three essays on family economics and early childhood ... · three essays on family economics and...

Three Essays on Family Economics and

Early Childhood Development

Hengheng Chen

Dissertation submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Economics, Science

Suqin Ge, ChairRichard A. Ashley

Hans H. HallerDjavad Salehi-Isfahani

Wen You

April 20, 2012Blacksburg, Virginia

Keywords: Children’s Cognitive Development, Black-White Test Score Gap,Parental Time, Collective Model, Simultaneous Equations Model, General Equilibrium

Copyright 2012, Hengheng Chen

Three Essays on Family Economics and

Early Childhood Development

Hengheng Chen

(ABSTRACT)

This dissertation consists of three essays studying the effects of collective household decisionson early childhood development from both empirical and theoretical perspectives. The firstchapter outlines the dissertation, by presenting the motivations, methods, conclusions, andpolicy implications for the entire dissertation.

Chapter two examines early childhood development using a collective model with children’scognitive production. We jointly estimate the home input demand with children’s cognitiveproduction functions based on a simultaneous equations model. Biases are considered thatare caused by the non-random selection of time inputs and possible correlations across inputsand outcomes functions. A direct measure of time inputs relying on children’s time diariesfrom the Child Development Supplement of the Panel Study of Income Dynamics (PSID-CDS) has been constructed. We thereby relax the assumption that there is no differencebetween parental time spent on children and leisure. Our results show that parental timeinputs, especially the active time interacting with children’s daily activities, have substantialeffects on both children’s math and reading test scores. The time inputs vary across parents’age, race, and eduction levels.

In chapter three, we conduct a standard Blinder-Oaxaca decomposition to evaluate the roleof home inputs in the black-white test score gaps based on the empirical model presented inchapter two. Aside from the finding that children’s ability accounts for a large proportion ofthe differences, we find that home inputs can also explain a significant portion of the gap.When the maternal time is equalized at the average levels of white children, the racial dif-ferences in children’s reading and math test scores can be closed by approximately 30%-50%.

The last chapter extends a collective model with household production to the general equi-librium framework. We concentrate on the impacts of a global bargaining power shift withinthe household on children’s cognitive achievement, especially on those who live with singlemothers. The model shows that a global bargaining power change in favor of the female may

not necessarily be beneficial to the children living with their single mothers. An increase offemale’s market equilibrium wage rate as a result of reduced labor supply by married womenmay induce single mothers to work longer hours, spend less time with children, and com-pensate them with more monetary investment compared with the case when the equilibriumwage rate stays constant.

iii

Acknowledgments

First, I am grateful to my advisor, Dr. Suqin Ge for her persistent guidance and support.Without her help, stimulating suggestions and encouragement during the past five years,the possibility to complete the dissertation would be very hard to estimate. It is indeed aprivilege to work with her, learn from her, and be friend with her.

I would also like to acknowledge insightful comments from all my committees: Dr. RichardAshley, Dr. Hans Haller, Dr. Djavad Salehi-Isfahani, and Dr. Wen You. Their advices onboth theories and techniques have nurtured my research from the very beginning.

I want to thank all my friends and colleagues at Virginia Tech for the wonderful time we spendtogether in Blacksburg, a cozy little town which surely changes and will continue to changemy views on life. Special thanks to Congwen Zhang, Shaojuan Liao, and Tian Liang for theunconditional sisterhood love, to Yue Meng and Haiyang Fu for the glory days living underone roof, to Ellen Green, Tiefeng Qian, Xiaojin Sun, Golnaz Taghvatalab, Ming Yi, ZhiyuanZheng, Beibei Zhu, and all the other Economics Ph.D. students at Virginia Tech for research.

The very last gratitude belongs to my dear families: my mother Rongzhen Wang, my fatherZhenxin Chen, and my husband Juqi Liu. Without your unlimited support and love, thedissertation topic probably would not even interests me in the first place. Indeed, I mayjust try to prove my father’s believe in parenting, but unfortunately, he never doubts therightness of it even without reading my dissertation.

iv

Contents

1 Introduction 1

2 The Effects of Parental Time on Children’s Cognitive Development 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Household Production of Child’s Cognitive Achievement . . . . . . . 11

2.2.2 Child’s Cognitive Production Function . . . . . . . . . . . . . . . . . 13

2.2.3 Empirical Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Variable Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Estimate Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Baseline Specification of Production Function . . . . . . . . . . . . . 24

2.4.2 Comparison with Alternative Estimation Methods . . . . . . . . . . . 29

2.4.3 Robustness Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Appendix A: Data Supplement . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.8 Appendix B: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 The Role of Home Inputs in the Black-White Test Score Gaps 59

v

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Empirical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.1 Decomposing Gaps in Test Scores: A Simple Model . . . . . . . . . . 68

3.3.2 Basic Results for the Benchmark Decomposition . . . . . . . . . . . . 71

3.3.3 Basic Results for Another Type of Decomposition . . . . . . . . . . . 75

3.3.4 A Closer Look at Home Inputs . . . . . . . . . . . . . . . . . . . . . 77

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.6 Appendix A: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Bargaining Power Shift and Children’s Cognitive Achievement 106

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2 Collective Model with Household Production:A Partial Equilibrium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.1 Basic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.2 Decision Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.3 Determination of Home Inputs . . . . . . . . . . . . . . . . . . . . . . 113

4.2.4 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.6 Identification Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.3 Collective Model with Household Production:A General Equilibrium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.1 Basic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.2 Children’s Cognitive Production Depends on Goods Only . . . . . . . 120

4.3.3 Children’s Cognitive Production Depends on Goods and Time . . . . 124

4.4 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

vi

4.5 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.6 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Bibliography 146

vii

List of Figures

Chapter 4.

Figure 4.1: Numerical Solutions for 4.3.2, I = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 4.2(a): Numerical Solutions for 4.3.3, I = 0.1, πp = 0.6, and πm = 0.8 . . . . . . . . . . . . . . . . 140

Figure 4.2(b): Numerical Solutions for 4.3.3, I = 0, πp = 0.6, and πm = 0.8 . . . . . . . . . . . . . . . . . . 141

Figure 4.2(c): Numerical Solutions for 4.3.3, I = 0.05, πp = 0.6, and πm = 0.8 . . . . . . . . . . . . . . . 142

Figure 4.2(d): Numerical Solutions for 4.3.3, I = 0.1, πp = 0.5, and πm = 0.9 . . . . . . . . . . . . . . . . 143

Figure 4.2(e): Numerical Solutions for 4.3.3, I = 0.1 and πp = πm = 0.7 . . . . . . . . . . . . . . . . . . . . . .144

Figure 4.2(f): Numerical Solutions for 4.3.3, I = 0.1 and πp = πm = 0.9 . . . . . . . . . . . . . . . . . . . . . .145

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List of Tables

Chapter 2.

TABLE 2.1: Variable Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

TABLE 2.2: Basic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

TABLE 2.3(a): Baseline Model - Production Function

Dependent Variable → Reading Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

TABLE 2.3(b): Baseline Model - Production Function

Dependent Variable → Math Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

TABLE 2.3(c): Baseline Model - Input Demand Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

TABLE 2.4: Robustness Check w.r.t. Different Time Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

TABLE 2.5(a): Heterogeneity in Effect of Parental Time Inputs (Married HH)


TABLE 2.5(b): Heterogeneity in Effect of Parental Time Inputs (Married HH)


TABLE 2.5(c): Heterogeneity in Effect of Maternal Time Inputs (Single-Mother HH)


TABLE 2.5(d): Heterogeneity in Effect of Maternal Time Inputs (Single-Mother HH)


TABLE 2.6: Robustness Check w.r.t. High Orders of Time Measures . . . . . . . . . . . . . . . . . . . . . . . . . 50

Appendix Tables for Chapter 2.

TABLE A2.1(a): Value-Added Specification - Production Function


TABLE A2.1(b): Value-Added Specification - Production Function


TABLE A2.1(c): Value-Added Specification Model - Input Demand Function . . . . . . . . . . . . . . . . . 55

ix

TABLE A2.2: Value-Added Specification Sample - Basic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

TABLE A2.3: Inputs Demand Functions w.r.t. Different Time Measures . . . . . . . . . . . . . . . . . . . . . . 57

TABLE A2.4: Inputs Demand Functions w.r.t. High Orders of Time Measures . . . . . . . . . . . . . . . .58

Chapter 3.

TABLE 3.1(a): Descriptive Statistics

PSID-CDS Standardized Applied Problem (Math) Test Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

TABLE 3.1(b): Descriptive Statistics

PSID-CDS Standardized Letter-Word Identification (Reading) Test Scores . . . . . . . . . . . . . . . . . . . . 83

TABLE 3.2(a): Descriptive Statistics - Married Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

TABLE 3.2(b): Descriptive Statistics - Single-Mother Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

TABLE 3.3(a) Determinants of Test Scores: Reading Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

TABLE 3.3(b) Determinants of Test Scores: Math Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

TABLE 3.4(a): Decomposition of the White-Black Test Gaps, Married Households

PSID-CDS WJ-R Applied Problem (Math) and Letter-Word Identification (Reading) Tests . . . 87

TABLE 3.4(b): Decomposition of the White-Black Test Gaps, Single-Mother Households


TABLE 3.5(a): Decomposition of the White-Black Test Gaps by Gender, Married Households


TABLE 3.5(b): Decomposition of the White-Black Test Gaps by Gender, Single-Mother House-

holds


TABLE 3.6(a): Decomposition of the White-Black Test Gaps by Age, Married Households


TABLE 3.6(b): Decomposition of the White-Black Test Gaps by Age, Single-Mother Households


TABLE 3.7(a): Decomposition of the White-Black Test Gaps, Active & Passive Time Separated

PSID-CDS WJ-R Applied Problem (Math) and Letter-Word Identification (Reading) Tests, Mar-

ried Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

TABLE 3.7(b): Decomposition of the White-Black Test Gaps, Active & Passive Time Separated

PSID-CDS WJ-R Applied Problem (Math) and Letter-Word Identification (Reading) Tests, Single-

Mother Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

TABLE 3.8(a): Decomposition of the White-Black Test Gaps, Married Households PSID-CDS WJ-

x

R Applied Problem (Math) and Letter-Word Identification (Reading) Tests . . . . . . . . . . . . . . . . . . . 93

TABLE 3.8(b): Decomposition of the White-Black Test Gaps, Single-Mother Households


TABLE 3.9(a): Descriptive Statistics of Parental Time, Married Households . . . . . . . . . . . . . . . . . . 95

TABLE 3.9(b): Descriptive Statistics of Parental Time, Single-Mother Households . . . . . . . . . . . . 96

TABLE 3.10(a): Decomposition of the White-Black Parental Time Differences, Married Households

97

TABLE 3.10(b): Decomposition of the White-Black Parental Time Differences (Active, Passive

Separated), Married Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

TABLE 3.10(c): Decomposition of the White-Black Parental Time Differences, Single-Mother

Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

TABLE 3.10(d): Decomposition of the White-Black Parental Time Differences (Active, Passive

Separated), Single-Mother Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Appendix Tables for Chapter 3.

TABLE A3.1: Variable Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

TABLE A3.2: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

TABLE A3.3: Estimation Results, Production Function, Married Households . . . . . . . . . . . . . . . . 102

TABLE A3.4: Estimation Results, Production Function, Single-Mother Households . . . . . . . . . . 103

TABLE A3.5: Estimation Results, Inputs Demand Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

TABLE A3.6: Estimation Results, Inputs Demand Function, Different Time Measures . . . . . . . 105

Chapter 4.

TABLE 4.1(a): Numerical Solutions for the Example in 4.3.2 (Non-labor income I = 0.1) . . . . 130

TABLE 4.1(b): Numerical Solutions for the Example in 4.3.2 (Non-labor income I = 0) . . . . . .131

TABLE 4.1(c): Numerical Solutions for the Example in 4.3.2 (Non-labor income I = 0.05) . . . 132

TABLE 4.2(a): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp = 0.6,

& πm = 0.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133

TABLE 4.2(b): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0, πp = 0.6,

& πm = 0.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134

TABLE 4.2(c): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.05, πp = 0.6,

& πm = 0.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

TABLE 4.2(d): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp = 0.5,

& πm = 0.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136

xi

TABLE 4.2(e): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp =

πm = 0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

TABLE 4.2(f): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp =

πm = 0.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

xii

Chapter 1

Introduction

An increasing body of recent literature has shed lights on the understanding of early child-

hood development. Most of them focus on three aspects: 1) home inputs, 2) schools, and

3) heredity. Since the first two can be largely affected by well-designed economic policies,

this dissertation tends to improve the efficiency of public policies by filling the gap between

studies on family economic decision making and children’s cognitive achievement. In partic-

ular, we concentrate on the effects of parental time inputs on children’s math and reading

test scores and aim to reinforce the importance of family resource allocation decisions on the

welfare of the next generation.

Our study bears policy concerns in the first place, given two astonishing facts: 1) dramatic

increases in both labor force participation rates of mothers with young children and female

wage rate during the recent several decades in the United States, and 2) a strong correlations

between children’s cognitive achievement and future labor market outcomes1. For example,

in recent years, high-income, college-graduate mothers (as well as fathers) tend to sell more

productive time on the market without depressing their time spent with children compared

with other social groups - Guryan, Hurst and Kearney (2008) have highlighted the impres-

sive findings using data from the American Time Use Surveys. This may be responsible

1Almond and Currie (2011) has surveyed the recent work showing the events even before age five can

have large long term effects on children’s future outcomes.

1

Hengheng Chen Chapter 1. 2

for the persistent social inequality across generations and the increasing income gap in the

United States. Therefore, policies that are concerned with social equality require a deep

understanding of parental behaviors and their impacts on children’s cognitive production.

Another major motivation for us to study the effects of home inputs on children’s cognitive

development has stemmed from the finding that the average academic performance of white

children exceeds that of the blacks persistently. The empirical results on the black-white test

gaps are statistically significant and robust across a wide variety of tests, subjects, samples,

and cohorts, and when a range of control variables are included. The explanation for the

gap, however, proves elusive in the previous literature. No consistent conclusion is achieved

about the role of home inputs in accounting for the disparity of test scores among different

racial groups. The efforts to locate the sources except for the genetic reasons are paramount

for policies that aim to further improve the social welfare of the black. Our study compares

the role of family inputs in determining racial differences of test scores, and therefore, makes

useful policy suggestions in this area.

Furthermore, a technical curiosity also motivates us after the children’s detailed time diaries

become available. The Child Development Supplement of the Panel Study of Income Dy-

namics (PSID-CDS) has collected assessments of children’s cognitive development, family

background information from both parents, and their 24-hour time diaries for one randomly

select weekday and one weekend day with duration, and activity code2. We thereby are able

to construct a series of measurements for maternal or paternal time inputs to their children

and no longer rely on proxies, such as employment data or an ambiguous “HOME” index.

The demand functions for parental time inputs are precisely estimated in the first stage to

deal with endogeneity bias resulting from parents’ non-random selection of resource alloca-

tions based on unobserved heterogeneity. Many interesting questions are investigated, such

as the effects of paternal time or the difference between parent’s active time and passive

time, when the time-use data are employed.

2We have discussed the data carefully in the next chapter.


The dissertation consists of three major parts, investigating the effects of home inputs on

children’s cognitive development from both theoretical and empirical perspectives. Chap-

ter 2 estimates children’s cognitive production and the approximated demand functions of

parental time input simultaneously using iterated three stage least squares. Based on the em-

pirical results in chapter 2, we conduct a standard Blinder-Oaxaca decomposition in chapter

3 to quantify the role of home inputs in the black-white test score gaps. Chapter 4 develops

a theoretical collective model with household production and extends the model to a general

equilibrium framework. It analyzes the specific impacts of bargaining power shift on chil-

dren’s welfare.

Children’s cognitive production function is jointly estimated with the demand functions of

parental time inputs using the data from the PSID-CDS in chapter 2. The non-random se-

lection of parental time inputs because of unobserved heterogeneity of parents and children

generates spurious correlation between home inputs and test score. We, therefore, approxi-

mate the decision rules on each parent’s time spent with children based on a collective model

with household production to help provide more precise estimates. The results show that

parental time inputs, especially their active time interacting with children’s daily activities,

have positive and sizeable impacts on both children’s reading and math test scores. The

optimal choices of time inputs depend on households characteristics, such as race, gender,

age, or parents’ education levels, and the parental time effects vary across these different

characteristics.

In chapter 3, a standard Blinder-Oaxaca decomposition has been implemented to study the

role of home inputs in the black-white test score gaps. We find that though children’s ability

accounts for a large proportion of the differences, home inputs explain the gap by approxi-

mately 30%-50%. Moreover, paternal time input is of paramount importance and has sizable

impacts on girl’s cognitive development. Therefore, policies that encourage divorced fathers

to increase the frequency of their visits may not only improve female children’s welfare but

have profound and lasting influence on the intergenerational persistence.


We develop a theoretical collective model with household production in chapter 4. The effects

of a global bargaining power shift are analyzed under both partial and general equilibrium

frameworks. The model indicates that the wage effects induced by the bargaining power

change on single mothers’ decision making may not be necessarily positive towards children’s

development. An increase of female’s market equilibrium wage rate resulting from less labor

supplied by married women may make single mothers work longer hours and spend less time

with their children.

Chapter 2

The Effects of Parental Time on

Children’s Cognitive Development

(ABSTRACT)

This chapter evaluates the effects of parental time on children’s cognitive development. The

non-random selection of parental time inputs created by unobserved heterogeneity of parents

and children generates spurious correlation between time inputs and cognitive achievement.

We develop a collective model of household resource allocation with altruistic parents and

derive decision rules on each parent’s time spent with children. Children’s cognitive produc-

tion function is jointly estimated with demand functions of time inputs using test scores and

time diaries from the Child Development Supplement of the Panel Study of Income Dynam-

ics (PSID-CDS). The estimates show that parental time inputs, especially their active time

interacting with children, have positive and sizeable effects on both children’s reading and

math test scores. We also find that time inputs vary substantially across parents’ age, race,

and education level.

5


2.1. Introduction

In recent years an extensive body of empirical work has evaluated the effects of parental time

inputs, specifically, maternal time input, on children’s cognitive production. The study itself

bears important policy implications given the dramatic increase in the labor force partici-

pation rates of mothers with young children and the strong correlation between children’s

cognitive achievement and future labor market outcomes1. If high-income, well-educated

mothers tend to spend more time and money on their children without depressing the labor

supply in the market compared with their low-income counterparts, the income gap may be

increasingly widened, leading to persistent social inequality across generations. Two major

concerns in labor economics, income inequality and its intergenerational persistence, thereby

can be traced back to children’s cognitive technology. Policies that are concerned with social

equality require a deep understanding of parental behaviors and children’s cognitive produc-

tion. This chapter examines the effects of time inputs from both parents on the cognitive

achievement for children who are aged above three years old, living with married parents as

well as single mothers.

The optimal decisions about labor supplied in the market, parental time for children’s well-

beings, and leisure can be extremely complicated implied by economic theories when we

consider the opportunity cost associated with time. Parents may reduce labor supply (and

hence, the labor income) or squeeze leisure to increase time spent with children. How they

balance the trade-offs is, therefore, an empirical question. In the previous empirical litera-

ture, however, the results are inconclusive. Very few studies take paternal time spent with

children into account, and the effects of maternal time on children’s outcome, usually mea-

sured by employment, appear mixed. As is summarized in Bernal and Keane (2010), among

the literature using NLSY, there is an almost equal amount of studies showing positive,

negative, and no impact between maternal employment and children’s outcomes.

The diversity of empirical results, mainly stemming from the deficient data available, chal-

1Almond and Currie (2011) has surveyed the recent work showing the events even before age five can

have large long term effects on children’s future outcomes.


lenges the estimation of parental time effects on children’s cognitive achievement from three

aspects. An individual survey, such as NLSY, usually provides limited information on fa-

thers. Meanwhile, an implicit assumption that there is no correlation between paternal and

maternal time choices in a married household may under- or over-estimate the true param-

eters associated with maternal time inputs when the two are substitutes or complements.

Though, in a single-mother household, the lack of fathers’ information sometimes does not

induce non-negligible biases, a poor measure of maternal time input may lead to implau-

sible results. In the previous literature, a wide range of measurements of maternal time

are employed. Their proxies of time inputs, such as maternal employment2 or “HOME”

environment index contaminate the estimates for multiple reasons. The maternal employ-

ment, for example, does not differentiate between leisure and non-marketable housework.

But the possible substitution between them may exaggerate the negative effect of maternal

employment. The “HOME” index, combining both time and monetary inputs, is even more

problematic for us to explain the impacts of different types of home inputs on children’s

cognitive development. Finally, the non-random selection of parental time inputs created

by unobserved heterogeneity of parents and children generates spurious correlation between

time inputs and cognitive achievement. A commonly used “hybrid” production function

may provide misleading results by including several home inputs and income and control

variables on the right-hand side when unobserved heterogeneity systematically affects par-

ents’ choices on parental time inputs. The estimates thereby are either upward biased or

downward biased, indeed causing unreliable inferences.

In this chapter, we employ a direct measure of time inputs. Children’s time diaries from

the Child Development Supplement of the Panel Study of Income Dynamics (PSID-CDS)3

allow us to obtain the duration that mothers or fathers participate in children’s daily ac-

tivities. Therefore, biases due to the non-equivalence between parental time and leisure, or

between maternal and paternal time are eliminated. As for the policy design, these are of

2Since NLSY provides no time-use data, many previous studies try to estimate the impact of maternal

employment or childcare use to deduce the effects of maternal time inputs on children’s outcomes, such as

Baydar and Brooks-Gunn (1991), Blau and Grossberg (1992), and Bernal and Keane (2011).3Panel Study of Income Dynamics, public use dataset, produced and distributed by the Institute for

Social Research, Survey Research Center, University of Michigan, Ann Arbor, MI (2011).


paramount importance. For instance, policies that tend to encourage parents to spend more

time with children by lowering the incentives for labor supply may not necessarily take effect,

if only leisure increases. Moreover, suggestions that value the maternal time inputs alone

are no longer sufficient, as we find fathers may play a leading role in children’s math skill

production. The void we fill in regarding a better evaluation of the impacts of parental time

inputs on children’s cognitive achievement may in general improve the policy efficiency and,

consequently, children’s total welfare.

