Endogenous Altruism:

Theory and Evidence from Chinese Twins∗

Junjian Yi†

November 18, 2013

AbstractIn this paper, I investigate the endogenous formation of inter-sibling altruism and its impli-cations for intra-household investment in children’s human capital. The theoretical analysisshows that parental fostering of inter-sibling altruism during childhood serves as a device toameliorate commitment constraints within families. The increase in inter-sibling altruism en-hances total returns from intra-household investment in children’s human capital by allocatingmore resources to better-endowed children, and decreases inequality in the distribution of con-sumption among children via inter-sibling transfers. Theoretical predictions are supported bythe empirical results that are based on the Chinese longitudinal child twin survey. I find thatparents are more likely to educate their children to be altruistic toward one another when gapsin children’s prenatal endowments are larger. Given gaps in prenatal endowments, parents in-vest more in better-endowed children’s human capital when they educate their children to bemore altruistic. When parents have more children, they educate children to be more altruis-tic. Hence, parental investment becomes more likely to reinforce gaps in children’s prenatalendowments in larger families. The empirical results suggest that the literature understatesthe degree of parental aversion to inequality among their children because of the omission ofinter-sibling altruism.

JEL classification: D10, D13, J24Keywords: Altruism, intra-household human capital investment, inter-sibling transfers

∗I am grateful to my supervisors, James Heckman, Gary Becker, and Junsen Zhang, who provided support, en-couragement, guidance, and extremely helpful suggestions. I am also thankful for the essential feedback providedby Richard Blundell, Ted Bergstrom, Pierre-Andre Chiappori, Steve Durlauf, Michael Keane, Robert Pollak, MarkRosenzweig, Zheng Michael Song, Alessandra Voena, as well as participants in the seminar of family economics atthe University of Chicago. The collection of survey data sets used in this paper was financially supported by theResearch Grant Council of Hong Kong and the Chinese University of Hong Kong, and has been carefully guided byJunsen Zhang, Mark Rosenzweig, and James Heckman. The idea of this paper is inspired by my early experiencewhen I lived with my parents and brother in a small village in rural China. Comments are welcomed. The usualdisclaimer applies.

†Department of Economics, University of Chicago, 1126 East 59th Street Chicago, IL 60637, USA; Email: jun-jian@uchicago.edu.

1 Introduction

This paper investigates the endogenous formation of inter-sibling altruism and its implications for

intra-household investment in children’s human capital. Although economists rely heavily on the

assumption that preferences are fixed and immutable, many would concede that preferences can be

influenced by examples and persuasion. It is increasingly recognized that families play a crucial

role not only in developing children’s human capital but also in fostering children’s preferences.

Levy-Garboua et al. (2006) review the social, biological, and psychological literature about family

influence on the formation of children’s social preferences such as altruism. Furthermore, children

are most malleable at their very early stages when living with their parents (Heckman, 2007). Thus,

parents can significantly influence children’s preference formation. However, the literature seldom

explores why parents have incentives to influence children’s preferences. This paper shows that

parents’ desire to influence children’s inter-sibling altruism is closely related to their decisions on

intra-household investment in children’s human capital.

We consider a conflict between efficiency and equality in intra-household investment in chil-

dren’s human capital (Becker & Tomes, 1976; Behrman et al., 1982; Sheshinski & Weiss, 1982).

Efficiency is defined as that total returns from intra-household investment in children’s human

capital are maximized, and inequality refers to the distribution of or gaps in consumption among

children throughout the paper. If parents invest more in better-endowed children’s human capital

trying to equalize marginal productivity of investment among children, on one hand, total returns

from investment in children’s human capital are maximized. In this case, efficiency increases

whereas equality deteriorates. On the other hand, if parents invest more in less-endowed children,

equality improves whereas efficiency decreases.1

The literature suggests parents invest more on better-endowed children’s human capital, and

then use inter-vivos transfers or bequests to compensate less-endowed children. Through this

scheme, both objectives of efficiency and equality can be achieved. However, the literature has

1Parental human capital investment and children’s prenatal endowments are assumed to be complementary inputsin the human capital production function. I further assume that the credit market is imperfect.

1

not found empirical evidence on compensatory inter-vivos transfers or bequests from parents to

children (Behrman, 1997).

If parents and children can make a binding commitment, there exists another mechanism

through which parents can improve efficiency and equality simultaneously. Specifically, parents,

who act as social planners, invest more in better-endowed children to equalize marginal produc-

tivity of family investment. When children become adults, these better-endowed children, who

have higher earnings because of better endowment and heavier human capital investment, make

transfers to their siblings. By doing so, total returns from investment in children’s human capital

are maximized. At the same time, the inequality in the distribution of consumption among children

decreases via inter-sibling transfers.

However, it is difficult to enforce such an agreement with intertemporal choices. The return

from investment in children’s human capital is received after the investment period. The children

with higher earnings may not make as much inter-sibling transfers as parents requested. Therefore,

a commitment problem arises. This paper proposes that parental fostering of inter-sibling altruism

during childhood serves as a device to ameliorate the commitment problem within families. In

particular, the increase in inter-sibling altruism enhances total returns from intra-household invest-

ment in children’s human capital by allocating more resources to better-endowed children, and

decreases inequality in the distribution of consumption among children via inter-sibling transfers.

Therefore, parental desire to foster children’s inter-sibling altruism is driven by both efficiency and

equality considerations.

Five testable predictions are derived from the theoretical analysis on the endogenous formation

of altruism and intra-household investment in children’s human capital. First, parents invest more

in educating children to be altruistic toward one another when gaps in children’s prenatal endow-

ments are bigger. This prediction is intuitively appealing because the marginal gain to parental

investment in fostering children’s altruism is larger when gaps in children’s prenatal endowments

are bigger. Second, given gaps in children’s prenatal endowments, parents invest more in better-

endowed children’s human capital when they educate children to be more altruistic. Therefore,

2

parental investment in children’s altruism enhances total returns from intra-household investment

in children’s human capital. The third prediction is derived by combining first two predictions.

That is, the degree of parental aversion to inequality among children is understated if parental in-

vestment in fostering children’s altruism is omitted from the commonly adopted regression equa-

tion of intra-household resource allocation. Two more predictions are derived when family size

is introduced into the analysis. Fourth, family size increases marginal gains from parental invest-

ment in fostering children’s altruism. Therefore, parents invest more in educating children to be

altruistic in larger families. Fifth, more parental investment in children’s altruism in larger families

causes larger gaps in intra-household investment in children’s human capital, reinforcing gaps in

children’s prenatal endowments more.

These five theoretical predictions are empirically tested by using data derived from the longitu-

dinal Chinese child twin survey (LCCTS). The LCCTS is probably the first longitudinally socioe-

conomic survey on twin households around the world. The first wave was conducted in 2002 with

children aged 9 on average. The second wave was conducted in 2013. The LCCTS covers detailed

information on intra-household investment in children’s human capital in the first wave, in addition

to an extensive array of variables on child-, parent-, and household-specific characteristics in both

waves.

The LCCTS is ideal for the present study for several reasons. First, we carefully design the

LCCTS questionnaire to solicit information about practices of family education on children’s inter-

sibling altruism, which is called as family altruistic education in the paper. In particular, we include

two questions about family altruistic education in the questionnaire. One is the frequency in which

parents brought children together to worship ancestors during the Qingming festival. The other is

the frequency in which parents told children that having fraternal love among siblings is a virtue,

such as telling children the story of “Kong Rong giving up pears.” The construction and validity

of these two measures of family altruistic education are discussed in a more detailed manner in

the data section below, in which I show that both measures are good proxy variables for family

altruistic education.

3

To directly examine the validity of these two measures of family altruistic education, second,

we further design an experiment to measure inter-sibling transfers in the LCCTS.2 Empirical re-

sults indicate that family altruistic education measured by either variable increases an individual’s

transfers to twin siblings, suggesting that these two variables are effective in predicting children’s

inter-sibling transfers.

Third, the LCCTS provides two natural experiments brought about by twinning. These exper-

iments facilitate identification in the empirical analysis. The first experiment is the within-twin

variation in birth weight which is used as a proxy variable for prenatal endowment. The second

is the variation in family size induced by twinning. These two types of variations are crucial in

empirically testing the theoretical predictions.

The empirical results are consistent with the theoretical predictions. First, the empirical ev-

idence shows that parents are more likely to educate their children to be altruistic when gaps in

children’s birth weight are larger. Given gaps in children’s birth weight, I find that parents invest

more on better-endowed children’s human capital when parents educate children to be more al-

truistic. If the variable of family altruistic education is omitted from the regression equation of

intra-household resource allocation, parental human capital investment is more likely to reinforce

gaps in children’s prenatal endowments. The result suggests that the degree of parental aversion

to inequality is underestimated if family altruistic education is omitted. When family size is larger

which is exclusively induced by twinning in the estimation sample, parents educate their children

to be more altruistic. Finally, intra-household investment in children’s human capital is more likely

to reinforce gaps in children’s prenatal endowments in larger families.

This paper conceptually links to four strands of economics literature. The first one is about

altruism, commitment problems, and resource allocation. Becker (1974) concludes that altruism

enhances resource allocation efficiency by internalizing externality. This conclusion is challenged

by later studies (Hirshleifer, 1977; Bernheim & Stark, 1988; Lindbeck & Weibull, 1988). For in-

stance, Hirshleifer (1977) notices that the effect of altruism on resource allocation crucially hinges

2We are not able to directly measure inter-sibling transfers in the LCCTS because most of the children are inschool in 2013.

4

on whether parents who are altruistic have the last word or not. If parents cannot control the last

action in temporal sequence, a commitment problem arises. In this case, altruism does not nec-

essarily improve resource allocation efficiency. In this regard, the literature focuses on how the

presence of commitment problems limits the effect of altruism on resource allocation. The current

paper approaches the relationship between altruism and commitment problems from the opposite

direction. That is, children’s inter-sibling altruism can be endogenously fostered by parents in

response to commitment constraints within families, enhancing total returns from intra-household

investment in children’s human capital.

Second, this paper links to the literature of the endogenous formation of preferences. Stable

preferences, maximizing behavior, and market equilibrium have once been regarded as a trinity of

fundamental assumptions in economics. The literature later admits individual differences in prefer-

ences and then explores economic rationales of endogenously forming an individual’s preferences

(Becker & Mulligan, 1997; Stark & Falk, 1998). Parents are found to exert significant influence on

children human capital development and preference formation (Heckman, 2008). Recently, Becker

et al. (2013) investigate incentives for parents to “manipulate” preferences of children to induce

them to support their parents in old age. The current paper contributes to this strand of literature

by investigating parental incentives to educate children to form a different type of preferences, that

is, the inter-sibling altruism.

Intra-household resource allocation is the third strand of economics literature to which the

current paper contributes to. Parental investment strategy of reinforcement versus compensation

has been an important research topic (Becker & Tomes, 1976; Griliches, 1979; Rosenzweig &

Wolpin, 1980; Behrman et al., 1982). However, the literature does not consider inter-sibling trans-

fers in studying parental investment decisions. The current paper suggests that parents consider

children’s inter-sibling transfers in adulthood by fostering children’s inter-sibling altruism when

they make investment decisions. Considering the endogenous formation of inter-sibling altruism,

inferences drawn from the observed intra-household resource allocation in the literature should be

re-evaluated. Specifically, because of the omission of the endogenous formation of inter-sibling al-

5

truism, this strand of economics literature understates the degree of parental aversion to inequality

among children.

