The One Child Policy and Family Formation in Urban China

Abstract

This paper estimates mortality and fertility rates prevailing in Ireland during the 25-year period before the Great Irish Famine of 1845-1849. A technique is developed to estimate the age-specific mortality level during the Famine and the number of Famine-related deaths. The paper concludes that fertility rates were declining during the period 1821-1845 and that the effects of the Famine were especially severe on the very young and the very old. Ignoring deaths among emigrants, it is estimated that one million individuals perished as a result of the Famine. The analysis permits year-by-year reconstruction of the Irish population age structure for the period 1821-1851.

Full text

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The One Child Policy and Family Formation in Urban China




The One Child Policy and Family Formation in Urban China


Gordon Anderson∗ ∗ and Teng Wah Leo♣ ♣♦ ♦


June 2007
(Working Paper)

Abstract

The Chinese government implemented the One Child Policy in an attempt to stave off population
explosion and its potential negative economic consequences on their infant economy in 1979.
This article examines the consequences of this policy on marital matching and family size
decisions. Using a simple General Equilibrium model, we show how by constraining marital
output on the quantity of children dimension raises the marginal benefits for increased positive
assortative matching, and greater investment in children. These theoretical predictions were next
verified empirically, by first verifying the prediction on positive assortative matching using
Distributional Overlap Test, which involves the comparisons of the joint density of spousal
educational attainment, and provides support for the hypothesis of increased positive assortative
matching among the urban population. To support this positive finding, we next examined if the
policy was indeed binding. Using Poisson regression, we found suggestive evidence that the One
Child Policy principally affected the quantity of children decision by suppressing parental gender
preference of their child. This suggest that births beyond the first child are purely accidental
among younger mothers, particularly those who were younger than 25 years of age when One
Child Policy was legislated, since marital match, and family size decisions would not have been
completed by then. This was affirmed using a pure Poisson model. In addition, we also found
some evidence of increased educational attainment among children, further verifying the
hypothesis that the One Child Policy altered familial decisions in urban China.



∗ University of Toronto, Department of Economics. Email: anderson@chass.utoronto.ca
♣ St. Francis Xavier University, Department of Economics. Email: tleo@stfx.ca
♦ We would like to thank Aloysius Siow, Loren Brandt, Lars Osberg, and Kuan Xu for their very helpful comments
and suggestions, and seminar participants at CEPA at The University of Toronto, Dalhousie University, and the
CEA Meeting.

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The One Child Policy and Family Formation in Urban China
One of the most controversial and far reaching population control policies in recent history is
China’s One Child Policy (OCP) introduced in 1979. Directed at China’s large population
growth rate vis-à-vis food production from the limited arable land, OCP represented a
considerable intervention in the household choice process. Achieved through fines and various
other forms of coercion, families were encouraged to limit the production of offspring. Such an
intervention could have changed fundamentally the nature of both existing and anticipated
marriage arrangements and can be expected to have influenced family formation decisions in
many dimensions; in the choice of partner, the family size and investments in children. However
there is a sense in which the policy was not at odds with the background against which it was
introduced. Fertility (number of live births per married woman aged 20-44) was already in
considerable decline prior to the OCP, in 1965 it was 6.4 and by 1980 it was 2.2. Therborn
(2004) suggests that this was in part a result of urbanization, which had been proceeding at a
tremendous pace1 since children are numerically more valuable in rural rather than urban
settings. Coupled with this is the apparent preference for sons in China the expression of which,
facilitated by the development of fetus gender detection and selective abortion, have generated a
somewhat skewed sex ratios at birth2. In a patriarchal society, patrilocal residence of married
sons is much more common than matrilocal residence providing considerable old age security
benefits for parents of sons as opposed to daughters.
With regard to the decline in family size prior to the OCP the demographer J.C. Caldwell
developed a theory which has a distinctly economic flavor (Caldwell (1982)). His view was that
fertility was high when children are an asset to their parents and low when they become a

1 In 1949 7.3 % of the population was urbanized, by 1990 20.1 % was urbanized Anderson and Ge (2006).
2 Therborn (2004) suggests that the decline was related to the increasing possibility of the number (and gender) of
children being a matter of choice with the development and spread of contraceptive and fetus gender detection
techniques. Usually boy/girl sex ratio’s at birth are around 105/100, in China in 1995 the ratio was 117/100 (Peng
and Guo (2000)).

