Supplement to: “The Value of Children: Inter-Generational Support, Fertility, and Human Capital”∗ Jaqueline Oliveira† Clemson University This Draft: October 28, 2015

FOR ONLINE PUBLICATION

∗

This paper is part of my Ph.D. research. I gratefully acknowledge graduate research funds from the Cowles Foundation, the Economic Growth Center, and the Sasakawa Foundation. I am extremely thankful to Mark Rosenzweig, T. Paul Schultz, and Christopher Udry for their guidance and support. I have also benefitted from comments by three anonymous referees, Bruno Badia, Dan Keniston, Tom Mroz, Albert Park, John Strauss, Nancy Qian, Michael Makowsky, Chuck Thomas, and all the participants of the Development Workshops at Yale University, of the NEUDC Conference (2012), and of the 6th CESI Meeting (2013). † Email: [email protected]

A A.1

Model Partial Effects of n on q, s, and cf in the Model without Transfers from Children

In this subsection, I derive the effects of an exogenous change in n on the demands for the other endogenous variables. Throughout the derivations, I assume that the parents’ utility function is separable in its arguments. The approach is to treat n as a parameter n, and derive comparative statics results around the optimal n. Totally differentiating the first-order conditions from parents’ optimization problem with respect to n and I, while imposing that cp = s, yields the following set of simultaneous linear differential equations written in matrix form:      0 −πq n −πs −1 dλ πq q −1    −πq n Uqq     0 0     dq  =  λπq 0  dn . (A.1)  −πs 0 Upp 0   ds   0 0  dI −1 0 0 Uf f dcf 0 0 The second-order conditions for utility maximization are ∆ < 0 and φ22 , φ33 , and φ44 > 0, where ∆ is the determinant of the bordered Hessian matrix and φ22 , φ33 , and φ44 are the cofactors of the elements of the principal diagonal. Using the Cramer’s rule to solve for dq, ds, and dcf , I obtain dq = ds = dcf =

1 {−φ12 (πq qdn − dI) + φ22 λπq dn} ∆

(A.2)

1 {φ13 (πq qdn − dI) − φ23 λπq dn} ∆

(A.3)

1 {−φ14 (πq qdn − dI) + φ24 λπq dn}, ∆

(A.4)

where

φ12 = −πq nUpp Uf f

<0

φ13 =

πs Uqq Uf f

>0

φ14 =

−Uqq Upp

<0

φ23 =

−πq nπs Uf f

>0

φ24 =

πq nUpp

< 0.

One can decompose the effect of an increase in n into two parts. The first part is the income effect from the increase in the price of q, which follows from the interaction between n and q in the budget constraint. The signs of the cofactors in this problem imply that the income effects are negative for all the variables. The “compensated price effect” (obtained by setting πq qdn = dI) is negative in (A.2) and positive in (A.3) and (A.4).

A.2

Partial Effects of n, q, and s on T in the Model with Transfers from Children

In this subsection, I present detailed derivation of the partial effects of n, q, and s on total transfers T . Assume that a child’s utility V (ck , cp ) is separable in its arguments, so that one can re-write

1

the first-order condition 1 as Vk (q − T ∗ ) = Vp (s + T ). Assuming Vk (.) and Vp (.) are continuous and strictly increasing, T ∗ = q − Summing over across all children yields T = nq − nVk−1 (Vp (s + T )) Vk (q −

T ) = Vp (s + T ). n

(A.5) Vk−1 (Vp (s

+ T )).

(A.6) (A.7)

From equation (A.7), it follows that the partial effects of T with respect to n, q, and s are ∂T = ∂n

1 Vpp Vkk

∂T = ∂q

1 n

+

T >0 n2

(A.8)

>0

(A.9)

< 0.

(A.10)

T >0 n2

(A.11)

>0

(A.12)

> 0.

(A.13)

1 Vpp Vkk

+

1 n

−1

∂T = ∂s

Vkk 1 Vpp n

+1

Consequently, because cp = s + T , ii follows that ∂cp = ∂n

1 Vpp Vkk

∂cp = ∂q ∂cp = ∂s

A.3

1 n

+ 1

Vpp Vkk

+

1 n

1 Vpp n Vkk

+1

Partial Effects of n on q, s, and cf in the Model with Transfers from Children

In this subsection, I derive the effects of an exogenous change in n on the demands for q, cf , and s when parents’ old-age consumption is determined by n and q, in addition to s. Let us denote the predetermined level of fertility by n. First, I define the utility function in the problem with transfers as U (n, q, cp (n, q, s), cf ) = U (n, q, s, cf ), where cp (n, q, s) is obtained from the adult children’s problem derived in subsection 2.2. The overbar indicates that I am now dealing with the problem where children make transfers to parents in the second period. Totally differentiating the first-order conditions from problem (2) with respect to n and I yields the following system of differential equations:

2



 0 −πq n −πs −1 dλ  −πq n U qq   U 0 qs    dq  −πs U sq Uss 0   ds dcf −1 0 0 U ff