Fourth, this paper is closely related to the literature of inter-household transfers. Altruism and

inter-household transfers have been extensively studied by economists (Cox, 1987, 1990; Altonji

et al., 1992; Laferrere & Wolff, 2006). The literature focuses on parent-child transfers. However,

inter-sibling transfers are important for families, in particular for those living in developing coun-

tries where extended families are prevalent (Zhang, 2013). The results of this paper have important

implications for future studies on inter-sibling transfers. That is, both the degree of altruism and

earnings of adult children are endogenous, which are determined by parental decisions simultane-

ously.

Empirically, this paper has two major contributions. To the best of my knowledge, on one hand,

this study is the first one empirically examining economic incentives for the endogenous formation

of inter-sibling altruism. On the other hand, this is also the first empirical study examining the

effect of children’s inter-sibling altruism on parental resource allocation. The empirical evidence of

this paper shows that inter-sibling altruism enhances total returns from intra-household investment

in children’s human capital.3

The remainder of the paper is organized as follows. Section 2 builds the theoretical model and

derives testable predictions. Section 3 describes the data. Section 4 tests the predictions. Section

5 discusses the robustness of the empirical results. Section 6 concludes.

2 Theory

This section explores theoretically parental decisions about intra-household investment in chil-

dren’s human capital, taking children’s inter-sibling transfers in adulthood into account. Moreover,

parents can influence children’s altruism toward one another, and thus, children’s altruism is en-

dogenously formed. I first characterize incentives for parents to influence children’s altruism, and

3Although economists are interested in altruism, few studies empirically examine altruism and intra-householdresource allocation. Foster & Rosenzweig (2001) may be exceptional by showing evidence that altruism increasesgains from income pooling in the presence of risk. But they do not consider the endogeneity of altruism.

6

then analyze implications of the endogenous formation of children’s altruism for intra-household

human capital investment. The model shows that parental fostering of children’s altruism serves as

a device to ameliorate commitment constraints within families. I demonstrate that fostering inter-

sibling altruism by parents may not only enhance total returns from intra-household investment

but also decrease gaps in children’s consumption. Finally, I introduce family size into the analy-

sis, and study family size, the endogenous formation of children’s altruism, and intra-household

human capital investment. Several theoretical predictions are derived from the model, which are

tested in following sections.

2.1 Model Set-up

Parents are assumed to be altruistic toward their children. They care about not only their own

consumption cp, but also children’s consumption cτ, where τ indexes the τth child (τ = 1, ..., n).

Therefore, parents have a utility function such that U = U(cp, c1, ..., cn). The parental utility

function has the following property:

Uci S Uc j if ci T c j, (1)

where i, j = 1, ..., n. Property (1) indicates that the ratio of marginal utilities derived from child

i′s consumption over that of child j is independent of or separable from parental consumption.

With this separability property, the intra-household resource allocation between parents and chil-

dren does not qualitatively affect the resource allocation among children. To focus on resource

allocation among children only, parental consumption is omitted from the utility function below.

Furthermore, Property (1) implies that, in addition to children’s total consumption, parents care

about the distribution of children’s consumption. That is, parents are averse to inequality in the

distribution of consumption among their children because parents derive higher marginal utility

from the child who consumes less than the other one. To capture the degree of parental aversion to

7

inequality, I specify the following parental utility function:

U = (cρi + cρj)1ρ , (2)

where ρ ≤ 1. With this specific function form, the parameter ρ can effectively capture the degree

of parental aversion toward inequality among children. For example, if ρ = 1, parents exhibit

no inequality aversion, and then U = ci + c j. When ρ = −∞, parents exhibit infinite inequality

aversion. The latter is the Rawlsian case, that is, U = min(ci, c j). For simplicity, each household is

assumed to have two children only (τ = i, j).4 Furthermore, children are assumed to be symmet-

rically treated. The relaxation of this assumption does not affect the comparative static analysis

below as discussed in Appendix A.

Children’s consumption depends on labor market earning E plus or minus inter-sibling transfers

T . That is, cτ = Eτ ± T . If child τ makes transfers to the sibling −τ, his or her consumption is

Eτ − T ; otherwise, it is Eτ + T . The labor market earning is determined by the wage rate ω and

children’s human capital h. The latter is a function of children’s prenatal endowment e and family

human capital investment I. Therefore,

Eτ = ωh(eτ, Iτ).

To simplify the analysis, I assume a Cobb-Douglas human capital production function form, such

that hτ = φeβeτ IβI

τ , where 0 < βe, βI < 1.

The inter-sibling transfers T are determined in equilibrium which is a function of children’s

earnings and the degree of inter-sibling altruism a, such that T = T (Ei, E j, a), where 0 ≤ a ≤ 1.

The household budget constraint is that pI(Ii + I j) = Y , where pI is the price of intra-household

investment in children’s human capital and Y is family income, which is assumed to be exogenous

in the analysis. Without loss of generality, I assume that ei > e j throughout the paper.

4The case with three or more children is analyzed below.

8

2.2 Intra-household Investment in Children’s Human Capital

I first analyze intra-household investment in children’s human capital without considering inter-

sibling transfers, which sets up a benchmark for future analysis. In this case, the degree of inter-

sibling altruism a is assumed to be zero. Children’s consumption depends solely on labor market

earning E. The optimal intra-household investment in children’s human capital is determined by

the following equilibrium condition:

Inequality aversion:(ln

Ii

I j

)ρ<1,a=0

=ρβe

1 − ρβIln

ei

e j(3)

If ρ = 1, parents are not averse to inequality. The optimal intra-household human capital invest-

ment in this special case is determined by

Efficient:(ln

Ii

I j

)ρ=1,a=0

=βe

1 − βIln

ei

e j(4)

Equation (4) sets the equilibrium condition that total returns from intra-household investment in

children’s human capital are maximized. Therefore, the equilibrium condition of Equation (4) is

defined as the efficient condition in intra-household human capital investment in the paper. In this

special case, children’s total earnings or consumption are also maximized.

Comparing Equation (3) with Equation (4), parents allocate fewer resources to the better-

endowed child when ρ < 1 than that in the case ρ = 1. Furthermore, when ρ is smaller, parents

invest less in the better-endowed children. Thus, total returns from intra-household human capital

investment decreases, and the gap in children’s earnings shrinks. This observation leads to my first

proposition:

Proposition 1 An efficiency versus equality trade-off exists in the intra-household investment in

children’s human capital when parens are averse to inequality in the distribution of earnings

among children.

This is a standard result in family economics.

9

I then analyze intra-household investment in children’s human capital when parents consider

children’s inter-sibling transfers such that a , 0. Children are assumed to be altruistic toward

one another and make inter-sibling transfers when they become adults. In this case, the problem

can be formulated into a two-stage game. In the second stage, children grow up into adulthood.

They make decisions about inter-sibling transfers by taking their earnings as given. In the first

stage, parents make decisions about intra-household investment in children’s human capital by

taking their adult children’s optimal inter-sibling transfers into account. By backward induction,

the subgame perfect equilibrium can be solved.

I begin with the second stage, investigating children’s transfer decisions when they are adults.

Assume the utility function of child τ to be

Vτ = ln cτ + a ln c−τ,

where a is the degree of child τ’s altruism toward his or her sibling −τ as discussed above. The

degree of altruism is assumed to be exogenous and be the same between two children. This as-

sumption is justified in the next section when the degree of children’s altruism is allowed to be

endogenous. Child τ derives higher utility from his or her sibling’s consumption when he or she

has a higher degree of altruism.

Transfers are made from the better-endowed child i to child j in equilibrium because Ei > E j.

That is, parental investment does not fully compensate the gap in children’s prenatal endowments

when ρ , −∞. Therefore, the consumption is Ei − T for child i, and it is E j + T for child j. If

E j ≤ aEi, I have an interior solution and the optimal transfer T is as follows:

T =a

1 + aEi −

11 + a

E j. (5)

∂T∂a > 0, ∂T

∂Ei> 0, and ∂T

∂E j< 0. If E j > aEi, I have a corner solution and the optimal transfer is zero.

With an interior solution, the optimal consumption is E1+a for child i, and it is aE

1+a for child j, where

E = Ei + E j.

10

I now explore the implication of inter-sibling altruism for intra-household investment in chil-

dren’s human capital in the first stage. Using first-order conditions, the subgame perfect equilib-

rium condition requires that

Altruism:(ln

Ii

I j

)ρ<1,a<1

=−ρ

1 − βIln a +

βe

1 − βIln

ei

e j. (6)

Equation (6) shows that parental human capital investment strategy depends on the degree of chil-

dren’s inter-sibling altruism. The introduction of inter-sibling altruism has significant implication

for intra-household investment in children’s human capital, which is embodied in the following

proposition.

Proposition 2 The increase in the degree of inter-sibling altruism not only enhances total returns

from intra-household investment in children’s human capital but also decreases the gap in chil-

dren’s consumption.

The intuition underlying the proposition is as follows. On one hand, by comparing Equations (6)

and (4), the efficiency distortion induced by parental aversion to inequality is embodied in the first

term in the right hand side of Equation (6). The absolute value of this term decreases when a

increases. In the extreme case when inter-sibling altruism is perfect, such that a = 1, this term

becomes zero. Under this scenario, total returns from intra-household investment in children’s

human capital are maximized, which is illustrated as below:

Perfect:(ln

Ii

I j

)ρ<1,a=1

=

(ln

Ii

I j

)ρ=1,a=0

=βe

1 − βIln

ei

e j

Therefore, the efficiency distortion brought about by parental aversion to inequality is fully inter-

nalized by inter-sibling altruism. On the other hand, c j = aci by Equation (5). As such, the gap in

children’s consumption shrinks when a increases.

If the degree of inter-sibling altruism was observable, I would be able to estimate Equation

(6). However, it is usually unobservable. Furthermore, parents may influence the formation of

11

children’s altruism. Parental desire to foster children’s altruism intertwines with their decisions

on intra-household investment in children’s human capital. Therefore, the degree of children’s

altruism may be endogenous. If this is the case, the econometric specification in the literature,

which is based on Equation (3), may be misspecified because of the omission of inter-sibling

altruism. Thus, the inferences about parental inequality aversion drawn from estimates in the

literature of intra-household resource allocation would be re-evaluated, which is discussed in the

section below.

2.3 The Endogenous Formation of Children’s Inter-sibling Altruism

This section explores the endogenous formation of children’s inter-sibling altruism. Proposition

1 shows that parents face an efficiency versus equality trade-off when they invest in children’s

human capital. Acting as a social planner, parents may hypothetically make decisions which en-

hance efficiency and equality simultaneously. Specifically, when children are young, parents make

decisions according to Equation (4) maximizing total returns from intra-household investment in

children’s human capital. Parents then ask child i, who has higher earnings, to transfer part of

his or her earnings to child j when children become adults. This scheme seems ideal, but suffers

from a commitment problem. The return to human capital is received after the investment period.

The money is in the hand of child i when he or she becomes an adult. His or her preference may

be different from that of his or her parents. Thus, he or she may not make as much inter-sibling

transfers as parents requested.