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The One Child Policy and Family Formation in Urban China
liability, although empirical verification of the idea encountered difficulties “..the marginal value
of each extra child is impossible to determine..” (Caldwell et. al.(1982)). Becker (1993)
formalized this in developing models where both number of and quality of children and the
quality of partners feature as part of the household decision process (since, in the following, the
term “quality” refers to both gender and educational attainment rendering it a somewhat
pejorative tone, the term “type” will be employed in it’s place). Becker’s model can be used to
rationalize the effect of urbanization and the preference for sons at birth. An important feature of
Becker’s analysis is that “quantity” and “type” choices are to some degree simultaneous, with
each influencing the other to an extent3. He demonstrates that while quantity and type are likely
to be substitutes they cannot be close substitutes (because the budget constraint between quantity
and type is convex, equilibrium would not exist if the indifference curve between quantity and
type were in some sense “less” convex).
Here the OCP is construed as a rationing policy constraining the quantity (but not the
quality) of children, it is anticipated that the demand equations for quantity and quality of
children will be affected accordingly (Neary and Roberts (1980), Deaton (1981)). Furthermore if
family formation decision makers are forward looking, the policy will affect the choice of
partner decision, rendering the owner of childrearing attributes less of a comparative advantage
relative to someone with income generating attributes all other things equal. Samples of family
cohorts for the years 1989 and 1991 to 2001 are used to examine the extent to which the OCP
impacted and constrained family formation decisions, and its consequent effect on family size,

3 Family formation has most frequently been discussed in the economics literature as an adjunct to the study of
female labor supply, the issue being whether fertility should or should not be an argument in the labor supply
equation. To some extent this hinges upon the nature of the planning horizon. One culture in developing female
labor supply models is to assume that lifetime fertility decisions are made early in life, “at marriage is the most
popular choice” observes Browning (1992). An alternative culture is to assume a simultaneous model where
attempting to have more children and supplying more female labor are jointly decided upon each period, the issue
being whether fertility is endogenous to the labor supply decision. Here we abstract from the labor supply decision
and focus on the realized family outcomes of child quantity and type.

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The One Child Policy and Family Formation in Urban China
and familial investments in children. The data are drawn from an urban survey of six provinces
in China; Shaanxi, Jilin, Hubei, Sichuan, Guangdong and Shandong4. The first two may be
considered low, the second two intermediate and the third pair high income provinces.
Table 1. GDP Comparison
Unit: billion RBM Yuan
1992
33.96
36.65
69.07
87.61
127.18
128.06
2001
133.33
149.14
339.89
354.26
751.88
678.63
Shaanxi
Jilin
Hubei
Sichuan
Guangdong
Shandong
Source: SSB
Note: data of 1992 is the average GDP from 1985
to 1992, that of 2001 is from 1994 to 2001

The partner choice decision is considered in terms of the cohort of males and females by
year of birth. Specifically, we divide the sample into 3 cohorts, the first with couples whose
oldest spouse in the marriage was born between 1940 and 1949, the second cohort from 1950 to
1959, and the last from 1960 to 1969. Considering that our sample consisted of families whose
first child was born when the mother was about 25 years of age, this marriages would consists of
families whose spousal choice would have been made prior to the OCP, while the latter two
cohorts would have made their spousal choice after the OCP. Sorting is considered to take place
over the educational attainment of partners where attainment is integer indexed from 0 to 5 with
5 being college graduates and above, 4 being individuals who obtained technical education, 3
being high school, 2 being middle school, and 1 being primary school and lower.
For quantity of children and type decisions the objective is to study those households who
for all intents and purposes completed their family size decisions, so we focused on households
whose mother’s are over the ages of 25. To examine differential in choices induced by OCP, we

4 These data were obtained from the National Bureau of Statistics as part of the project on Income Inequality during
China's Transition organized by Dwayne Benjamin, Loren Brandt, John Giles and Sangui Wang.