We deal with the endogeneity issue by a joint estimation of parental time inputs and chil-

dren’s cognitive production functions. The underlying strategy extends the empirical method

introduced by Bernal and Keane (2010). Rather than approximating the decision rules of

parental time inputs in a unitary fashion, we develop a collective model to derive household

optimal choices across intra-household time allocations. The demand for each parent’s time

input therefore responds to individual preferences and exogenous distribution factors. The

attempt to disentangle children’s cognitive production from each member’s distinct taste of

time allocation appears highly desirable, in particular when the quality of each parent’s time

is non-homogeneous. For example, comparative advantages across multiple members result

in intra-household specialization, which may account for a portion of the test gap between

genders. The theoretical foundation, first introduced by Chiappori (1988), Chiappori (1992),

and Browning and Chiappori (1998a), also generalizes the intra-household resource alloca-

tion process for different types of households. Hence, a unified framework can be applied for

both married households and single-mother households.

Children’s cognitive production functions and functions for parental time inputs are esti-

mated simultaneously using the iterated three stage least squares (3SLS). The simultaneous

equations model accommodates non-random selection of time inputs and possible corre-

lations across equations. In theory, the 3SLS estimates are unbiased and asymptotically

consistent compared with the OLS estimates when some variables are endogenous, and they

are more efficient than the two-stage least squares (2SLS) estimates if the error terms are

correlated. According to the constrained optimization program, the distribution factors,

along with some taste variables, do not affect children’s cognitive technology, but enter the


demand functions of parental time inputs. Therefore, instead of searching for exogenous

variables as instruments based on some policy reforms or natural experiments, the collective

model provides a new direction to locate the instruments and meet the identification condi-

tions for the simultaneous system of equations.

The estimates of our baseline specification show substantial effects of maternal time input

on children’s cognitive achievement and positive effects of paternal time input on children’s

math scores. For example, in married households, one additional hour of total maternal time

input per day enhances children’s reading test scores by roughly 6.55% and math scores by

4.22% on average. The same amount of time increase in paternal time input improves chil-

dren’s math scores by around 2.39%. The general findings on total time inputs, however,

may mask important differences between active and passive parental time inputs. We fur-

ther estimate a model with inputs functions for both parents’ active and passive time inputs.

The results are remarkable. Parents’ active time inputs have substantial effects on children’s

cognitive development. One additional hour of direct maternal time increases reading scores

by 14.5%, and as for active paternal time, it improves math scores by 7.15%, ceteris paribus.

The policy implications are crucial towards the concerns of substitution among active and

passive time inputs and pure leisure. Moreover, the optimal choices of time inputs vary

across household preferences, such as race, gender, age, or parents’ education levels. The

marginal effects of different types of parental time inputs also change with heterogeneity.

Our work is mainly related to several recent papers. Blundell, Chiappori and Meghir (2005)

have developed a collective household model with children where children are assumed to be

public goods with unobserved utilities. In our model, we consider a household utility with

observed children’s cognitive achievement and individual heterogeneity. Bernal and Keane

(2010) have initiated a quasi-structural method to estimate the effects of child-care choices

on children’s cognitive production. Todd and Wolpin (2004) have estimated children’s cog-

nitive technology using value-added method when facing imperfect data. Both papers have

nurtured our empirical work to a great extent. Boca, Flinn and Wiswall (2010) have started

to consider a more sophisticated dynamic process of children’s cognitive development using

the PSID-CDS and its time diaries. Our cross-sectional analyses, however, can be seen as


a complement to their work for researchers who plan to further studying the role of home

inputs on the human capital accumulation. Numerous studies on maternal employment (e.g.,

Blau and Grossberg (1992) and Liu, Mroz and van der Klaauw (2010)), household income

(e.g., Blau (1999) and Shea (2000)), or school inputs (e.g., Canon (2010)) on children’s

cognitive development, as well as studies on children’s health production (e.g., Rosenzweig

and Schultz (1983)) and non-cognitive production (e.g., Cunha, Heckman and Schennach

(2010)), have enriched our knowledge from multiple perspectives.

The chapter is organized as follows. Section 2.2 derives a collective model with children’s

cognitive achievement and describes the empirical strategies that we use to estimate chil-

dren’s cognitive technology. Section 2.3 discusses the data and variable measurements. In

section 2.4 we present the estimation results, and section 2.5 concludes.

2.2. The Model

In this section, we present an extended collective model with household production of child’s

cognitive achievement. According to the model, the decision rules of intra-household time

and goods allocations are non-randomly selected and correlated with individual preferences.

When the heterogeneity, either of child or of parents themselves, cannot be fully observed by

the researchers, a regular OLS estimation of child’s cognitive production function may suffer

from the endogeneity bias, which leads to the estimates inconsistent and biased. We there-

fore approximate the demand functions of parental time inputs from the collective model and

estimate them with child’s cognitive production function simultaneously using iterated three

stage least squares (3SLS). With the instruments that affect the child’s cognitive achieve-

ment indirectly through the channels of parental time inputs, the Simultaneous Equations

Model (SEM) can help recover the structural parameters in the production function. Some

parents’ preferences and local distribution factors implied by the model are able to perform

the role as exogenous instruments.


2.2.1 Household Production of Child’s Cognitive Achievement

We extend a static collective model of intra-household resource allocation to include a pro-

duction process of child’s cognitive achievement. A household has two members (i = p,m)4

and each member’s preference over child’s cognitive achievement5, A, leisure, li, a Hicksian

composite good, Ci, and a set of preference factors, Si (e.g., age and eduction), can be

characterized by a utility function,

Ui = U(Ci, li, A;Si), i = p,m. (2.1)

Child is a public and time-intensive good. Her cognitive achievement is assumed to be

“produced” by inputs of paternal time, tp, and maternal time, tm, as well as other non-time

goods inputs, X (e.g., toys and private school). Define the technology of child’s cognitive

production,

A = F (tp, tm, X, µ), (2.2)

where µ represents the child-specific endowed ability, which is assumed to be determined at

the time of conception and known to the parents.

The decision process of intra-household resource allocation is assumed to be Pareto-efficient.

The household thereby allocates the resource by maximizing a weighted sum of two individual

utilities:

H(C, lp, lm, A;Sp, Sm, λ) = λ · Up(Cp, lp, A;Sp) + (1− λ) · Um(Cm, lm, A;Sm), (2.3)

where Cp+Cm = C is the total household private consumption. The Pareto weight, λ ∈ [0, 1],

depends on individual wages, wp and wm, preference factors, Sp and Sm, non-labor income,

I, and a set of distribution factors, Z. The distribution factors, by definition, are variables

that only alter each member’s respective bargaining position without shifting her prefer-

ence. Examples of Z include sex ratio and divorce legislation (Chiappori, Fortin and Lacroix

4We use “p” for father and “m” for mother to avoid such an ambiguity as “m” for male and “f” female.5The model simplifies the fertility decision and therefore the child achievement A can be interpreted as

the average achievement within a household. In our empirical analyses, we take household fertility choice

into consideration and assume it endogenous for married households.


(2002)), non-labor income and wealth at the time of marriage (Thomas (1990); Thomas,

Contreras and Frankenberg (1997)).

The household faces the following budget constraint:

Cp + Cm +n∑j=1

Xjpj = wphp + wmhm + I, (2.4)

where the price of the composite good is normalized to one, and pj’s are exogenous prices

of n non-time inputs. Each spouse i has a time endowment of one, which can be used for

leisure li, market labor hi, and parental time input ti. Parent’s time budget constraints are

li + hi + ti = 1, i = p,m. (2.5)

The optimal time allocations (lp, lm, tp, tm) and goods consumption (Cp, Cm, X) within a

household are determined by an optimization program (P) in which the household utility

(2.3) is maximized subject to conditions (2.2), (2.4), and (2.5), given the parameter space (p,

w, I, S, Z, µ). Under certain regularity conditions, the solution to (P) can be proved equiv-

alent to a Subgame Perfect Nash Equilibrium in a two-stage game. Each member within a

household chooses her own time and goods consumption freely in the second stage to max-

imize the individual utility (2.1) subject to a conditional sharing rule. The rule, mutually

consented, is an allocation of the residual non-labor income after determining the parental

time and goods inputs in the first stage. A welfare analysis based on the two-stage game

shows that changes in the bargaining position (and thereby the Pareto weight λ) shift the

optimal choices in a predictable direction when we impose some weak conditions on child’s

cognitive technology. This also generates testable hypotheses for the empirical studies. A

detailed discussion with welfare analyses is in section 4.2.

In addition, single mothers face a similar decision process as married parents do, but with

a constant Pareto weight, i.e., λ = 1. They maximize the individual utility (2.1) subject to


the following constraints:

Cm +n∑j=1

Xjpj = wmhm + I,

lm + hm + tm = 1

A = F (tm, X, µ).

In single-mother households, paternal time is negligible and therefore is assumed not to enter

child’s cognitive production function6.

The model is the extension of a collective household model by introducing a household

production of child’s cognitive achievement. Blundell, Chiappori and Meghir (2005) first

consider a collective household labor supply model with public consumption. Our specifi-

cation of the model, however, has two distinctive features. First, the outcome of household

production is child’s utility which is unobservable in Blundell, Chiappori and Meghir (2005),

whereas the outcome of household production in our model is child’s cognitive achievement

and can be well measured by various achievement test scores. Second, Blundell, Chiappori

and Meghir (2005) assume that child’s utility only depends on parental time and child-specific

expenditures. In our formulation, we allow child’s cognitive achievement to be correlated

with child-specific endowments known to the families but not controlled by them. As we

discuss subsequently, these two distinctions are far from innocuous in the estimation of the

household production function.

2.2.2 Child’s Cognitive Production Function

The reduced-form demand functions for n input goods on child’s cognitive development can

be derived from the optimization program (P) as

Xj = Xj(p, w, I, S, Z, µ), j = 1, ..., n. (2.6)

6Fathers on average spend 2.3 hours per week with their children who live with single mothers, as compared

with 23.6 hours per week for those in married households. Child support from fathers in single-mother families

is captured in non-labor income I.


Similarly, demand functions for parental time inputs are

ti = ti(p, w, I, S, Z, µ), i = p,m. (2.7)

Therefore, according to (2), the reduced-form production function of child’s cognitive achieve-

ment can be derived as

A = Γ(tp(p, w, I, S, Z, µ), tm(p, w, I, S, Z, µ), X(p, w, I, S, Z, µ), µ) (2.8)

= Γ(p, w, I, S, Z, µ). (2.9)

Ideally, a complete analysis on child’s cognitive production (2.2) requires an access to data

on all current and past home and school inputs as well as information on child’s genetic

endowments. Since the existing data are typically deficient in one or more dimensions,

many previous studies on child’s cognitive production focus on the estimation of a reduced-

form child’s production function, such as equation (2.9)7. The production function (2.9)

assumes that parental time inputs enter the technology and the bargaining position matters,

but the estimation of it does not really make use of the notion of an underlying cognitive

production technology, nor does it recover parameter estimates of the technology. Some

empirical applications, concerned with the effect of a specific input (for example, school

input or childcare) on child’s cognitive development, have estimated a “hybrid” production

function with the form like

A = Ψ(Y, p, w, I, S, Z, µ). (2.10)

In equation (2.10), a measure of cognitive achievement is regressed on one observed input,

Y , and all other parameters. The estimated coefficient of Y , however, evaluates the total

7Some applications of household production of child outcomes are based on a unitary household model

without time allocation decisions. Household utility is defined over household consumption C and child’s

cognitive achievement A, which is produced by child-specific expenditures X, and some preference shifters

S. The problem is to maximize a household utility H(C,A;S) subject to a monetary budget constraint

C+∑n

j=1Xjpj = I and child production function A = F (X,µ). The reduced-form production function thus

becomes A = Γ(p, I, S, µ) and is independent of market wage rates and distribution factors.


effect of an exogenous change in Y on achievement without holding other inputs constant8,

and therefore the estimate is biased when those endogenous inputs are interdependent.

Our goal is to estimate the technology of child’s cognitive achievement, like (2.2), evaluated

at the optimal parental time and goods inputs, i.e., the equation (2.8)9. Several difficulties

remain even when child’s cognitive achievement and some relevant inputs are observed. First,

the estimation of a complete cognitive development process, which involves all the previous

time and goods inputs, is infeasible given the data available. For example, the PSID-CDS

has collected data every five years and currently has only three waves. The five-year time

interval cannot capture the true dynamic essence of child’s cognitive development, let alone

a decreased sample size as young children transit to adulthood. Moreover, current inputs

and past ones are usually highly correlated, inducing the issue of multi-collinearity. In

our married sample, for example, the correlation between two-wave maternal time inputs

is over 0.9. We simplify the production function by assuming that: 1) only current inputs

matter, and 2) inputs affect child’s cognitive achievement in a linear fashion. The restricted

production function for each child k therefore takes the form as

Ak = α0 + α1tp,k + α2tm,k + α3Xk + µk. (2.11)

The second difficulty is the selection issue. Based on (2.6) and (2.7), there is a clear correla-

8In the model, the total effect of an increase in input Y on achievement A can be shown as

ΨY =∂A

∂Y=∂F

∂Y+∂F

∂X

∂X

∂Y,

where X = {X, tp, tm}\Y includes all other inputs besides Y . The effect does not correspond to any single

technological parameter in the production function. The bias term ∂F∂X

∂X∂Y depends on the decision rules of

household resource allocation, and its sign and magnitude are determined by the underlying preferences and

constraints.9If the goal is to estimate the entire structure, i.e., both the preference parameters in the utility and the

underlying production technology, the data requirement is much more demanding and more restrictions need

to be imposed on the model. See Blundell, Chiappori and Meghir (2005) for a discussion on identifiability

of a model without unobserved heterogeneity.


tion between tp (and tm, X) and µ, which implies that parents may choose the home inputs

according to their child’s endowed ability. If a child’s ability cannot be fully observed, the

endogeneity of home inputs may lead to inconsistent and biased estimates in the production

technology (2.11). The issue can be even more complicated when parent’s own endowment

cannot be fully observed. For example, a high ability mother, more likely to have a high

ability child, may choose to spend more time with her child because she knows her time

input would be very effective and “produce” good achievement. In this case, the observed

relationship between maternal time input and child’s cognitive achievement does not cor-

respond to the true marginal effect of time input. The estimated effect of maternal time

input is exaggerated because of the positive correlation between maternal time input tm and

her own unobserved heterogeneity. Or this high ability mother may invest less time to her

child’s development because she knows the time input from her high ability spouse would

be more productive. This circumstance results in a correlation across different time inputs,

which can cost an efficiency loss during the estimation of a system of equations.

To clarify the problem, let us assume

µk = β0 + β1Scm,k + β2µk + εk. (2.12)

Here Scm,k is a subset of mother’s preference factors Sm,k10, which is correlated with child’s

endowment (e.g., education) and may not be fully observed, i.e, Scm,k = {Sc,om,k, Sc,uom,k }. µk is

the part of child’s endowment, mean independent of Scm,k and µk = {µko, µkuo}. εk is the

idiosyncratic shock. µk then can be written as

µk = β0 + β1Sc,om,k + β2µk

o + εk, (2.13)

where εk = β1Sc,uom,k + β2µk

uo + εk.

10We implicitly assume that child’s endowed ability is only correlated with some maternal factors, such

as mother’s education and test scores. The reason why we impose the condition lies in the concerns that 1)

parents’ eduction levels are highly correlated with each other; and 2) mother usually has a larger influence

on child’s endowed ability.


If we approximate the decision rule for the parental time inputs based on (2.7)11, the demand

functions of parental time inputs for child k are

ti,k = γ0 + γ1IHk (D) + γ2Si,k + γ3Zk + γ4µk, i = p,m, (2.14)

where IHk includes household labor and non-labor incomes and is assumed to capture all

the price information. Since IHk is likely to be endogenous, it depends on some local labor

demand conditions D. If we substitute (2.13) into (2.14) and assume part of Si,k cannot be

fully observed (i.e., Si,k = {Soi,k, Suoi,k}), the time-input demand functions become

ti,k = γ0 + γ1IHk (D) + γ2Si,k + γ3Zk + γ4(β0 + β1S

c,om,k + β2µk

o + εk)

= δ0 + δ1IHk (D) + δ2S

oi,k + δ3Zk + δ4S

c,om,k + δ5µk

o + ˆεi,k, i = p,m, (2.15)

where

ˆεi,k = γ4εk + δ2Suoi,k

= γ4β1Sc,uom,k + γ4β2µk

uo + δ2Suoi,k + γ4εk.

The presence of ˆεi,k in the decision rule of ti,k implies that parental time inputs are endoge-

nous in (2.11) and instruments are required to recover the parameters in (2.11). We further

argue that the local labor demand conditions D, some preference factors which do not affect

child’s cognitive development, and distribution factors Z, which are uncorrelated with Sc,uom,k

and µkuo, can play the role of exogenous instruments. Moreover, Suoi,k

12 can be correlated for

i = p,m, and therefore, the error terms in two time-input demand functions are potentially

correlated with each other. We later deal with the issue by estimating them together using

3SLS.

The last difficulty is measurement of other inputs in X. Following the previous literature,

we assume there are two types of inputs in X, the school inputs Xsch and goods inputs XG.

11In theory, both parents’ preference factors Sm and Sp should enter each time-input function simulta-

neously. However, since spouses’ characteristics, such as age and education, usually are highly correlated

with each other, which causes multi-collinearity issue in the empirical work, we assume that there are some

preference variables which only affect i’s time-input choice.12For Suo

m,k, it does not include information which is already captured by Sc,uom,k , and therefore, you could

think it is Sc,uom,k

⋂Suom,k.


The school inputs Xsch are measured by two local school-quality variables, and in our cross-

sectional analysis, they are assumed to be exogenous, independent of parents current-year

labor market outcomes, preferences, and bargaining position. The goods inputs XG unfortu-

nately are mostly unobserved. In the PSID-CDS, information on child-specific expenditures

is limited and incomplete. For example, it only provides expenditures on 8 categories (e.g.,

food, cloth, school supplies, and etc.) with questionably underestimated data. We use

household income IH as the proxies of goods inputs. The equation (2.11) therefore becomes

Ak = θ0 + θ1tp,k + θ2tm,k + θ3Xsch,k + θ4IHk (D) + θ5S

c,om,k + θ6µk

o + εk, (2.16)

which is estimable. Household income IH , however, is probably endogenous, correlated with

individuals and conditioned on the household decision rules of intra-household resource allo-

cation. For example, labor income is the product of market wage rate and individual labor

supply. We thereupon argue that the local labor demand conditions D are uncorrelated with

either child’s or parents’ heterogeneity but affect household member’s labor supply, which

lends them to be instruments. The same logic is applied to distribution factors Z.

2.2.3 Empirical Strategies

The system of equations (2.15) and (2.16) are estimable with appropriate data. Instead of

estimating a production function alone, a simultaneous equations model is more desirable in

that parameter estimates in a time-input demand function like (2.15) convey additional in-

formation of interest. The indirect channels, through which children’s cognitive achievement

is affected, can be identified in the system. For example, we usually use child’s gender to

partially capture the heterogeneity µ. The coefficient estimate for gender in (2.15) helps us

test whether parents spend more time with their same-gender children. Together with the

estimates in (2.16), we may decompose the gender test gap by nature versus nurture. We

estimate the equations (2.15) and (2.16) jointly using iterated 3SLS, which also allows the

correlation between error terms across the equations. In contrast to 2SLS, an adjustment of

variance-covariance structure in the third stage increases the efficiency of the estimates and

lends the inferences much credits.


We employ two test scores to measure different types of children’s cognitive achievement,

and estimate two production functions jointly with inputs demand functions. The underlying

assumption is that an input may have different marginal effects across different production

functions, but their errors can be correlated. In our empirical studies, we also include a

set of child’s characteristics and parents characteristics to control observed heterogeneity (or

preference shifters). A complete variable description presents in Table 2.1. Finally, while we

consider a particular functional form to estimate our baseline model, we implement many

generalizations and alternatives of (2.15) and (2.16) to let the data inform us which matters

most for child’s cognitive development. For instance, we estimate a model with parental

active and passive time inputs separately, as well as a set of models that allow for interac-

tions between parental time inputs and observed characteristics. All important results are

discussed in section 2.4.

2.3. Data

2.3.1 Sample Selection

Our data primarily come from two waves of the Child Development Supplement (CDS-I

and CDS-II) of the Panel Study of Income Dynamics (PSID). The PSID is a longitudinal

study of a representative sample of U.S. individuals and families, with an oversample of

African-American and low-income households. The study once has collected data annually

between 1968 and 1997, and subsequently through biennial surveys. In 1997, the CDS, one

of major PSID research components, has fielded. Detailed assessments of children’s cognitive

development along with children’s time diaries were collected in the CDS-I for 3,563 children

aged between 0 and 12 from 2,394 randomly select PSID families (with a response rate of

88%). Up to two children per family were surveyed. Later in 2002-2003 (CDS-II) 2,907

children aged below 18 from 2,021 original PSID-CDS I families were re-interviewed and no

new children were added. The most important and unique component in the PSID-CDS is

children’s 24-hour time diaries for one randomly select weekday and one weekend day, in

which all activities, duration, and who else participated or stayed around in each activity


have been recorded. In the CDS-I, 2,904 children have complete time diaries (with a 82%

response rate) and in the CDS-II, 2,569 children (with a 88% response rate).