 πq q −1     λπq − U qn 0  dn  = ,   −U sn 0  dI 0 0 



(A.14)

where

U qq

∂ 2 cp = Uqq + Up 2 + Upp ∂q {z |



p Uqq

U qn

∂cp ∂q

2 (A.15) }

   ∂ 2 cp ∂cp ∂cp = Up + Upp ∂q∂n ∂q ∂n | {z }

(A.16)

   ∂cp ∂cp ∂ 2 cp + Upp = Up ∂q∂s ∂q ∂s | {z }

(A.17)

" #  ∂ 2 cp ∂cp 2 = Upp + Up 2 + Upp −1 ∂s ∂s {z } |

(A.18)

   ∂cp ∂cp ∂ 2 cp + Upp . = Up ∂s∂n ∂s ∂n | {z }

(A.19)

p Uqn

U qs = U sq

p Uqs

U ss

p Uss

U sn

p Usn

Solving for dq, ds, and dcf , yields the following: dq = ds = dcf =

1 {−φ12 (πq qdn − dI) + [φ22 λπq + (−φ22 U qn + φ32 U sn )]dn} ∆

(A.20)

1 {φ13 (πq qdn − dI) + [−φ23 λπq + (φ23 U qn − φ33 U sn )]dn} ∆

(A.21)

1 {−φ14 (πq qdn − dI) + [φ24 λπq + (−φ24 U qn + φ34 U sn )]dn}, ∆

(A.22)

where ∆ is the determinant of the bordered Hessian matrix in (A.14) and φij s are the cofactors of the bordered Hessian. The second-order conditions for utility maximization imply that ∆ < 0 and φ22 , φ33 , and φ44 > 0. The signs of the effects in (A.20)–(A.22) depend not only on the assumptions about parents’ preferences but also on the adult children’s preferences. From equations (A.11), (A.12), and (A.13), one can see that the signs of the second derivatives of the old-age consumption function, cp (n, q, s), Vpp play an important role and depend closely on Vkk . Notice that in order to determine these signs, I need to make assumptions about the signs and the magnitudes of the third derivatives of children’s utility from own consumption and parents’ consumption. To simplify the analysis, I consider the

3

cases in which Vppp = Vkkk = 0.1 In this case, it follows that

∂ 2 cp ∂ 2 cp , ∂n2 ∂n∂s

< 0,

∂ 2 cp ∂n∂q

> 0, and

∂2c ∂ 2 cp , and ∂q∂sp = 0.2 Therefore, I can sign the last terms in equations (A.15)–(A.19) ∂q 2 p p p p p p Uqq , Uqs , Uqq , and Usn < 0, and Uss > 0. The sign of Uqn cannot be determined.

∂ 2 cp , ∂s2

as follows:

The relationships between the cofactors of the problem with transfers and the problem without are as follows: p φ12 = φ12 − πq nUss Uf f + πs U qs Uf f

≶0 ≶0

φ22 =

p φ13 − πq nU qs Uf f + πs Uqq Uf f p φ14 + (U qs )2 − U ss Uqq p φ22 − Uss

φ23 =

φ23 − U qs

>0

φ24 =

p πq nUss

≶0

φ13 = φ14 =

φ24 +

φ33 −

φ33 =

− πs U qs

p Uqq

≶0 >0

> 0.

In conclusion, when the model is extended to account for the role of children in parents’ oldage consumption, it is not clear that changes in the economic environment that lower fertility consequently increase parental investments in children’s human capital, even when preferences do not imply complementarity between the quantity and the quality of children. The ambiguity arises from the interaction between the quantity and the quality of children in children’s behavior ∂2c regarding old-age transfers. This interaction is embodied in the derivative ∂q∂np , which is positive under the assumption that Vppp = Vkkk = 0. The result also holds for Cobb-Douglas utility functions.

A.4

Cobb-Douglas Case

In this subsection, I derive the second own and cross derivatives of cp with respect P to n, q, and s. P Assume that V (q − Tk , s + Tk + j6=k Tj ) = αk log(q − Tk ) + αp log(s + Tk + j6=k Tj ), where αk and αp are parameters satisfying αk > 0, αp > 0. Assuming an interior solution, one can derive the utility with respect to Tk and find the symmetric equilibrium transfer, T ∗ , which is given by the following equation: T∗ =

αp q − αk s . αp + αk n

Parents’ old-age consumption can be written as follows:   αp q − αk s . cp = s + n αp + αk n It follows from these functional forms that ∂cp αp T ∗ = >0 ∂n (αp + αk n) 1

(A.23)

In fact, this assumption is stronger than required for the following results to hold. For instance, they hold in the case of Cobb-Douglas utility. 2

From Young’s theorem,

∂ 2 cp ∂s∂q

< 0,

∂ 2 cp ∂q∂n

> 0, and

∂ 2 cp ∂s∂q

= 0.