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The One Child Policy and Family Formation in Urban China
distinguish between mothers who were 25 and younger when OCP was legislated, versus
mothers who were older than 25 in 1979. For these examinations, we use data from the odd years
from 1989 to 2001.
In the following, Section 1 formulates a simple model and develops some comparative
statics for the various family formation decisions. Partner choice decisions are examined
empirically in section 2, and child quantity and gender decisions are examined in section 3. This
is followed by child quality decisions in section 4, and a brief discussion and conclusion in
section 5.
1
A Simple Model
Consider a model where an individual lives for 2 periods, one as a child, and one as an
adult. At the beginning of the adult period, agents choose to marry or remain single (there is no
divorce in this model)5. The rate at which an adult meets someone of the opposite gender is
random. Marriage is dependent on the type of male and that of their potential spouse. We assume
transferable utility. Let h denote a husband or a male, and w denote the wife or female. Let the
agent’s type be t, continuous on a support defined within [ ] tt, , and distributed with density f(.)
and distribution F(.) for both men and women. If they find a match, they will then choose the
number of children to have, and the amount of investment in each child. The aspect of utility
derived from children is described by a function q dependent on the type of parents, the number
of children, and the amount of investment per child,
() n,kttqq
wh
,,
≡
, where The .
other aspect of a married individual’s utility is derived from personal consumption. We assume
that utility derived from child consumption and own consumption to be multiplicatively
{ }
0
+
+∈
Rq

5 The focus is the gains from marriage and how it affects matching and child investment decisions thus, without loss
of generality we solve the problem from the perspective of men, apportioning all the rents from marriage to them.
The imposition of other sharing rules will not affect the essence of the results presented below.

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The One Child Policy and Family Formation in Urban China
separable,
()h
c
hw
cknt
h
tkntqu
hw
|,, max
,,,
=
. If instead the individual chooses to remain single, utility will
only be derived from personal consumption which in turn is dependent on her own type as well,
.
{}
whics
i
c
i
i
,,max
∈=
Income realization of the family or individual is assumed to be dependent on the type of
match and the individual’s type respectively. Specifically, we assume family income to be
, and income for a single individual to be
(
whtt yx, )( )
t
i
{}
whi yv,, ∈
, where y is the average
income within the economy and . This setup thus abstracts from redistributive concerns
arising from policy. Further, this formulation of income together with the range of q ensures that
for some matches and individual types, the choice to remain single will be chosen. That is the set
of single individuals by type is non-empty.
+
∈Rvx,
We will make the following functional assumptions,
Assumption 1: Investment in children, k, and the choice of the number of children, n, are
substitutes in the function q(.), which parents derive from having children in their marriage. That
is qkn(tw,n,k|th) ≤ 0.
Assumption 2: for
0
ˆ , 0ˆ
≤≥
iii
t tt
uu
[ ]
t,
{}
whitti
, ,
∈∈
.
Assumption 3: (Complementarity of Types)
{ wh,jij , ,iu
ˆ
jit t
, 0
}
∈≠≥
. Further, let
[ ](
,
∈
tt
)
{ } t , t, on , ,|,, maxarg
**
∈==⇔=
whwhhw
t
ttttttkntut
w
.
Assumption 4: (Convex in Types When Single)
{}
whivv
iii
t tt
,, 0, 0
∈≥≥
.
Assumption 5: (Single Crossing)
0,
>
∂
∂
∂
∂
y
s
y
u
hh
,
()() 00
=≥=
ysyu
hh
, and that
y
s
y
u
∂
hh
∂
∂
≥
∂
.
Assumption 1 creates the tradeoff between the choice of investment per child, and the
number of children in a family. Assumption 2 ensures that is well behaved on the support of u ˆ

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The One Child Policy and Family Formation in Urban China
the agents’ type. Assumption 3 says that given an agent’s type, they would prefer to be matched
with someone of the same type or better. Together with assumption 4, this ensures that agents
would always prefer to match with someone closer to their own type, since the concavity of u
and the convexity of v(.) ensures that gross marital output attains a maxima for agents of a
sufficiently low type on the support. An example of a function that would meet these
assumptions is when q(.) and x(.) are quadratic functions with respect to
ˆ
()
wh
tt −
on
6
[ 1 , 0,
∈
whtt
]

