university of colorado march 3, 2015 support for this research was provided by the u.s. national...

University of ColoradoMarch 3, 2015

Support for this research was provided by the U.S. National Institute of Child Health and Human Development.

Family Size of Women and Children during the Demographic Transition

David LamUniversity of MichiganLetícia MarteletoUniversity of Texas

Family size and cohort size• This paper is a sequel to previous papers looking at family size and

cohort size during the demographic transition:• David Lam and Letícia Marteleto, “Stages of the Demographic

Transition from a Child’s Perspective: Family Size, Cohort Size, and Children’s Resources,” Population and Development Review, June 2008, 34(2): 225-252.

• David Lam and Letícia Marteleto, “Small Families and Large Cohorts: The Impact of the Demographic Transition on Schooling in Brazil,” in The Changing Transitions to Adulthood in Developing Countries, National Academies Press, Washington, DC, 2006, pp. 56-83.

• We showed that family size declines while cohort size increases during a long period of the demographic transition – 20 to 30 years in most countries. During this period competition for resources within families is declining, while competition with cohort members is increasing.

Number of surviving siblings of 9-11-year-olds and total number of 9-11-year-olds in population, Brazilian censuses 1960-2000

0.0

1.0

2.0

3.0

4.0

5.0

6.0

1960 1965 1970 1975 1980 1985 1990 1995 2000

Nu

mb

er o

f si

bli

ng

s

0

50

100

150

200

250

Co

ho

rt s

ize

(196

0=10

0)

Stage 1, rising family size and

cohort size

Stage 2, falling family size and

rising cohort size

Stage 3, falling family size and

cohort size

Number of surviving siblings of 9-11 year-olds

Total number aged 9-11 in population

David Lam and Letícia Marteleto, “Stages of the Demographic Transition from a Child’s Perspective: Family Size, Cohort Size, and Children’s Resources,” Population and Development Review, June 2008.

Purpose of this paper• How are changes in the family size of children

related to changes in fertility during the demographic transition?

• This is important if children are competing for resources with siblings.

• While it may seem obvious that children’s family size will fall when fertility declines, they do not need to fall at the same rate or even change in the same direction.

• We expand on Preston (1976) and study a large number of countries during the demographic transition.

• We also look at the inequality in family size, extending Preston’s results.

Distribution of family size for womenNumber of children surviving to women aged 45-49,

Brazil 1960

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.02

0.04

0.06

0.08

0.1

0.12

Women (mean=4.5)

Number of children

Den

sity

Children’s family size as a function of women’s family size

fw(s)=pdf for family size of women aged 45-49

fc(s)=pdf for family size of children of women aged 45-49

Ws = mean family size for women aged 45-49

0

( ) ( )( )

( )

( ) ( ) if

( ) ( ) if

( ) ( ) if

W WC n

WW

C W W

C W W

C W W

sf s sf sf s

ssf s ds

f s f s s s

f s f s s s

f s f s s s

Mean family size of women and childrenNumber of children surviving to women aged 45-49,

Brazil 1960

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.02

0.04

0.06

0.08

0.1

0.12

0.14Women (Mean=4.5)Children (Mean=7.1)

Number of children

Den

sity

Sam Preston, Family Sizes of Children and Family Sizes of Women,” Demography 1976


W = standard deviation of family size for women aged 45-49

Cs = mean family size of their children

02

0

0 00

2 2 2

20 0

( )

( )( )

( )( )

( ) ( )

n

W W

n

Wn nW

C C nWW

n n

W W W W

WC W

W W

s s f s ds

s f s dssf s

s s f s ds s dsssf s ds

s s f s ds s f s ds

s ss s

( )Cf s

Sam Preston, Family Sizes of Children and Family Sizes of Women,” Demography 1976


W = standard deviation of family size for women

Cs = mean family size of their children

2W

C WW

s ss

(1)

CVW = coefficient of variation of women’s family size ( / )W Ws

22

21 (1 )W

C W W WW

s s s CVs

(2)

Fertility can decline without a decrease in mean family size for children

50% of women have no children 50% of women have 6 children

Mean family size for women = 1/2*2 +1/2*6 = 4.0Mean family size for children = 2/8*2 + 6/8*6=0.5+4.5 = 5.0