4

∂cp αp n = >0 ∂q (αp + αk n)

(A.24)

αp ∂cp = > 0. ∂s (αp + αk n)

(A.25)

Deriving equations (A.23)–(A.25) with respect to n, q, and s gives ∂ 2 cp 2αk αp T ∗ = − <0 ∂n2 (αp + αk n)2

(A.26)

αp2 ∂ 2 cp ∂ 2 cp >0 = = ∂n∂q ∂q∂n (αp + αk n)2

(A.27)

∂c2p ∂c2p αp αk = =− <0 ∂n∂s ∂s∂n (αp + αk n)2

(A.28)

∂ 2 cp =0 ∂q 2

(A.29)

∂ 2 cp ∂ 2 cp = =0 ∂q∂s ∂s∂q

(A.30)

∂ 2 cp = 0. ∂s2

(A.31)

The signs of the derivatives in equations (A.26)–(A.28) follow from the fact that T = nT ∗ and ∗ ∂T ∗ ∂T ∗ that ∂T ∂n < 0, ∂q > 0, and ∂s < 0.

A.5

Heterogenous Children

In this subsection, I derive the effect of an increase in the number of children on total transfers to senior parents in the case where there is heterogeneity in children’s quality. I maintain the assumption that children have same preferences over own consumption and parents’ consumption. Assuming that a child’s utility V (ck , cp ) is separable in its arguments, the equation that determines the equilibrium transfer of child i with quality endowment qi is Vk (qi −

Ti∗ )

= Vp (s +

n X

Tj∗ ).

(A.32)

j=1

Assuming Vk (.) and Vp (.) are continuous and strictly increasing, the optimal transfer of child i can be written as Ti∗ = qi − Vk−1 (Vp (s +

n X j=1

Summing over i = 1, 2, ..., n yields

5

Tj∗ )).

(A.33)

T = q − nVk−1 (Vp (s + T )) Vk (

(A.34)

q−T ) = Vp (s + T ), n

(A.35)

P P where T = ni=1 Ti∗ and q = ni=1 qi∗ . From equation A.35, T can be written as a function of n, (q1 , q2 , ..., qn ), and s. Notice that T depends on the vector of children’s quality only through Q = nq , where Q is average children’s quality. What matters for determining total transfers to parents is the average children’s quality (income), not its distribution. Differentiating with respect to n while holding Q and s constant yields3 ∂T = ∂n

1 Vpp Vkk

+

1 n

T > 0. n2

There are two important conclusions from this extension: 1. Heterogeneity in children’s quality does not change the partial effect of the number of children on total transfers to senior parents. 2. The partial effect of the number of children on total transfers does not depend on the quality of the additional child, provided that average children quality is held constant. In equilibrium, the number of children will affect transfers through its effect on parents’ optimal choices of Q and s. The total effect of n on T is ! ! ! ∂Q 1 T 1 −1 ∂s dn + Vpp dn + dn. dT = Vpp V 2 1 + V kkn ∂n +1n + 1 ∂n Vkk

n

Vkk

n

pp

Consider an increase in the number of daughters versus an increase in the number of sons. Holding Q and s constant, the effect on total transfers to senior parents is the same. The overall effect on total transfers, however, depends on how parental investment in children’s quality and old-age savings respond to the gender of the additional child. The additional child’s gender might lead to different effects on the optimal choice of Q depending on parental preferences over sons’ quality and daughters’ quality. Additionally, parents might save more in response to an increase in the number of sons compared to an increase in the number of daughters because sons need additional resources to make them competitive in the marriage markets (Wei and Zhang (2011)). 3

An increase in n that keeps Q constant is such that quality.

6

Pn

∂qi i=1 ∂n

= Q0 , where Q0 is the initial level of average

B

Endogeneity in Age First Birth

Rosenzweig and Wolpin (1980) present a formal proof for why the effect of first-born twins is still unbiased, despite the fact that mother’s age at first birth is a choice. The idea is as follows. Let N be the number of children, F BT be an indicator for first-born twins, and AF B be mother’s age at first birth. Let u represent the purely random component of the incidence of first-born twins. If we could observe u we would estimate N = βu + η. A simple OLS regression would yield unbiased estimates of β since u and η are uncorrelated. However, we can decompose the first-born twin incidence as: F BT = αAF B + u, where AF B and u are independent. This yields u = F BT − αAF B. Substituting the latter equation into the first one: N = βF BT + βαAF B + η. Despite AF B being correlated with η, because all variation in F BT after controlling for AF B is through u, and u is independent of AF B, OLS will yield consistent estimate of β. In my analysis, however, including age at first birth does not change the main estimates, as seen in Table B.1. The first stage coefficient on the first-born dummy increases, reflecting the inverse association between age at first birth and twinning. However, all the estimates of the fertilitytransfer relationship are virtually unchanged when we exclude age at first birth from the estimated model.