We use children in the PSID-CDS with observed parental time (either non-zero maternal

time or non-zero paternal time) to estimate the parental time effects on children’s cognitive

achievement. It decreases the sample size to 2,837 children in the CDS-I and 2,480 in the

CDS-II. Since married parents may behave differently from single mothers do on parental

time inputs, we construct a married-household sample and a single-mother sample to eval-

uate the time effects respectively. The married-household sample restricts children who live

with their bio-parents to eliminate the possible noise from family reconstitution13. There

are 1,781 and 1,316 children living with bio-parents in the CDS-I and the CDS-II, respec-

tively. Since family information, especially on fathers’ backgrounds and household income,

is available in the main PSID data, the married sample merely consists of children whose

parents are household heads and wives in the PSID. After excluding observations with miss-

ing data, in particular invalid test scores, the final married sample has 819 children from the

CDS-I and 873 from the CDS-II. The single-mother sample conditions children living with

their bio-mother alone and thereby consists of 921 children from the CDS-I and 832 from

the CDS-II 14. Similar to the married sample, we require these single mothers are household

heads in the main PSID, reducing the sample size to 654 in the CDS-I and 572 in the CDS-II.

We finally exclude the observations with missing data and leave 362 and 401 children in the

CDS-I and the CDS-II, respectively.

In our empirical work, we pool the data from two waves of the PSID-CDS to estimate a

baseline model for the married households and single-mother households. The sample size

thereby increases 1,692 for the married sample and 763 for the single-mothers. Both samples

are weighted by the primary caregiver/child weight provided by the PSID-CDS to adjust

unequal probabilities of sample selection and non-response or data missing at random15.

13Our estimation results do not show significant difference if we include children who live with one step

parent and one bio-parent.14The CDS has less than 3% (around 90) children in each wave coming from single-father households.

Given the extremely small size, these children are not taken into our consideration.15Detailed information on sample weights can refer to Description of the 1997 PSID Child Supplement


2.3.2 Variable Measurement

Cognitive Achievement Measures: We employ the standardized scores of two sub-tests,

the Letter-Word Identification (LW) and the Applied Problems (AP), from the Woodcock-

Johnson Revised (WJ-R) Tests of Achievement as the measures of children’s cognitive

achievement. The WJ-R is a well-established and respected measure for cognitive achieve-

ment. In the CDS, children aged above 3 received LW and AP sub-tests to provide their

assessment of reading and math skills. For children aged above 6, another two sub-tests,

the Passage Comprehension and the Calculation (only in the CDS-I), were administered,

whereas given the wide range of ages to which LW and AP were administered, we only use

LW and AP test scores as our measure of children’s math and reading achievement. As for

the test scores, the CDS provides raw and standardized scores. The raw scores is the num-

ber of correct items that children completed, and the standardized scores display the sample

child’s reading and math abilities compared with national average for the child’s age. Since

the standardized score is normalized and well represents sample children’s relative cognitive

abilities cross-sectionally, it is a proper measure for our analyses16.

Parental Time Inputs Measures: The PSID-CDS provides time diary activity-level files

in which each record includes activity code, duration of the activity, and who else partici-

pated in or stayed around with the child in the activity. Each child has two 24-hour time

diaries for one randomly select weekday and one weekend day. We therefore aggregate the

duration of every activity (in hours per day) in which a child’s bio-mother participated as

“active” maternal time, and similarly the duration of activities in which a child’s bio-mother

did not involve but stayed around as “passive” maternal time. A weekly measure of time is

Weights and PSID Technical Report, The 2002 PSID Child Development Supplement (CDS-II) Weights,

Elena Gouskova, Ph.D., available online http://psidonline.isr.umich.edu/Guide/documents.aspx16According to the CDS user guide, for more information about standardized scoring and interpretation,

please refer to: Woodcock, R.W., & Mather, N. (1989, 1990). WJ-R Tests of Achievement: Examiner’s

Manual. R. W. Woodcock & M. B. Johnson, Woodcock-Johnson Psycho-Educational Battery - Revised.

Allen, TX: DLM Teaching Resources.

http://psidonline.isr.umich.edu/Guide/documents.aspx


constructed by a weighted summation with a weekday weight 5 and a weekend weight 2 for

each category of time. The same procedure applies to paternal time. The weekly “total”

maternal (or paternal) time input is a simple summation of active and passive maternal (or

paternal) time.

Other Inputs Measures: Child-specific expenditure should be a better measure of goods

inputs on children’s cognitive achievement, but, as we discussed in section 2.2, the PSID-CDS

lacks this information17. Following the previous literature, we thereby use household income

variables to measure the goods inputs. Both parents’ labor income and household non-labor

income are included and treated as endogenous variables. Instead of using current value

of household assets as non-labor income, we use a five-year average asset value to measure

non-labor income to comply with the permanent income hypothesis18.

We use two local school-quality variables to measure current school inputs, the county-level

pupil-teacher ratio and the state-level average teacher salaries. The data on the pupil-teacher

ratio are extracted from the Common Core of Data (CCD) by the National Center for Educa-

tion Statistics (NCES) of the U.S. Department of Education, and state-level average teacher

salaries from the American Federation of Teachers, Table II-4 in Survey and Analysis of

Teacher Salary Trends 2007. They are merged with our samples using 5-digit FIPS code

provided by the PSID GEOCODE19.

Other Variables: For better capturing child’s endowment, we include two specific variables

in children’s cognitive production functions, child’s WISC Digit Span (DS) Test for Short

17The CDS-II expanded the measures on child-specific expenditures by including the amount of money

spent by household and non-household members on gifts, vacations, school supplies, clothing, car insurance,

car payments, car maintenance, and food. The data, however, suffer from severe underestimation. For exam-

ple, the average annual child-specific food expenditure in the CDS-II is about $895 for married households,

while the Consumer Expenditure Survey (CEX) in 2002 reports an average annual food expenditure around

$7785 for a typical married household with two children ($1946 per person). We compare each category for

child-specific expenditures in the CDS-II with those in 2002 CEX and find them consistently underestimated.18Detailed information, please see Appendix A Data Supplement19The PSID GEOCODE is confidential, requiring a specific data-use application.


Term Memory20 and mother’s raw scores of Passage Comprehension (PC) test. As DS test

is a component of individual’s IQ test and insensitive to the acquired factors, we employ it

to control part of children’s unobserved heterogeneity.

Furthermore, since local labor demand conditions affect individual’s labor supply choice,

and consequently parental time inputs without a direct impact on children’s test scores,

three state-level labor demand conditions are applied, the annual unemployment rate, the

percentage of employment in service sector21, and the median hourly wage rate22. A divorce

legislation index, summation of three divorce law dummies, is used as a distribution factor

following the previous literature23.

2.3.3 Descriptive Statistics

Table 2.2 provides the descriptive statistics for our select samples. Since we pool the data

from two waves of the CDS, the sample children are aged from 3 to 18 years old with average

of 10 years. In general, children from married households outperform their counterparts

who live with single mothers. The average gap of reading scores is 8.23 standardized points

(about 8.19%), and for math scores it is 6.41 standardized points (about 6.24%). Many fac-

tors may cause the gap. First, married mothers on average spend 4.5 hours more per week

on children’s daily activities than those single mothers do. Given that it accounts for around

0.25 standard deviation, the maternal time loss for the single-mother children is substan-

tial. Moreover, a large portion of the loss is active maternal time. Married mothers invest

2.669 hours more (16.03%) active time and 1.835 (10.14%) passive time per week on average

in their children compared with single mothers. Second, married fathers on average spend

27.502 hours per week (roughly 4 hours a day) with children either actively or passively.

20Wechsler Intelligence Scale for children - Revised. Copyright 1974 by the Psychological Corporation.21Detailed information can refer to the Appendix A Data Supplement22Labor demand conditions are obtained from the Occupational Employment Statistics (OES) Survey

conducted by Bureau of Labor Statistics, Department of Labor. Please refer to http://stats.bls.gov/oes/.23The data sources are: 1) the Wolfer’s AER data appendix, and 2) the Divorce Law Table extracted from

www.divorcesource.com.

http://stats.bls.gov/oes/

www.divorcesource.com


In our single-mother sample, we almost observe no paternal time inputs. Hence, children

living with mothers alone suffer from the extensive loss of both maternal time (especially

active time) and paternal time. In addition, the average labor and non-labor income for

the single-mother families are low. Children from the single-mother sample tend to have

more behavior problems (i.e., larger BPI), worse health status at birth, and lower birth

weight than children from married households do. Married mothers on average have much

higher raw scores of PC test and around one more year of schooling than those single mothers.

No significant differences are displayed towards school inputs between two samples, probably

because the measures of school inputs are at state level or county level, which to some extent

eliminates the variations across each household. Meanwhile, the local exogenous variables

do not show much of the distinction between two samples, except that more single-mothers

families are from the South24. Another striking finding is that African-American children ac-

count for 50.6% in the single-mother sample in contrast to 6.8% in married households. Given

that married parents spend significant large amount of time with children, African-American

children living with single-mothers may suffer the most across different racial groups.

2.4. Estimate Results

2.4.1 Baseline Specification of Production Function

Before arriving at a baseline specification, we first assess whether it is appropriate to pool

the sample and specify the production function of child’s cognitive achievement in a con-

temporaneous fashion. Sample pooling, if appropriate, is advantageous for an increasing

randomness and a huge efficiency gain. Given that we have more than 80 parameters to

estimate in our simultaneous equations model, pooling samples from two waves of the PSID-

CDS is desirable. In addition, the CDS-I has 819 qualified children from married households

24We construct four census region dummies based on U.S. Census Bureau. Please see Appendix A Data

Supplement.


and 362 from single-mother households aged between 3-12, and the CDS-II has 873 and 401

aged between 6-18. Sample pooling expands the age of select children from 3 to 1825. We

once have considered to pool data from two tests by implicitly assuming the effects of all the

inputs are independent of different tests, but the results reject the hypothesis. The different

patterns of children’s math and reading skills development fail to support a child’s cognitive

production function that is irrelevant across different test scores. We consequently estimate

the reading and math production functions separately.

Pooling data from two waves, on the other hand, makes it impossible for us to estimate a

cumulated cognitive production function, as only two waves of data are available and appro-

priate for our empirical studies. In order to justify the contemporaneous specification, we

have estimated a value-added version in which lagged test scores and parental time inputs

from CDS I are used. The results are presented in section 2.8 appendix B table 1. Aside

from the issue that a small sample size (around 600 observations) causes huge efficiency loss,

the estimates themselves suffer from bias and inconsistency. The test scores provided by

the PSID-CDS I clearly account for a large portion of scores in the PSID-CDS II and affect

the parents’ choices of time inputs in the second wave, but many coefficients of parental

time inputs become biased, economically insignificant, or wrong in signs. The possible rea-

sons are: 1) a small and selected sample (e.g., the sample children are aged between 9-18

and many of them are considered adolescents); 2) highly questionable proxies of the lagged

variables (there is a five-year interval between two waves of the PSID-CDS); and 3) a multi-

collinearity issue (a piece of solid evidence is that the correlation between two-wave maternal

time is more than 0.5). After a careful examination of all these issues, we finally settle on

the baseline specification and report the results in Table 2.3. For both types of households,

we report 3SLS results for reading and math scores separately in panel (a) and panel (b).

All the maternal time effects are positive and significant. In general, the marginal effect

of maternal time input for married mothers is more valuable than that for single mothers.

25Considering the possibility that children’s cognitive production function has a structural change after

certain age, we have estimated the model for different age groups. The results are either severely biased

because of small sample size or very close to our pooled sample estimates.


Based on our 3SLS estimates in the married sample, every one additional hour increase in

maternal time within a day (seven hours per week) raises the child’s reading scores by 7.126

standardized points (approximately 0.39 s.d.) and math scores by 4.606 standardized points

(approximately 0.29 s.d.) on average. If evaluated at the mean test scores, it accounts

for almost a 6.55% increase in reading and a 4.22% increase in math, respectively. The

marginal effects of total maternal time inputs are quite substantial in magnitude, though in

our baseline specification we assume there is no difference between active and passive time.

Given that on average only about 49.21% of the maternal time are active time, children’s

test scores, either reading or math, may be improved a lot by very few direct time invest-

ment from their mothers. For the single-mother sample, one additional hour of maternal

time investment per day on average increases child’s reading and math test scores by 2.072

(2.06% or 0.10 s.d.) and 2.646 (2.58% or 0.14 s.d.) standardized points, respectively. The

effects are much smaller than married mothers’ (29.07% for reading and 57.44% for math).

The underlying reasons may reside in three major aspects: 1) single mothers on average

have less schooling and worse passage comprehension skill that devalue the marginal effects

of maternal time; 2) children living with single mothers in general lie in inferior status - low

birth weight, bad health, and severe behavior problems - which leads to the same amount

of maternal time inputs less efficient; and 3) less intense maternal time inputs are provided

by single mothers - 47.91% of total maternal time comes from direct time investment for

our single-mother sample. Later during our robustness checks, we find strong evidence that

supports the second and third arguments. Therefore, children living with single mothers may

suffer from fewer amount of maternal time inputs as well as poorer quality of maternal time.

Policies concerning children’s cognitive development for those from single-mother families

probably need to create even more incentives for the mothers to increase their active time

inputs along with a general improvement towards children’s well-being.

Paternal time is surprisingly important to children’s math skill development. The marginal

effect of paternal time input is positive and sizable on children’s math scores in our married

sample. The 3SLS estimate is significantly different from zero at 10% significance level and

imply a 2.611-standardized-point (2.39% or 0.16 s.d.) increase in math scores by fathers

spending one more hour a day with children, actively or passively. The impact is smaller


than the maternal time effect on average (only accounts for 56.69%), but should not be

ignored. Paternal time effect on children’s reading scores are negative but insignificant. In

reality, fathers may invest more time in children’s reading skill development when they re-

alize their children experience some reading disorders that are not observed by researchers.

But we consider the selection issue in the first stage estimation and evaluate the true effect

of parental time on children’s cognitive achievement. In summary, for children’s cognitive

production, the maternal time input is crucial especially for reading, and the paternal time

input is also indispensable for math. The interesting pattern later is reinforced when we

separate the effects of active and passive time inputs in the production functions.

For married households, non-labor income effects are insignificant for both math and reading

scores26. It is consistent with the results in some previous literature27 that wealth does not

play a leading role in children’s cognitive development. However, an increase in fathers’ labor

income has a positive impact on math scores, whereas an increase in mothers’ labor income

affects child’s reading scores negatively. Every one-thousand-dollar increase in fathers’ labor

income raises children’s math scores by 0.105 standardized points if everything else constant.

Given that the inter-percentile range between the 25th quantile to the 75th quantile of fa-

thers’ labor income is around 35.45 thousand dollars, a gap of math scores caused by an

increase of fathers’ labor income from the 25th to the 75th is approximately 3.72 standard-

ized points (3.41% or 0.23 s.d.). The same inter-percentile of paternal time between the 25th

and the 75th is around 21.5 hours per week, which can induce a gap of children’s math scores

over 8.02 standardized points, so paternal time effect is considerably more substantial than

the fathers’ labor income effect. A-thousand-dollar increase in mothers’ labor income impairs

children’s reading scores by 0.302 standardized points on average. Though mothers’ labor

income is relatively lower and condensed than fathers’, an increase of mothers’ labor income

from the 25th quantile to the 75th may still generate a gap of 7.79 standardized points on

reading scores for children whose mothers work. In our single-mother sample, the nega-

26We have tried current household asset income, five-year average asset income with higher orders, and log

of five-year average asset income to measure the non-labor income. None of them shows significant income

effects.27For example, Blau (1999) has found a small effect of income that makes income transfer a probably

ineffective approach to achieving substantial improvements in children’s outcomes.


tive effects of mothers’ labor income become silent, but instead household non-labor income

significantly affects children’s cognitive achievement. In general, 0.032 standardized points

for reading test (or 0.028 for math) arise for an one-thousand-dollar increase in household

non-labor income. One point that should be mentioned is that these income effects are not

equal to child-specific monetary inputs effects since the portion of child-specific expenditure

may change as household income changes. The effects of household income can overestimate

or underestimate the actual monetary inputs effects on children’s cognitive development.

School inputs are effective in child’s cognitive development. Broadly speaking, a large pupil-

teacher ratio depresses children’s cognitive achievement, whereas higher average teachers’

salary improves children’s reading scores. The effects are also more substantial for married

households, and probably more crucial towards children’s math skill development. The im-

pact of the county-level pupil-teacher ratio prevails. For example, in our married sample,

the average pupil-teacher ratio is 17.133, and our results suggest that if it increases to 18.133

the child’s reading and math scores reduce by 0.416 and 0.784 standardized points, respec-

tively. The impact may be striking when the ratio increases to a certain mediocre level, for

instance 25, which on average may lower the reading scores by 3.273 and math scores by

6.168 standardized points. Therefore, policies aiming to shrink the gap of test scores may

still need to consider creating a better school system and reducing the regional dispersion of

school inputs. In contrast to married households, the school inputs effects in single-mother

households are much more subtle. A higher pupil-teacher ratio decreases children’s reading

scores, and thereby the endeavors to promote small classes may be of paramount impor-

tance. The state-level average teachers’ salary also affects child’s reading test scores, but at

a quantitatively smaller level28.

Comparing the results of two samples, we find that the marginal effects of inputs are usually

larger for children in married households than those for children living with single mothers.

Though a smaller maternal time effect may result from single mothers’ unobserved ability,

the impacts of school inputs imply that children’s unobserved heterogeneity is probably the

28For the child’s reading scores, teachers’ average salary has a small and negative coefficient, but it is only

statistically significant at 10% significance level.


main force that drives the marginal effects of inputs less valuable in our single-mother sample.

Finally, children’s IQ test scores are positively correlated with their cognitive achievement.

High-ability children have relatively higher reading and math test scores if everything else

is constant. For children who live with single mothers, the mothers’ passage comprehen-

sion skills affect the cognitive production significantly. We further find that children’s non-

cognitive factors impact the test scores as well. For example, in single-mother households,

children with worse health status at birth have lower reading test scores, but math scores

are higher for those who have larger birth weight.

2.4.2 Comparison with Alternative Estimation Methods

In theory, a simultaneous equations model is advantageous towards the identification of

indirect channels, through which preference factors or distribution factors affect children’s

cognitive achievement. The OLS estimator for the production functions, however, is still un-

biased, consistent, and even more efficient than the IV-base estimators, as long as observed

variables can capture child’s heterogeneity thoroughly. We thereby start from estimating

the production functions using OLS method and test the endogeneity for different sets of

questionable variables. Both individual and joint endogeneity tests based on the difference-

in-Sargan statistic reject the null hypothesis that time inputs are exogenous. Thus, either

the 2SLS estimates or the iterated 3SLS estimates with valid instruments must be superior to

the OLS estimates. We present two columns of OLS results in Table 2.3 for our married and

single-mother samples, respectively. For each sample, the first column reports the estimates

for a production specification based on equation (2) in which inputs are implicitly assumed

exogenous, and the second column reports the results of a mixed reduce-form production

function based on equation (10) in which several inputs and a set of control variables are

included to somewhat capture the child’s unobserved heterogeneity.

The OLS results unsurprisingly suffer from the plausible downward bias. The effects of pa-


ternal time input are significantly negative, and the maternal time input has extremely small

impacts on children’s cognitive development. For example, the OLS results imply that the

marginal effect of maternal time for the reading test scores is between 0.4-0.9 in married

households, but our 3SLS estimate is above 7! The downward bias may imply a negative

correlation between children’s unobserved ability and parents’ time investment29. In other

words, parents may choose to spend more time with children if they experience some dif-

ficulty. Moreover, the OLS estimates report a negative effect of non-labor income in some

cases, against either theoretical implication or basic intuition. Therefore, conclusions drawn

from the contaminated OLS results must be deceiving, and policies relying on the estimates

without dealing with endogeneity could be misleading.

We also present 2SLS estimates in Table 2.3. They are very close to the results estimated

by the iterated 3SLS. The theory suggests that a 3SLS estimator would be more efficient

than a 2SLS estimator when the error terms across equations are correlated with each other.

Our results in Table 2.3 appear consistent with the theory as some 3SLS estimates now are

significant and have the right signs. In addition, 2SLS tends to report imprecise estimates

for the time inputs demand functions with a possibly inconsistent, less efficient first stage

estimation. The F statistics for the inputs functions are relatively small according to a larger

p-value, which makes the null hypothesis of overall insignificance hard to reject. Parents’

time input choices respond to both parents’ and children’s unobserved heterogeneity, and as

a result, the error terms in the system are likely to be correlated with each other.

We employ the iterated 3SLS method to estimate the simultaneous equations model for an-

other windfall that the parameter estimates in the demand functions are available and allow

us to analyze the decision process of parental time inputs. We present the 3SLS results for

the demand functions of time inputs in Table 2.3(c). The estimates show us several remark-

able points. First, for married households African-American parents spend much less time

with their children. Compared with the reference group, the non-Caucasian non-African-

29For a classic linear model, y = Xβ + ε, the expectation of OLS estimate β is E(β) = β +

E[(X ′X)−1X ′ε|X]. If the OLS estimator is downward biased, we could have E(β) < β which means

E(X ′ε) < 0.