50% of women have 6 children50% of women have 2 children

Mean family size for women = 1/2*0 +1/2*6 = 3.0 (decrease of 1)Mean family size for children = 6.0 (increase of 1)

Case 2: After fertility decline

Case 1: Before fertility decline

Implications of Preston’s result

• Mean children’s family size is always greater than mean women’s family size, as long as there is any variance in fertility

• Fertility decline and family size:– If CV goes up while mean fertility declines, children’s

family size will decline more slowly than women’s family size

• Comparing subgroups:– If a group with higher fertility also has a higher CV,

then the difference in family size of children will be larger than the difference in family size of women.

2(1 )C W Ws s CV (2)

Preston’s analysis of U.S. data• Preston found exactly these two patterns in historical U.S.

data for women aged 45-49

Year

Women's mean

family size

Children's mean

family sizeRatio

(2) / (1)CV for

women(1) (2) (3) (4)

1890 4.99 7.78 1.56 0.751910 4.09 7.17 1.75 0.871940 2.66 5.36 2.02 1.011950 2.29 4.91 2.14 1.071960 2.25 4.41 1.96 0.981970 2.71 4.46 1.65 0.81

Women’s family size fell 54% from 1890-1950, but children’s family size fell only 37%

Due to rising CV

Year

Women's mean

family size

Children's mean

family sizeRatio

(2) / (1)CV for

women(1) (2) (3) (4)

1970 White 2.65 4.20 1.58 0.761970 Nonwhite 3.16 6.32 2.00 1.00

Nonwhite women had 19% higher family size than white women, but nonwhite children had 50% higher family size than white children

Due to larger CV for nonwhites

Implications for family size during demographic transition

• “These patterns are a disconcerting precedent for those concerned with issues of population quality in less developed countries; the pace of reductions in family size for children can be expected to lag behind that for women in the process of fertility transition” (Preston 1976: 108).

• One purpose of this paper is to test this prediction across a wide range of countries.

We extend Preston’s results to look at the standard deviation of children’s family size

C = standard deviation of family size for children

W = standard deviation of family size for women

CVW = coefficient of variation of women’s family size ( / )W Ws

SW = skewness of family size for women

2 2 1 ( )C W W W WCV S CV (3)

Empirically, skewness is slightly negative at high levels of fertility and becomes increasingly positive as fertility declines. The term in brackets tends to be less than 1 at high levels of fertility and greater than 1 at low levels of fertility.

Increasing skewness causes children’s standard deviation to fall more slowly than women’s standard deviation as fertility declines.

Number of children surviving to women aged 45-49, Brazil 1960-2010

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.05

0.1

0.15

0.2

0.25

0.3

0.351960

1970

1980

1991

2000

2010

Number of children

Den

sity

Distribution becomes more skewed as fertility declines

Surviving family size of children of women aged 45-49, Brazil 1960-2010

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.05

0.1

0.15

0.2

0.25

0.3

0.351960

1970

1980

1991

2000

2010

Number of children

Den

sity

This distribution also becomes more skewed as fertility declines, but is shifted to right compared to distribution for women.

Distribution of surviving family size for women aged 45-49 and

children of women aged 45-49, Brazil 1960-2010

Note that bottom figure is a reweighted

version of the top figure

Women’s family size

Children’s family size

Distribution of family size for women aged 45-49 and

children of women aged 45-49

Thailand 1970 and 2000

0.0

5.1

.15

.2.2

5.3

.35

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Women

Children

Thailand 2000

0.0

5.1

.15

.2.2

5.3

.35

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Women

Children

Thailand 1970

Mean for women=6.53Mean for children=8.21

(ratio 1.26)

Mean for women=2.55Mean for children=3.47

(ratio 1.36)

Empirical Analysis• We use IPUMS-International census data and

Demographic and Health Surveys• We use 101 countries, 310 total data sets, including

high-income countries with fertility data• Scatterplots include both cross-sectional variation

across countries and time series variation within countries

• We show estimates of OLS regression and Fixed Effect (FE) regression using only within-country variation over time.