7

8

0.77*** (0.095)

2.76 8818

0.72** (0.35) 0.34*** (0.034) -0.032*** (0.012) -0.022 (0.037)

1786.96 1786.96 8818 8818

0.70* (0.37) 0.34*** (0.035) -0.035** (0.014)

0.33 8783

0.085** (0.040) 0.055*** (0.0065) -0.0054*** (0.0019)

0.33 8783

0.085** (0.038) 0.053*** (0.0060) -0.0050*** (0.0016) -0.0025 (0.0046)

0.32 8818

0.10*** (0.034) 0.038*** (0.0066) -0.0021 (0.0018)

0.32 8818

0.098*** (0.034) 0.039*** (0.0062) -0.0035** (0.0016) 0.0083** (0.0038)

Robust standard errors in parentheses. Mother’s education was imputed to missing observations using the

2.76 8818

0.049*** (0.014) -0.0052 (0.0033) -0.11*** (0.0035)

0.81*** (0.087)

Transfer amount Got transfers Co-reside (3) (4) (5) (6) (7) (8) IV IV IV Probit IV Probit IV Probit IV Probit b/se b/se b/se b/se b/se b/se

Stars indicate statistical significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

columns (7) and (8) it is a dummy for whether parents live with at least one married child aged 22 years or older.

children; in columns (5) and (6) is an indicator for whether parents received positive net transfers from children; in

variable in columns (3) and (4) is the (log) total transfer amount received from children net of transfers made to

had first-born twins. Dependent variable in columns (1) and (2) is the total number of children ever born. Dependent

regressions but not reported. Number of children is instrumented using an indicator variable for whether the mother

of birth, county FE, and dummy for whether information on mother’s education is imputed) included in all level

controls (mother’s current age squared, parent is married, agricultural hukou, indicator for missing children’s month

predicted values from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional

Notes:

Mean dep. var. Number of obs

0.051*** (0.015) Mother’s education -0.021*** (0.0035) Age 1st birth

Mother’s age

N. children

First-born twins

N. children (1) (2) OLS OLS b/se b/se

Table B.1: Number of children and old-age support, controlling for mother’s age at first birth

C

Additional Results

This section presents additional results which complement the findings discussed in the paper.

Mother’s Health Outcomes and First-Born Twins Table C.1 shows regressions of each health outcome variable on the first-born twin instrument separately. These additional estimates confirm that there is no significant relationship between maternal health and the incidence of first-born twins.

LATE and First-Born Twin Instrument The first-born twin instrument is likely to induce changes in fertility close to the first parity. Indeed, this is confirmed by the results from Table C.2 below. This characteristic of the first-born twin instrument needs to be considered when interpreting the IV estimates.

Adult Children The CHARLS provides data on basic demographic and economic characteristics of non-resident and resident children of senior respondents such as gender, age, education, working status, and income brackets.4 Among the children of senior households in the main sample, I select resident and nonresident, who have at least one sibling. Including first-born twins, there are 23,135 observations, 533 of which belong to households with first-born twins. When I restrict the sample to singletons, there is a total of 21,310 observations, 193 of which belong to households with first-born twins. I show summary statistics for this sample in Table C.3. It is clear that the sample of children from twin mothers differs considerably depending on whether or not first-born twins are included. Tables C.4 and C.5 present additional evidence of the quantity-quality trade-off. The estimating sample used in the results from Table C.4 includes only singleton children with at least two siblings. The estimating sample in Table C.5 includes first-born singletons and instruments the number of siblings using an indicator for whether the child’s mother had second-born twins.

Rural and Urban Samples Table C.6 presents descriptive statistics for samples of senior mothers with urban and rural registration (hukou).

Payment for Childcare Services Tables C.7 and C.8 present evidence that rule out the payment-for-service hypothesis as a mechanism underlying the fertility-transfer relationship.

Offspring Sex and Birth Spacing Table C.9 shows evidence that first-born twins do not alter the gender composition of the offspring but, as expected, lead to shorter average children’s birth spacing. The estimates of the fertilitytransfer relationship are robust to adding these two variables as controls. 4

In the CHARLS survey, the total income of a child and his/her spouse is reported in brackets (None, < 2,000 yuans, 2,000 to 5,000 yuans, 5,000 to 10,000 yuans, 10,000 to 20,000 yuans, 20,000 to 50,000 yuans, 50,000 to 100,000 yuans, 100,000 to 150,000 yuans, 150,000 to 200,000 yuans, 200,000 to 300,000 yuans, > 300,000. Income bracket corresponds to the midpoint of the reported income brackets.

9

Gender differentials in savings and human capital Table C.10 presents estimates for average children’s education and asset holdings while separating the number of children into the number of daughters and the number of sons.

10

Table C.1: Health outcomes of senior mothers and first-born twins

Dep. variable

Health outcomes on Twins b/se n.obs

Good health

0.0095 (0.034) Health problems -0.039 (0.037) Difficulty in ADL 0.026 (0.020) Physical disability 0.0062 (0.015) Infectious disease -0.014 (0.014) Notes:

8329 8327 8335 8326 8335

Table reports coefficient on first-born twin

dummy. Dependent variable in left hand column. Robust standard errors in parentheses. Additional controls (mother’s current age and age squared, age at first birth, parent is married, agricultural hukou, mother’s education and income, and county FE) included in all regressions but not reported. Good health is an indicator variable for whether the individual’s self-reported health status is good, very good, or excellent.