American group, the gap is 4.858 hours per week for mothers and 5.466 hours per week

for fathers. Consider the case that the effect of maternal time on children’s reading skill

development is 1.018. Reading scores of African-American children can be 4.95 standardized

points lower than the average simply because of less maternal time inputs. African-American

children’s math skill development may be further hindered since paternal time plays an im-

portant role on math skill production. If we compare these African-American children with

the Caucasian ones, the gap of paternal time inputs is strikingly 11.392 hours per week. It

means, with everything else constant, the math scores for a typical African-American child

may be lower than her Caucasian peer by 4.249 standardized points (4%) purely for the

lack of father’s company. The test score gap would be further expanding after adding the

maternal time effects. For our married household sample, a typical African-American child

has lower test scores by around 10 standardized points than other groups. The discrepancy

almost totally results from the low level of parental time inputs. Second, girls spend more

time with mothers while boys stay longer with their fathers. It may because of role mod-

els, preferences, household specialization, or comparative advantage toward the opportunity

cost of parental time. We, however, are not quite certain about whether the gender-specific

time input choices causes the gender gap in test scores or the opposite. A further investiga-

tion in our later robustness checks provides plausible insights. Third, well-educated fathers

choose to spend more time with their children. On average, one additional year of schooling

increases the paternal time input by 1.168 hours a week. In other words, a 4-year-college-

graduate father is willing to spend 0.7 hours a day more than a high school graduate is,

which contributes to child’s math scores by around 2 standardized points increase. There-

fore, though male’s labor supply is insensitive to the wage rate and fairly constant for the

majority, high-education fathers tend to crowd out their leisure time by investing more time

in children’s cognitive development.


2.4.3 Robustness Checks

Robustness check with respect to different time measures

The empirical analyses so far focus on the effects of total parental time inputs on children’s

cognitive achievement. We implicitly assume that the quality of maternal or paternal active

and passive time are of no difference. The estimates, however, may underestimate the true

effects of parental time inputs if only active parental time matters while passive time has

either no impact or negligible effects. Policies that aim to increase parents involvement with

children’s daily activities without considering possible substitute effects between active and

passive time may still be less efficient. As we discussed before, approximately half of the total

parental time is passive time (e.g., 50.78% for married mothers, 52.09% for single mothers,

and 50.08% for married fathers). If the substitution of active time involvement with passive

one hinders children’s cognitive development and active parental time inputs are insensitive

to the labor supply, children’s welfare may hardly be improved merely through some reforms

targeting the female labor market. Hence, we have estimated a system of equations in which

time inputs are separated by active and passive time, and presented the estimates of pro-

duction functions in Table 2.4. The first two columns in Table 2.4 report reading and math

production functions for the married households and the last two columns report estimates

for the single-mother households.

The results are remarkable. The estimated coefficients of maternal active time input are

statistically significant at the regular significance level, and much larger in magnitude than

those in the baseline specification. In particular, the coefficients for the reading-skill produc-

tion function are doubled, either in married households or in single-mother households. One

additional hour per day that married mothers participate in their children’s daily activities

directly increases children’s reading scores by 15.281 standardized points (14.05%) holding

everything else constant. Given the standard deviation is around 18.491, it accounts for 83%

of standard deviation, a substantial improvement in children’s reading skill when married

mothers spend more active time with children. The maternal active time effect is slightly

smaller in magnitude for children’s math achievement. It increases around 5.817 standard-


ized points (5.33% or 0.35 s.d.) for the married sample and 3.591 standardized points (3.50%

or 0.19 s.d.) for the single-mother sample. The effect from maternal passive time inputs

is not significant for children’s math skill development. It probably implies that children’s

reading skill production requires more direct involvement of mothers, such as reading books

or telling stories. A substitute between mothers’ active time input and passive time input

may impede children’s cognitive development on reading. We calculate the correlation be-

tween two types of maternal time inputs, obtaining -0.0132 and -0.1784 for married and

single mothers, respectively. It implies a possible tendency that mothers may compensate

their children with more passive time inputs when they participate less in children’s daily

activities directly. Our results reinforce the initial concern that parental time, especially the

active time, is distinct from leisure. A labor supply shock does not necessarily shift parents

choices on time inputs in a predictable direction, and consequently lend employment being

an imprecise measure of parental time.

For married households, active paternal time is crucial towards children’s cognitive devel-

opment on math. The effect almost triples our baseline estimate. One additional hour per

day that fathers spend with their children actively increases math scores by 7.798 standard-

ized points (7.15% or 0.47 s.d.) on average, larger than the active time effect from married

mothers by approximately 34.06%. An interesting pattern appears that active maternal time

input is more efficient towards children’s reading skill production while active paternal time

input more valuable toward math. Multiple reasons may explain this, for example, the role

models, household specialization, and/or comparative advantage, but as for the concern on

deteriorating math performance of U.S. current young generation, the importance to pa-

ternal time inputs must be attached. Policies regarding children’s welfare, not limited to

cognitive outcomes, should cross beyond the scope of female labor participation.

Household non-labor income becomes positively related to children’s cognitive achievement

in our married sample and the effects are fairly consistent across different cognitive pro-

duction processes and types of households. Mothers’ raw scores of PC test and children’s

IQ test scores are positively correlated with current cognitive outcomes, implying a sizable

genetic impact on children’s cognitive development. Interestingly, we no longer observe a


gender discrepancy in children’s math scores for the married sample after time inputs are

measured separately by active and passive ones. In general, boys’ outperformance on math

may result from genetic factors and/or nurture tradition that fathers usually spend more

time with boys. Our results imply that nurture tradition may dominate the genetic factors

in that math-score gap disappears as the inputs are well measured. We attach separate

time inputs functions in the section 2.8 appendix B Table 3 in which girls appear to enjoy

around 2.022 more hours of active time with mothers whereas boys have 2.317 more hours

of active time with fathers every week. Moreover, white fathers spend a significantly larger

amount of time with children actively and passively than other racial groups do. In contrast,

African-American children have the least active time with their fathers. Single mothers

merely compensate their boys with more indirect time inputs.

Heterogeneity in the parental time inputs

In Table 2.5 we assess how the effects of parental time inputs varies with children’s or parents’

basic characteristics. In general, the signs of the interactions imply the direction of changes

in marginal time effects caused by changes in children’s or parents’ characteristics. We in-

clude interactions between various parental time inputs and children’s age, gender, three

non-cognitive indices, parents’ age, education levels, and mothers’ raw scores of passage

comprehension test. The marginal effects of parental time inputs are affected by children’s

characteristics, but are consistently insensitive to parents’ traits, in particular for our mar-

ried sample.

In married households, the marginal effect of maternal time on children’s reading scores is

lower for with older children, boys, children who suffer from more behavior problems, or chil-

dren who have a worse health status at birth. On average, the marginal effect of maternal

time declines by 0.096 when children are one year older. As a result, maternal time inputs

during early childhood are crucial and effective to improve children’s cognitive achievement.

It also supports many results presented in extensive previous literature (for example, Cunha,

Heckman and Schennach (2010)) that parental investment is more effective at early ages.


Maternal time input is more productive towards girls’ reading skill development. The gen-

der gap in general is around 1.527, which means the same amount of maternal time input

improves girls’ reading scores on average 1.527 standardized points higher than boys’. Chil-

dren’s problem behavior and worse health status at birth generally devalue the marginal

effect of maternal time. Similarly, for children’s math skill development, the marginal effect

of married mothers’ time depreciates when children are boys, have more behavior problems,

or lower birth weight. The gender gap for math is around 1.005, slightly smaller than the

reading case. The marginal effect of paternal time input, however, is much less sensitive to

the heterogeneity. It becomes less valuable when children’s mothers have higher raw scores

of passage comprehension test, but more valuable when children are boys (for reading) or

when children have lower birth weight.

The marginal effect of maternal time in single-mother households varies as children’s char-

acteristics change, but it also depends on her education levels. One more year of schooling

increases the marginal effect by 0.160 on reading and 0.283 on math for the single mothers.

It implies that quality of maternal time affects children’s cognitive production only for those

who are from absent-father families if we assume individual’s eduction levels have a positive

correlation with quality of their parental time. Two reasons probably account for the results.

First, the quantity effect dominates, in particular when children living with both bio-parents

obtain abundant parental time. Given that a small portion of total parental time is directly

related to cognitive-specific activities, quantity of time may be more important than qual-

ity. Second, married parents in our sample have relatively higher average education levels

(around 13.5 years) with standard deviations around 2.7, so one extra year of schooling for

married mothers may have an insignificant impact on quality of their time inputs.


Robustness check with high orders of time inputs

In Table 2.6 we present child’s cognitive production functions with high orders of parental

time inputs 30. We include the square terms of both parental time inputs with an intersection

between them in each production function to check the curvature of parental time effects as

more and more units of time inputs are invested.

The marginal effects of maternal time input are positive and significant when evaluated at

the mean, and follow the law of diminishing marginal productivity. For example, one addi-

tional hour of maternal time within a day decreases the improvement in children’s reading

scores from 7.312 standardized points (6.72% or 0.40 s.d.) to 0.782 standardized points

(0.72% or 0.04 s.d.) as maternal time climbs from the 25th percentile to the 75th percentile

in our married sample, holding everything else constant. The same amount of maternal time

inputs benefits children a lot at the lower percentile, and the marginal effect declines at the

rate around 0.042 for married households or 0.028 for single-mother households. Regard-

ing policy design, a possible extra cost associated with lower female labor supply or even

participation rate may be worth considering. In general, the well-educated females spend

more time with children. The policy that motivates all mothers to further increase maternal

time by disencouraging their labor supply may not be necessarily profitable to the whole

society given the tradeoff between loss and gain. A true concern should be laid on creating

incentives of time inputs for those who currently spend very few active hours with children.

The marginal effect of paternal time does not display an expected pattern. Possible expla-

nations for the unusual pattern are as follows. First, a critical error induced by the misspeci-

fication of demand functions of paternal time inputs may be responsible for the results. The

F test, with a test statistic 1.05 and 0.393 p-value, cannot reject the null hypothesis that

the paternal time input equation is overall statistically insignificant. And second, fathers

in reality may pay more attention to children who reside in the tails of the distribution.

They may spend more time with children who suffer from severe reading disorder or display

30The corresponding inputs demand functions estimated by 3SLS are shown in section 2.8 appendix B

Table 4.


uncommon talents on reading skill development. We once have plotted the paternal time

against the different test scores by child’s or parent’s characteristics. Some paternal time

indeed appears somewhat “U-shaped”, consistent with the results. Furthermore, as we have

discussed before, less valuable paternal passive time may obscure the curvature of marginal

effect of paternal time inputs since it may impact children’s cognitive production in an op-

posite direction to effective active paternal time, and thereby make total effects misleading.

2.5. Conclusion

Previous literature has emphasized the importance of parental time inputs to children’s

cognitive achievement, but the common non-experimental data that link parents’ - espe-

cially mothers’ - time spent with children to cognitive outcomes only reflect the correlations

between the two. The insight developed by a household collective model with children’s cog-

nitive production indicates that children’s cognitive achievement is produced by time inputs

which are selected by each parent individually. Consequently, if heterogeneity exists and

cannot be fully captured by data, an endogeneity issue arises, causing the OLS estimates bi-

ased and inconsistent. Our approach is motivated by the fact that parental time inputs that

affect children’s cognitive achievement are themselves solutions to a household optimization

program conditional on a mutual consent towards household non-labor income and child-

related consumption. It provides a guide to approximate the rules of parental time inputs and

estimate the quasi-structure system including both inputs demand and production functions.

We use the PSID-CDS I and II to assess the impact of parental time inputs on children’s cog-

nitive achievement for children aged above three from married households and single-mother

households. Direct measures of parental time inputs using child time diaries from the PSID-

CDS are used to capture the actual parents’ time spent with children. Earlier time-inputs

proxies, for instance maternal employment, appear to mask the differences across parental

time, leisure, and household work, and thereby lead to imprecise evaluation of the effects of

parental time inputs. This chapter improves the estimation of children’s cognitive produc-


tion by using a more accurate time-input measure and a more advantageous method.

More precise estimates are obtained after we deal with the endogeneity issue for the parental

time inputs. The main results show a substantial positive effect of maternal time inputs on

children’s cognitive development. For example, estimates of our baseline specification for

married households indicate that one additional hour per day of total maternal time input

enhances children’s reading test scores by roughly 7% and math test scores by 4% on average.

The effects are even striking when the active maternal time input is evaluated separately

from the passive time inputs - one more hour of the active maternal time input increases

reading scores by 14%. The impact of paternal time input on children’s math ability pro-

duction is quite significant as well. One additional hour per day of active paternal time

input improves children’s math scores by approximately 7% on average. The active time

effects almost double the impacts of total parental time inputs, that reinforces the concern

that indirect measures of parental time inputs may conceal the importance of parental time

effects on children’s cognitive production. The optimal choices of time inputs depend on

households characteristics, such as race, gender, age, or parents’ education levels, and the

parental time effects vary across these different characteristics.

Two research directions would be worth considering in the future. First, the dynamic essence

of children’s cognitive production may require a revision of the empirical studies when de-

tailed panel data are available. Either a pure structural analysis or an extended systematic

approach can be of empirical interest for estimation of the effects of home inputs on human

capital accumulation. Second, children’s health technology may be considered. The interde-

pendence between the two can improve our understanding of the total effects of home inputs

on children’s well-being.

2.6. Tables


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1997

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Oth

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Married HH Single-M HH Married HH Single-M HH

Reading 108.762 100.532 Child White 0.815 0.339

( 18.491 ) ( 20.198 ) ( 0.389 ) ( 0.474 )

Math 109.044 102.638 Black 0.068 0.506

( 16.519 ) ( 18.666 ) ( 0.252 ) ( 0.500 )

Boy 0.488 0.520

Maternal Time ( 0.500 ) ( 0.500 )

Total 39.247 34.743 No. siblings 1.536 1.378

( 18.586 ) ( 17.787 ) ( 1.035 ) ( 1.181 )

Active 19.316 16.647 Age (month) 128.864 131.866

( 12.522 ) ( 14.067 ) ( 44.874 ) ( 44.029 )

Passive 19.931 18.096 IQ 13.494 13.146

( 13.843 ) ( 13.081 ) ( 5.058 ) ( 5.161 )

Paternal Time BPI 7.601 9.267

Total 27.502 - ( 5.657 ) ( 6.402 )

( 16.707 ) - Health at birth (worse) 0.079 0.117

Active 13.728 - ( 0.270 ) ( 0.321 )

( 10.963 ) - Birth weight (ounce) 120.510 116.234

Passive 13.774 - ( 19.622 ) ( 22.748 )

( 12.247 ) - Mother Age (year) 38.958 36.977

Income ( 6.221 ) ( 7.772 )

Mother's Labor Income 17.321 15.627 Education 13.420 12.257

( 21.012 ) ( 14.314 ) ( 2.661 ) ( 2.137 )

Father's Labor Income 53.248 - PC Test Scores 33.101 29.519

( 80.940 ) - ( 4.654 ) ( 5.608 )

Non-Labor Income 170.282 32.251 Drink 0.387 0.251

( 878.372 ) ( 331.329 ) ( 0.487 ) ( 0.434 )

School Smoke 0.096 0.276

Pupil-Teacher Ratio 17.133 17.098 ( 0.295 ) ( 0.447 )

( 2.871 ) ( 2.693 ) Father Age (year) 41.258 -

Average Teacher Salaries 51.411 50.568 ( 6.750 ) -

( 7.864 ) ( 8.533 ) Education 13.547 -

( 2.828 ) -

MSA 0.753 0.739 Unemployment Rate 4.824 5.163

( 0.432 ) ( 0.439 ) ( 1.812 ) ( 1.770 )

Northeast 0.169 0.160 Percentage of service 0.172 0.170

( 0.375 ) ( 0.367 ) ( 0.016 ) ( 0.014 )

Midwest 0.269 0.208 Hourly wage rate 13.347 13.037

( 0.444 ) ( 0.406 ) ( 1.250 ) ( 1.406 )

South 0.262 0.458 Div_ind 1.286 -

( 0.440 ) ( 0.499 ) ( 0.781 ) ‐

Observations 1692 763 1692 763

Test Scores

Inputs

Others

TABLE 2.2: Basic Statistics

▪ Both samples are weighted using PSID-CDS PCG/Child level weights; all standard deviations are in the brackets.

Local Exogenous Variables


OLS1 OLS22SLS 3SLS OLS1 OLS2

2SLS 3SLS

Maternal Time 0.123 0.058 1.017 1.018 0.118 0.112 0.216 0.296

(0.027)*** (0.026)** (0.146)*** (0.140)*** (0.060)** (0.056)** (0.172) (0.172)*

Paternal Time -0.086 -0.059 -0.621 -0.353 - - - -

(0.030)*** (0.029)** (0.243)** (0.215) - - - -

Mother's Labor Income -0.054 -0.045 -0.330 -0.302 -0.031 -0.048 -0.229 -0.213

(0.024)** (0.023)* (0.167)** (0.165)* (0.085) (0.078) (0.232) (0.235)

Father's Labor Income 0.011 0.009 -0.033 -0.026 - - - -

(0.007) (0.007) (0.042) (0.042) - - - -

Non-Labor Income -0.001 -0.001 0.001 -0.001 -0.003 -0.004 0.031 0.032

(0.000)*** (0.000)*** (0.005) (0.005) (0.002) (0.002)** (0.012)*** (0.012)***

Pupil-Teacher Ratio -0.238 -0.330 -0.299 -0.416 0.215 -0.468 0.424 -0.180

(0.184) (0.249) (0.252) (0.247)* (0.401) (0.468) (0.333) (0.097)*

Average Teacher Salaries 0.044 0.167 0.172 0.169 -0.210 0.260 -0.188 0.421

(0.059) (0.118) (0.086)** (0.085)** (0.125)* (0.204) (0.096)** (0.337)

Mother's Education 1.046 0.713 1.721 1.514 0.153 0.041 0.524 0.438

(0.292)*** (0.332)** (0.600)*** (0.590)*** (0.523) (0.508) (0.646) (0.653)

Mother's PC Test Scores 0.409 0.250 0.131 0.161 1.086 0.507 0.933 0.916

(0.128)*** (0.123)** (0.147) (0.145) (0.202)*** (0.196)*** (0.215)*** (0.217)***

IQ 1.448 2.087 2.199 2.254 1.600 2.339 1.828 1.889

(0.115)*** (0.151)*** (0.183)*** (0.179)*** (0.281)*** (0.343)*** (0.223)*** (0.225)***

Boy -3.519 -3.195 -0.745 -1.127 1.319 0.799 0.077 0.062

(0.920)*** (0.861)*** (1.261) (1.237) (1.917) (1.741) (1.691) (1.710)

BPI -0.242 -0.222 0.059 0.073 0.051 -0.065 -0.009 -0.018

(0.092)*** (0.087)*** (0.115) (0.113) (0.176) (0.161) (0.127) (0.128)

Health at birth -1.676 -2.000 -0.401 -0.546 -3.425 -5.416 -4.171 -4.544

(1.624) (1.578) (2.171) (2.149) (3.528) (2.911)* (2.665) (2.692)*

Birth weight 0.011 0.017 0.004 0.013 -0.009 -0.012 -0.016 -0.013

(0.023) (0.023) (0.031) (0.031) (0.043) (0.039) (0.035) (0.036)

No. of Obs 1692 1692 1692 1692 763 763 763 763

Controls Included - X - - - X - -

F-Stat 24.31 16.47 23.44 24.37 5.36 5.13 15.74 15.5

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

▪ All standard deviations are in the brackets. *** indicates significance at 1%, ** at 5%, and * at 10%.

▪ Year dummy is included for all the specifications.

▪ OLS2 also includes: black, white, no. siblings, child's age (in month) and its squares, both parents' ages, age squares, and education levels (only mothers' for single-mother households), drink, smoke, MSA, Northeast, Midwest, South, and three local labor demand conditions.

TABLE 2.3 (a) : Baseline Model - Production Function

Dependent Variable -> Reading Scores

Married HH Single-Mother HH

▪ Both samples are weighted using PSID-CDS PCG/Child level weights.A38



2SLS 3SLS

Maternal Time 0.059 0.008 0.658 0.658 0.064 0.048 0.304 0.378

(0.026)** (0.024) (0.123)*** (0.135)*** (0.043) (0.042) (0.155)** (0.156)**

Paternal Time -0.059 -0.054 0.055 0.373 - - - -

(0.029)** (0.027)** (0.205) (0.200)* - - - -

Mother's Labor Income -0.029 -0.026 -0.047 0.087 -0.001 -0.041 0.204 0.219

(0.027) (0.024) (0.141) (0.144) (0.070) (0.064) (0.209) (0.213)

Father's Labor Income 0.008 0.003 0.075 0.105 - - - -

(0.006) (0.004) (0.036)** (0.038)*** - - - -

Non-Labor Income 0.001 0.001 0.003 0.000 -0.001 -0.002 0.027 0.028

(0.000)* (0.000)*** (0.004) (0.005) (0.001) (0.001)* (0.010)*** (0.011)***

Pupil-Teacher Ratio -0.293 -0.236 -0.618 -0.784 0.096 -0.490 0.086 -0.087

(0.152)* (0.225) (0.213)*** (0.236)*** (0.368) (0.402) (0.299) (0.088)

Average Teacher Salaries -0.005 -0.025 -0.005 -0.036 -0.145 0.339 -0.095 0.083

(0.054) (0.095) (0.073) (0.081) (0.117) (0.222) (0.086) (0.305)

Mother's Education 1.422 0.291 0.560 -0.128 0.844 0.850 0.293 0.213

(0.204)*** (0.249) (0.507) (0.477) (0.498)* (0.455)* (0.580) (0.591)


(0.103)** (0.095) (0.124) (0.140) (0.190)*** (0.196)** (0.193)*** (0.196)***

IQ 1.155 1.496 1.607 1.623 0.990 1.577 1.197 1.253

(0.096)*** (0.121)*** (0.155)*** (0.172)*** (0.240)*** (0.286)*** (0.200)*** (0.203)***

Boy 3.525 3.741 3.992 3.387 2.573 3.200 2.435 2.422

(0.800)*** (0.760)*** (1.065)*** (1.188)*** (1.903) (1.666)* (1.518) (1.549)

BPI -0.432 -0.395 -0.240 -0.237 -0.073 -0.222 -0.105 -0.114

(0.073)*** (0.069)*** (0.097)** (0.109)** (0.140) (0.129)* (0.114) (0.116)

Health at birth -1.614 -1.820 -1.160 -1.512 -3.140 -6.634 -3.567 -3.911

(1.552) (1.408) (1.833) (2.073) (3.110) (2.499)*** (2.392) (2.439)

Birth weight 0.001 -0.009 -0.011 -0.001 0.123 0.090 0.128 0.130

(0.022) (0.021) (0.026) (0.030) (0.039)*** (0.032)*** (0.032)*** (0.032)***

No. of Obs 1692 1692 1692 1692 763 763 763 763


F-Stat 28.84 21.93 28.84 23.76 13.72 10.76 18.78 18.2

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

▪ OLS2 also includes: black, white, no. siblings, child's age (in month) and its squares, both parents' ages, age squares, and education levels (only mothers' for single-mother households), drink, smoke, MSA, Northeast, Midwest, South, and three local labor demand conditions.