Summary of datasets with children ever born variable

Total IPUMS-I DHSTotal datasets 310 144 166Number of countries 101 59 68Number of countries with multiple years 74 41 43

Coefficient of variation in women’s family size, by mean family size, children ever born to women aged 45-49

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Women’s mean family size

Co

effi

cien

t o

f va

riat

ion

Slope=-0.051 (0.003)

b(FE)=-0.026 (0.005)N=310 (101 countries)

CV rises as fertility falls

Coefficient of variation in women’s family size, by mean family size, children ever born to women aged 45-49

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Chile 1960

China 1990 Jordan 1990

Slovenia 2002

Ukraine 2007

Uruguay 1975


Co

effi

cien

t o

f va

riat

ion

Slope=-0.051 (0.003)

b(FE)=-0.026 (0.005)N=310 (101 countries)

CV rises as fertility falls

Skewness in women’s family size, by mean family size, children ever born to women aged 45-49

1 2 3 4 5 6 7 8 9-1

-0.5

0

0.5

1

1.5

2

2.5


Ske

wn

ess

Slope=-0.307 (0.010)

b(FE)=-0.271 (0.017)

Skewness is negative at high fertility levels, increases to highly positive at low fertility levels

Mean children’s family size by mean women’s family size, children ever born to women aged 45-49

1 2 3 4 5 6 7 8 91

2

3

4

5

6

7

8

9

10

45-degree lineChildren's meanOLS line


Ch

ild

ren

’s m

ean

fam

ily

size

Slope=1.041 (0.017)

b(FE)=1.191 (0.029)

Standard deviation in children’s family size, by mean number of children ever born to women aged 45-49

1 2 3 4 5 6 7 8 90

1

2

3

4

5

6


S.D

. ch

ild

ren

’s f

amil

y si

ze

Slope=-0.059 (0.022)

b(FE)= 0.031 (0.036)

Standard deviation of children’s family size rises slightly with falling fertility, in spite of the large decline in the mean

Multipliers for mean and standard deviation

2(1 )C W W Ws s CV s k

2 2

1/2

1 ( )

1 ( )

C W W W W

C W W W W

C W

CV S CV

CV S CV

m

If k and m are constant, then mean and standard deviation of children’s family size will decline at same rate as the mean and standard deviation of women’s family size. We look at how k and m vary with the mean of women’s family size.

Ratio of children’s mean family size to women’s mean family size, by mean family size of women aged 45-49, children ever born

1 2 3 4 5 6 7 8 91

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


k=C

hil

dre

n’s

mea

n /

Wo

men

’s m

ean

Slope= -0.060 (0.004)

b(FE)= -0.028 (0.006)

Preston was right that children’s family size falls more slowly than women’s family size

South Africa2007, 2001, 1996


1 2 3 4 5 6 7 8 91

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

China 1982, 5.36*1.17=6.29

Mexico 1960, 5.16*1.67=8.62


k=C

hil

dre

n’s

mea

n /

Wo

men

’s m

ean

Mexico 1960 and China 1982 had similar fertility levels, but mean family size of children was 37% higher in Mexico (2.4 extra siblings)

Ratio of children’s mean family size to women’s mean family size, by mean family size of women aged 45-49, surviving children

1 2 3 4 5 6 7 8 91

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


k=C

hil

dre

n’s

mea

n /

Wo

men

’s m

ean

Slope= -0.059 (0.006)

b(FE)= -0.040 (0.005)

Pattern using surviving children is very similar to pattern using children ever born

N=266 (87 countries)

1.1

1.2

1.3

1.4

1.5

1.6

1.1

1.2

1.3

1.4

1.5

1.6

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

A. Brazil B. Colombia C. Mexico D. Chile

E. DominicanRepublic F. Ecuador G. Panama H. Peru

Ch

ildre

n's

me

an

/ W

ome

n's

mea

n

Mean number of surviving children, women 45-49

Figure X. Ratio of children's family size to women's family size

1.1

1.2

1.3

1.4

1.5

1.6

1.1

1.2

1.3

1.4

1.5

1.6

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

I. Thailand J. Indonesia K. Philippines

L. Bangladesh M. Jordan N. Morocco

Ch

ildre

n's

me

an

/ W

ome

n's

mea

n



1.1

1.2

1.3

1.4

1.5

1.6

1.1

1.2

1.3

1.4

1.5

1.6

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

O. Kenya P. Ghana Q. Rwanda

R. Tanzania S. Zambia T. Zimbabwe

Ch

ildre

n's

me

an

/ W

ome

n's

mea

n



Implications for children’s resources

• Suppose some fixed resource, like mother’s time, is divided by the number of children.