Health problems is an indi-

cator for whether the individual reported at least one of the following conditions: hypertension, high cholesterol, diabetes, cancer, lung or liver disease, heart problems, kidney disease, stomach problems, emotional or psychiatric problems, memory-related problems, arthritis or rheumatism, or asthma. Difficulty in ADL indicates whether the individual has difficulties with activities of daily living. Physical disability and Infectious disease are also indicator variables for having any physical disability and infectious disease, respectively. Stars indicate statistical significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

11

Table C.2: First-born twins and family size probabilities N>1 (1) OLS b/se First-born twins

0.22*** (0.019) Age 1st birth -0.021*** (0.0011) Mother’s age 0.039*** (0.0035) Mother’s education -0.0053*** (0.0013) Mean dep. var. Number of obs

0.83 8818

N>2 (2) OLS b/se

N>3 (3) OLS b/se

N>4 (4) OLS b/se

N>5 (5) OLS b/se

N>6 (6) OLS b/se

0.21*** (0.029) -0.031*** (0.0011) 0.051*** (0.0041) -0.0036** (0.0015)

0.16*** (0.031) -0.025*** (0.0010) 0.0085** (0.0038) 0.0012 (0.0012)

0.092*** (0.026) -0.015*** (0.00088) -0.016*** (0.0035) 0.00080 (0.00080)

0.071*** (0.022) -0.0078*** (0.00064) -0.018*** (0.0028) 0.0012** (0.00052)

0.037** (0.016) -0.0031*** (0.00043) -0.0096*** (0.0020) 0.00073** (0.00032)

0.48 8818

0.25 8818

0.12 8818

0.05 8818

0.02 8818

Notes: Standard errors clustered at the household level in parentheses. Mother’s education was imputed to missing observations using the predicted values from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional controls (mother’s current age squared, parent is married, agricultural hukou, indicator for missing children’s month of birth, county FE, and dummy for whether information on mother’s education is imputed) included in all regressions but not reported. Dependent variables in columns (1), (2), (3), (4), (5), and (6) are indicators for whether the number of children is larger than one, two, three, four, five, and six, respectively. Stars indicate statistical significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

12

Table C.3: Sample of children: summary statistics, by mother’s twinning status

Variable N. siblings Some schooling Elementary school or higher Middle school or higher Income bracket Income > 20,000 CNY Child currently works Child’s age Male child Co-resident child Number of obs

Including first-born twins Excluding first-born twins (1) (2) (3) (4) (5) Whole sample Non-twin mothers Twin mothers Whole sample Twin mothers 2.48 (1.62) 0.91 (0.29) 0.88 (0.32) 0.63 (0.48) 23117.3 (30816.1) 0.35 (0.48) 0.87 (0.33) 34.9 (10.2) 0.54 (0.50) 0.29 (0.45)

2.47 (1.62) 0.91 (0.29) 0.88 (0.32) 0.63 (0.48) 23270.3 (31041.3) 0.36 (0.48) 0.87 (0.33) 34.9 (10.2) 0.53 (0.50) 0.29 (0.45)

3.05 (1.83) 0.87 (0.34) 0.85 (0.36) 0.61 (0.49) 16505.7 (17428.2) 0.24 (0.43) 0.78 (0.41) 34.9 (11.2) 0.57 (0.49) 0.29 (0.46)

2.65 (1.54) 0.90 (0.30) 0.88 (0.33) 0.61 (0.49) 23143.4 (30377.7) 0.35 (0.48) 0.87 (0.33) 35.4 (10.2) 0.53 (0.50) 0.26 (0.44)

4.02 (1.81) 0.81 (0.39) 0.78 (0.41) 0.48 (0.50) 17820.9 (19927.8) 0.26 (0.44) 0.81 (0.39) 37.3 (11.0) 0.56 (0.50) 0.26 (0.44)

23135

22602

533

21310

193

Notes: Summary statistics calculated from estimation sample. Table presents means and standard deviation (in parentheses). Financial values reported in CNY.

13

Table C.4: Number of siblings and adult children’s human capital, children with at least two siblings Less than elementary Less than middle Income (1) (2) (3) IV Probit IV Probit IV b/se b/se b/se N. siblings Age 1st birth Mother’s age Mother’s education Child’s age Mean dep. var. Number of obs

0.077** (0.037) 0.0031 (0.0035) 0.020*** (0.0052) -0.014*** (0.0018) 0.0057*** (0.00086)

0.090* (0.052) 0.0029 (0.0051) 0.021*** (0.0067) -0.016*** (0.0023) 0.0069*** (0.0012)

0.15 15498

0.46 15498

-6523.9** (3242.0) -646.0** (304.8) 3034.2*** (347.5) 376.0*** (135.4) -42.1 (100.6)

Income (4) IV IR b/se -6447.7* (3445.9) -627.4** (320.1) 2212.3*** (358.3) 485.7*** (131.3) -138.9 (101.9)

23179.27 23179.27 12626 12626

Notes: Standard errors clustered at the household level in parentheses. Sample includes nontwin children from senior CHARLS respondents, resident or non-resident, with at least two siblings. Missing data on child’s educational attainment were imputed using predicted values from a regression of non-missing educational attainment on child’s age, gender, birth order, mother’s age, age squared, and county dummy. Mother’s education was imputed to missing observations using the predicted values from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional controls (mother’s current age squared, parent is married, agricultural hukou, adult child’s gender and first-born dummy, indicator for missing month of birth, mother’s county FE, and indicator for whether mother’s and child’s education are imputed) included in all regressions but not reported. The dependent variable in column (1) is an indicator for whether the child has less than elementary education; in column (2), it is an indicator for whether the child has less than middle school. The dependent variable in column (3) is the midpoint of the income bracket to which the income of the child and his/her spouse belongs; in column (4), I use the lower and upper bounds of the income brackets in a interval regression estimation with number of siblings instrumented by first-born twins. The differences in sample sizes are due to missing income brackets data for most resident children. Stars indicate statistical significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