TABLE 2.3 (b): Baseline Model - Production Function

Dependent Variable -> Math Scores

▪ Both samples are weighted using PSID-CDS child level weights.




S-M HH S-M HH

Dependent Variable -> MT PT MT Dependent Variable -> MT PT MT

Mother's Labor Income 0.108 0.184 0.542 Age^2 (month) 0.000 -0.001 -0.000

(0.203) (0.134) (0.892) (0.000) (0.000)*** (0.000)

Father's Labor Income 0.044 -0.145 - IQ -0.193 -0.201 -0.374

(0.042) (0.081)* - (0.194) (0.196) (0.213)*

Non-Labor Income 0.009 -0.001 -0.002 BPI -0.296 -0.017 0.043

(0.007) (0.007) (0.018) (0.093)*** (0.101) (0.173)

Pupil-Teacher Ratio -0.031 0.494 0.412 Health at birth -0.739 0.164 6.472

(0.286) (0.328) (0.403) (1.955) (2.007) (4.935)

Average Teacher Salaries 0.141 0.037 0.269 Birth weight -0.028 -0.004 0.004

(0.100) (0.098) (0.152)* (0.027) (0.030) (0.040)

Black -4.858 -5.466 3.989 Mother's Education -0.719 - -0.572

(2.856)* (2.874)* (6.127) (0.586) - (1.991)

White 1.846 5.926 4.540 Mother's Age 0.056 - -0.166

(1.898) (1.977)*** (5.643) (0.085) - (0.113)

Boy -2.296 2.451 -0.865 Mother's PC Test Scores 0.133 -0.267 -0.498

(1.040)** (1.077)** (2.741) (0.141) (0.142)* (0.378)

No. of Siblings -0.523 1.287 -2.942 Father's Education 1.168 -

(1.473) (1.602) (4.772) (0.576)** -

Age (month) -0.189 0.218 0.006 Father's Age -0.013 -

(0.065)*** (0.064)*** (0.125) (0.079) -

No. of Obs 1692 1692 763

Estimation Method 3SLS 3SLS 3SLS

F-Stat 10.35 3.74 4.94

P-value 0.000 0.000 0.000

Married HH

TABLE 2.3 (c): Baseline Model - Input Demand Function

▪ Both samples are weighted using PSID-CDS PCG/Child level weights.



Married HH

▪ The inputs demand functions also include: MSA, Northeast, Midwest, South, and divorce index (only for the married households).


Reading Math Reading Math

Maternal Time

Active 2.183 0.831 0.522 0.513

(0.427)*** (0.341)** (0.170)*** (0.150)***

Passive -1.350 -0.099 -0.490 -0.265

(0.637)** (0.513) (0.246)** (0.217)

Paternal Time

Active -0.487 1.114 - -

(0.480) (0.394)*** - -

Passive 0.758 -0.347 - -

(0.480) (0.428) - -

Mother's Labor Income -0.491 -0.075 -0.088 0.174

(0.224)** (0.160) (0.192) (0.169)

Father's Labor Income -0.064 0.058 - -

(0.058) (0.042) - -

Non-Labor Income 0.022 0.015 0.022 0.021

(0.009)** (0.007)** (0.011)* (0.010)**

Pupil-Teacher Ratio -0.464 -0.617 0.231 0.009

(0.335) (0.258)** (0.319) (0.281)

Average Teacher Salaries 0.465 0.164 -0.109 -0.043

(0.140)*** (0.108) (0.094) (0.083)

Mother's Education 0.856 -0.595 0.272 0.399

(0.768) (0.485) (0.571) (0.503)

Mother's PC Test Scores 0.849 0.406 0.900 0.826

(0.252)*** (0.198)** (0.202)*** (0.178)***

IQ 2.563 1.997 1.888 1.263

(0.275)*** (0.211)*** (0.218)*** (0.192)***

Boy 1.111 1.801 3.870 4.992

(2.196) (1.739) (1.756)** (1.546)***

BPI 0.103 -0.223 -0.041 -0.140

(0.156) (0.120)* (0.124) (0.109)

Health at birth -0.379 -2.191 -5.079 -4.514

(2.983) (2.286) (2.617)* (2.304)**

Birth weight -0.012 0.018 -0.025 0.118

(0.046) (0.035) (0.034) (0.030)***

No. of Obs 1692 1692 763 763

F-Stat 12.11 18.19 16.63 20.32

P-value 0.000 0.000 0.000 0.000


▪ The models are estimated using three stage least squares.


TABLE 2.4 : Robustness Check w.r.t. Different Time Measures




Maternal Time (MT) 0.306 0.189 1.441 0.823 1.661 1.880 -0.239 0.980

(0.140)** (0.569) (1.855) (0.405)** (0.356)*** (0.406)*** (1.240) (0.139)***

Paternal Time (FT) 0.015 0.168 6.765 -0.682 -1.019 -0.559 0.279 -0.329

(0.169) (0.224) (2.412)*** (0.610) (0.414)** (0.625) (1.084) (0.226)

Mother's Age*MT -0.004

(0.003)

Father's Age*FT -0.001

(0.004)

Mother's Edu.*MT -0.004

(0.040)

Father's Edu.*FT -0.012

(0.014)

PC Test Scores*MT -0.037

(0.054)

PC Test Scores*FT -0.205

(0.072)***

Child's Age*MT -0.008

(0.003)***

Child's Age*FT 0.005

(0.004)

Child's Gender*MT -1.527

(0.364)***

Child's Gender*FT 0.910

(0.412)**

BPI*MT -0.152

(0.037)***

BPI*FT 0.052

(0.056)

Birth weight*MT 0.003

(0.010)

Birth weight*FT -0.002

(0.009)

Health Status*MT -0.286

(0.167)*

Health Status*FT 0.359

(0.245)

F-Stat 32.30 24.32 11.57 16.36 15.71 16.25 26.20 25.88

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000




▪ The cognitive production functions also include: school inputs, income variables, noncognitive indices, boy, IQ, mother's education levels, and mother's PC test scores.

TABLE 2,5 (a) : Heterogeneity in Effect of Parental Time Inputs (Married HH)


By Parents' Age

By Parents' Education

By Mother's PC Test Scores

By Child's Age

By Child's Gender

By Child's BPI

By Child's Birth Weight

By Child's Health Status


Maternal Time (MT) -0.028 0.438 1.641 -0.279 1.086 1.194 -3.668 0.691

(0.133) (0.551) (1.816) (0.392) (0.269)*** (0.332)*** (1.537)** (0.135)***

Paternal Time (FT) 0.301 0.137 7.240 0.915 -0.153 0.034 4.299 0.155

(0.176)* (0.225) (2.330)*** (0.591) (0.309) (0.489) (1.397)*** (0.207)

Mother's Age*MT 0.002

(0.003)

Father's Age*FT -0.006

(0.004)

Mother's Edu.*MT -0.026

(0.039)

Father's Edu.*FT -0.007

(0.014)


(0.053)

PC Test Scores*FT -0.217

(0.069)***

Child's Age*MT 0.000

(0.003)

Child's Age*FT -0.006

(0.004)


(0.272)***

Child's Gender*FT 0.185

(0.307)

BPI*MT -0.098

(0.030)***

BPI*FT 0.009

(0.043)


(0.012)**

Birth weight*FT -0.034

(0.011)***


(0.158)

Health Status*FT 0.113

(0.218)

F-Stat 24.57 21.90 11.51 12.94 23.13 20.20 14.52 24.56

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

All standard deviations are in the brackets. *** indicates significance at 1%, ** at 5%, and * at 10%.

Year dummy is included for all the specifications.

The cognitive production functions also include: school inputs, income variables, noncognitive indices, boy, IQ, mother's education levels, and mother's PC test scores.

Both samples are weighted using PSID-CDS PCG/Child level weights.

TABLE 2.5 (b) : Heterogeneity in Effect of Paternal Time Inputs (Married HH)


By Parents' Age

By Parents' Education


By Child's Age

By Child's Gender

By Child's BPI




Maternal Time (MT) 0.033 -1.940 0.138 0.770 0.431 0.585 0.709 0.389

(0.116) (1.024)* (1.275) (0.118)*** (0.234)* (0.236)** (1.201) (0.191)**


(0.003)

Mother's Edu.*MT 0.160

(0.082)*


(0.042)


(0.001)***


(0.245)

BPI*MT -0.037

(0.017)**

Birth weight*MT -0.006

(0.010)


(0.231)

F-Stat 17.57 15.66 18.04 18.75 16.82 15.48 16.53 14.55

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000


TABLE 2.5 (c) : Heterogeneity in Effect of Maternal Time Input (Single-Mother HH)


By Mother's Age

By Mother's Education


By Child's Age

By Child's Gender

By Child's BPI







Maternal Time (MT) 0.033 -3.524 0.491 0.665 0.414 0.609 -0.97 0.496

(0.104) (1.036)*** (1.122) (0.108)*** (0.207)** (0.211)*** (1.041) (0.181)***


(0.003)

Mother's Edu.*MT 0.283

(0.083)***


(0.037)


(0.001)***


(0.216)**

BPI*MT -0.042

(0.015)***


(0.009)


(0.220)**

F-Stat 20.67 15.12 22.26 20.82 20.49 18.50 20.62 15.84

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000







By Mother's Age

By Mother's Education


By Child's Age

By Child's Gender

By Child's BPI

TABLE 2.5 (d) : Heterogeneity in Effect of Maternal Time Input (Single-Mother HH)



Reading Math Reading Math

Maternal Time 2.182 2.588 1.438 1.206

(0.424)*** (0.447)*** (0.412)*** (0.375)***

Maternal Time^2 -0.021 -0.025 -0.014 -0.012

(0.005)*** (0.005)*** (0.004)*** (0.004)***

Paternal Time -0.996 -0.329 - -

(0.587)* (0.607) - -

Paternal Time^2 0.011 0.007 - -

(0.005)** (0.005) - -

Maternal T × Paternal T -0.000 -0.005 - -

(0.004) (0.004) - -

Mother's Labor Income -0.292 -0.069 0.193 0.398

(0.149)** (0.155) (0.192) (0.175)**

Father's Labor Income -0.018 0.078 - -

(0.043) (0.045)* - -

Non-Labor Income -0.002 0.008 0.027 0.022

(0.004) (0.004)* (0.011)** (0.010)**

Pupil-Teacher Ratio -0.131 -0.520 0.109 -0.105

(0.210) (0.223)** (0.329) (0.299)

Average Teacher Salaries 0.080 -0.055 -0.121 -0.058

(0.073) (0.077) (0.097) (0.088)

Mother's Education 1.795 0.597 -0.550 -0.204

(0.573)*** (0.595) (0.584) (0.531)

Mother's PC Test Scores 0.399 0.179 0.694 0.679

(0.128)*** (0.135) (0.207)*** (0.188)***

IQ 1.722 1.382 1.854 1.167

(0.133)*** (0.142)*** (0.197)*** (0.179)***

Boy -1.065 4.646 -0.331 1.771

(1.200) (1.270)*** (1.721) (1.564)

BPI -0.156 -0.330 0.068 -0.040

(0.092)* (0.098)*** (0.125) (0.113)

Health Status at birth -1.644 -1.733 -5.610 -4.600

(1.865) (1.985) (2.725)** (2.476)*

Birth Weight 0.026 -0.005 -0.008 0.128

(0.026) (0.028) (0.035) (0.032)***

No. of Obs 1692 1692 763 763

F-Stat 25.00 21.30 15.24 17.56

P-value 0.000 0.000 0.000 0.000

▪ The models are estimated using three stage least squares.


TABLE 2.6 : Robustness Check w.r.t. High Orders of Time Measures





2.7. Appendix A: Data Supplement

U.S. Census Region: According to the definition by the U.S. Census Bureau 31, we con-

struct four census-region dummies - Northeast, Midwest, South, and West (reference group).

Puerto Rico and the Island Areas are not part of any census region or census division. The

constituent states see following:

Northeast Region

Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, Connecticut, New York,

New Jersey, Pennsylvania.

Midwest Region

Ohio, Indiana, Illinois, Michigan, Wisconsin, Minnesota, Iowa, Missouri, North Dakota,

South Dakota, Nebraska, Kansas.

South Region

Delaware, Maryland, District of Columbia, Virginia, West Virginia, North Carolina, South

Carolina, Georgia, Florida, Kentucky, Tennessee, Alabama, Mississippi, Arkansas, Louisiana,

Oklahoma, Texas.

West Region (reference)

Montana, Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah, Nevada, Washington,

Oregon, California, Alaska, Hawaii.

Non-Labor Income: The PSID main data provide detailed assets values for 94, 99, 01,

03 survey year. We first generate the assets income for each survey year available using the

formula:

Assets Income = Main House Value + Net Value of Other Real Estate

+Net Value of Vehicle + Net Value of Farm or Business

+Net Value of Stocks + Value of All Cash Accounts

+Net Value of Other Assets

−Remaining Mortgage Principle for the Main House−Other Debts,

31U.S. Census Bereau, http://www.census.gov/index.html


and convert into 1997 dollar. The 1997 non-labor income is the average of assets income in

1994 and in 1999, and the 2001 non-labor income is the average of assets income in 1999,

2001, and 2003.

The Percentage of Employment in Service Section: OES survey conducted by the

Bureau of Labor Statistics provides a unique, five-digit numerical identifier for each OES

occupation, called “occ code”. The code from 61000 to 69999 identifies the occupational

division for service. We thereby aggregate the total employment for both the state and the

state-level service division using the “occ code”, and then divide the total state-level em-

ployment for service division by the total state-level employment to generate the variable of

the percentage of employment in service section.

2.8. Appendix B: Tables



2SLS 3SLS

Reading Test _1 0.092 0.131 0.096 0.095 0.124 0.194 0.120 0.116

(0.009)*** (0.025)*** (0.009)*** (0.009)*** (0.015)*** (0.045)*** (0.012)*** (0.011)***

Maternal Time 0.001 0.000 0.004 0.004 0.001 0.001 0.001 0.000

(0.000)** (0.000) (0.002)** (0.002)*** (0.001)* (0.000)** (0.001) (0.001)

Paternal Time -0.001 -0.000 -0.004 -0.005 - - - -

(0.000) (0.000) (0.002)* (0.002)** - - - -

Maternal Time _1 0.001 0.000 -0.000 -0.000 -0.000 -0.000 -0.000 0.000

(0.000)* (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

Paternal Time _1 -0.000 -0.000 0.001 0.001 - - - -

(0.000) (0.000) (0.000) (0.000) - - - -

Mother's Labor Income -0.000 -0.000 -0.004 -0.004 0.001 0.000 0.001 0.001

(0.000) (0.000) (0.002)** (0.002)** (0.001) (0.001) (0.002) (0.002)

Father's Labor Income 0.000 0.000 -0.001 -0.000 - - - -

(0.000) (0.000) (0.000) (0.000) - - - -

Non-Labor Income -0.000 -0.000 0.000 0.000 0.000 0.000 0.001 0.001

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001) (0.001)

Pupil-Teacher Ratio -0.004 -0.001 -0.002 -0.002 0.004 -0.004 0.003 0.003

(0.002)* (0.003) (0.003) (0.003) (0.005) (0.006) (0.004) (0.004)

Average Teacher Salaries 0.000 0.001 -0.002 0.001 -0.001 0.001 -0.000 -0.000

(0.001) (0.001) (0.003) (0.001) (0.002) (0.002) (0.001) (0.001)

Mother's Education 0.002 0.004 0.015 0.014 -0.008 -0.006 -0.009 -0.009

(0.002) (0.003) (0.007)** (0.007)** (0.004)* (0.004) (0.005)* (0.005)*


(0.002)** (0.001) (0.002)*** (0.002)*** (0.002)*** (0.002) (0.002)*** (0.002)***

IQ 0.010 0.010 0.013 0.013 0.013 0.011 0.013 0.013

(0.001)*** (0.001)*** (0.002)*** (0.002)*** (0.003)*** (0.003)*** (0.002)*** (0.002)***

Boy -0.023 -0.017 -0.008 -0.007 -0.002 0.000 -0.003 -0.003

(0.011)** (0.009)* (0.014) (0.014) (0.021) (0.018) (0.017) (0.017)

BPI -0.001 -0.001 -0.001 -0.001 -0.002 -0.003 -0.002 -0.002

(0.001) (0.000)* (0.001) (0.001)* (0.001) (0.001)** (0.001)* (0.001)*

Health at birth -0.023 -0.023 -0.008 -0.011 -0.012 -0.017 -0.010 -0.009

(0.020) (0.019) (0.024) (0.024) (0.035) (0.033) (0.025) (0.024)

Birth weight 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

No. of Obs 662 662 662 662 307 313 307 307


F-Stat 23.31 18.66 23.90 24.45 15.98 16.92 25.06 26.31

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

▪ OLS2 also includes: black, white, no. siblings, child's age (in month) and its squares, both parents' ages, education levels, MSA, Northeast, Midwest, South.


TABLE A2.1 (a) : Value-Added Specification - Production Function






2SLS 3SLS

Math Test _1 0.144 0.125 0.132 0.132 0.147 0.219 0.143 0.134

(0.015)*** (0.035)*** (0.011)*** (0.011)*** (0.025)*** (0.040)*** (0.014)*** (0.014)***

Maternal Time -0.000 -0.000 -0.001 -0.001 0.000 0.000 0.001 0.000

(0.000) (0.000) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001)

Paternal Time -0.000 -0.000 0.002 0.003 - - - -

(0.000) (0.000) (0.002) (0.002) - - - -

Maternal Time _1 0.000 0.000 0.000 0.000 0.000 -0.000 0.000 0.000

(0.000) (0.000) (0.000) (0.000) (0.001) (0.000) (0.000) (0.000)

Paternal Time _1 -0.001 -0.001 -0.001 -0.001 - - - -

(0.000)** (0.000)** (0.000)* (0.000)* - - - -

Mother's Labor Income -0.001 -0.001 -0.001 -0.001 0.000 -0.000 -0.000 0.000

(0.000)*** (0.000)** (0.002) (0.002) (0.001) (0.001) (0.002) (0.002)

Father's Labor Income 0.000 0.000 0.000 0.000 - - - -

(0.000) (0.000) (0.000) (0.000) - - - -

Non-Labor Income 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.002

(0.000)*** (0.000)* (0.000) (0.000) (0.000)* (0.000) (0.001)*** (0.001)***

Pupil-Teacher Ratio -0.004 -0.001 -0.007 -0.007 -0.001 -0.007 -0.005 -0.005

(0.002)** (0.003) (0.003)** (0.003)*** (0.005) (0.005) (0.004) (0.004)

Average Teacher Salaries -0.000 0.000 -0.000 -0.000 -0.001 0.004 0.001 0.001

(0.001) (0.001) (0.001) (0.001) (0.002) (0.002)** (0.001) (0.001)


(0.003)*** (0.003)** (0.006) (0.006) (0.006) (0.005) (0.006) (0.006)

Mother's PC Test Scores 0.003 0.001 0.003 0.003 0.005 -0.001 0.004 0.004

(0.002)** (0.002) (0.002) (0.002)* (0.002)** (0.002) (0.002)** (0.002)**

IQ 0.012 0.011 0.013 0.013 0.016 0.015 0.016 0.016

(0.002)*** (0.002)*** (0.002)*** (0.002)*** (0.003)*** (0.004)*** (0.002)*** (0.002)***

Boy 0.025 0.033 0.017 0.015 0.029 0.024 0.024 0.025

(0.011)** (0.011)*** (0.012) (0.012) (0.024) (0.020) (0.018) (0.018)

BPI -0.001 -0.001 -0.001 -0.001 -0.004 -0.005 -0.004 -0.004

(0.001)** (0.001)*** (0.001)** (0.001)** (0.002)** (0.002)*** (0.001)*** (0.001)***

Health at birth 0.002 0.000 0.000 0.002 0.008 0.010 0.004 0.005

(0.018) (0.018) (0.022) (0.021) (0.036) (0.035) (0.027) (0.026)

Birth weight 0.000 0.000 0.000 0.000 0.002 0.002 0.002 0.002

(0.000) (0.000) (0.000) (0.000) (0.001)*** (0.000)*** (0.000)*** (0.000)***

No. of Obs 662 662 662 662 307 307 307 307


F-Stat 29.80 31.05 42.46 43.68 14.74 15.00 28.05 29.17

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

TABLE A2.1 (b) : Value-Added Specification - Production Function



▪ OLS2 also includes: black, white, no. siblings, child's age (in month) and its squares, both parents' ages, education levels, MSA, Northeast, Midwest, South.