• Suppose women’s family size falls from 8 to 2.• If children’s family size also fell to 1/4 its original size, then

resources per child would go up 4 times.• Alternatively, suppose the ratio of children’s family size to

women’s size increases from 1.2 to 1.5 (children’s family size falls from 9.6 to 3.0).

• Resources per child increase by 3.2 times instead of 4 times. The increase is 20% smaller than it would have been if children’s family size had fallen as fast as women’s family size.


1 2 3 4 5 6 7 8 91

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2


k=C

hil

dre

n’s

mea

n /

Wo

men

’s m

ean

Increase in ratio from 1.2 to 1.5 when fertility falls from 7 to 2. This implies a 20% smaller increase in resources per child than if ratio remained constant. (1.2/1.5=0.8)

“Leakage” in resources per child2(1 )

ln( ) ln( ) ln( )

ln( ) ln( ) ln( )

C W W W

C W

C W

s s CV s k

s s k

s s k

t t t

If k is constant, then a 10% decline in mean women’s family size will lead to a 10% decline in mean children’s family size, and a 10% increase in resources per child. An increase in k by 10% will completely offset this. The percentage change in k can be thought of as a “tax” or “leakage” in the increase in resources per child that would have occurred from the decline in fertility.

“Leakage” in resources per child – Regression of log(k) on log(children ever born)

OLS FE OLS FElog(CEB-4549) -0.179 -0.084

[0.012]** [0.021]**log(Surv-4549) -0.167 -0.118

[0.017]** [0.017]**Constant 0.57 0.419 0.523 0.449

[0.020]** [0.033]** [0.026]** [0.026]**

Observations 310 310 266 266R-squared 0.42 0.07 0.27 0.21Number of countries 101 87

Standard errors in brackets* significant at 5%; ** significant at 1%

log(k) using surviving children

log(k) using children ever born

Family size of school-aged children• The children of women aged 45-49 are not

necessarily an interesting group.– Many of them are not children– They span a large age range.

• We may be more interested in a particular age range of children, say school-age children.

• We will focus on children aged 10 (or 9-11).• We show empirically that the following is a good

approximation for describing the mean family size of ten-year old children:

2(10) (25 49) (25 49)1C W Ws s CV (4)

Family size of school-aged children

• Calculate the family size of children aged 9-11:– Match them to their mothers in the censuses and

DHS surveys– Take the number of surviving children of their mothers

• Generate the surviving family size of women 25-49 using the same reweighting used above

• A comparison shows that these distributions are very similar in all countries.

• Children aged 9-11 are roughly representative of the children of women aged 25-49

Family size of children aged 9-11 and children of women aged 25-49, Brazil 1960

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16 Children 9-11

Children of women 25-49

Number of children

Den

sity

Mean family size of children aged 9-11 and children of women aged 25-49

12

34

56

7M

ean

fam

ily s

ize

of c

hild

ren

9-1

1

1 2 3 4 5 6 7Mean family size of children of women 25-49

Children 9-1145-degree line

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion

0 1 2 3 4 5 6 7 8 9 10 11Number of surviving children

20001991198019701960

Women 25-49, Brazil

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


20001991198019701960

Children 9-11, Brazil

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974

Women 25-49, Ecuador

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974

Children 9-11, Ecuador

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


200019841980

Women 25-49, Costa Rica

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


200019841973

Children 9-11, Costa Rica

Figure 3. Distribution of family size, women 25-49 and children 9-11

Cumulative distribution of family size for

women aged 25-49 and children aged

9-11, Brazil, Ecuador, and Costa

Rica

Large families are much more common for children than for

women.