6 We suspect a model with search costs that fall as agent types rises may generate similar results we present below.
.
Assumption 5 requires a more detailed explanation. First diagrammatically, it implies the
following,
u/s
y
u(y)
s(y)
y’ y’’
Figure 1: Attractiveness of Marriage

Essentially what we need is that the utility an individual gets from remaining single as a function
of average income within the economy, must intersect the utility in the marriage state from
below, for an individual given his type, and the type of his spouse. Consider first as starting point


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The One Child Policy and Family Formation in Urban China
that the curve for utility from marriage, u, and that if the individual chooses to remain single, s,
has the same vertical intercept. Ignoring intra-household bargaining, and examining the choice
for marriage from the point of view of a man, than by virtue that the choice set open to a married
man is larger, and that any choice available to a single man is likewise available to a married
man, by revealed preference, the utility from marriage must be at least as high as that derived
from the single state. Should we include intra-household bargaining, such that it implies a
reduction in the utility from marriage, it would be equivalent to a reduction of utility in the
marriage state (Assuming intra-household bargaining involves a monotonic transformation.).
This would then give us the justification for assumption 5.
In addition, we also abstract from intra-household bargaining, and focus instead on the
total value of marital output. Without loss of generality then, the solution to the individual’s
problem will be solved from the perspective of the man choosing a prospective wife.
1.1
Single Man
If an individual of type ti, chooses to remain single, he solves the following problem,
i
c
i
cs
i
max
=

subject to
( )
i
t
i
c yv
≥

where y is the average income of all individuals within the economy, and
()
{ }
0jiRttx
ji
≠+=
+,0,:
?
. Then an individual’s income is described by the product of
()
{ wh, i,jjittx
ji
, ,0,
∈≠=
} and y, which means that her income is a proportion of the average
income, dependent ultimately on the individual’s type. The optimal consumption choice is that

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The One Child Policy and Family Formation in Urban China
the individual spends all his income on herself
() 0,
==
jii
tt yxc . Let the utility of the single
individual be
()
() 0,,
==
jii
tt yxyts

1.2
Married Man
If the individual finds a suitable match and chooses marriage, he solves the following
problem subject to his budget constraint and the participation constraint in order for his
prospective spouse to enter into matrimony with him.
()h
c
hw
kccn
tkntqu
wh
|,, max
,,,
=

subject to
()
()
wwhw
w
t
( )
hwh
yvctkntq
tt yx
≥
nk
|
c
,
c
≤++
,
,

Where ch, and cw, are the consumption choices, and th and tw are the types for the husband and
wife respectively.
Further, we assume q(.), the aspect of a parent's utility derived from having children, is
increasing and concave in n (number of children) and k (investment per child). By the usual non-
satiation argument, the budget constraint holds with equality, and since the husband can always
make himself better off by just meeting the participation constraint, the participation constraint
holds with equality as well. Thus
( )
w
kn
,
()
()
( )
w
kn,
()
hw
whh
hw
w
ttq
t yv
,
nktt yxc
ttq
tyv
,
c
|
,
|
−−=⇒
=

and he solves,
()()() ( )
w
t
whhw
kn
h
yv nktt yxtkntqu
−−=
,|,,max
,


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The One Child Policy and Family Formation in Urban China
The first order conditions are,
()
()
()()
*******
|,,,|,,ktkntqkntt yxtkntq
hwwhhwn
=−
(1)
()
()
()()
*******
|,,,|,,ntkntqkntt yxtkntq
hwwhhwk
=−
(2)
Where k* and n* are the optimal values for investment per child, and number of children
respectively. In equilibrium,
()()
*
**
*
**
|,,|,,
n
tkntq
k
tkntq
hwkhwn
=