S-M HH S-M HH

Dependent Variable -> MT PT MT Dependent Variable -> MT PT MT

Reading Test_1 3.927 13.205 -8.869 Child's Characteristics

(3.995) (8.259) (6.933) Black 5.343 4.896 -2.541

Math Test_1 5.005 -3.731 0.921 (5.646) (8.417) (9.660)

(4.041) (5.153) (7.921) White 8.411 10.094 -10.182

Maternal Time_1 0.140 - 0.178 (4.252)** (5.236)* (13.410)

(0.067)** - (0.108)* Boy -6.180 2.171 3.243

Paternal Time_1 - 0.013 - (2.556)** (4.062) (4.621)

- (0.143) - No. of Siblings 0.962 -0.377 5.961

Mother's Labor Income 0.362 -0.091 1.455 (4.863) (5.002) (3.166)*

(0.417) (0.237) (0.983) Age (month) -0.353 0.205 0.811

Father's Labor Income 0.011 -0.334 - (0.345) (0.444) (0.731)

(0.076) (0.214) - Age^2 (month) 0.000 -0.002 -0.004

Non-Labor Income 0.020 0.016 0.417 (0.001) (0.002) (0.003)

(0.009)** (0.010)* (0.361) IQ -0.409 0.171 -0.077

Pupil-Teacher Ratio 0.719 1.488 0.126 (0.392) (0.507) (0.521)

(0.766) (1.051) (1.320) BPI 0.044 0.138 0.508

Average Teacher Salaries 0.296 0.315 -0.134 (0.106) (0.189) (0.331)

(0.234) (0.306) (0.380) Health at birth 4.405 5.128 8.720

Parents' Characteristics (4.165) (5.490) (8.447)

Mother's Education -2.240 - -3.815 Birth weight -0.049 0.076 0.074

(1.769) - (2.771) (0.067) (0.118) (0.112)

Mother's Age -0.313 - -0.781

(0.229) - (0.465)*

Mother's PC Test Scores 0.570 0.355 -0.146

(0.332)* (0.445) (0.772)

Father's Education - 2.332 -

- (1.991) -

Father's Age - -0.626 -

- (0.330)* -

No. of Obs 662 662 307


F-Stat 1.97 0.95 1.22

P-value 0.002 0.538 0.218

Married HH

TABLE A2.1 (c): Value-Added Specification Model - Input Demand Function



▪ The inputs demand functions also include: MSA, Northeast, Midwest, South, and divorce index (only for the married households)

Married HH


Married HH S-M HH Married HH S-M HH

Test Scores Reading 108.508 100.914 Others Child Characteristics

(19.191) (21.617) White 0.761 0.330

Math 107.741 100.622 (0.427) (0.471)

(15.782) (19.082) Black 0.058 0.513

Inputs (0.234) (0.501)

MT Current 34.988 31.458 Boy 0.475 0.524

(18.558) (16.681) (0.500) (0.500)

Lag 42.89823 35.35849 No. siblings 1.613 1.336

(19.040) (21.859) (1.044) (1.217)

PT Current 27.151 - Age (month) 164.004 164.661

(18.068) - (33.683) (33.908)

Lag 27.52387 - IQ 16.036 16.059

(15.584) - (4.231) (4.231)

Income Mother's Labor Income 20.366 19.923 Noncognitive Indices

(20.495) (15.173) BPI 8.619 9.875

Father's Labor Income 62.448 - (10.244) (6.672)

(105.544) - Health at birth (worse) 0.082 0.155

Non-Labor Income 155.310 8.710 (0.274) (0.363)

(873.531) (37.524) Birth weight (ounce) 120.519 114.698

School Pupil-Teacher Ratio 16.862 16.455 (19.663) (23.652)

(2.913) (2.643) Mother Characteristics

Average Teacher Salaries 52.310 50.944 Age (year) 42.165 40.754

(2.913) (8.007) (5.563) (7.283)

Father Characteristics Education 13.188 12.350

Age (year) 44.486 ‐ (2.974) (2.183)

(6.080) ‐ PC Test Scores 33.052 29.669

Education 13.321 ‐ (4.462) (5.624)

(3.116) ‐

Observations 662 307

TABLE A2.2: Value-Added Specification Sample - Basic Statistics


▪ All standard deviations are in the brackets.


Dependent Variable -> MT_Active PT_Passive PT_Active PT_Passive MT_Active MT_Passive

Mother's Labor Income 0.109 -0.179 0.225 -0.123 1.383 -0.544

(0.131) (0.172) (0.087)*** (0.102) (1.154) (0.793)

Father's Labor Income 0.042 0.022 -0.046 -0.074 - -

(0.028) (0.035) (0.048) (0.060) - -

Non-Labor Income -0.001 0.008 -0.004 0.002 0.018 -0.011

(0.004) (0.006) (0.004) (0.005) (0.024) (0.016)

Pupil-Teacher Ratio -0.033 0.001 0.145 0.209 0.841 -0.169

(0.188) (0.260) (0.208) (0.247) (0.571) (0.395)

Average Teacher Salaries -0.010 0.195 -0.045 0.118 0.127 -0.019

(0.066) (0.090)** (0.063) (0.073) (0.219) (0.151)

Child's Characteristics

Black -2.853 3.656 -4.250 1.880 10.435 -3.189

(1.942) (2.849) (1.941)** (2.325) (8.808) (6.050)

White 1.013 4.014 3.083 3.151 8.051 -2.257

(1.204) (1.866)** (1.273)** (1.593)** (7.897) (5.422)

Boy -2.022 -0.029 2.317 0.266 -3.141 3.584

(0.722)*** (0.885) (0.697)*** (0.718) (2.765) (1.910)*

No. of Siblings 0.001 0.859 0.663 0.374 0.336 0.389

(0.925) (1.463) (1.020) (1.319) (2.616) (1.794)

Age (month) -0.149 -0.161 0.132 -0.045 -0.078 0.095

(0.042)*** (0.060)*** (0.042)*** (0.048) (0.174) (0.119)

Age^2 (month) 0.000 0.000 -0.001 0.000 -0.000 -0.000

(0.000) (0.000)** (0.000)*** (0.000) (0.001) (0.000)

IQ -0.217 0.022 -0.312 0.083 -0.291 -0.217

(0.127)* (0.166) (0.125)** (0.143) (0.297) (0.206)

BPI -0.201 -0.094 -0.044 0.023 0.224 -0.069

(0.064)*** (0.079) (0.065) (0.070) (0.208) (0.144)

Health at birth -0.279 -0.516 0.293 -0.038 11.070 -2.530

(1.354) (1.673) (1.302) (1.366) (6.355)* (4.377)

Birth weight -0.014 -0.020 -0.021 0.015 0.055 -0.050

(0.019) (0.023) (0.019) (0.020) (0.057) (0.039)

Parents' Characteristics

Mother's Education -0.142 -0.156 - - -2.847 1.773

(0.367) (0.438) - - (2.717) (1.868)

Mother's Age 0.009 0.087 - - -0.047 -0.247

(0.052) (0.080) - - (0.163) (0.112)**

Mother's PC Test Scores -0.104 0.213 -0.204 -0.034 -0.710 0.334

(0.096) (0.125)* (0.093)** (0.101) (0.542) (0.373)

Father's Education - - 0.584 0.614 - -

- - (0.327)* (0.426) - -

Father's Age - - -0.030 0.095 - -

- - (0.050) (0.061) - -

No. of Obs 1692 1692 1692 1692 763 763

Estimation Method 3SLS 3SLS 3SLS 3SLS 3SLS 3SLS

F-Stat 10.61 2.83 5.07 2.51 2.21 1.73

P-value 0.000 0.000 0.000 0.000 0.000 0.000




TABLE A2.3: Inputs Demand Functions w.r.t. Different Time Measures




S-M HH

Dependent Variable -> Maternal Time Paternal Time Maternal Time

Mother's Labor Income -2.864 -0.399 -1.946

(0.568)*** (0.513) (2.168)

Father's Labor Income -0.427 -1.332 -

(0.175)** (0.331)*** -

Non-Labor Income 0.081 0.038 0.128

(0.027)*** (0.031) (0.039)***

Pupil-Teacher Ratio -0.101 2.627 2.453

(1.489) (1.708) (1.048)**

Average Teacher Salaries 0.757 0.493 0.225

(0.481) (0.530) (0.421)


Black 31.379 -7.629 -10.589

(13.854)** (15.268) (16.678)

White 1.026 21.096 -14.729

(11.286) (12.178)* (14.775)

Boy -3.263 6.842 -7.850

(4.842) (5.409) (5.093)

No. of Siblings -7.697 3.400 -5.845

(8.544) (9.119) (4.908)

Age (month) -0.377 0.360 0.507

(0.345) (0.371) (0.323)

Age^2 (month) 0.001 -0.001 -0.002

(0.001) (0.001) (0.001)

IQ 1.366 1.252 -0.396

(0.818)* (0.977) (0.570)

BPI 0.186 0.724 -0.350

(0.418) (0.503) (0.389)

Health at birth 10.380 10.200 -1.961

(8.751) (10.120) (12.085)

Birth weight -0.055 0.180 -0.072

(0.123) (0.142) (0.108)


Mother's Education 5.450 - 5.225

(2.228)** - (5.113)

Mother's Age 0.033 - -0.408

(0.486) - (0.314)

Mother's PC Test Scores 1.253 0.093 0.099

(0.648)* (0.695) (1.040)

Father's Education - 8.559 -

- (2.422)*** -

Father's Age - -0.441 -

- (0.440) -

No. of Obs 1692 1692 763


F-Stat 1.65 1.05 2.42

P-value 0.023 0.393 0.000

TABLE A2.4: Inputs Demand Functions w.r.t. High Orders of Time Measures

Married HH





Chapter 3

The Role of Home Inputs in the

Black-White Test Score Gaps

(ABSTRACT)

This chapter studies the role of home inputs in the black-white test score gaps. Relying

on rich household survey data from the PSID-CDS, we estimate the children’s cognitive

production function and conduct standard Blinder-Oaxaca decomposition to quantify the

sources of disparity in children’s cognitive achievement between two races. We find that

children’s ability accounts for a large proportion of the differences. Home inputs, especially

maternal time input, also explain a significant portion of the gap. When maternal time is

equalized at the average levels of white children, the differences in children’s reading and

math test scores can be closed by approximately 30%-50%.

3.1. Introduction

The average academic performance of white children exceeds that of the blacks persistently.

At almost any point in time, the white children outperform the blacks, and the disparity

59


tends to widen with age1. The empirical results are statistically significant and robust across

a wide variety of tests, subjects, samples, and cohorts, and even when a range of control

variables are included, although the mean group differences have decreased slowly over time2.

As a common premarket factor, the early childhood achievement is a strong predictor of in-

dividual’s future labor market outcomes, such as education attainment and labor income3. A

large black-white test score gap thereby may be correlated with the wage differences among

racial groups which highlight many major labor economic issues, for example, racial dis-

crimination, income inequality, and intergenerational persistence. Thus, studies that help

understand the sources of racial difference in cognitive achievement are crucial in many ways

to improve the efficiency of policies concerning the overall social equality.

The explanation for the black-white test score gap, however, proves elusive in the previous

empirical studies. Neither school quality nor family investment achieves a consistent con-

clusion about its role in accounting for the disparity quantitatively. For instance, Cook and

Evans (2000) have argued that changes within schools attribute nearly 75% of the conver-

gence in black/white test scores in the 1970-88 time period, but numerous empirical work

based on cross-sectional or short panel data has found much smaller effects of school inputs

on children’s math and reading test scores, e.g., Todd and Wolpin (2004), Krueger (2003),

and a series of influential literature written by Hanushek et al4. Similar disagreements exist

1Fryer and Levitt (2004) have demonstrated that the black-white test score gap among incoming kinder-

gartners is not significant using a newly available data set from Early Childhood Longitudinal Study, but

in their another paper Fryer and Levitt (2006) they have found that the blacks lose their ground relative to

other races substantially after entering the schools. An extensive body of previous literature has found the

gap persists over time even for the noncognitive skills, such as Carneiro, Heckman and Masterov (2005).2Cook and Evans (2000) have found that the difference in test scores has fallen by about one-half since

1970 by only looking at the National Assessment of Educational Progress (NAEP) data. Hedges and Nowell

(1999), however, have argued that the rate of change in the test score gap declines, and given the rate over

the past 30 years, the non-NAEP data would suggest another more than 50 years to close the gap.3Neal and Johnson (1995), Carneiro, Heckman and Masterov (2005), and Murnane, Willett and Levy

(1995a) have discussed the role of premarket factors in wage determination among races. Almond and

Currie (2011) have surveyed the recent work and shown that even the events before children’s age five can

have long-run effects on their future outcomes.4For example, Hanushek (2003), Hanushek and Raymond (2005), Rivkin, Hanushek and Kain (2005).


in the literature over whether home inputs, like maternal employment or family incomes,

matter in producing higher test scores5. Child ability, on the other hand, explains a large

fraction of the black-white test score differences at a point in time by about 50% (at most!).

As is discussed in Jencks and Phillips (1998), however, no genetic evidence supports that

black children have less innate intellectual ability than their white counterparts. Further-

more, policies that aim to narrow the black-white test score gap can hardly affect ability

directly to improve the academic performance of the blacks.

The diversity in the literature on the roles of different inputs, especially the home inputs, on

children’s test score gap are mainly stemming from two reasons: 1) the deficiency of data,

and 2) the endogeneity of inputs. Most nation-wide surveys, such as the National Assess-

ment of Educational Progress (NAEP), the Early Childhood Longitudinal Study (ECLS),

and the Children of the National Longitudinal Survey of Youth (CNLSY), commonly used

in the empirical analyses of black/white test score gap, provide very few variables of home

inputs. For example, the only variable for all years between 1970 to 1988 in the NAEP is

the highest education level of each parent, which cannot map parental time or monetary

inputs monotonically. The newly available data set ECLS includes a socioeconomic compos-

ite measure6 to capture the impacts of home inputs on children’s cognitive production, but,

like the “HOME” index offered by the CNLSY, such a composite measure combines both

household time and monetary inputs, which may appear to be substitutes. Moreover, home

inputs are endogenous because of the unobserved heterogeneity for either parents or children.

As long as parents select the time and goods inputs based on their children’s unobserved

heterogeneity, the spurious correlations between the home inputs and test scores cost the

regular OLS estimates biased and inconsistent. Any empirical analysis which fails to deal

with the endogeneity produces implausible results in explaining the test score differences

between the blacks and the whites.

In this chapter, we use the Child Development Supplement of the Panel Study of Income Dy-

5A good review of the literature can refer to Bernal and Keane (2010).6The variable was computed by ECLS and detailed description can refer to the data appendix in Fryer

and Levitt (2004).


namics (PSID-CDS)7 to shed new light on the role of home inputs in black-white test score

gaps. An enormous amount of information is obtained for each targeted child, including

both child’s and parent’s basic characteristics, family background, school inputs, and test

scores. In particular, the children’s time diaries from the PSID-CDS provide the duration

of child’s each daily activity for both a randomly selected weekday and a weekend day, as

well as who else participate in or is around the child. A direct parental time measure and

both parents’ labor/non-labor incomes are constructed to better capture the effects of home

inputs on children’s cognitive production. Therefore, we can fill the gap by evaluating the

effects of many omitted family variables in the previous studies and estimating the effects of

school inputs more precisely given the potential correlation between schools and families.

Considering the existence of non-random selection for the home inputs, we estimate the

children’s cognitive production functions and the parental time demand functions simulta-

neously to deal with the endogeneity. The simultaneous equations model estimated by the

iterated three-stage least squares offers consistent and efficient estimates when the home in-

puts are selected based on unobserved heterogeneity, and it also allows non-zero correlations

among the errors. We conduct the standard mean decomposition thereafter by equalizing

each input at the average level of white children to calculate how many percentages of the

black-white test score gap can be closed if there is no difference of that input between the

blacks and the whites. Our empirical strategy can help separate the contribution of different

inputs to the black-white test score gap and identify both the direct and indirect sources

that may cause the black children to lose ground relative to the whites.

Our standard decomposition indicates that maternal time input explains a very large pro-

portion of the black-white gaps for both reading and math test scores in married-household

sample. Almost half of the reading test score gap can be closed by equalizing the black

children’s maternal time input at the average level of the whites. The results are informa-

tive for two reasons. First, different from some previous literature which argues that family

background is not responsible for or only explains a small fraction of the black-white test

7Panel Study of Income Dynamics, public use dataset, produced and distributed by the Institute for

Social Research, Survey Research Center, University of Michigan, Ann Arbor, MI (2011).


score gaps, we have found that the role of maternal time input in the disparity is important.

Second, compared with the direct income effects, an increase in maternal time input for the

blacks may be a more efficient way to close the gap.

The paternal time input accounts for about 5% of the math test score gap. Given the fact

that the average paternal time gap between the blacks and whites is only about 2.2 hours

per week, the paternal time input plays a relatively small role in black-white test score

gaps quantitatively. Moreover, its explanation power increases to 13.58% for the girls’ math

scores for married households. Considering the finding that the average test scores decrease

significantly if girls do not have fathers, we argue that black girls’ cognitive development

may be improved substantially by spending more time with their fathers. Children’s abil-

ity explains about half of the differences in test scores in general, but the roles of school

inputs in black-white test score gaps are not obvious. Income effects are worth noting in

closing the gaps. Though direct impacts from various of incomes are small, incomes ex-

plain about 20% of the differences in parental time inputs. The indirect income effects are

substantial, and therefore, a simple statement that money is not of importance may need a

revision. Policies based on cash transfers can still play a role in lowering the social inequality.

The chapter is organized as follows. In the section 3.2, we discuss the data and descriptive

sample statistics. The decomposition procedure and empirical results are presented and de-

tailed analyzed in the section 3.3. The section 3.4 concludes.

3.2. Data

3.2.1 Sample Selection

The data we use primarily come from the first two waves of the Child Development Supple-

ment (CDS-I and CDS-II) of the Panel Study of Income Dynamics (PSID). The PSID is a

longitudinal study of a representative sample of U.S. individuals and families, with an over-


sample of African-American and low-income households. The study once has collected data

annually between 1968 and 1997, and subsequently through biennial surveys. The PSID-

CDS I has fielded in 1997, including detailed assessments of children’s cognitive development

as well as home and school information for 3,563 children aged between 0 and 12 from 2,394

randomly select PSID families (with a response rate of 88%). Up to two children per family

were surveyed. Later in 2002-2003 (CDS-II) 2,907 children aged below 18 from 2,021 original

PSID-CDS I families were re-interviewed and no new children were added. There are 6470

observations in total from two waves of the PSID-CDS.

In both waves, two subsets of the Woodcock-Johnson Revised Tests of Achievement (WJ-R),

the Letter-Word Identification (LW) and the Applied Problems (AP), were administered by

the interviewers for children aged three years and older. For children aged above 6 years old,

another two sub-tests, the Passage Comprehension and the Calculation (only in the CDS-I),

were administered. Given the wide range of ages to which LW and AP were covered, later

in the empirical analyses we employ LW and AP scores as our primary measures of chil-

dren’s reading and math achievement. Both raw and standardized test scores are provided

by the PSID-CDS. The standardized score, however, is normalized and well represents sam-

ple children’s relative cognitive abilities cross the population, it is more appropriate for our

cross-section analyses8. There are in total 4155 observations of either black or white children

who have observed math test scores and 4172 who have observed reading test scores in both

waves. The black-white test score gap is around 13.149 standardized points for the math

test and around 12.069 for the reading test in our pooled full sample (see Table 3.1 (a) and

3.1 (b)). The average test score gaps by household types are around 13 standardized points,

slightly less a standard deviation. The within-group differences between two household types

are much smaller for both races compared with the racial test score gaps.

The most important and unique component in the PSID-CDS is children’s 24-hour time di-

8According to the CDS user guide, for more information about standardized scoring and interpretation,

please refer to: Woodcock, R.W., & Mather, N. (1989, 1990). WJ-R Tests of Achievement: Examiner’s

Manual. R. W. Woodcock & M. B. Johnson, Woodcock-Johnson Psycho-Educational Battery - Revised.

Allen, TX: DLM Teaching Resources.


aries for one randomly select weekday and one weekend day. The time diary survey collects

a targeted child’s daily activity, as well as the type, the duration, and who else participated

in or stayed around the child. In the CDS-I, 2,904 children have complete time diaries (with

a 82% response rate), and in the CDS-II, 2,569 children (with a 88% response rate). For un-

derstanding the role of home inputs, especially the parental time inputs, in the black-white

test score gaps, we restrict the sample children with observed parental time (either non-zero

maternal time or non-zero paternal time). Moreover, we merge each parent’s characteristics

including age, education, and labor/non-labor incomes, available in the PSID core surveys

to estimate the children’s cognitive production function and then decompose the differences

in the test scores.

We construct a married-household sample and a single-mother sample to conduct the anal-

yses conditional on household type, given the notable differences in household behaviors

for married and single parents. For example, almost none positive paternal time input is

observed for children living with their mothers only. Our married-household sample further

constrains children who live with their bio-parents to eliminate the possible noise from fam-

ily reconstitution9, and the single-mother-household sample requires children who live with

their bio-mother alone10. Finally, the selected married sample consists of 1692 observations,

including 1161 white children and 128 black children. The selected single-mother sample has

763 children where 404 children are reported white and 600 black.