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


20001991198019701960

Women 25-49, Brazil

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


20001991198019701960


0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974


0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974


0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


200019841980

Women 25-49, Costa Rica

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


200019841973

Children 9-11, Costa Rica

Figure 3. Distribution of family size, women 25-49 and children 9-11

In Brazil 80% of women aged 25-49 had less than four surviving children in

2000; Only 58% of children were in families with less than four surviving children in

2000, similar to the proportion for women in

1960.

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974


0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974


In Ecuador in 2001 only 3.2% of women were in a family with 8 or more children, but 10.6%

of children were in a family with 8 or more surviving children.

Inequality in children’s family size• Women’s standard deviation falls at slightly lower rate

than the mean• As a result, children’s mean falls at slightly lower rate

than women’s mean• But children’s standard deviation falls much more slowly

– it tends to stay roughly constant due to increase in skewness in women’s family size

• These imply that the coefficient of variation in children’s family size must increase as mean family size declines

• In general there is a substantial increase in inequality in children’s family size as mean family size declines

.2.3

.4.5

.6.7

.8.9

1C

oe

ffic

ien

t o

f v

ari

ati

on

1 2 3 4 5 6 7 8Mean number of children ever born, Women 45-49

Coefficient of variation of surviving family size of children by mean family size of women age 45-49

Coefficient of Variation of children’s family size increases as fertility declines

Slope= -0.046 (0.005)

b(FE)= -0.045 (0.005)

0.1

.2.3

.4.5

.6.7

.8.9

1C

um

ula

tiv

e p

rop

ort

ion

0 1 2 3 4 5 6 7 8 9 10 11Number of surv iv ing children

20001991198019701960


Brazil:90th percentile fell from 9 to 7 from

1960 to 2000

10th percentile fell from 2 to 1.

So 90/10 ratio rose from 4.5 to 7

If there are fixed resources in families, the 10th percentile child goes from getting 4.5

times as much as the 90th percentile child to getting 7 times as much. This is in addition to inequality in the amount of the fixed resource

between small and large families.

0.1

.2.3

.4.5

.6.7

.8.9

1C

umul

ativ

e pr

opor

tion


2001199019821974


Ecuador:80th percentile fell from 8 to 6 from

1974 to 2001

20th percentile fell from 4 to 2; 80/20

ratio rose from 2 to 3

12

34

56

78

Ra

tio

s o

f p

erc

en

tile

s

1 2 3 4 5 6 7 8Mean number of children ever born, Women 45-49

P90/P10

P80/P20

90/10 and 80/20 ratios for children’s surviving family size, by mean family size of women 45-49

Slope= -0.138 (0.026)

b(FE)= -0.151 (0.030)

Slope= -0.325 (0.097)

b(FE)= -0.426 (0.082)

12

34

56

78

12

34

56

78

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

A. Brazil B. Colombia C. Mexico D. Chile

E. DominicanRepublic F. Ecuador G. Panama H. Peru

P90/P10 P80/P20

Ra

tio o

f pe

rcen

tile

s


Figure X2. Measures of inequality of children's family size

12

34

56

78

12

34

56

78

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

I. Thailand J. Indonesia K. Philippines

L. Bangladesh M. Jordan N. Morocco

P90/P10 P80/P20

Ra

tio o

f pe

rcen

tile

s



12

34

56

78

12

34

56

78

2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8

O. Kenya P. Ghana Q. Rwanda

R. Tanzania S. Zambia T. Zimbabwe

P90/P10 P80/P20

Ra

tio o

f pe

rcen

tile

s



Conclusions• Mean family size of children tends to fall more slowly

than mean family size of women as fertility declines.• The ratio of mean children’s family size to mean women’s

family size rises from about 1.2 to 1.5 as fertility falls, the result of an increase in the coefficient of variation in women’s family size.

• There is roughly a 20% smaller increase in resources per child during the demographic transition than would be implied by fertility decline alone.

• Inequality in children’s family size tends to increase with the decline in fertility, the result of increasing skewness in women’s family size. The gap in resources per child between large and small families increases as fertility declines.

university of colorado march 3, 2015 support for this research was provided by the u.s. national...

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