Under a situation where n is no longer a choice variable, we would have only (2). We can hence
examine changes in n on the optimal choice of k as if n is a parameter. Let when we
examine cases where the number of children is exogenously determined and let the optimal
choice of investment for each child be k′.
n
~
n
=
1.3
Comparative Statics
The One Child Policy in China coincided with the Chinese Economic Reforms in 1979
which precipitated considerable economic growth. Should the impact on familial choices yield
the same outcomes, it would not be possible to identify the policy at work. This section examines
the impact derived from both policies on quantity and quality of children, followed by spousal
matching decisions. The following four propositions relate to how the One Child Policy and
economic growth affect spousal and family size choices.
Let
()()() ( )
w
t
whhw
kn
h
yvnktt yxtkntqu
ˆ
−−=
,|,,max
,
and
( )
h
t
h
yvs =
ˆ
, then a man's second
period utility is,
{}
hhs
ˆ ,
u
ˆ
U max
=

The reservation type of the potential spouse is determined by

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The One Child Policy and Family Formation in Urban China
( ) (
yx
)() ( )
w
t
( )
h
t
RR
whh
R
w
hh
yvyvkntttkntq
s
ˆ
u
⇒=−−
=
,|,,
ˆ
(3)
Where n and k are the optimal values for match between a man of type and woman of type
ht
R
wt . Letting
() n,ttt
h
R
w
R
w
≡
, from figure 2 it may be observed that (3) determines only the lower
bound of the reservation at point A. For spousal types below
R
wt , although he may be collecting
all the rents, he obtains no net benefit from marriage. It is only above
R
wt that marital utility
would exceed his utility from remaining single.
Men of a sufficiently low type may have an upper bound on the type of his spouse,
beyond which the marital gains from the match may not be sufficient for him to ensure she
obtains at least , which is reflected by the red line in figure 2. This may be determined
from
( )
wts
( ) (
yx
)() ( )
s
−
( )
h
t
R
w
R
whh
R
w
stkntttkntq
=−
,|,,
(4)
The lower bound is point B in the above diagram. The type of woman that would present
as the optimal spousal type occurs when the marginal gain in gross marital utility from choosing
a higher type spouse, equates with the marginal increase in cost he would have to pay to meet her
participation constraint. This is depicted on the diagram where the slope of the gross utility and
are the same. Beyond this optimal type, his own marital gains start decreasing, and fall
below his value of remaining single beyond
( )
wt yv
R
wt . Note that by construction,
R
wh
R
w
ttt
≤≤
.


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The One Child Policy and Family Formation in Urban China
su/
wt
R
wt

R
wt
( )
hts

t
t
( )
wts

A
B
Figure 2: Reservation Values Given Type
()
hw
tnktu
ˆ
|,,

() ( )
w
t
hw
stnktu
ˆ
+
|,,

Intuitively, given that quantity and quality of children are substitutable, a binding policy
that impinges on a family’s choice in one dimension would yield an increase in the remaining
dimension. This is shown in proposition 1 below,
Proposition 1: An exogenously enforced reduction in the number of children raises equilibrium
investment in children.
Proof:
Let k’ be the optimal level of investment per child with n~ children in the family.
Differentiating k’ with respect to n~ from (2),

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The One Child Policy and Family Formation in Urban China
(
~
n
)
()
0
~
n
'
~
n
'
~
n
'
~
n
~
n
'
≤
−−−
−
q
−++
=
∂
∂
qk yxq
k yxqkqqq
k
nk kk
knnn

Given assumption 1, a binding constraint on the number of children, i.e. one that is lower than
what the parents would have chosen, would increase investments in children. ■
Yet, the success of the Economic Reform of 1979, which raised the income, and
consequently the quality of lives among the Chinese populace, should similarly raise familial
investments in children, assuming children are “normal goods”. That the reform came at the
same time at the OCP, would accentuate the increase in investments (holding the nominal cost of
investments constant), and consequently child quality.
Proposition 2: An exogenous increase in income would increase the number of children born
into the family and the level of investment per child.
Proof:
Differentiating (1) and (2) with respect to y respectively gives,
()()
()()
0
2
0
2
***
*
***
*
≥
−−
−=
∂
∂
≥
−−
−=
∂
∂
nqkn yxq
xq
y
k
kqkn yxq
xq
y
n
k kk
k
n nn
n