The variable description is available in Table A-1 in appendix A. All the samples are weighted

by the primary caregiver/child weight provided by the PSID-CDS to adjust unequal proba-

bilities of sample selection and non-response or data missing at random11.

9Our estimation results do not show significant difference if we include children who live with one step

parent and one bio-parent.10The CDS has less than 3% (around 90) children in each wave coming from single-father households.

Given the extremely small size, these children are not taken into our consideration.11Detailed information on sample weights can refer to Description of the 1997 PSID Child Supplement

Weights and PSID Technical Report, The 2002 PSID Child Development Supplement (CDS-II) Weights,

Elena Gouskova, Ph.D., available online http://psidonline.isr.umich.edu/Guide/documents.aspx

http://psidonline.isr.umich.edu/Guide/documents.aspx


3.2.2 Descriptive Statistics

Summary statistics of the test scores by race, age groups, and gender are displayed in the

Table 3.1 (a) and 3.1 (b), with white referring solely to non-Hispanic whites. The selected

samples have very close basic statistics as the full samples. We do not find significant test

score gaps within each racial group between two different household types, especially for the

math test scores. The weighted average math test scores for the white children living with

their bio-parents are almost the same as those living with single mothers. The results are

quite consistent in different age groups. Girls living with bio-mothers alone, however, appear

to suffer from the lack of fathers. In general, both black and white girls in single-mother

households are approximately 5 standardized points lower in math test and 10 standardized

points lower in reading test than those in married households. Later we conduct an inves-

tigation on the role of home inputs in girls’ black-white test score differences in order to

examine the effects of missing fathers on girls’ cognitive development. A separate analysis

conditional on the household type thereby explains girls’ relatively large test score gaps be-

tween and within racial groups.

Table 3.2 (a) and 3.2 (b) provide descriptive statistics by race for all the variables we use in

the estimation12. White children on average outperform the blacks by over 13 standardized

points for both math and reading tests, regardless of the household type. In the married-

household sample, white mothers spend more hours, either active or passive, with their

children. The average gap in total maternal time is around 7 hours per week (0.4 s.d. and

about one hour per day) between two racial groups, where about half of the differences are

from active maternal time. The gap of paternal time between the blacks and the whites is

not as large as the maternal time gap, but white fathers still spend about 2 hours more per

week with their children by directly participating in children’s daily activities.

Married African-American mothers on average earn 6,000 dollars more annually in the labor

market, whereas white couples have much higher non-labor incomes and labor incomes from

the males. Therefore, black children in general live in the households with lower level of

12Descriptive statistics for the whole samples please see the Table A-2 in section 3.6 appendix A.


total incomes than the whites. No significant differences are observed in school inputs be-

tween two races, which is consistent with the fact that school quality converges among racial

groups over time. Children’s characteristics on average are similar, except that white chil-

dren have relatively larger birth weight by around 9 ounces. White parents, however, have

some advantages relative to the blacks. They are more educated and have higher Passage

Comprehension (PC) test scores by about 5 points. Geographically, more black children live

in the south region but fewer live in the northeast region.

White children outperform the blacks in the single-mother households by around 13-15 stan-

dardized points on average. In contrast to the mild disparity in test scores within the racial

group for different household types, the black-white test score gaps across racial groups con-

ditional on the same household type are substantial. Therefore, only a small fraction of the

black-white test score gaps may be explained by the fact that a relatively larger proportion

of black children living with single mothers compared with the whites (for example, 0.79

in our selected single-mother sample), but there must be other factors that play important

roles to account for the disparity.

As for the time inputs, black single-mothers spend slightly more time with their children

than the whites do by about 0.408 hours per week. The excess maternal time comes from

participating in children’s daily activities. The labor income earned by the black single moth-

ers, however, is lower than the whites by around 5,000 dollars a year, just opposite to the

case for married households in which more maternal time but lower average level of female

labor income are observed. Moreover, the household non-labor income for the white group is

much higher than that for the blacks by more than 75,000 dollars on average. No significant

advantages toward school inputs are found in the data again for the white children, though

the whites have slightly larger county-level pupil-teacher ratio and higher state-level average

teacher salaries. Surprisingly, white children who live with single mothers suffer from more

problem behaviors and worse health status at birth, but their average level of birth weight

is still higher compared with the blacks. White single mothers have higher PC test scores

by more than 5 points on average than the blacks but have similar years of schooling.


3.3. Empirical Analysis

The questions we wish to address are rather straightforward: What are the source of the

disparity in black and white cognitive achievement and how important are they? We use

a decomposition technique attributed to Oaxaca (1973) to answer the questions. In this

section, we first present the theoretic framework for two types of Blinder-Oaxaca decom-

positions. All the empirical results are displayed subsequently to explain the differences in

children’s test scores across the blacks and whites. Moreover, we conduct another decomposi-

tion towards parental time inputs choices in the end of the section to further understand the

differences in the optimal choices of parental time inputs. Since a relatively large proportion

of black-white test score gaps can be attributed to the parental time inputs, an analysis on

the choices of parental time inputs is necessary to answer our questions thoroughly.

3.3.1 Decomposing Gaps in Test Scores: A Simple Model

Before we estimate the children’s cognitive production functions to examine the role of home

inputs in the black-white test score gaps, we estimate the test score equations with differ-

ent covariates in the Table 3.3 (a) and 3.3 (b) using the OLS method to see whether the

conditional test score gaps are affected by controlling home inputs. The first three columns

show the results based on a full sample and the next six columns are based on selected

samples. The coefficients of a dummy variable “Black” indicate the differences in test scores

across black and white groups given different sets of covariates. The conditional disparity

of test scores between the blacks and whites decreases dramatically when the home inputs

are included, indicating that the potential test score gap may be closed to some extent hold-

ing the same home inputs for two racial groups. Since the OLS results support that the

traditional inputs - home, school, and ability - in children’s cognitive production function

are likely responsible for the black-white test score gaps, the consistent and unbiased esti-

mates for the effects of the inputs are required for further answering our questions. The

partial effects of home inputs, however, estimated by the OLS are biased and inconsistent

when parents’ choices depend on children’s ability which cannot be fully observed by the

economists. As a consequence of the contaminated estimates, the Blinder-Oaxaca decom-


position produces misleading results to quantify the contributions from each input to the

test score gaps. Therefore, we first need to choose a proper model to estimate children’s

cognitive production functions. According to the discussion in chapter 2, the Simultaneous

Equation Model (SEM), which jointly estimates both children’s cognitive production func-

tions and parental time inputs demand functions, offers parameter estimates more precisely,

given that the choices of parental time inputs are endogenous.

We specify the benchmark model based on the previous chapter. A child k’s cognitive

production function is:

Ak = θ0 + θ1tp,k + θ2tm,k + θ3Xsch,k + θ4IHk (D) + θ5Wk + µk, (3.1)

where ti,k denotes the parent i’s time input (i = p,m, p for father and m for mother), Xsch,k

includes school inputs variables, IHk (D) measures the household incomes which depend on a

set of labor market demand conditions D, and Wk are variables that may affect children’s

cognitive development directly, such as children’s noncognitive measures, children’s observed

ability, and children’s basic characteristics. µk represents the error term that captures all

the unobserved factors. According to the decision rules for the optimal home inputs derived

from a collective household model with children’s cognitive production, we obtain the demand

function for parental time inputs by approximating the rules, i.e.,

ti,k = δ0 + δ1IHk (D) + δ2Wk + δ3Zi,k + εi,k, i = p,m, (3.2)

where Zi,k includes parent i’s basic characteristics and some other control variables, such as

census region dummies and distribution factors. εi,k is the idiosyncratic shock.

The system of equations (3.1) and (3.2) are estimated simultaneously using iterated three

stage least squares (3SLS). Later in our empirical work, the household income variables

and the number of children in the married households are also treated as endogenous, in-

strumented by the local labor market demand conditions and the dummies on whether the

mother is a smoker or alcoholic. The iterated 3SLS method allows for the correlation be-

tween error terms across different equations and adjusts the variance-covariance structure in

the third stage to increase the efficiency of the parameter estimates. Thus, the inferences


are more credible compared with the two stage least squares.

Since the system of equations consists of more than 80 parameters, we first estimate the

model based on our whole selected samples and conduct the benchmark decomposition by

assuming that the marginal effects of the inputs are constant across all racial groups. We

construct a series of counterfactuals by equalizing the inputs of the blacks at the average

levels of the whites to calculate how much percentage of the black-white test score gap can

be closed. For example, if the black children living with their bio-parents enjoy the same

amount of paternal time as the whites do, a counterfactual for the black children is:

AB = θ0 + θ1tpW + θ2tm

B + θ3 ¯XschB

+ θ4 ¯IH(D)B

+ θ5WB, (3.3)

where, except the tpW is the white sample average of total paternal time input, all the

other independent variables are the averages for the blacks. Then the difference between the

counterfactual and the predicted test scores for the white group is:

AW − AB = θ2(tmW − tmB) + θ3( ¯Xsch

W − ¯XschB

) + θ4( ¯IH(D)W − ¯IH(D)

B) + θ5(W

W − WB).

(3.4)

In other words, the product of θ1 and tpW , which is the average paternal time input in the

white group, represents the increase or the decrease of the possible standardized points for

the black children if spending the same amount of time with their fathers. The ratio of the

product to the actual difference of test scores between the blacks and whites accounts for

the percentage changes of the total test score gap that is attributable to the paternal time

input. Following the method, we can decompose the black-white test score gaps by different

children’s characteristics, such as gender and age. The product of each estimated marginal

effect and the whites’ average input (or children’s observed ability) can be obtained to cal-

culate the possible percentage changes of the total test score differences. One thing worth

mentioning is that the reliability of the decomposition relies heavily on the assumption that

the marginal effects are constant across different racial groups. If the marginal effects change

with race, the estimates may under- or over-state the total effects.

Another type of the decomposition is also implemented after estimating the system of equa-

tions for the black and white samples separately. The advantage of this decomposition


method is that we somewhat relax the hypothesis that the production technology is the

same between two racial groups, which is weakly rejected by the data. However, the sample

size becomes much smaller when separate white and black samples are used. For example,

there are only 128 observations left in our white single-mother household sample.

We construct the counterfactuals by equalizing the marginal effects for two groups to account

for the percentage changes of the total test score gap that attributes to the differences in

the marginal returns of the covariates. For example, if we estimate the model for different

racial groups separately, the estimated marginal effect of paternal time input θ1 is different

between the blacks and whites. The disparity of black-white test score can result from two

sources: 1) the difference in the average paternal time input between the blacks and whites,

and 2) the difference in the marginal return of paternal time input on different children’s

test scores. We thereby can assume that the blacks have the same amount of paternal time

input as the whites to see how much of the test score gap can be closed or can consider

a counterfactual when the marginal effect of paternal time input for the blacks is equal to

the whites in order to calculate the percentage changes of the total test score differences.

Furthermore, the interaction term can be obtained to account for the fact that two differences

exist simultaneously between the blacks and whites. In our example, they can be written as:

E = [tpW − tpB]θB1 , C = tp

B[ ˆθW1 − θB1 ], I = [tpW − tpB][ ˆθW1 − θB1 ].

3.3.2 Basic Results for the Benchmark Decomposition

The results of the benchmark decomposition are presented in the Table 3.4 (a) and Table 3.4

(b) for the married-household sample and the single-mother household sample, respectively.

In general, the variables representing children’s ability account for a substantial proportion

of the overall differences in test scores across the blacks and whites, especially in the single-

mother households. For the reading test, these ability measures explain almost half of the

gap, consistent with many previous empirical results13. The contribution of children’s abil-

13For example, Todd and Wolpin (2004).


ity, however, is much smaller for the math test compared with the reading test. Though it

is slightly counter-intuitive, a possible reason is that the measures of children’s ability are

more correlated with their reading skill development. For example, mother’s passage com-

prehension scores tend to affect children’s reading skill more than their math skill. Since the

PSID-CDS does not provide the AFQT scores used by much previous literature to measure

ability, it is reasonable that our measures generate slightly different results in this regard. In

addition, the partial effect of increasing children’s ability usually may understate the total

effect because of two factors: 1) high-ability children may receive more or less parental time

inputs than those low-ability ones, and 2) the marginal effects of inputs may be higher for

the high-ability children compared with others. We, however, only take care of the first

scenario by estimating the time input demand functions simultaneously and thereby may

still underestimate the total effects of children’s ability.

Approximately, 54.22% (or 30.53%) of the black-white reading (or math) test score gap can

be closed if black mothers spend the same amount of time as the average white mothers do

either actively or passively for the married-household sample. In other words, an increase of

7 hours per week of total maternal time for a typical black mother can raise their children’s

average reading (math) test scores by more than 7 (around 4) standardized points, hold-

ing everything else constant at the observed levels. Compared with previous literature, our

results indicate a relatively more important role of maternal time input in the black-white

test score gaps for children living with their parents. For example, the empirical results are

much larger than that in Todd and Wolpin (2006), which have concluded about 10%-15%

of the gap closed by home inputs. Their results are based on two different empirical model

settings: 1) it does not separate the household type to control the differences in optimal

choices of home inputs, and 2) the measure for the home inputs “HOME” combines mon-

etary and time inputs. If the maternal time inputs cannot explain as much of the gap for

the single-mother households as that for the married ones, a pooling effect reported in the

Todd and Wolpin (2004) does not contradict ours. Not surprisingly, in our single-mother

household sample, the test score gaps between black and white groups increase by equalizing

maternal time input because black mothers spend more time on average with their children

compared with the whites. Therefore, our empirical implications can be seen as a detailed


analysis supplemented to the previous literature.

The math test score gap shrinks by around 4.75% when black fathers spend 2.7 hours more

per week on average with their children. Considering that a paternal time increase of 24

minutes a day is easily attainable, policies that encourage black fathers to spend more time

with children may be very effective to improve the welfare of black children. In contrast to

the large effects of parental time inputs for the married households, the differences of various

incomes only explain a very small fraction of black-white test score, approximately 7-8%

for the married households and 1-3% for the single-mother households. Given the fact that

there is a huge disparity of household incomes, especially the non-labor income, between

the blacks and the whites (for example, the gap of average total household incomes, both

labor and non-labor, for married households is over 160 thousand dollars), policies aiming

to eliminate the test score gap through the channel of incomes may not be effective. The

school effects are vague, accounting for negligible proportion for the differences, consistent

with previous cross-section analyses on the effects of school inputs on children’s cognitive

development.

The benchmark decompositions of both test score gaps lead to three basic conclusions. First,

the differences in children’s ability14 account for a substantial proportion of black-white test

score gaps. Almost half of the reading test score gap can be closed by equalizing the level

of black children’s ability at the average white ones. Second, maternal time inputs explain

a significant fraction of the test score gap for the children living with bio-parents. It con-

tributes more than children’s ability for both reading and math skill development. The

reading test gap, for example, can be narrowed by more than 50% when the black children

enjoy the same amount of maternal time as the whites do. Third, household incomes and

school inputs can explain little of the disparity in black-white test scores.

We replicate the analyses by gender and by age groups to test the sensitivity of these results.

14Following the measures in chapter 2, we use mother’s PC test scores, education levels, and children’s DS

test scores to measure children’s ability.


Meanwhile, another model specification in which parent’s active time input and passive time

input enter the production functions individually is also estimated to help further analyze

the contributions of different time inputs to the test score gaps. We present the results in

Table 3.5 (a) to Table 3.7 (b).

Tables 3.5 (a) and 3.5 (b) display the decomposition results based on the averages for the

blacks and the whites separately by gender. We present the actual average test score gaps

as well as the predicted ones according to the baseline model specification. The predictions

for the girls are not as precise as the boys’, and in general, our model tends to over-estimate

the gaps for the girls but under-estimate for the boys. Both parents’ time inputs have larger

impacts on girls’ test score gaps, in both married and single-mother households. In partic-

ular, the difference in paternal time input for the girls accounts for around 13.58% of the

total actual math test score gap between the blacks and the whites. Given the fact that

girls’ within-racial-group test score gaps between two household types are non-negligible

on average, the black girls living with their single-mothers may especially suffer from the

loss of paternal company. Policies that encourage fathers to increase the frequency of their

visits to children after divorce may have significant effects on the black girls’ cognitive de-

velopment, and consequently on the next black generations. Another interesting finding is

that, though the black single-mothers on average spend more time with their children, only

black girls benefit from it. White single-mothers spend more time with their boys compared

with the blacks, and therefore, a small fraction of the black-white test score gaps can be ex-

plained by the maternal time inputs. The contribution, however, is not sizable in magnitude.

The income and school inputs tend to have larger effects for the samples separated by gen-

der. In general, they explain a larger fraction of the differences in girls test scores than those

of boys. It may result from that our model over-predict the gaps for the girls. Children’s

ability as we have discussed above accounts for almost half of the disparity, except for the

math test score gaps for the married households.

Tables 3.6 (a) and 3.6 (b) present the decomposition results for the black-white test score


gaps by two age groups. Unfortunately, the model does not predict test score gaps for chil-

dren aged between 3-6 years old very precisely, especially the reading score. Our estimated

test scores tend to exaggerate the disparity of test scores between the blacks and whites.

Though the marginal returns of the inputs on small children’s cognitive production may

behave differently from the older ones, our estimates based on the sub-samples are likely to

be less credible because the sample size for children less than 7 years old is extremely small

in our sample. For those aged 7 years old and above, the decompositions report very similar

results as our baseline model does.

Tables 3.7 (a) and 3.7 (b) display the results for the model specification where the active

and passive parental time inputs enter the cognitive production separately. For the married

households, the differences in active maternal time explain a large fraction of the black-

white test score gaps. An increase in maternal time, especially a direct participation in

children’s daily activities, closes the disparity of the reading (math) test scores by 62.98%

(24.60%). Since children’s cognitive achievement is a great predictor for their future labor

market outcomes, policies that concentrate on the black-white labor market disparities (for

example, the wage gap) need to pay much attention to the tradeoff between female’s market

work and maternal time allocations. Fathers’ direct time investment on their children are

also very important. In our sample, we only observe an average gap of 2.212 hours per week

in active paternal time input between the blacks and the whites. The difference, however,

accounts for 17.43% of the total disparity in black-white math test scores. We thereby believe

that policies trying to equalize the active paternal time input at the average level of white

children may be a very efficient way to close the test gaps, and moreover, improve the total

welfare of the blacks in general.

3.3.3 Basic Results for Another Type of Decomposition

We conduct another type of decomposition after estimating the benchmark model separately

for the blacks and whites. The results are presented in Tables 3.8 (a) and 3.8 (b). They are

informative to some extent though the relatively small size of the sample may hinder the

efficiency and consistency of the parameter estimates. The production function estimates


are shown in section 3.6 appendix A Table A-3 and Table A-4.

For the married households, equalizing the maternal time at the average level of the whites

still accounts for a sizable proportion. It explains about 17.96% of the total gap for the

reading test and 6.08% for the math test, indicating the importance of the maternal evolve-

ment in children’s daily activities. The marginal effects of maternal time input, however, are

much higher for the whites compared with the blacks in married households. We observe

a substantial improvement on black children’s test scores if black mothers’ time input has

the same returns as the whites do. The marginal returns of the maternal time input depend

on both children’s heterogeneity and parents’ heterogeneity. In our samples, white children

have superior health status at birth and less severe problem behaviors which may increase

the marginal returns of maternal time input, and furthermore, white females on average have

higher education levels leading to higher time quality in educating their children. Policies

that are designed to increase the overall test scores of the blacks may need to consider the

differences in the marginal returns of the maternal time input. For the single-mother house-

holds, even though the average black single-mothers spend more time with their children,

white single-mothers’ still have higher returns on their time investment to children’s cognitive

development. The effect is extremely outstanding on children’s reading skill development.

That is, if the black single-mothers have the same returns of maternal time input, the test

score gap can be closed.

The differences in the levels of household incomes account for a large fraction of the black-

white test score gaps. Since on average the black families have much lower household incomes

than the whites do, the returns of the household incomes for the black families are higher.

The finding is consistent with the law of diminishing marginal returns. From the policy

perspective, it may imply that some initial monetary investment to children’s cognitive de-

velopment could be very profitable, and a direct money transfer to those low-income black

families may increase the test scores significantly. Children’s ability as usual plays an im-

portant role in the black-white test score gaps.


3.3.4 A Closer Look at Home Inputs

Results from the standard decomposition of children’s cognitive production functions show

that the parental time inputs explain a large proportion of the black-white test score gaps.

For children’s reading skill development, white married mothers’ higher time inputs account

for more than half of the disparity. Even for the single-mother households, those black moth-

ers spend more time with their children, but the lower marginal returns of maternal time

still widen the differences in children’s test scores. In this part, we further investigate the

differences in parental time inputs between the blacks and the whites, in order to see the

sources that account for the time input gaps based on our simultaneous equations model

estimation. A standard decomposition is employed for the demand functions of parental

time inputs, and the results are displayed in Tables 3.10 (a) to 3.10 (d).

In order to understand the patterns of parental time inputs across different racial groups, we

present the descriptive statistics of the time inputs by children’s age groups and gender sep-

arately for the blacks and whites first in Tables 3.9 (a) and 3.9 (b). In the married-household

sample, we find that on average white mothers spend more active/passive time with their

children by about 3.382/3.530 hours per week compared with the blacks. The difference in

the active/passive maternal time input has narrowed as children reach 11 years old and then

somewhat widened when their children transit to adolescent. The differences in the paternal

time inputs, however, do not display a similar pattern. The average active paternal time in

the white group is 2.213 hours per week higher than that in the blacks, but the disparity has

monotonically widened as children get older. For those who aged above 15 years old, white

fathers spend 4.351 hours per week more with their children compared with black fathers.