To the ameliorate the commitment problem, parents are motivated to educate their children to

be altruistic toward one another. Proposition 2 shows that fostering children’s altruism by parents

not only enhances total returns from intra-household investment in children’s human capital but

also decreases the gap in children’s consumption. Therefore, parental desire to foster children’s

inter-sibling altruism can be driven by both efficiency and equity considerations. In this manner

three factors, namely intra-household investment in children’s human capital, parental fostering

children’s inter-sibling altruism during childhood, and inter-sibling transfers in adulthood, are log-

12

ically correlated.

Children’s altruism, for simplicity, is assumed to be generated by the following production

function:

aτ = ϕtστ ,

where 0 < σ < 1. The variable t denotes family investment in fostering children’s altruism, which

may include parental time, energy, money, and so on.5 The “altruism production function” has two

notable characteristics. On one hand, family investment in children’s altruism is a choice variable

only for parents.

On the other hand, family investment in children’s altruism is assumed to be a public good.

Therefore, ti = t j = t and ai = a j = a. This assumption is a simplification for analytic convenience,

which can be justified from multiple dimensions. First, the scale effect of educating children to be

altruistic may be sufficiently large. This justification is supported by empirical evidence presented

below. Second, inter-sibling transfers are made from the better-endowed child i to the other child

j. Therefore, parents may need only to educate child i to be altruistic. Third, the relaxation of

this assumption does not affect the analysis below. If children’s prenatal endowments and parental

investment are complementary inputs in producing children’s altruism, the better-endowed child

who has higher earnings also has higher degree of altruism. Thus, transfers are still made from

child i to child j.

When inter-sibling altruism is endogenized, children’s decisions on inter-sibling transfers are

similar to those as before because both the degree of inter-sibling altruism and their earnings are

given in adulthood. However, the problem that parents face is different from what they face when

children’s altruism is exogenous. On one hand, the cost of fostering children’s altruism is added

into the household budget constraint, such that pI(Ii + I j) + ptt = Y , where pt is the price of family

investment in influencing children’s altruism. On the other hand, parents have to decide how much

to invest in children’s altruism, in addition to their investment decisions about children’s human

capital.

5The section on data below discusses practices that parents adopted to influence children’s preference formation.

13

I first investigate parental decisions on fostering children’s altruism. Owing to the nonlinearity

of the model setting, it is impossible to derive a closed-form analytic solution for the optimal

family investment in children’s altruism. However, considering that the utility function, human

capital production function, and production function of children’s altruism are strictly concave and

twice-differentiable, the existence, uniqueness, and continuity of the solution to family investment

in children’s altruism directly follows, which is denoted as a function below

t∗ = t(Y, pI , pt, ei, e j, ω).

The comparative static analysis shows an important relationship between parental investment in

children’s altruism and the gap in children’s prenatal endowments, which is embodied in the fol-

lowing corollary:

Corollary 1 Parental investment in fostering children’s inter-sibling altruism increases when the

gap in children’s prenatal endowments widens.

The intuition underlying this corollary is that the marginal gain from investment in children’s

altruism increases with the gap in children’s prenatal endowments. Specifically, the marginal gain

from parental investment in children’s altruism is Tt(Uc j − Uci). Tt = ∂T∂a

∂a∂t , which is the marginal

effect of investment in children’s altruism on inter-sibling transfers. Tt > 0 and Ttt < 0. Ucτ = ∂U∂cτ

(τ = i, j). Considering that e j < ei, c j < ci when ρ , −∞ or a , 1, Uc j−Uci > 0. The marginal gain

from parental investment in children’s altruism is positive. If ρ , −∞, the difference in children’s

consumption increases when the gap in children’s prenatal endowments widens. Thus, Uc j − Uci

increases.

I now examine parental decisions about intra-household investment in children’s human capital

when altruism is endogenously fostered. In this case, the subgame perfect equilibrium requires the

14

following condition:6

Endogenous Altruism:(ln

Ii

I j

)ρ<1,a=ϕtσ

=−ϕρ

1 − βI+−σρ

1 − βIln t +

βe

1 − βIln

ei

e j(7)

Equation (7) is the core of the analysis in the current paper, which has following implications. First,

parental human capital investment strategy is related to their investment decisions on fostering

children’s altruism. This relationship is embodied in the following corollary.

Corollary 2 If parents are sufficiently averse to inequality among children, such that ρ < 0, par-

ents invest more on the human capital of the better-endowed child when they invest more on fos-

tering children’s inter-sibling altruism.

Second, Equation (7) shows that the coefficient before ln eie j

is the same as the one in Equation

(4) when parents are not averse to inequality. This observation leads to the third proposition.

Proposition 3 The parameters of human capital production technology are separable from pa-

rameters of parental preference when family investment in children’s altruism is included into the

intra-household human capital investment equation.

To understand the intuition underlying this proposition, the intra-household resource allocation can

be conceptually considered as a two-step process. In the first step, parents divide total investment

between children’s human capital and altruism. The proportion allocated to foster children’s altru-

ism determines the gap in children’s consumption. In the second step, parents divide human capital

investment between the two children. Since the gap in children’s consumption is only determined

by the degree of altruism, the intra-household human capital investment is purely determined by

the production technology. Therefore, the intra-household human capital investment between the

two children reflects the human capital production technology only when parental investment in

children’s altruism is controlled for in Equation (7). On the other hand, the parental preference

6This condition is based on the assumption that parents expect an interior solution for inter-sibling transfers inthe second stage at the equilibrium. If parents expect a corner solution, they will not invest in fostering children’sinter-sibling altruism, and the intra-household human capital investment decisions follows Equation (3).

15

parameter of inequality aversion is embodied in intra-household investment in fostering children’s

altruism.

Finally, I derive a corollary by Equation (7) which has significant implications for the empirical

analysis in the literature of intra-household investment in children’s human capital.

Corollary 3 If parents are sufficiently averse to inequality in the distribution of consumption

among their children, such that ρ < 0, the omission of family investment in children’s altruism

biases upward of the estimate of parental preference parameter ρ. That is, the degree of parental

aversion to inequality is understated.

The econometric specification in the literature is usually based on Equation (3), which is misspec-

ified because of the omission of parental investment in children’s altruism t. When ρ < 0, t is

positively correlated with ln IiI j

. By Corollary 1, t is also positively correlated with ln eie j

. The esti-

mated coefficient before ln eie j

in Equation (3) is overstated. Therefore, inferences about the parental

reference parameter ρ drawn from estimates of Equation (3) in the literature are misleading.

2.4 Family Size, Inter-sibling Altruism, and Intra-household Human Capi-

tal Investment

This section considers that households have more than two children. For simplicity, each family

is assumed to have three children.7 The parental utility function is that U = (cρi + cρj + cρk)1ρ ,

where the subscript k indexes the third child. The human capital production function is the same

as before. I assume that ei > e j and ei > ek. Children make transfer decisions in adulthood. Their

utility function is the same as before. The optimal transfer T from child i to child ι (ι = j, k) is

a1+2a Ei−

a1+2a Eι. When e j = ek, T j = Tk. The consumption for child i is E

1+2a , where E = Ei+E j+Ek.

The other two children have a same level of consumption, which is aE1+2a .

Parents make decisions on investing in children’s human capital and altruism, respectively,

7The case that households have more than three children can be analyzed in a similar manner. To be consistentwith the empirical analysis where the variation in family size is entirely attributed to twinning, family size is assumedto be exogenous in the analysis.

16

taking children’s optimal transfers in adulthood into account. The following corollary is derived

by examining parental decisions on investing in children’s altruism:

Corollary 4 Parental investment in children’s inter-sibling altruism is higher when family size is

larger.

The intuition underlying this corollary is that the marginal gain from parental investment in chil-

dren’s altruism increases with family size. Specifically, when parents have three children, the

marginal gain is Tt(Uc j +Uck −Uci). Remember that the marginal gain is Tt(Uc j −Uci) when parents

have two children. Therefore, the marginal gain from parental investment in children’s altruism is

larger if parents have three children than two. The cost of parental investment in children’s altru-

ism may also increase with family size. But the scale effect of educating children to be altruistic is

assumed to be sufficiently large. Therefore, the marginal cost changes little with family size.

Combining Corollaries 2 with 4, I derive the following corollary about family size and intra-

household investment in children’s human capital:

Corollary 5 The intra-household human capital investment is more likely to reinforce gaps in

children’s prenatal endowments in larger families.

Corollary 2 states that, when parents are sufficiently averse to inequality, the intra-household in-

vestment in children’s human capital is more likely to reinforce gaps in children’s prenatal endow-

ments if parents invest more children’s altruism. Corollary 4 derives a positive effect of family size

on parental investment in children’s altruism. Thus, Corollary 5 directly follows.

Corollary 5 has important implications for the literature on family size and intra-household

investment in children’s human capital (Becker & Tomes, 1976; Rosenzweig & Wolpin, 1980).

When inter-sibling altruism enters into the model, intra-household human capital investment is

more likely to reinforce gaps in children’s prenatal endowments in larger families. Correspond-

ingly, gaps in investment in children’s human capital is bigger in larger families. The theoretical

prediction is consistent with empirical findings in Rosenzweig & Zhang (2009) and Aizer & Cunha

(2012). However, the mechanism they propose is different from mine. They assume that gaps in

17

children’s prenatal endowment are larger in families with more children. Considering that prena-

tal endowment and family investment are complementary inputs in the human capital production

function, gaps in investment in children’s human capital are bigger in larger families. I test my

theory against theirs in the sections below.

Corollary 5 also has important implications for understanding the relationship between family

size and inequality in a society. Sheshinski & Weiss (1982) demonstrate the importance of intra-

household resource allocation in accounting for inequality in a society. When we consider the

endogenous formation of children’s inter-sibling altruism and intra-household investment in chil-

dren’s human capital, inequality in the distribution of human capital and earnings may be larger in

societies with larger families. On the contrary, inequality in the distribution of consumption may

be smaller in these societies because of higher degree of altruism and larger transfers.

All theoretical predictions are summarized in the five corollaries above, which are tested in the

empirical analysis. Before reporting empirical results, the data set used in the paper is presented

below.

3 Data

This section describes the data set used in the empirical analysis. I first introduce the LCCTS,

which provides two types of variations brought about by twinning to facilitate causal identification.

The first one is the within-twin variation in prenatal endowments (Yi, 2013a). The other is the

variation in family size (Rosenzweig & Wolpin, 2000). I then discuss the construction of the

key variable in the empirical analysis, that is, family altruistic education. The validity of the

constructed proxy variables for family altruistic education in predicting children’s inter-sibling

transfers is also examined. Finally, I present summary statistics of main variables.

18

3.1 The Longitudinal Chinese Child Twin Survey

The data used for the empirical analysis are derived from the LCCTS. The first wave of the survey

was conducted by the Urban Survey Unit (USU) of the National Bureau of Statistics in late 2002

and early 2003 in Kunming, China. The children were aged 9 on average in the first wave. The

follow-up survey was conducted by the same unit in 2013. Kunming is the capital city of Yunnan

and has a total population of approximately 5 million. Yunnan is a less developed province located

at the far southwestern corner of China.

The LCCTS is the first longitudinal household survey on twin children of which I am aware.