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Table C.5: Number of siblings and adult children’s human capital, sample of first-born singletons with at least one sibling Less than elementary Less than middle Income (1) (2) (3) IV Probit IV Probit IV b/se b/se b/se N. siblings Age 1st birth Mother’s age Mother’s education Child’s age Mean dep. var. Number of obs Notes:

0.047 (0.038) 0.00092 (0.0036) 0.017*** (0.0046) -0.013*** (0.0018) 0.0029** (0.0015)

0.030 (0.053) -0.011** (0.0052) 0.021*** (0.0058) -0.016*** (0.0018) -0.0012 (0.0020)

0.11 7062

0.35 7062

-2923.4 (2962.4) -670.3** (284.8) 4222.5*** (335.0) 491.1*** (130.7) -367.8*** (97.3)

Income (4) IV IR b/se -3177.8 (2820.2) -489.4* (282.3) 3076.3*** (344.1) 546.5*** (124.8) -270.5** (117.4)

23935.60 23935.60 5947 5947

Standard errors clustered at the household level in parentheses. Sample includes

first-born singleton children from senior CHARLS respondents, resident or non-resident, with at least one sibling. Missing data on child’s educational attainment were imputed using predicted values from a regression of non-missing educational attainment on child’s age, gender, mother’s age, age squared, and county dummy. Mother’s education was imputed to missing observations using the predicted values from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional controls (mother’s current age squared, parent is married, agricultural hukou, adult child’s gender, indicator for missing month of birth, mother’s county FE, and indicator variables for whether mother’s and child’s education are imputed) included in all regressions but not reported. Number of siblings is instrumented using an indicator variable for whether the mother had second-born twins. The dependent variable in column (1) is an indicator for whether the child has less than elementary education; in column (2), it is an indicator for whether the child has less than middle school. The dependent variable in column (3) is the midpoint of the income bracket to which the income of the child and his/her spouse belongs; in column (4), I use the lower and upper bounds of the income brackets in a interval regression estimation with number of siblings instrumented by second-born twins. The differences in sample sizes are due to missing income brackets data for most resident children. Stars indicate statistical significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

15

Table C.6: Sample of parents: summary statistics, by mother’s current hukou status

Variable N. children Received time transfers Financial transfer amount Positive transfer amount Received financial transfers Co-reside with married child HH per capita income HH per capita labor income Parents’ financial assets Parents own house HH per capita expenditures Mother’s current age Age at first birth Mother is married

Whole sample Urban Rural (1) (2) (3) Mean/SD Mean/SD Mean/SD 2.76 (1.45) 0.049 (0.22) 1787.0 (17416.9) 5392.7 (29936.7) 0.34 (0.47) 0.32 (0.46) 9973.1 (16095.8) 6891.4 (13076.9) 7562.1 (45270.4) 0.74 (0.44) 13468.9 (16831.4) 58.2 (10.3) 23.1 (4.37) 0.81 (0.39)

2.24 (1.30) 0.034 (0.18) 1388.9 (7662.2) 7065.3 (16094.9) 0.22 (0.41) 0.24 (0.43) 20817.4 (23608.0) 10016.3 (19121.2) 18868.6 (76110.2) 0.82 (0.38) 18002.4 (17971.4) 59.0 (10.3) 24.1 (4.35) 0.82 (0.38)

2.91 (1.45) 0.053 (0.22) 1907.2 (19420.9) 5125.9 (31582.8) 0.38 (0.49) 0.34 (0.47) 6731.1 (11149.0) 5957.1 (10443.5) 4187.7 (29721.8) 0.72 (0.45) 12120.2 (16235.9) 58.0 (10.3) 22.7 (4.33) 0.81 (0.39)

8818

2045

6773

Number of obs

Notes: Summary statistics calculated from estimation sample. Financial values reported in 2011 CNY.

16

17 636.58 8784

0.31 8784

-0.015 (0.042) -0.0041 (0.0046) 0.19*** (0.0084) -0.0048*** (0.0016) 0.26 8805

0.21 8713

0.19 8805

0.15 8658

0.029 0.017 0.064 0.029 (0.051) (0.038) (0.048) (0.030) 0.0036 0.0022 0.0025 0.000085 (0.0055) (0.0042) (0.0052) (0.0033) 0.058*** 0.076*** 0.077*** 0.096*** (0.0051) (0.0069) (0.0048) (0.0060) -0.0020 -0.0026* -0.0039*** -0.0039*** (0.0020) (0.0015) (0.0015) (0.0013) 0.05 8805

0.015 (0.022) -0.00099 (0.0024) 0.018*** (0.0025) -0.00070 (0.00071)

