improve welfare and healthcare, to reduce mortality rates, and
to extend longevity thereby. If everyone lives longer, the aver-
age age (median) and the proportion of elders and seniors will
increase inevitably. Aging of people does not mean senescence
of the whole society after all. A society will become more ma-
ture, more experienced, knowledgeable and intelligent (Ogawa,
2008). The youths can share their elders’ accumulated knowl-
edge and wisdom, and the old is keen to care the young. This
has been the triumphant way of evolution of man and it should
be a best frame for an ideal harmonious society in the future.
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