The households were initially identified by the USU based on the 2000 population census ac-

cording to whether children have the same birth year and month and whether they have the same

relationship with the household head. The addresses of these households were then obtained from

the census office, and the presence of twins was verified by a visit to the household. For purposes

of comparison, households with non-twin children in the same age group were also surveyed.8 See

Rosenzweig & Zhang (2009) for a detailed description of the first wave of LCCTS. The second

wave followed households surveyed in the first wave.

The LCCTS provides an ideal opportunity to test the theoretical predictions for several reasons.

First, two types of “natural experiments” brought about by twinning facilitate causal identification.

Second, the survey questionnaire is carefully designed to solicit key information needed in the

empirical analysis, such as family altruistic education practice when children were young. Third,

we carefully design an experiment to measure inter-sibling transfers. Using the experimentally

measured inter-sibling transfers, we can directly test the validity of the constructed proxy variables

for family altruistic education. Finally, the survey covers an extensive range of information about

intra-household human capital investment and children outcomes, in addition to a wide range of

demographic, social, and economic information at the household level.

8For every child twin household identified, the fourth household on the right-hand side of the same block waschosen to locate a non-twin child household. If the fourth household was not an eligible household, interviewerswould continue to go to the fifth, sixth, and so on.

19

3.2 Variations in Birth Weight and Family Size

Two natural experiments brought about by twinning are used to carry out identification in the

empirical analysis. The first one is the within-twin variation in birth weight. Birth weight is the

only variable in the LCCTS data which can be used as a proxy for children’s prenatal endowments.

Using a within-twin fixed-effects estimator, Behrman & Rosenzweig (2004), Almond et al. (2005),

Black et al. (2007), and Oreopoulos et al. (2008) find that, relative to his or her twin sibling, lower

birth weight babies have worse outcomes, both short-run in terms of one-year mortality rates and

longer run in terms of educational attainment and earnings.

Using the LCCTS data, Yi (2013a) systematically examines the assumption that within-twin

variations in birth weight are exogenous in socioeconomic analysis, and find no evidence to re-

ject it.9 Considering that institutional, socioeconomic, and nutritional differences are substantial

between rural and urban areas in China, for instance, I examine the distributions of birth weight

and within-twin differences in birth weight by rural versus urban areas. Figure A1 in Appendix

B shows significant difference in birth weight between rural and urban areas. In contrast, the

difference in within-twin variations in birth weight between rural and urban areas is very minor

(Figure A2). In Table A1, I regress within-twin differences in birth weight on a series of child-,

parent-, and household-specific characteristics. No variable is found which can explain variations

in within-twin difference in birth weight.

The variation in family size induced by twinning plus the Chinese “one-child policy” is the

second natural experiment, which is crucial to test Corollaries 4 and 5. The specifics of the one-

child policy and its enforcement vary from one place to another in China. In Kunming, as in the

rest of China, the one-child policy is strictly enforced in the urban area. Rural households in the

sample site, however, are encouraged to have one child but are exempted from the strict one-child

policy. Table A2 tabulates the distribution of family size based on the LCCTS for five groups:

urban households with and without twins on the first birth; rural households with and without

twins on the first birth; and rural households with twins on the second birth. As can be seen, the

9By contrast, the within-sibling difference in birth weight could be affected by parental characteristics and prefer-ence if children are not twins (Rosenzweig & Wolpin, 1995).

20

policy is strictly enforced.10 Using twinning to assess the effects of family size on intra-household

investment in children’s altruism and human capital requires twinning to be randomly distributed in

the population. Table A3 provides descriptive statistics of parental characteristics by urban versus

rural and by twinning. If twinning is random, the characteristics of the parents with and without

twins on the first birth should be the same within each group. This hypothesis is confirmed by the

statistics shown in the table.

3.3 Family Altruistic Education

In the empirical analysis, I use family altruistic education to measure parental investment in chil-

dren’s altruism. Specifically, two variables described below are constructed as proxies for family

altruistic education. The validity of these two variables in predicting children’s inter-sibling trans-

fers is also examined.

3.3.1 Constructing Variables of Family Altruistic Education

The LCCTS survey questionnaire is carefully designed to solicit information on the practice of

family altruistic education. After consulting several recognized sociologists, psychologists, and

educators before the survey, two questions are included in the questionnaire. The first is about

the frequency in which parents brought all children together to worship ancestors or visit ancestral

tombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.

The second is about the frequency in which parents taught children to have fraternal love among

siblings as a virtue, such as telling their children about the story of “Kong Rong giving up pears”

when children are younger than 12.11 Both indicators are measured by a 5-point Likert scale from

1 (never) to 5 (always).

The two questions can effectively solicit information on the practice of family altruistic edu-

cation. First, worshiping ancestors or visiting ancestral tombs during the Qingming festival is a10Owning to strict enforcement and a policy that is gender neutral, twinning on the first birth, but not child gender,

is a powerful predictor of family size in our sample. See the statistic evidence shown in Rosenzweig & Zhang (2009).11See the discussion in Appendix C about the textbook example of “Kong Rong giving up pears” used to educate

children from Song dynasty (960 A.D. to 1279 A.D.) on the value of comity and fraternal love.

21

typical mechanism to transmit the Confucian culture, which claims that fraternal love is one root of

humanity. Levy-Garboua et al. (2006) suggest that cultural transmission is one major mechanism

to form an individual’s social preferences. Second, similar to religious rites, the activity during

the Qingming festival creates family values that help foster fraternal love within families. Third,

worshiping ancestors or visiting ancestral tombs with all family members facilitates interactions

among children, which help foster children’s inter-sibling altruism (Hoffman, 1975; Eisenberg

et al., 1992). Finally, one of the best methods to educate children is using examples. The story

of Kong Rong giving up bigger pears to his brothers is the well-known example used to educate

children about the value of comity and fraternal love.

These two questions are answered by parents and each child, respectively. Table 1 presents the

correlation matrix of the practice of family altruistic education reported by all family members.

First, correlation coefficients of family altruistic education practice reported by each twin are as

high as 0.92 and 0.93. The high correlation coefficients suggest that parents may educate their

children to be altruistic in same occasions at the same time. This fact justifies the assumption that

the scale effect of fostering children’s altruism by parents is sufficiently larger in the theoretical

analysis. Second, correlation coefficients of two types of practice by twin and parents are all above

0.60. Third, correlation coefficients between two types of family altruistic education practice are

lower than 0.50.

In the empirical analysis, for simplicity, I use the mean of the reported frequencies of each

family altruistic education practice as a measure. Therefore, I have two measures for the intensity

of family altruistic education. I also use the factor analysis to generate a score of the intensity

of family altruistic education. The results remain qualitatively the same as reported in the paper.

Finally, the two measures of family altruistic education practice may have functions other than

educating children to be altruistic toward one another. The implications of alternative interpretation

of family altruistic education practice for the analysis in the current paper are explored in a detailed

manner below in Section 5.

22

3.3.2 Family Altruistic Education and Inter-sibling Transfers

This section examines the validity of the constructed variables of family altruistic education in pre-

dicting children’s inter-sibling transfers. Given that most children are still in school in the second

wave of the survey, we carefully design a dictatorship game to measure inter-sibling transfers or

the degree of inter-sibling altruism. In particular, an arithmetic test is included in the interview. All

twin children are requested to take this test. However, only one twin is randomly chosen to have

monetary reward for taking this test. The amount of the reward is based on the test performance.

After receiving the money, the twin was asked how much he or she would like to transfer to his or

her twin sibling. Transfers are actually made by our interviewers during the survey. I construct a

variable that represents the percentage of the individual’s total reward that he or she transfers to his

or she twin sibling, which is used as a measure of inter-sibling transfers in the empirical analysis.

To empirically examine the validity of the constructed variables of family altruistic education,

I estimate the following regression equation:

Tτ,ι = η0 + η1tι + κτ,ιη2 + κ−τ,ιη3 + υι, (8)

where Tτι is the transfer from child τ to his or her sibling −τ in household ι; t is a measure of family

altruistic education discussed above; κ measures individual characteristics; and υ is an error term.

If family altruistic education is effective in fostering children’s altruism and then in enhancing

inter-sibling transfers, η1 is predicted to be positive.

Considering that family altruistic education is a parental choice, it may be endogenous in Equa-

tion (8). I employ a two-stage least squares (2SLS) method to deal with the endogeneity problem.

The theoretical analysis suggests that the within-twin difference in birth weight is an ideal instru-

mental variable for family altruistic education. Corollary 1 states that, on one hand, the gap in

children’s birth weight, which is used as a proxy variable for prenatal endowment, affects parental

investment in fostering children’s altruism. This corollary is empirically tested in the section be-

low. On the other hand, the gap in birth weight does not directly affect inter-sibling transfers when

23

both twin children’s characteristics are controlled for in the regression equation.

The OLS and 2SLS estimation results of Equation (8) are reported in Table 2. Column (1)

reports OLS estimates. The estimated coefficient on family altruistic education is consistent with

the prediction. An individual’s transfers to his or her twin sibling increase when the intensity

of family altruistic education is higher. The estimate is statistically significant at the 10% level.

Regarding other covariates, an individual’s own years of schooling, on one hand, increase transfers

to his or her twin sibling, and the estimate is statistically significant. On the other hand, his or her

sibling’s years of schooling decrease an individual’s transfers, although the estimate is statistically

insignificant.

The 2SLS estimates are reported in column (2) by considering the endogeneity of family al-

truistic education. The estimated coefficients undergo minimal change. I add individual charac-

teristics, such as age, gender, and birth place, in column (3). The estimation results for family

altruistic education and individual’s own and his or her sibling’s schooling years remain the same

with reasonable variation. I replicate estimations reported in columns (1) to (3) by using a second

measure of family altruistic education. The results are qualitatively the same, which are reported

in columns (4)-(6). In summary, the results in Table 2 suggest that the variables of family altruistic

education effectively predict children’s inter-sibling transfers. Therefore, these two variables are

valid for the empirical analysis below.

3.4 Descriptive Statistics of Main Variables

This section describes other main variables, which are empirical counterparts to those in the theo-

retical analysis. Table 3 reports descriptive statistics. One major dependent variable in the empir-

ical analysis is intra-household investment in children’s human capital, which is measured in the

first wave of the survey in 2002 with children aged 9 on average. This variable includes school

tuition fees and money spent on purchasing books and stationery, hiring home tutors, and attend-

ing tutorial class. Educational investment on children constitutes a substantial fraction of family

income. In particular, educational investment on one child alone amounts to RMB 986 per year,

24

out of a per capita family income of RMB 3,030 per year. Birth weight is used as a proxy vari-

able for children’s prenatal endowments. Family income Y is proxied by household asset surveyed

in the first wave, which is a score generated by using the factor analysis (Kolenikov & Angeles,

2009). The durable goods used to generate the household asset score include washing machine,

refrigerator, air-conditioner, shower heater, water dispenser, sterilized cupboard, motor cycle, care,

radio/recorder, color television, video recorder, video displayer, hi-fi, camera, telephone, and mo-

bile phone. Children’s human capital is measured by their years of schooling in the second wave

of the survey in 2013.