0.05 7336

0.028 (0.023) 0.00071 (0.0025) 0.055*** (0.0055) -0.0017 (0.0011)

significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

number of co-resident grandchildren aged 15 to 18, respectively. Stars indicate statistical significance. Stars indicate statistical

variables for the number of co-resident grandchildren aged 0 to 6, the number of co-resident grandchildren aged 7 to 14, and the

the number of co-resident grandchildren aged 15 to 18, respectively. Dependent variables in columns (4), (6), and (8) are indicator

(5), and (7) are the number of co-resident grandchildren aged 0 to 6, the number of co-resident grandchildren aged 7 to 14, and

indicator for whether respondent or spouse took care of grandchildren in the previous year. Dependent variables in columns (3),

average number of hours the senior respondent and spouse spent caring for granchildren in the previous year; in column (2) it is an

instrumented using an indicator variable for whether the mother had first-born twins. Dependent variable in column (1) is the

whether information on mother’s education is imputed) included in all level regressions but not reported. Number of children is

age squared, parent is married, agricultural hukou, indicator for missing children’s month of birth, county FE, and dummy for

from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional controls (mother’s current

Notes: Robust standard errors in parentheses. Mother’s education was imputed to missing observations using the predicted values

Mean dep. var. Number of obs

-96.4 (129.4) Age 1st birth -16.6 (14.0) Mother’s age 221.1*** (14.7) Mother’s education -6.33 (5.79)

N. children

Cared for grandchildren Grandchildren 0 to 6 Grandchildren 7 to 14 Grandchildren 15 to 18 (1) (2) (3) (4) (5) (6) (7) (8) IV IV Probit IV IV Probit IV IV Probit IV IV Probit b/se b/se b/se b/se b/se b/se b/se b/se

Table C.7: The effects of number of children on hours of childcare and number of co-resident grandchildren

18 No No 2.76 8771

-0.11*** (0.0035) 0.050*** (0.014) -0.0049 (0.0033) Yes Yes 2.76 8771

-0.11*** (0.0035) 0.062*** (0.014) -0.0054* (0.0033)

0.81*** 0.82*** (0.087) (0.087)

No No 1794.97 8771

0.68* (0.35) -0.027 (0.038) 0.34*** (0.034) -0.032*** (0.012) Yes Yes 1794.97 8771

0.70** (0.35) -0.024 (0.037) 0.34*** (0.037) -0.032*** (0.012)

Transfer amount (3) (4) IV IV b/se b/se

No No 0.33 8737

0.082** (0.039) -0.0028 (0.0047) 0.054*** (0.0060) -0.0051*** (0.0016) Yes Yes 0.33 8737

0.083** (0.038) -0.0027 (0.0046) 0.053*** (0.0064) -0.0051*** (0.0016)

No No 0.31 8771

0.10*** (0.033) 0.0086** (0.0038) 0.038*** (0.0062) -0.0034** (0.0016)

Yes Yes 0.31 8771

0.085*** (0.028) 0.0088*** (0.0032) -0.010** (0.0041) -0.0011 (0.0014)

Got transfers Co-reside (5) (6) (7) (8) IV Probit IV Probit IV Probit IV Probit b/se b/se b/se b/se

∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

least one married child aged 22 years or older. Stars indicate statistical significance. Stars indicate statistical significance.

parents received positive net transfers from children; in columns (7) and (8) it is a dummy for whether parents live with at

transfer amount received from children net of transfers made to children; in columns (5) and (6) is an indicator for whether

columns (1) and (2) is the total number of children ever born. Dependent variable in columns (3) and (4) is the (log) total

of children is instrumented using an indicator variable for whether the mother had first-born twins. Dependent variable in

dummy for whether information on mother’s education is imputed) included in all level regressions but not reported. Number

current age squared, parent is married, agricultural hukou, indicator for missing children’s month of birth, county FE, and

values from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional controls (mother’s

Notes: Robust standard errors in parentheses. Mother’s education was imputed to missing observations using the predicted

N. co-res. grandchildren Hours of childcare Mean dep. var. Number of obs

Mother’s education

Mother’s age

Age 1st birth

N. children

First-born twins

N. children (1) (2) OLS OLS b/se b/se

Table C.8: Number of children and old-age support, controlling for number of co-resident grandchildren and hours of childcare

19 No No 1.91 8818

No No 1786.96 8818

Yes Yes 1786.96 8818

0.67** (0.32) -0.035 (0.032) 0.34*** (0.033) -0.032*** (0.012) No No 0.33 8818

0.076* (0.044) -0.0050 (0.0048) 0.045*** (0.0044) -0.0050*** (0.0015) Yes Yes 0.33 8818

0.072* (0.041) -0.0060 (0.0041) 0.045*** (0.0043) -0.0051*** (0.0015)

Got transfers (5) (6) IV IV b/se b/se

No No 0.32 8818

0.11** (0.047) 0.0100** (0.0051) 0.039*** (0.0053) -0.0038** (0.0017)

Yes Yes 0.32 8818

0.10** (0.042) 0.0075* (0.0043) 0.040*** (0.0051) -0.0039** (0.0017)

