Age Distribution of a Zero-Growth Population: Implications for China

doi:10.4236/chnstd.2013.22011

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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

Email: [email protected]

Received December 2nd, 2012; revised February 17th, 2013; accepted March 7th, 2013

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) =



t and mi(t) =







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) =



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) =



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:

 



 



 

1000

11 ,

1,2,; 0,1,

iiii

Nt mtNtQt

tmtNtNtB

izt

 

 



(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



































 



 



1111

111 2

111

iii iii

ii i

ii z

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













 

111

11 1

iiii i

ii z

ENmm m

mm mN





 





 





(3)

That is,













 

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

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



, we can make an approximate estimation:



000

EmmE







 (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

. We

here also drop the time dimension by assuming no changes in

the total population and age-specific mortality rates over time.

Obviously,





= 1. Since b = B/N and m are crude birth

and death rates, from Equations (3) and (5) we can write



000

zzz

ii iii

iii

EB Ebm PPmPm

N



 

 (7)

or abbreviated this as



 (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) =





 

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,

 

Ttf t



 (9)

where and are the lower and upper bounds of the

childbearing age for females. Let hi(t) =



Tt , = 1,







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

SONG J.

 

ii i

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

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

 

011

11 1

ii i

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,

 

01 1

11 1

rii i

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):









 

1, 0,1,

ii ,

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,

1, .

bBNb





(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

be the number of females

in Ni, and i

F be the total number of girls born in i

that

year, i

m be female mortality in i

, and i

h be the daugh-

ter-bearing age distribution of women in i

. We then have



011

11 1

nf ii

ia iif

Tmmm

NNT

























(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.







01 1

11 1

111.

Tmmm

mm m











 









 





(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),













01 1

11 1

Tmm mR

mRT







 











n

(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

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).

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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