Twin child, twin pair, parental, and household characteristics are used as proxy variables for

wage, as well as prices of human capital investment and family altruistic education in the empir-

ical analysis. Specifically, the rural indicator is commonly used as a proxy variable for wage in

studies about developing countries because of the large wage gap between rural and urban areas

(Rosenzweig, 1980). I use the household registration status (hukou) as a proxy variable for prices.

At the time of the survey, although public education is not free in both urban and rural areas, the

quality of education in urban areas is higher. The sector of maternal employment is also a good

proxy variable for the price of educational investment. Children’s education is usually subsidized

by the government if mothers are employed in the public sector. Moreover, job turnovers from the

public to the private sectors are rare for Chinese women (Cai et al., 2008).

4 Empirical Results

In this section, I test the theoretical predictions which are embodied in the five corollaries. First, I

test the effect of gaps in children’s birth weight on family altruistic education. I then test the effect

of family altruistic education on the gap in intra-household investment in children’s human capital.

The implication of the omission of family altruistic education for the bias with the estimates of

the degree of parental aversion to inequality is also examined. Finally, I test the effects of family

size on family altruistic education and on gaps in intra-household investment in children’s human

25

capital.

4.1 Birth Weight and Family Altruistic Education

This section tests Corollary 1 about the effect of the gap in children’s prenatal endowments on

family altruistic education. Children’s birth weight is used as a proxy variable for prenatal endow-

ments. The following regression equation is estimated:

tι = γ0 + γ1∆eι + πtιγ2 + πp

ι γ3 + πhι γ4 + ει, (9)

where the dependent variable tι is a measure of the intensity in family altruistic education in house-

hold ι; ∆ is an operator of within-twin difference, and thus, ∆eι = ei − e j; πt, πp, and πh are twin

pair-, parent-, and household-specific characteristics; ε is an error term indicating measurement

errors with family altruistic education. Corollary 1 predicts that parental desire to foster children’s

inter-sibling altruism increases when gaps in children’s prenatal endowments are bigger. There-

fore, γ1 is expected to be positive. Equation (9) is estimated by using the sample including twin

children only.

The identification assumption in estimating Equation (9) is that within-twin variations in birth

weight are uncorrelated with the error term, which is discussed in the section above. This as-

sumption can also be tested by estimating Equation (9). In particular, if the assumption holds,

the estimated coefficient of γ1 would be unaffected by the inclusion of twin pair-, parent-, and

household-specific characteristics in the regression equation.

The estimation results, which are reported in Table 4, are consistent with the prediction by

Corollary 1. Across all specifications, the estimated coefficients on the gap in children’s birth

weight are positive and statistically significant. The results suggest that parents are more likely

to educate children to be altruistic when gaps in children’s birth weight are bigger. Further-

more, as expected, the estimated coefficients on gaps in children’s birth weight change little when

parent- and household-specific characteristics are added in the regression equation. The Durbin-

26

Wu-Hausman test statistics between specifications (1) and (2) and between specifications (2) and

(3) are statistically insignificant. The results suggest that within-twin differences in birth weight

are plausibly exogenous. The possibility of any remaining unobservable variables in the error term,

which may correlate with gaps in children’s birth weight and family altruistic education simulta-

neously, is discussed in the section below.

Regarding other covariates, parents are more likely to educate children to be altruistic when

both twin children are males, and when the mothers are highly educated, Han Chinese, and not

employed in the public sector.

4.2 Inter-sibling Altruism and Intra-household Human Capital Investment

This section tests Corollaries 2 and 3 and empirically examines the implications of inter-sibling

altruism for intra-household investment in children’s human capital. Specifically, I estimate the

following regression equation:

∆Iι = δ0 + δ1∆eι + δ2tι + ∆κιδ3 + ∆µι, (10)

where the dependent variable ∆Iι is the within-twin difference in parental investment in children’s

human capital in household ι; ∆κ is a vector of within-twin differences in individual characteristics;

∆e is the within-twin difference in birth weight; t is a measure of the intensity in family altruistic

education; and ∆µ is an error term. Equation (10) is estimated by using the sample including

twin children only. The regression specification above is a standard one adopted in the empirical

literature of intra-household investment in children’s human capital, except the inclusion of family

altruistic education.

Considering that family altruistic education is a parental choice, I employ a 2SLS method to

perform the estimation. The theoretical model, once again, guides the choice about instrumental

variables for family altruistic education. In particular, the theoretical analysis shows that, on one

hand, household income (Y) and prices of intra-household human capital investment (pI) and fam-

27

ily altruistic education (pt) are theoretical determinants of family investment in fostering children’s

altruism. On the other hand, Y , pI , and pt do not enter Equation (7) in difference form, which is

the theoretical counterpart of the regression equation above. Thus, these variables are qualified

as instrumental variables for t in estimating Equation (10). Empirically, twin pair-, parent-, and

household-specific characteristics are used as proxy variables for Y , pI , and pt.

Corollary 2 predicts that parents allocate more resources to the better-endowed child when they

educate their children to be more altruistic toward one another and they are sufficiently averse to

inequality. Therefore, δ2 is positive if ρ < 0.12 In addition, Proposition 3 states that δ1 reflects

only human capital production technology when family altruistic education is controlled for in

the regression equation. Therefore, δ1 is expected to be positive when prenatal endowments and

parental investment are complementary inputs in producing human capital. Finally, Corollary 3

predicts that the degree of parental aversion to inequality is under-estimated if family altruistic

education is omitted from Equation (10). That is, ρ is over-estimated. To test this prediction,

Equation (10) is estimated without and with controlling for family altruistic education, respectively.

If parents are sufficiently averse to inequality, such that ρ < 0, Corollary 3 predicts that δ1 is larger

when t is omitted from Equation (10) because both corr(∆e, t) > 0 as predicted by Corollary 1,

which is verified in Table 4, and δ2 > 0 as predicted by Corollary 2.

The estimation results, which are reported in Table 5, are consistent with theoretical predic-

tions. Following Behrman et al. (1982, 1986), column (1) includes the differences in children’s

birth weight and gender as independent variables. The estimated coefficient on the difference in

children’s birth weight is positive and statistically significant. Based on Behrman et al. (1982), this

positive estimate indicates that the preference parameter of parental aversion to inequality among

their children ρ is positive. In contrast, Yi (2013b) demonstrates that this estimated coefficient in

column (1) reflects a combination of parameters from both parental preference and human capital

production technology. Thus, the estimated coefficient in column (1) does not necessarily inform

us the sign of ρ. Moreover, the theoretical model in the current paper shows that the regression

12Note that ∆Iι = Ii− I j and ∆eι = ei−e j. I further rank twin children by ei > e j within each household in compilingthe data.

28

specification in column (1) is misspecified because it disregards the consideration of inter-sibling

transfers when parents make decisions on intra-household investment in children’s human capital.

Family altruistic education is added as an independent variable in column (2). Four major

findings are noted. First, the estimated coefficient on family altruistic education is positive and

statistically significant at the 5% level. This result is consistent with the prediction by Corollary

2, suggesting that parents take children’s inter-sibling transfers into account when they make deci-

sions on intra-household investment in children’s human capital. Second, the estimated coefficient

on the gap in birth weight remains positive after controlling for family altruistic education. Propo-

sition 3 states that the effect of the gap in birth weight on the gap in intra-household human capital

investment is determined by human capital production technology only when family altruistic ed-

ucation is controlled for in the regression. Therefore, the positive estimate of δ1 in column (2)

suggests that prenatal endowments measured by birth weight and intra-household human capi-

tal investment are complementary inputs in the human capital production function. The result is

consistent with the recent literature of human capital production technology (Cunha & Heckman,

2007; Heckman, 2007).

Third, the positive estimate of δ2 implies ρ to be negative by Corollary 2. As discussed above,

it is wrongly concluded that ρ was positive by the positive estimate of δ1 in column (1) when I

adopt the traditional specification in the literature. Therefore, ρ is over-estimated if t is omitted

from Equation (10). That is, the degree of parental aversion to inequality among their children is

understated if researchers fail to take inter-sibling transfers into account. This result is consistent

with the prediction by Corollary 3. Fourth, comparing columns (2) and (1), the magnitude of

the estimated coefficient on the gap in children’s birth weight decreases when family altruistic

education is controlled for in the regression equation. This result is, once again, consistent with

the prediction by Corollary 3. The omission of inter-sibling altruism or family altruistic education

biases up the estimate of ρ, and then understates the degree of parental aversion to inequality.

The 2SLS estimation results are reported in column (3). Given that family altruistic educa-

tion may be an endogenous variable, household assets, a rural dummy, as well as maternal age,

29

schooling years, and sector of employment are used as instrumental variables. These instrumental

variables are proxies for household income, wage, and prices of human capital investment and

family altruistic education in the theoretical model. The 2SLS estimates are qualitatively the same

as the OLS estimates. The Durbin-Wu-Hausman test statistics are significant at the 1% level, sug-

gesting that family altruistic education is an endogenous variable in the intra-household human

capital investment equation. Columns (4)-(5) replicate columns (2)-(3), except that I use an alter-

native measure of family altruistic education. The results almost remain the same when the second

measure of family altruistic education is used. Finally, estimated coefficients on the difference in

gender between twin siblings are consistently positive across all columns. The result is consistent

with Behrman et al. (1986), demonstrating the existence of gender preference in intra-household

investment in children’s human capital.

Estimating Equations (9) and (10) by rural and urban areas separately, I also explore the differ-

ence in the endogenous formation of inter-sibling altruism and intra-household investment in chil-

dren’s human capital between areas. I find weak evidence that rural households are more likely to

rely on the mechanism of fostering children’s inter-sibling altruism to address the conflict between

efficiency and equality in intra-household human capital investment. The results are discussed in

detail in Appendix D.

In summary, the results reported in Table 5 are consistent with the theoretical predictions by

Corollaries 2 and 3. Parents take children’s inter-sibling transfers into account when they make

decisions on intra-household investment in children’s human capital. Parents are more likely to

take a reinforcement strategy in investing children’s human capital if they educate children to be

more altruistic. The degree of parental aversion to inequality among their children is understated

if family altruistic education is neglected. The robustness of the estimates reported in Table 5 with

respect to cross-household heterogeneities in preferences and measurement errors is discussed in

Section 5 below.

30

4.3 Family Size, Family Altruistic Education, and Intra-household Human

Capital Investment

This section tests Corollaries 4 and 5 about family size, family altruistic education, and intra-

household investment in children’s human capital. I first estimate the following regression equation

to test Corollary 4 about the effect of family size on family altruistic education:

tι = ζ0 + ζ1nι + ζ2∆eι + πpι ζ3 + πh

ι ζ4 + vι, (11)

where nι is family size or the number of children in household ι. The variations in family size

are exclusively induced by twinning in the estimation sample as discussed below. The variables

of t, πp, and πh are defined above; v is an error term. Equation (11) is similar to Equation (9),

except for two changes. First, the number of children is added as an additional covariate. Second,

∆e measures the difference (in terms of absolute value) of birth weight between children born in

the first and second birth parity. Two children are born at the second parity for twin households.

Therefore, the average birth weight of twin children is used as the birth weight at second parity for

twin households. Corollary 4 predicts that parents are more likely to educate their children to be

altruistic in larger families. Thus, ζ1 is predicted to be positive.