Co-reside (7) (8) IV IV b/se b/se

Robust standard errors in parentheses. Mother’s education was imputed to missing observations using the

No No 0.56 8818

0.72** (0.35) -0.022 (0.037) 0.34*** (0.034) -0.032*** (0.012)

Transfer amount (3) (4) IV IV b/se b/se

statistical significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

parents live with at least one married child aged 22 years or older. Stars indicate statistical significance. Stars indicate

for whether parents received positive net transfers from children; in columns (7) and (8) it is a dummy for whether

total transfer amount received from children net of transfers made to children; in columns (5) and (6) it is an indicator

children (for one-child households, the average birth spacing was set to zero); in columns (3) and (4) it is the (log)

of children; in column (2) it is the sum of the age difference between subsequent children divided by the number of

had first-born twins. Dependent variable in column (1) is the number of male children as a fraction of the total number

regressions but not reported. Number of children is instrumented using an indicator variable for whether the mother

missing month of birth, and dummy for whether information on mother’s education is imputed) included in all level

controls (mother’s current age squared, parent is married, agricultural hukou, county FE, indicator for children’s

predicted values from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional

Notes:

Sex mix Spacing Mean dep. var. Number of obs

-0.11*** (0.0059) -0.014 (0.019) -0.0025 (0.0059)

0.033 -0.75*** (0.027) (0.077)

0.0012 (0.00088) Mother’s age -0.0016 (0.0033) Mother’s education 0.00014 (0.0013)

Age 1st birth

N. children

First-born twins

Sex mix Spacing (1) (2) OLS OLS b/se b/se

Table C.9: Number of children and old-age support, controlling for sex mix and spacing

20

0.60*** (0.14) 0.51*** (0.12) -0.69* (0.40)

8745

8745

0.6655 8745

2.76

8265

9.29 0.4772

0.045 (0.040) 0.042 (0.033) 0.074*** (0.012)

0.028 (0.050)

7851

0.74

0.6988 7851

0.74

0.019 (0.054) 0.033 (0.051) 0.0033 (0.0051) 0.0035 (0.0050) 0.0012 (0.0016)

Owns house (8) (9) IV Probit IV Probit b/se b/se

0.35 (0.41) 0.40 (0.38) 0.044 0.0035 (0.039) (0.0052) 0.042 0.0034 (0.033) (0.0050) 0.074*** 0.0013 (0.012) (0.0016) 7562.05 7562.05 0.8258 0.4781 0.8394 8265 8745 8745

9.29

-0.83* (0.45) -0.62 (0.41) 0.0020 (0.043) 0.011 (0.044) 0.21*** (0.012)

0.39 (0.37)

Assets (6) (7) IV IV b/se b/se

Robust standard errors in parentheses. Mother’s education was imputed to missing observations using the predicted values

1.48

-0.053*** -0.11*** 0.0045 (0.0026) (0.0035) (0.043) 0.025** 0.049*** 0.0070 (0.010) (0.014) (0.044) 0.00035 -0.0051 0.21*** (0.0029) (0.0033) (0.012)

-0.70*** (0.090) 1.27*** (0.086)

1.27

-0.052*** (0.0028) 0.026** (0.011) -0.0054 (0.0035)

1.30*** (0.11) -0.74*** (0.085)

Avg. Educ (4) (5) IV IV b/se b/se

significance. ∗ ∗ ∗ < 0.01, ∗∗ < 0.05, ∗ < 0.1.

columns (8) and (9) it is a dummy for whether the mother or her spouse own the house they currently live in. Stars indicate statistical

(5) is the average children’s years of schooling; in column (6) and (7) it is the (log) financial assets legally owned by the parents and in

children ever born, the total number of daughters, and the total number of sons, respectively. Dependent variable in columns(4) and

indicator for whether at least one of the twins is a boy. Dependent variable in columns (1), (2), and (3) are the total number of

daughters, and number of sons are instrumented using an indicator variable for whether at least one of the twins is a girl, and another

information on mother’s education is imputed) included in all level regressions but not reported. Number of children, number of

squared, parent is married, agricultural hukou, county FE, indicator for missing children’s month of birth, and dummy for whether

from a regression of mother’s years of schooling on age, age squared, and county dummy. Additional controls (mother’s current age

Notes:

Mean dep. var. Overid test; p-val Daughter = Son; p-val Number of obs

Mother’s education

Mother’s age

Age 1st birth

N. sons

N. daughters

N. children

First-born boy

First-born girl

N. daughters N. sons N. children (1) (2) (3) OLS OLS OLS b/se b/se b/se

Table C.10: Number of children, educational attainment, and parental asset holdings, by child’s gender

References Rosenzweig, Mark R., and Kenneth Wolpin. 1980. “Life-Cycle Labor Supply and Fertility : Causal Inferences from Household Models.” Journal of Political Economy, 88(2): 328–348. Wei, Shang-Jin, and Xiaobo Zhang. 2011. “The Competitive Saving Motive: Evidence from Rising Sex Ratios and Savings Rates in China.” Journal of Political Economy, 119(3): 511–564.

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