Therefore, an increase in income would increase the number of children in the family, and the
level of investment per child.■
Propositions 1 and 2 implies that the OCP and Economics Reform of 1979 would have
reinforced each other, consequently denying identification of the true cause of changes in
investment in children if any if we were to consider the impact of the OCP solely from the
perspective of child outcome. We can however examine how either policy could have effected
changes in which individuals chose their spouses. Intuitively, an additional dimension available
to individuals to adjust is spousal choice with the enactment of the OCP, the type of a spouse.

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The One Child Policy and Family Formation in Urban China
The outcome of children is dependent on both investments, as well as genetically transmitted
qualities from their parents. The exogenous imposition on quantity decision by the OCP could
have also accentuated the importance of good spousal match, assuming positive assortative
matching is the norm.
Proposition 3: When the number of children is fixed below the optimal choice that a married
couple would have chosen given their types then:
1.
for all men, the lower bound on the reservation type of a prospective spouse would rise,
while the upper bound would fall, and
2.
Agents who choose to marry would exhibit increased assortative matching.
Proof:
For the proof of point 1, differentiating
R
wt in (3) with respect to the number of children
n~,
(
)
)
−
(
0
'
~
n
'
~
n
'
~
n
~
n
≤
+−
−−
=
∂
∂
R
w
R
w
R
w
ttt
R
w
yvqyxk yxq
k yxqqk
t

Where k’ is the optimal choice of k given
h
R
ww
ttt,
=
and n~. Since n~ is binding from
below, by revealed preference the marginal benefit would be greater than the marginal cost, and
the numerator is non-positive. By assumption 3, and
h
R
w
tt ≤
, the greater the type of an
individual, the greater the gains to marriage, so the denominator is positive.
For the upper bound on the reservation value, we differentiate
R
wt in (4) with respect to
n~ as above.
(
)
)
(
0
' '
~
n
' '
~
n
' '
−
~
n
~
n
≥
−+
−
qyx
−
=
∂
∂
R
w
R
w
R
w
ttt
R
w
yvk yxq
k yxq qkt


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The One Child Policy and Family Formation in Urban China
Where k’’ is the optimal choice of k given
h
R
ww
ttt,
=
and . The numerator as before is non-
positive. By assumption 3, and
n~
h
R
w
tt ≥
, the denominator is negative. Point 1 follows.
Since there is a narrowing in the range of potential matches around the agents type,
incidences of assortative matches rise. Formally, let a man of type be matched with and
married to a woman of type . Then
ht
*
wt
*
w
()
ww
P ttt 1;
≤≤=

It follows that
*
w
∫
*
w
∫
*
w
*
w
*
w
*
w
*
w
*
w
*
w
*
w
1
( )
h
f t
1
( )
h
f t
( | )
h
t dt
( , )
f t t dt
( , )
F t t
( , )
F t t
1
tt
hhh
tt
f t
⎡
⎣
⎤
⎦
==−
.
=

The total differential w.r.t. n~ may be written as
?
n
?
n
?
?
n
?
( , )
w
F t t( , )
w
F t t
( , )
w
F t t
t
∂
( , )
w
F t t
t
∂
1
( )
h
f t
1
( )
h
f t
..
hh
hw
hw
w
w
t
t
dn dn
⎡
⎣
⎤
⎦
∂−
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
∂∂
∂
∂∂
∂
−+
∂
0
=
Since
?
n
?
n
( , )
w
F t t
t
∂
1
( )
h
f t
0,0,00
w
hw
w
t
t
and
∂
∂
∂∂
∂
>>>
<
It may be observed that
?
n
( , )
w
F t t ( , )
w
F t t
0.
hh
⎡
⎣
⎤
⎦
∂−
<
∂
■
Essentially there are three types of men, those who would always prefer not to marry (L),
those who benefit from marriage, but who would never be able to attract high type spouses (M)
relative to their own type and those who are coveted by all spousal types (H). Each of these is
depicted in figure 3.

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