Considering that family size is an endogenous variable, twinning in combination with the Chi-

nese population policy are used as a natural experiment to carry out the identification. In particular,

only two types of rural households are included in the estimation sample. The first type includes

households with two singletons. The second one includes households with three children and

twinning at the second birth parity. Since each family is allowed to have at most two births by

the population policy in rural Kunming, variations in family size in estimating Equation (11) are

entirely attributed to twinning, and n equals 2 or 3.

To test Corollary 5 about family size, family altruistic education, and gaps in intra-household

31

investment in children’s human capital, the following regression equation is estimated:

∆Iι = θ0 + θ1nι + θ2tι + θ3∆eι + ζι, (12)

where ∆Iι is the gap in intra-household human capital investment among children born at differ-

ent parities. The sample used to estimate Equation (12) is the same as the one used to estimate

Equation (11). Therefore, variations in n in estimating Equation (12) are entirely attributable to

twinning. The average human capital investment of the two children is used as the human capital

investment at the second birth parity for twin households. The definition of ∆e is the same as

Equation (11).13 Considering that family altruistic education is an endogenous variable in Equa-

tion (12), I use a 2SLS estimator to perform the estimation. Similar to the estimation of Equation

(10), parents- and household-specific characteristics are used as instrumental variables for family

altruistic education.

Corollary 5 predicts that intra-household investment in children’s human capital is more likely

to reinforce gaps in children’s prenatal endowment in larger families. Therefore, θ1 is predicted to

be positive. In addition, θ2 is predicted to be positive by Corollary 2. Furthermore, the magnitude of

the estimated θ1 is predicted to be larger when family altruistic education is included into Equation

(12) than that when it is not included. The reason is that corr(n, t) > 0 by Corollary 4 and θ2

is predicted to be positive by Corollary 2. Finally, Proposition 3 states the estimate of θ3 reflects

human capital production technology only when family altruistic education is included in Equation

(12). Thus, θ3 is predicted to be positive when children’s prenatal endowments and parental human

capital investment are complementary inputs in the human capital production function.

The estimation results of Equation (11) are reported in Table 6. In column (1), the gap in

children’s birth weight is included in the regression equation as the independent variable only.

Consistent with the theoretical prediction by Corollary 1, parents are more likely to educate the

children to be altruistic when gaps in children’s birth weight are bigger. Family size is added as

an additional covariate in column (2). Consistent with the prediction by Corollary 4, the estimated

13I rank birth parity in each household by ei > e j in compiling the data set.

32

coefficient on family size is positive and statistically significant at the 5% level, suggesting that

parents are more likely to educate children to be altruistic in larger families. Furthermore, the

magnitude of the estimated coefficient on the gap in children’s birth weight decreases when family

size is added in the regression equation. The result is expected because gaps in children’s prenatal

endowments, which are measured by birth weight, is bigger when family size is larger (Rosen-

zweig & Zhang, 2009), and the effect of family size on family altruistic education is positive by

Corollary 4. The results in column (2) remain qualitatively the same when parental and household

characteristics are added in the regression equation, which are reported in column (3). Finally, the

estimation results remain robust when the dependent variable is replaced by an alternative measure

of family altruistic education, which are reported in columns (4)-(6).

The estimation results of Equation (12) are reported in Table 7. The estimates reported in

column (1) show that family size increases gaps in intra-household investment in children’s human

capital. This result is consistent with the prediction by Corollary 5 that family size increases the

intensity of family altruistic education, and thus enhances gaps in intra-household investment in

children’s human capital. However, this result is also consistent with the interpretation proposed

by Rosenzweig & Zhang (2009) and Aizer & Cunha (2012). They suggest that the positive effect of

family size on gaps in intra-household investment in children’s human capital is attributed to bigger

gaps in children’s prenatal endowments in larger families. If parental human capital investment

reinforces differences in children’s prenatal endowments, gaps in intra-household investment in

children’s human capital are bigger in larger families.

To test these two alternative hypotheses, the difference in children’s birth weight is firstly added

in column (2). The estimated coefficient on the difference in birth weight is positive and statisti-

cally significant, suggesting that parental human capital investment reinforces gaps in children’s

prenatal endowments. Furthermore, the magnitude of the estimated coefficient on family size de-

creases when the difference in birth weight is added in the regression equation. Therefore, the

positive effect of family size on gaps in intra-household investment in children’s human capital

is partly attributed to larger differences in children’s prenatal endowments, which is proposed by

33

Rosenzweig & Zhang (2009) and Aizer & Cunha (2012).

In addition to the mechanism proposed by Rosenzweig & Zhang (2009) and Aizer & Cunha

(2012), Corollary 5 suggests another one through which family size increases gaps in intra-household

investment in children’s human capital. To test this mechanism against that of Rosenzweig &

Zhang (2009) and Aizer & Cunha (2012), family altruistic education is added as an additional

covariate in column (3). Consistent with the prediction by Corollary 5, the magnitude of the es-

timated coefficient on family size decreases after controlling for family altruistic education. The

result suggests that part of the positive effect of family size on gaps in intra-household investment

in children’s human capital is attributable to the increase in family altruistic education. Therefore,

both mechanisms proposed by Rosenzweig & Zhang (2009) and Aizer & Cunha (2012) and by

this paper co-exist. The results remain robust when the endogeneity of family altruistic education

is considered (column (4)) and family altruistic education is replaced by an alternative measure

(columns (5) and (6)).

In summary, the estimation results reported in Tables 6 and 7 support predictions by Corollaries

4 and 5. Family size increases the intensity of family altruistic education, consequently causing

intra-household investment in children’s human capital being more likely to reinforce gaps in chil-

dren’s prenatal endowments in larger families.

5 Heterogeneities, Measures Errors, and Alternative Interpre-

tations

This section discusses the robustness of the observed empirical pattern with respect to potential

cross-household heterogeneities and measurement errors with main variables. The latter two fac-

tors could invalidate the used instrumental variables, and thus alternative interpretations arise. I

first consider cross-household heterogeneity in parental preference of inequality aversion among

children, which is unobservable and contained in the error terms from Equations (9) to (12). If

this is the case, parents invest more in better-endowed children’s human capital when they are

34

less averse to inequality. At the same time, these parents are less likely to educate their children

to be altruistic. Thus, we should observe a negative correlation between gaps in intra-household

investment in children’s human capital and family altruistic education. This prediction contradicts

the estimated positive correlation in the current paper. Thus, the estimated empirical pattern is un-

likely driven by unobservable cross-household heterogeneity in parental preference of inequality

aversion.

Second, the measures of family altruistic altruism may be constructed with errors. For ex-

ample, worshiping ancestors has functions such as transmitting Confucius values in addition to

fraternal love. Furthermore, households may be heterogeneously influenced by traditional Con-

fucianism. In this case, more traditional parents should bring their children to worship ancestors

more frequently. At the same time, these families should more equally invest in children’s human

capital. As an old Chinese saying puts it, “sons should be treated as a pair of chopsticks.” Further-

more, Appendix D shows that rural families which are more traditional are less likely to take an

enforcement human capital investment strategy than urban families. This prediction, once again,

contradicts the reported positive correlation between gaps in intra-household investment in chil-

dren’s human capital and family altruistic education in the current paper. Therefore, the estimated

empirical pattern is less likely driven by measurement errors with practices of family altruistic

education or cross-household heterogeneity in Confucius cultural background.

Third, birth weight is not a perfect measure for prenatal endowments. Furthermore, within-twin

variations in birth weight may reflect unobservable cross-household heterogeneity in human capital

production technology. Assume that the observed empirical pattern is driven by the unobservable

human capital production technology and that within-twin differences in birth weight are larger in

households with inferior human capital production technology. I find difficult to justify why these

households more frequently bring children to worship ancestors and, at the same time, more likely

to adopt a reinforcement human capital investment strategy. Traditional families in rural areas may

have inferior human capital production technology and more frequently bring children to worship

ancestors. However, these families should more equally divide investment in children’s human

35

capital as discussed above.

Finally, I consider an alternative interpretation of parental preference parameter ρ in the utility

function. In the theoretical analysis above, ρ is interpreted as the degree of parental aversion to

inequality in the distribution of children’s consumption. In the presence of risk and uncertainty

regarding either the production of children’s human capital or the return to human capital in the

labor market, ρ can be re-interpreted as the degree of parental risk aversion. When ρ is smaller,

parents are more averse to risk, and then they are more likely to evenly invest children’s human

capital.

In this case, I have a theory on parental risk aversion and the endogenous formation of inter-

sibling altruism. Specifically, when parents are more risk averse and inter-sibling transfers do not

exist, they invest more on less-endowed children’s human capital to diversify risk than that when

parents are risk neutral. This practice decreases total returns from intra-household human capital

investment. Thus, parents have incentives to foster children’s inter-sibling altruism in the presence

of risk aversion. The increase in altruism acts as a device of income pooling for children. With

a higher degree of inter-sibling altruism, parents invest more in better-endowed children’s human

capital to enhance total returns from intra-household investment. With this scenario, the empirical

results in this paper can be re-interpreted. Future work is needed to distinguish these two motives

for parents to foster children’s inter-sibling altruism. In reality, both motives of risk aversion and

inequality aversion may co-exist when parents make decisions on investment in children’s human

capital.

6 Conclusion

This paper investigates parental incentives to educate children to be altruistic toward one another.

I show that parental desire to foster children’s altruism is closely related to their decisions on intra-

household investment in children’s human capital. Specifically, the theoretical analysis demon-

strates that fostering inter-sibling altruism in childhood by parents not only enhances total returns

36

from intra-household investment in children’s human capital but also decreases inequality in the

distribution of consumption among children. The theoretical predictions are empirically tested by

using a unique twin household survey data set. Empirical results are consistent with theoretical

predictions.

The results of this paper has important implications for understanding the relationships be-

tween economic development, cultural evolution, transformation of social structure, and public

policies, particularly in China and other East Asian countries. The paper shows that parental de-

sire to foster children’s preferences could be framed into an incentive structure that is determined

by socioeconomic factors. Economic development may change the incentive structure, and cor-

respondingly, parental desire to foster children’s preferences. Specifically, based on the current

study, the decrease in family size and the implementation of public education system may weaken

parental incentive to educate their children to be altruistic. Therefore, social networks which are

linked through altruism may shrink. In this case, public policies is requested to provide services

which are traditionally supplied by these social networks.

Future exploration on the endogenous formation of altruism is highly desirable. Two potential

topics are in my current research agenda. One is to study another type of altruism within families,

that is, filial piety, which is a virtue of respect for one’s parents and ancestors. Based on the same

data set used in this paper, I plan to test predictions derived from a theory of endogenous formation

of filial piety. Another topic is about the endogenous formation of altruism in organizations aside

from families, such as firms. I plan to investigate whether establishing employees’ altruism can

ameliorate commitment problems and then enhance productivity in work places or not.

37

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Table 1: Correlation Matrix of Reported Family Altruistic Education Practices

(1) (2) (3) (4) (5) (6)Reported by parents (1) 1

Practice 1 Reported by twin 1 (2) 0.692 1Reported by twin 2 (3) 0.704 0.915 1Reported by parents (4) 0.456 0.392 0.391 1

Practice 2 Reported by twin 1 (5) 0.297 0.378 0.351 0.660 1Reported by twin 2 (6) 0.327 0.395 0.376 0.640 0.930 1

Note: Practice 1 measures the frequency in which parents brought children together to worshipancestors or visit ancestral tombs during the Qingming festival (Tomb-Sweeping Day) when chil-dren were younger than 12. Practice 2 measures the frequency in which parents told children thathaving fraternal love among siblings is a virtue, such as telling children the story of “Kong Ronggiving up pears” when children were younger than 12. See appendix about the textbook exampleof “Kong Rong giving up pears” used to educate children on the value of comity and fraternal lovefrom Song dynasty (960 A.D. to 1279 A.D.). Both practices are measured by a 5-point Likert scalefrom 1 (never) to 5 (always).

42

Table 2: OLS and 2SLS Estimates Examining the Effects of Family AltruisticEducation on Inter-sibling Transfers

Dependent variable:Inter-sibling transfersa

OLS 2SLS 2SLS OLS 2SLS 2SLS(1) (2) (3) (4) (5) (6)

Family altruistic education 0.152∗∗ 0.127∗∗ 0.139∗∗

(Measure 1)b (η1) (2.19) (2.07) (2.11)Family altruistic education 0.072 0.068∗ 0.054∗

(Measure 2)c (η1) (1.01) (1.88) (1.78)Years of schooling (own) 0.025∗∗ 0.029∗ 0.026∗ 0.025∗ 0.025∗ 0.023∗

(1.99) (1.68) (1.69) (1.94) (1.71) (1.77)Years of schooling (sibling’s) -0.016 -0.010 -0.010 -0.016 -0.011 -0.012

(1.05) (1.01) (1.03) (1.11) (1.30) (1.24)Age -0.007 0.000

(0.48) (0.01)Male 0.091∗∗ 0.063∗

(1.96) (1.71)Rural 0.027 0.025

(0.98) (0.21)

R-squared 0.08 . . 0.07 . .# Obs. 508 508 508 508 508 508

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05,*** p < 0.01. (3) Family altruistic education is the endogenous variable in columns (2)-(3) and(5)-(6). The instrumental variable is the absolute value of the gap in children’s birth weight. (4)The estimation sample includes twin children who participated into the arithmetic test and thedictatorship game.a : The percentage of the amount of reward that the child transfers to his or her twin siblingb : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).c : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).

43

Table 3: Descriptive Statistics of Main Variables

Theoretical Empirical Mean SDVariables Counterpartst Family altruistic education (Measure 1)a 4.106 1.063

Family altruistic education (Measure 2)b 4.417 0.572T Inter-sibling transfersa 0.583 0.255I Educational investment (U/year) 985.806 1395.095e Birth weight (kg) 2.723 0.576h Years of schooling 13.935 3.418Y Household asset (score) -0.027 1.740ω, pI , ph

Child characteristics (κ)Male (dummy) 0.498 0.500Twin-pair characteristics (π)Age 20.377 3.101Both male (dummy) 0.375 0.484Both female (dummy) 0.377 0.484Parental characteristics (ξ)Maternal age 46.581 4.701Maternal ethnicity (dummy: Han=1) 0.844 0.363Maternal schooling years 8.639 3.273Maternal working section (Public=1) 0.078 0.268Household characteristics ($)Rural (dummy) 0.568 0.495

Note: The sample includes 508 pairs of twins.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).

44

Table 4: OLS Estimates Examining the Effects of Differences in Children’s Birth Weight onFamily Altruistic Education

Dependent variable: Family altruistic educationMeasure 1a Measure 2b

(1) (2) (3) (4) (5) (6)Difference in birth weightc (γ1) 0.321∗∗∗ 0.318∗∗∗ 0.314∗∗∗ 0.108∗ 0.104∗ 0.103∗

(4.27) (3.18) (3.14) (1.89) (1.78) (1.68)Age 0.018 0.046 0.038 0.009 -0.021 -0.019

(0.20) (0.50) (0.41) (0.19) (0.41) (0.38)Male twins 0.025∗∗ 0.031∗∗ 0.032∗∗ 0.013 0.021∗∗∗ 0.020∗∗∗

(2.08) (2.19) (2.30) (1.93) (2.80) (2.64)Female twins -0.077 -0.069 -0.072 -0.038 -0.047 -0.041

(0.89) (0.79) (0.83) (0.80) (0.99) (0.86)Maternal age 0.011 0.011 0.011∗∗ 0.010∗∗

(1.24) (1.24) (2.33) (2.13)Maternal years of schooling 0.044∗ 0.034 0.045∗∗∗ 0.034∗

(1.86) (0.91) (3.55) (1.74)Maternal ethnicity 0.260∗∗∗ 0.277∗∗∗ 0.115∗∗ 0.126∗

(Han=1) (3.07) (3.22) (2.31) (1.76)Maternal employment sector -0.341∗ -0.319∗ -0.185∗ -0.192∗

(Public=1) (1.86) (1.78) (1.83) (1.90)Household asset (score) 0.025 0.036∗∗

(0.86) (2.24)Rural 0.165∗ 0.086

(1.74) (1.07)

R-squared 0.09 0.18 0.23 0.07 0.24 0.29# Obs. 508 508 508 508 508 508

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05, ***p < 0.01. (3) The estimation sample includes only households with twin children.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).c : The absolute value of the difference in children’s birth weight.

45

Table 5: OLS and 2SLS Estimates Examining the Effects of Family Altruistic Education onIntra-household Human Capital Investment

Dependent variable:Difference in educational investment

OLS OLS 2SLS OLS 2SLS(1) (2) (3) (4) (5)

Difference in birth weight 54.624∗∗∗ 43.684∗∗∗ 34.645∗∗∗ 54.471∗∗∗ 34.518∗∗∗

(δ1) (3.82) (3.78) (2.81) (3.82) (3.75)Family altruistic education 18.960∗∗ 37.828∗∗

(Measure 1)a (δ2) (2.37) (2.20)Family altruistic education 10.365∗ 27.376∗

(Measure 2)b (δ2) (1.72) (1.67)Difference in gender 12.298∗ 12.493∗ 16.465∗∗ 11.798∗ 19.712∗

(Male=1) (1.93) (1.94) (2.05) (1.90) (1.69)

R-squared 0.06 0.08 . 0.07 .# Obs. 508 508 508 508 508

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05, ***p < 0.01. (3) Family altruistic education is the endogenous variable in columns (3) and (5). Theinstrumental variables are household asset, a rural dummy, and maternal age, schooling years, andworking sector. (4) The estimation sample includes only households with twin children.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).

46

Table 6: OLS Estimates Examining the Effects of Family Size on Family AltruisticEducation

Dependent variable: Family altruistic educationMeasure 1a Measure 2b

(1) (2) (3) (4) (5) (6)Difference in birth weight 0.249∗∗∗ 0.231∗∗∗ 0.219∗∗∗ 0.125∗ 0.113∗ 0.108(ζ1) (3.12) (3.05) (2.80) (1.81) (1.74) (1.64)Family size (ζ2) 0.283∗∗ 0.390∗∗∗ 0.257∗ 0.223∗

(2.27) (3.02) (1.96) (1.77)Maternal age -0.011 0.002

(0.95) (0.30)Maternal years of schooling 0.061∗∗ 0.079∗∗

(2.46) (1.68)Maternal ethnicity (Han=1) 0.530∗∗∗ 0.041

(4.73) (0.75)Household asset (score) 0.157∗∗∗ 0.027

(2.93) (1.03)

R-squared 0.09 0.18 0.23 0.07 0.24 0.29# Obs. 477 477 477 477 477 4

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05, ***p < 0.01. (3) The estimation sample includes only rural non-twin households with twin childrenand rural twin households with three children twinning at the second birth parity. Therefore, fer-tility equals to only two or three in our estimation sample.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).

47

Table 7: OLS and 2SLS Estimates Examining Family Size, Family Altruistic Education, andIntra-household Human Capital Investment

Dependent variable:Difference in educational investment

OLS OLS OLS 2SLS OLS 2SLS(1) (2) (3) (4) (5) (6)

Family size (θ1) 166.241∗∗ 125.834∗∗ 98.050∗∗ 95.120∗∗ 110.601∗∗ 108.887∗

(2.18) (2.16) (1.93) (2.31) (2.19) (1.76)Difference in birth weight 48.846∗∗∗ 34.557∗∗∗ 32.542∗∗∗ 40.327∗∗ 38.819∗∗

(θ2) (3.06) (3.04) (2.07) (2.08) (2.31)Family altruistic education 67.811∗ 80.975∗∗

(Measure 1)a (θ3) (1.69) (2.02)Family altruistic education 31.389∗ 56.965∗∗

(Measure 2)b (θ3) (1.83) (2.19)

R-squared 0.02 0.03 0.05 . 0.03 .# Obs. 477 477 477 477 477 477

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05,*** p < 0.01. (3) Family altruistic education is the endogenous variable in columns (4) and(6). The instrumental variables are household asset, a rural dummy, and maternal age, schoolingyears, and working sector. (4) The estimation sample includes only rural non-twin householdswith twin children and rural twin households with three children twinning at the second birthparity. Therefore, family size equals to two or three in our estimation sample.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).

48

Appendix A: Relax the Symmetry Assumption with Parental Utility Function

This appendix considers that children are treated asymmetrically in parental utility function. As-

sume parental utility function to be

U = [αcρi + (1 − α)cρj]1ρ ,

where 0 < α < 1. This function indicates that parents prefer child i if α > 0.5. Otherwise,

they prefer child j. With this generalized parental utility function, the equilibrium conditions for

various cases considered in the paper are as follows.

Inequality aversion:(ln

Ii

I j

)ρ<1,a=0

=1

1 − ρβIln

α

1 − α+

ρβe

1 − ρβIln

ei

e j,

Efficient:(ln

Ii

I j

)ρ=1,a=0

=1

1 − βIln

α

1 − α+

βe

1 − βIln

ei

e j,

Altruism:(ln

Ii

I j

)ρ<1,a<1

=1

1 − ρβIln

α

1 − α+−ρ

1 − βIln a +

βe

1 − βIln

ei

e j,

Endogenous Altruism:(ln

Ii

I j

)ρ<1,a=ϕtσ

=1

1 − ρβIln

α

1 − α+−ϕρ

1 − βI+−σρ

1 − βIln t +

βe

1 − βIln

ei

e j.

Therefore, relaxing the symmetric assumption does not affect the elasticities of IiI j

with respect to

eie j

, a, and h.

49

Appendix B: Supplementary Tables and FiguresTable A1: OLS Estimates of the Determinants of Birth Weights and Within-Twin

Differences in Birth Weights

Birth Weight(1) (2) (3) (4) (5)

Birth year 0.005* 0.005* 0.005* 0.005* 0.005*(0.003) (0.003) (0.003) (0.003) (0.003)

Birth order 0.098*** 0.097*** 0.100*** 0.105*** 0.088***(0.023) (0.023) (0.021) (0.022) (0.023)

Maternal age at twin birth 0.002 0.002 0.002 0.002 0.002(0.002) (0.002) (0.002) (0.002) (0.002)

Twin born in rural areas 0.025 0.009(0.019) (0.020)

Maternal educational years -0.005 -0.003(0.003) (0.003)

Maternal ethnic indicator -0.067*** -0.062**(0.025) (0.025)

Maternal occupation -0.036 -0.014(0.031) (0.034)

Joint F-statistics 7.90 8.38 8.97 7.91 5.61p-values 0.000 0.000 0.000 0.000 0.000Observations 2,940 2,940 2,940 2,940 2,940

Within-Twin Differences in Birth Weights(6) (7) (8) (9) (10)

Birth year 0.001 0.001 0.001 0.001 0.001(0.003) (0.003) (0.003) (0.003) (0.003)

Birth order 0.005 0 0.005 0.002 0.006(0.022) (0.022) (0.021) (0.021) (0.023)

Maternal age at twin birth 0.002 0.002 0.002 0.002 0.002(0.002) (0.002) (0.002) (0.002) (0.002)

Twin born in rural areas -0.01 -0.007(0.019) (0.020)

Maternal educational years 0 -0.002(0.003) (0.004)

Maternal ethnic indicator 0.035 0.034(0.023) (0.023)

Maternal occupation 0.017 0.023(0.031) (0.037)

Joint F-statistics 0.35 0.26 0.81 0.34 0.55p-values 0.844 0.901 0.520 0.853 0.795Observations 1,470 1,470 1,470 1,470 1,470

Note: Robust standard errors are in parentheses; * significant at 10%; ** significant at 5%; ***significant at 1%.Source: Yi (2013a) 50

Table A2: Distribution of Family Size by Rural vs. Urban and Twinning

Urban RuralNo first-birth First-birth No first- and First-birth Second-birth

twin twin second-birth twin twin only twin onlyOne child 94.4% 0 59.0 0 0Two children 5.6 99.1 40.3 92.2 0Three children 0 0.9 0.7 7.8 99.2Four children 0 0 0 0 0.8

Source: Rosenzweig and Zhang (2009)

51

Table A3: Parental Characteristics by Rural vs. Urban and Twinning

Urban RuralNo twin Twin No twin Twin

Maternal schooling years 10.8 10.5 8.23 7.69(3.15) (3.27) (2.98) (2.90)

Paternal schooling years 11.3 11.0 8.92 8.37(3.33) (3.39) (3.08) (3.00)

Maternal age 36.3 37.4 35.3 36.6(4.28) (4.92) (4.55) (4.98)

Maternal age at first birth 24.8 25.6 23.5 24.8(3.06) (3.54) (3.05) (3.96)

Note: Standard deviations are in brackets.Source: Rosenzweig and Zhang (2009)

52

0.2

.4.6

.81

Den

sity

1 2 3 4 5Birth Weight (kilograms)

Rural Urban

Figure A1: Kernel density estimates of birth weight (rural vs. urban)

Note: t-test (equality-of-mean): p-value = 0.003; two-sample Kolmogorov-Smirnov test for equal-ity of distribution functions: p-value = 0.018Source: Yi (2013a)

0.5

11.

5D

ensi

ty

−2 −1 0 1 2Within−twin Differences in Birth Weight (kilograms)

Rural Urban

Figure A2: Kernel density estimates of within-twin differences in birth weight (rural vs.urban)

Note: t-test (equality-of-mean): p-value = 0.763; two-sample Kolmogorov-Smirnov test for equal-ity of distribution functions: p-value = 1.000Source: Yi (2013a)

53

Appendix C: The Story of Kong Rong Giving up Pears

Kong Rong (153 A.D. to 208 A.D.) was a politician and scholar at the end of Eastern Han Dynasty

(25 A.D. to 220 A.D.). He was a 20th generation descendant of Confucius. A well-known story

commonly used to educate children on the value of comity and fraternal love involves a four-year-

old Kong Rong giving up the bigger pears to his older and younger brothers. Kong Rong had five

elder brothers and one younger brother. One day, his father bought some pears and asked him to

choose one first. Kong Rong picked up the smallest pear. His father was pleasantly surprised to see

it and asked: “There are plenty of pears here, I asked you to choose first. Why did you only choose

the smallest?” Kong Rong answered: “I am younger, so I should have a smaller one, please give

the bigger ones to my elder brothers.” His father was surprised and challenged him again, “Your

little brother is younger than you.” Kong Rong said: “My little brother is younger, so I should give

up the bigger one to him.” His father laughed loudly. Four-year-old Kong Rong knew the courtesy

of giving precedence to others which is a very important Chinese virtue.

54

Appendix D: The Endogenous Formation of Inter-sibling Altruism and Intra-household Hu-

man Capital Investment in Rural versus Urban Areas

In China, a rural versus urban divide for any area is made by the administration, and there is little

a household can do to take advantage of the policy difference across the boundary (Yusuf & Saich,

2008). The differences in institutional and socioeconomic backgrounds between rural and urban

areas are substantial. First, rural families are much poorer than urban ones. For example, the per

capita income was RMB 2,460 for rural residents in 2002. In contrast, it was RMB 3,680 for urban

residents. The ratio of educational expenditure per child between rural and urban areas is even

larger than that of per capita income. Specifically, the education expenditure per child was RMB

630 in rural areas, but RMB 1,227 in urban areas in 2002. Second, residents in urban areas were

partly covered by public health insurance, educational subsidization, and old age pension in the

first wave of the survey in 2002. But these public facilities did not exist in rural areas at the time.

Finally, rural households are more traditional being influenced by Confucianism, whereas urban

households are more modern being influenced by Western culture. With these institutional and

socioeconomic differences in mind, it is interesting to explore how these differences affects the

endogenous formation of altruism and intra-household human capital investment in rural versus

urban areas.

Table A4 reports estimates of Equation (10) in rural and urban areas, respectively, testing the

effects of gaps in children’s birth weight on family altruistic education. With the first measure, the

estimated effects of gaps in children’s birth weight on family altruistic education are consistently

larger in rural than those in urban areas (columns (1)-(3) versus (7)-(9)). The Durbin-Wu-Hausman

test results show that the differences are statistically significant at the 10% level. When the second

measure is used, the significant differences between rural and urban areas disappear (columns (4)-

(6) versus (10)-(12)).

Table A5 reports estimates of Equation (11) by rural versus urban areas, testing the effect

of family altruistic education on gaps in intra-household investment in children’s human capital.

First, intra-household investment in children’s human capital is less likely to reinforce gaps in

55

children’s prenatal endowment in rural than in urban areas (columns (1) versus (6)). Second,

the effects of family altruistic education on gaps in intra-household human capital investment are

statistically significant for rural areas only. Finally, the Durbin-Wu-Hausman test results show that

the differences in estimates reported in Table A5 are statistically insignificant between rural and

urban areas across all specifications.

Summarizing results reported in Tables A4 and A5, rural households more evidently rely on

the mechanism of fostering children’s inter-sibling altruism to address the conflict between effi-

ciency and equality in intra-household investment in children’s human capital. There are several

suggestive causes. First, rural households are more traditional, and thus, they may be more effec-

tive in fostering children’s inter-sibling altruism. Second, some urban households are rich enough,

and then the conflict between efficiency and equality is less severe for these households. Third,

the less-endowed children who have less earnings are supposed to be partly supported by public

pension system in urban areas, and then they may be less likely to rely on inter-sibling transfers

in adulthood. Therefore, parents have less incentives to foster children’s inter-sibling altruism in

urban areas.

Caution is exercised about the rural versus urban comparison. On one hand, the differences

in the estimated effects of gaps in children’s birth weight on family altruistic education between

rural and urban areas are sensitive to the choice of measures for family altruistic education. On the

other hand, the differences in the estimated effects of family altruistic education on intra-household

human capital investment between rural and urban areas are statistically insignificant.

56

Table A4: OLS Estimates Examining the Effects of Differences in Children’s Birth Weighton Family Altruistic Education in Rural versus Urban Areas

Dependent variable: Family altruistic educationMeasure 1a Measure 2b

Rural(1) (2) (3) (4) (5) (6)

Difference in birth weightc (γ1) 0.412∗∗∗ 0.397∗∗∗ 0.387∗∗∗ 0.124∗ 0.122 0.107(3.12) (2.80) (2.77) (1.67) (1.62) (1.45)

Twin-pair characteristics Y Y Y Y Y YMaternal characteristics N Y Y N Y YHousehold characteristics N N Y N N Y# Obs. 288 288 288 288 288 288

Urban(7) (8) (9) (10) (11) (12)

Difference in birth weightc (γ1) 0.212∗∗ 0.208∗∗ 0.197∗∗ 0.089 0.084 0.076(2.21) (2.08) (2.04) (1.33) (1.29) (1.01)

Twin-pair characteristics Y Y Y Y Y YMaternal characteristics N Y Y N Y YHousehold characteristic N N Y N N Y# Obs. 220 220 220 220 220 220

Durbin-Wu-Hausman test 0.08 0.08 0.07 0.39 0.51 0.54(p-value)

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05,*** p < 0.01. (3) The estimation sample includes only households with twin children. (4) Thetwin-pair characteristics include two dummy variables. One indicates that both twin children areboys, and the other both are girls. (5) The maternal characteristics include maternal age, years ofschooling, and ethnicity. (6) The household characteristic is a measure of household asset.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).c : The absolute value of the difference in children’s birth weight.

57

Table A5: OLS and 2SLS Estimates Examining the Effects of Family Altruistic Educationon Intra-household Human Capital Investment in Rural versus Urban Areas

Dependent variable:Difference in educational investment

OLS OLS 2SLS OLS 2SLSRural

(1) (2) (3) (4) (5)Difference in birth weight 30.012∗ 18.726 20.314∗ 22.747∗∗ 28.716∗∗

(δ1) (1.714) (1.54) (1.82) (2.50) (2.47)Family altruistic education 17.874∗∗ 28.716∗∗

(Measure 1)a (δ2) (2.16) (2.08)Family altruistic education 12.414∗ 19.576∗

(Measure 2)b (δ2) (1.68) (1.88)# Obs. 228 228 228 228 228

Urban(6) (7) (8) (9) (10)

Difference in birth weight 80.012∗∗ 52.442∗ 62.777∗∗ 58.878∗∗∗ 77.667∗∗∗

(δ1) (2.21) (1.68) (1.98) (2.98) (2.77)Family altruistic education 19.207 40.014(Measure 1)a (δ2) (1.17) (1.40)Family altruistic education 11.446 17.624(Measure 2)b (δ2) (1.22) (1.08)# Obs. 220 220 220 220 220

Durbin-Wu-Hausman test 0.16 0.46 0.55 0.59 0.72(p-value)

Note: (1) Absolute values of robust t-statistics are in parentheses. (2) * p < 0.10, ** p < 0.05, ***p < 0.01. (3) Differences in gender are controlled for in all specifications. (4) Family altruisticeducation is the endogenous variable in columns (3)-(5) and (8)-(10). The instrumental variablesare household asset, a rural dummy, and maternal age, schooling years, and working sector. (5)The estimation sample includes only households with twin children.a : The frequency in which parents brought children together to worship ancestors or visit ancestraltombs during the Qingming festival (Tomb-Sweeping Day) when children were younger than 12.It is measured by a 5-point Likert scale from 1 (never) to 5 (always).b : The frequency in which parents told children that having fraternal love among siblings is avirtue, such as telling children the story of “Kong Rong giving up pears” when children wereyounger than 12. It is measured by a 5-point Likert scale from 1 (never) to 5 (always).

58