The differences in the passive paternal time are even negative between the blacks and the

whites when children are below 12 years old, but again increase to around 6 hours per week

for the adolescents. Since the empirical results that the black-white test score gaps increase

after three grades has been commonly noticed, our findings about the interesting paternal

time input pattern may be another available explanation towards the issue. Parental time

inputs may have different contributions during children’s different development phrases. For

example, fathers’ company can be of paramount importance after certain age not only to


children’s cognitive development but also non-cognitive development.

White married mothers spend 3.9 hours more of active and passive time with boys per week

, and white fathers spend 2 hours more of active time with their boys per week . Com-

pared with white boys, black boys have slightly more passive paternal time. White girls

consistently spend more time with their parents, but they enjoy 3.126 hours more per week

of passive maternal time and 3.515 hours more per week of active paternal time, compared

with black girls. This interesting pattern may indicate that black fathers tend to substitute

their direct time input with the passive one and so do white mothers. Since in the previous

analyses, our results support that girls suffer more from the missing-father effect, we may

further argue that the active paternal time can be very productive during girls’ cognitive

development. Policies that concern the welfare of our next generation probably need to

thoroughly understand the girls’ development because in general mothers play a key role in

raising the next generation.

In the single-mother households, white mothers spend significantly less time with their chil-

dren aged below 6 years old. Since our previous results report a higher labor income from the

single mothers in the white group, we may conclude that in general white mothers may work

more in the market to compensate their less maternal time input with goods investment. It

may result from the relatively higher opportunity cost of time for the white single mothers

who on average are more educated. White girls suffer most from the missing fathers. Their

mothers spend less time actively participating in their daily activities by about 2.671 hours

per week.

Based on our standard decomposition results, the differences in the household incomes are

the major sources that have widen the gaps of parental time inputs between the blacks and

whites. The effects, however, are opposite for the maternal time versus the paternal time

in the married households. In general, equalizing the income at the levels of the average

whites can close the gap of maternal time input by 37.08%, but widen the gap of paternal

time input by more than 2 folds. In the decomposition of the black-white test score gaps,


the direct contributions from income effects to close the gap are very small. For example, in

our benchmark case, the differences in the household incomes can explain about 8.21% and

6.99% for the reading and math test score gaps respectively in married households, and 1.05%

and 3.31% for the reading and math test score gaps in single-mother households. However,

after examining the black-white parental time input gaps, we obtain the substantial indirect

contributions from income effects to close the test score gap. The results imply that we may

need to reconsider the role of household incomes in black-white test score gaps.

For married household, since 54.22% reading score gap can be closed by equalizing the mater-

nal time input and income effects can close the maternal time gap by about 37.08%, income

effects actually indirectly accounts for about 20.10% reading test score gap through the chan-

nel of maternal time input. Similarly, the differences in incomes explain 2.4% the reading

test score gap through the channel of paternal time input. Therefore, incomes account for

30.71% of the gap directly and indirectly, which is a large proportion of the disparity in

black-white reading test score gap, and 8.46% of the math test score gap. For single-mother

households, they are 21.30% and 23.46% of reading and math test score gaps. An extensive

amount of empirical literature has shown that current incomes usually do not have significant

effects on household resource allocations, whereas the permanent incomes do. Our results,

from another perspective, support this result in that people usually smooth their incomes

to finance some long-run investment, such as children’s goods, time, and school inputs and

current incomes thereby play an important role mainly through the indirect channel of home

inputs.

Fathers’ education level also accounts for a large fraction of the paternal time input gap.

Almost half of the paternal time difference can be closed if an average black father has the

same education level as the white one (30.84% of the active paternal time gap and 111.49%

of the passive paternal time gap). School inputs explain about 5% for both time gaps, but

they are almost all for passive time gaps. It may imply that black parents compensate their

children for a relatively lower school quality by spending more time around them.


3.4. Conclusion

This chapter studies the role of home inputs in black-white test score gaps using the standard

Blinder-Oaxaca mean decomposition method. The children’s cognitive production function

is estimated jointly with the demand function of parental time input to take into account the

endogenous home inputs caused by non-random household resource allocation. Children’s

detailed time diaries collected by the PSID-CDS surveys are employed to construct direct

measures of parental time inputs, and we evaluate the individual contributions from each

type of time input towards the differences in the black-white test scores. Compared with the

previous literature that mostly relies on the OLS regression, our empirical model improves

the parameter estimates by using an iterated three stage least squares and provides more

precise estimates for the marginal effects of parental time inputs and household incomes.

Therefore, the decomposition results based on the system of simultaneous equations are re-

liable and informative.

Our empirical results indicate that maternal time input, especially the active maternal time

input, explains a very large proportion of the black-white gaps for both reading and math

test scores in married-household sample. Almost half of the reading test score gap can be

closed by equalizing the black children’s maternal time input at the average level of the

whites. The paternal time input, especially the active one, accounts for about 5% of the

math test score gap, but its explanation power increases to 13.58% for the girls. The findings

imply that black girls’ cognitive development can be improved substantially if their fathers

spend more time with them. Both income and school inputs have small direct effects on

closing the test score gap between the two racial groups. Children’s ability accounts for

about half of the differences in test scores.

From the policy perspective, our decomposition results are very informative. Any labor

market regulations that may affect the household optimal time allocations should be care-

ful because of the tradeoffs among parental time, labor supply, and leisure. Policies that

encourage mothers to spend more time with their children, especially for the married black

mothers and single white mothers, may be effective towards the closure of black-white test


score gaps. Moreover, fathers’ company is important and has substantial impacts on girls’

cognitive development. Since girls who live with single mothers suffer from the missing fa-

thers, some incentives to increase the frequency of divorced fathers’ visits may improve girls’

welfare and have profound and lasting effects on intergenerational persistence.

Our studies on the black-white test score gaps are based on the mean decomposition, and

therefore, a further research can extend to understand the disparity of test scores among

racial groups on different quantiles. The simultaneous equations model can be estimated

by a quantile regression based on some recent development in numerical optimization algo-

rithm. The assumption that the behaviors of children’s cognitive production and parents’

time input decision are stable at each quantile can be relaxed in this case. We leave the

empirical investigations into the parents and children at lower or higher quantiles for future

research.

3.5. Tables


Full1 Sample

N = 4155


Full2 Sample 111.100 110.064 98.046 96.514

N = 3290 (16.246) (15.997) (14.666) (14.519)

Selected Sample 111.831 111.898 98.687 96.481

(15.773) (16.546) (14.056) (14.256)

Obs 1161 128 404 600

By Age Groups

3 - 6 107.304 104.175 97.612 94.358

(15.320) (16.173) (15.999) (15.595)

7 - 9 115.414 112.111 101.760 102.323

(16.613) (14.480) (13.787) (15.493)

10 - 12 114.240 118.823 99.976 95.138

(14.324) (18.178) (13.008) (14.836)

13 - 15 110.425 113.669 94.053 93.233

(15.825) (14.719) (10.793) (8.944)

16+ 106.977 102.859 98.054 96.196

(14.423) (15.299) (15.984) (10.968)

By Gender

Boys 113.580 115.647 97.251 97.380

(16.539) (13.773) (14.144) (13.765)

Girls 110.204 105.773 101.371 95.481

(14.855) (18.856) (13.495) (14.723)

● All the samples are weighted using PSID-CDS PCG/Child level weights.

● All standard deviations are in the brackets.

109.845 96.696

(16.457) (14.944)

TABLE 3.1(a) DESCRIPTIVE STATISTICS

PSID-CDS Standardized Applied Problem (Math) Test Scores

White Black


Full1 Sample

N = 4172


Full2 Sample 110.073 106.246 97.539 95.386

N = 3306 (17.596) (20.188) (19.128) (15.396)

Selected Sample 110.639 108.956 97.637 95.172

(17.954) (22.912) (20.876) (15.372)

Obs 1161 128 404 600

By Age Groups

3 - 6 104.251 95.071 101.369 93.216

(14.887) (13.726) (17.938) (12.712)

7 - 9 113.181 107.444 99.432 101.587

(17.187) (20.141) (22.092) (16.588)

10 - 12 111.842 112.459 101.236 93.662

(17.627) (22.434) (18.257) (16.066)

13 - 15 111.902 120.835 87.046 91.892

(19.971) (29.842) (21.259) (11.719)

16+ 111.147 102.477 96.773 93.840

(20.124) (8.363) (21.451) (15.684)

By Gender

Boys 108.486 112.572 93.592 94.985

(17.627) (24.705) (22.713) (14.851)

Girls 112.643 103.048 105.199 95.379

(18.039) (18.399) (14.098) (15.933)

● All the samples are weighted using PSID-CDS PCG/Child level weights.


108.738 96.669

(18.126) (16.837)

TABLE 3.1(b) DESCRIPTIVE STATISTICS

PSID-CDS Standardized Letter-Word Identification (Reading) Test Scores

White Black


WHITE BLACK WHITE BLACK

Reading 110.639 97.637 Child Boy 0.482 0.652

(17.954) (20.897) (0.500) (0.477)

Math 111.831 98.687 No. siblings 1.416 1.654

(40.313) (14.070) (0.913) (1.382)

Age (month) 127.861 128.840

Maternal Time (44.235) (43.553)

Total 40.313 33.400 IQ 13.767 12.545

(18.987) (16.536) (5.106) (4.448)

Active 19.939 16.557 BPI 7.500 7.588

(12.625) (11.069) (5.567) (6.394)

Passive 20.374 16.844 Health at birth (worse) 0.076 0.065

(14.086) (12.741) (0.265) (0.246)

Paternal Time Birth weight (ounce) 121.411 112.395

Total 27.992 25.277 (19.200) (22.928)

(17.027) (16.067) Mother Age (year) 39.200 39.606

Active 14.243 12.031 (6.124) (7.069)

(11.109) (11.161) Education 14.038 13.077

Passive 13.749 13.246 (1.953) (1.604)

(12.386) (10.939) PC Test Scores 33.936 28.593

Income (4.087) (5.634)

Mother's Labor Income 18.010 24.284 Father Age (year) 41.373 42.748

(21.855) (20.883) (6.607) (8.327)

Father's Labor Income 59.862 26.849 Education 14.165 13.136

(87.851) (22.245) (2.164) (2.154)

Non-Labor Income 199.674 51.825

(967.950) (238.271) MSA 0.719 0.800

School (0.449) (0.400)

Pupil-Teacher Ratio 16.775 16.909 Northeast 0.201 0.074

(2.733) (2.951) (0.401) (0.262)

Average Teacher Salaries 50.999 48.643 Midwest 0.294 0.215

(7.563) (7.765) (0.456) (0.411)

South 0.252 0.559

(0.434) (0.497)


● Both samples are weighted using PSID-CDS PCG/Child level weights.



TABLE 3.2(a): Descriptive Statistics - Married Households

Test Scores Others

Inputs


WHITE BLACK WHITE BLACK

Reading 108.956 95.172 Child Boy 0.620 0.527

(22.912) (15.383) (0.487) (0.500)

Math 111.898 96.481 No. siblings 1.030 1.627

(16.546) (14.267) (0.856) (1.274)

Age (month) 128.605 132.954

Maternal Time (44.419) (42.721)

Total 34.983 35.391 IQ 14.011 12.609

(15.720) (19.347) (5.807) (4.406)

Active 16.325 16.985 BPI 10.445 8.167

(10.616) (15.796) (5.602) (6.550)

Passive 18.659 18.406 Health at birth (worse) 0.121 0.079

(11.532) (13.795) (0.327) (0.270)

Income Birth weight (ounce) 120.106 111.949

Mother's Labor Income 17.983 13.028 (23.362) (23.334)

(14.716) (13.209) Mother Age (year) 37.209 36.602

Non-Labor Income 83.697 6.503 (7.472) (8.040)

(566.078) (25.873) Education 12.802 12.211

School (1.577) (2.250)

Pupil-Teacher Ratio 17.423 16.360 PC Test Scores 33.487 27.284

(2.645) (2.277) (4.262) (4.572)

Average Teacher Salaries 50.665 49.586

(7.947) (8.995) MSA 0.676 0.758

(0.470) (0.429)

Northeast 0.111 0.202

(0.315) (0.402)

Midwest 0.302 0.188

(0.461) (0.391)

South 0.394 0.563

(0.491) (0.496)



Inputs


● All samples are weighted using PSID-CDS PCG/Child level weights.

TABLE 3.2(b): Descriptive Statistics - Single-Mother Households

Test Scores Others


(1) (2) (3) (4) (5) (6) (7) (8) (9)

Black -12.047 -11.978 -11.815 -12.329 -6.996 -6.951 -13.808 -7.716 -7.662

(0.788)*** (0.798)*** (1.276)*** (2.367)*** (1.830)*** (1.821)*** (2.646)*** (2.163)*** (2.162)***

Age 0.218 0.272 0.316 -0.111 -0.115 0.302 -0.283 -0.280

(0.041)*** (0.046)*** (0.063)*** (0.066)* (0.067)* (0.108)*** (0.120)** (0.121)**

Age^2 -0.001 -0.001 -0.001 0.000 0.000 -0.001 0.000 0.000

(0.0002)*** (0.0002)*** (0.0002)*** (0.0002) (0.0002) (0.0004)*** (0.0004) (0.0004)

Gender -2.568 -2.429 -4.516 -3.071 -3.061 3.352 -0.125 -0.034

(0.741)*** (0.814)*** (1.062)*** (0.898)*** (0.894)*** (2.030) (1.695) (1.719)

Married HH ( =1 ) 3.124

(1.327)***

Home Inputs X X X X

School Inputs X X

Other Controls X X X X

No. of Obs 4172 4172 3306 1565 1565 1565 728 728 728

R20.0704 0.0846 0.1074 0.0710 0.3407 0.3421 0.1387 0.4039 0.4056

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Black -13.220 -13.563 -13.908 -13.776 -9.778 -9.415 -15.026 -10.908 -10.927

(0.703)*** (0.700)*** (0.866)*** (1.221)*** (1.258)*** (1.272)*** (1.832)*** (2.048)*** (2.057)***

Age 0.356 0.379 0.452 0.125 0.123 0.311 0.051 0.050

(0.040)*** (0.042)*** (0.056)*** (0.060)** (0.060)** (0.113) (0.103) (0.101)

Age^2 -0.001 -0.001 -0.002 -0.001 -0.001 -0.001 -0.001 -0.001

(0.0001)*** (0.0002)*** (0.0002)*** (0.0002)*** (0.0002)*** (0.0004)*** (0.0004) (0.0003)

Gender 2.950 3.204 2.854 3.934 3.825 4.968 2.161 1.978

(0.652)*** (0.702)*** (0.901)*** (0.798)*** (0.794)*** (1.749)*** (1.599) (1.570)

Married HH ( =1 ) 1.181

(0.911)

Home Inputs X X X X

School Inputs X X

Other Controls X X X X

No. of Obs 4155 4155 3290 1565 1565 1565 728 728 728

R20.1060 0.1418 0.1556 0.1077 0.3172 0.3202 0.2440 0.4114 0.4205

TABLE 3.3 (a)

Determinants of Test Scores: Reading Test

Full SampleSelected Sample

Married HH Single-M HH

TABLE 3.3 (b)

Determinants of Test Scores: Math Test

Full SampleSelected Sample

Married HH Single-M HH


Difference in actual test scores 110.639 97.637 13.003 100 111.831 98.687 13.144 100

Difference attributable to:

Time Inputs

Maternal 7.050 54.22 4.014 30.53

Paternal -0.151 -1.16 0.625 4.75

Incomes 1.068 8.21 0.919 6.99

School Inputs 0.363 2.79 -0.123 -0.93

Abilities

Child's IQ 2.781 21.39 1.757 13.37

Mother's Edu. 1.287 9.90 -0.392 -2.99

Mother's PC Score 0.981 7.54 -0.074 -0.56



Time Inputs

Maternal -0.204 -1.48 -0.215 -1.40

Incomes 0.144 1.05 0.510 3.31

School Inputs -0.093 -0.67 -0.283 -1.83

Abilities

Child's IQ 2.806 20.36 1.950 12.65

Mother's Edu. 0.517 3.75 0.769 4.99

Mother's PC Score 4.828 35.03 1.950 12.65

TABLE 3.4 (a): Decomposition of the White-Black Test Gaps, Married Households

PSID-CDS WJ-R Applied Problem (Math) and Letter-Word Identification (Reading) Tests

PSID-CDS WJ-R Reading Test PSID-CDS WJ-R Math Test

White BlackChange, W-B

% of Total Change


% of Total Change

TABLE 3.4 (b): Decomposition of the White-Black Test Gaps, Single-Mother Households


Change, W-B

% of Total Change


% of Total Change

White Black



Girls Difference in actual test scores 112.643 105.199 7.445 100 110.204 101.371 8.832 100

Difference in predicted test scores 113.186 103.199 9.987 134.15 110.793 97.669 13.124 148.59


Time Inputs

Maternal 4.956 66.58 2.822 31.95

Paternal -0.289 -3.88 1.200 13.58

Incomes 0.236 3.17 2.546 28.83

School Inputs 0.274 3.68 -0.776 -8.79

Abilities

Child's IQ 2.609 35.05 1.649 18.67

Mother's Edu. 1.656 22.24 -0.505 -5.71

Mother's PC Score 1.083 14.55 -0.081 -0.92

Boys Difference in actual test scores 108.486 93.592 14.894 100 113.580 97.251 16.329 100



Time Inputs

Maternal 7.889 52.97 4.491 27.503

Paternal -0.089 -0.60 0.369 2.260

Incomes 1.450 9.74 0.056 0.343

School Inputs 0.419 2.81 0.262 1.604

Abilities

Child's IQ 2.250 15.11 1.422 8.707

Mother's Edu. 1.103 7.41 -0.336 -2.059

Mother's PC Score 0.932 6.26 -0.070 -0.429

Change, W-B

% of Total Change

TABLE 3.5 (a): Decomposition of the White-Black Test Gaps by Gender, Married Households




% of Total Change

White Black


Girls Difference in actual test scores 103.048 95.379 7.669 100 105.773 95.481 10.292 100



Time Inputs

Maternal -0.845 -11.01 -0.893 -8.67

Incomes -1.884 -24.57 -0.819 -7.96

School Inputs -0.652 -8.50 -0.681 -6.62

Abilities

Child's IQ 2.059 26.85 1.431 13.90

Mother's Edu. 0.902 11.76 1.342 13.04

Mother's PC Score 3.449 44.97 3.235 31.43

Boys Difference in actual test scores 112.572 94.985 17.587 100 115.647 97.380 18.267 100



Time Inputs

Maternal 0.329 1.87 0.347 1.90

Incomes 1.581 8.99 1.430 7.83

School Inputs 0.365 2.08 0.042 0.23

Abilities

Child's IQ 3.056 17.38 2.123 11.62

Mother's Edu. 0.210 1.19 0.313 1.71

Mother's PC Score 5.597 31.83 5.249 28.74

TABLE 3.5 (b): Decomposition of the White-Black Test Gaps by Gender, Single-Mother Households



Change, W-B

% of Total Change


% of Total Change

White Black


3-6 yrs Difference in actual test scores 104.414 99.792 4.622 100 107.462 96.829 10.633 100



Time Inputs

Maternal 13.434 290.66 7.648 71.92

Paternal -0.133 -2.87 0.551 5.18

Incomes 0.177 3.83 2.559 24.07

School Inputs -0.026 -0.56 -1.005 -9.45

Abilities

Child's IQ -0.916 -19.82 -0.579 -5.44

Mother's Edu. 1.131 24.47 -0.345 -3.24

Mother's PC Score 0.971 21.02 -0.073 -0.69

7+ yrs Difference in actual test scores 112.135 97.156 14.979 100 112.881 99.101 13.779 100



Time Inputs

Maternal 5.431 36.26 3.092 22.44

Paternal -0.153 -1.02 0.633 4.59

Incomes 1.259 8.40 0.582 4.22

School Inputs 0.455 3.04 0.078 0.57

Abilities

Child's IQ 3.842 25.65 2.428 17.62

Mother's Edu. 1.323 8.83 -0.403 -2.93

Mother's PC Score 0.983 6.56 -0.074 -0.54

Change, W-B

% of Total Change

TABLE 3.6 (a): Decomposition of the White-Black Test Gaps by Age, Married Households




% of Total Change

White Black


3-6 yrs Difference in actual test scores 95.0709 93.4507 1.620 100 104.175 93.2558 10.919 100



Time Inputs

Maternal -1.959 -120.94 -2.071 -18.97

Incomes 6.377 393.60 5.272 48.28

School Inputs -0.834 -51.48 -0.705 -6.46

Abilities

Child's IQ -2.122 -130.99 -1.474 -13.50

Mother's Edu. 0.343 21.17 0.510 4.67

Mother's PC Score 4.644 286.63 4.355 39.89

7+ yrs Difference in actual test scores 111.865 95.4359 16.429 100 113.516 96.9765 16.540 100



Time Inputs

Maternal -0.116 -0.71 -0.123 -0.74

Incomes -1.254 -7.63 -0.537 -3.25

School Inputs 0.009 0.05 -0.219 -1.32

Abilities

Child's IQ 4.480 27.27 3.112 18.82

Mother's Edu. 0.556 3.38 0.827 5.00

Mother's PC Score 4.871 29.65 4.568 27.62

TABLE 3.6 (b): Decomposition of the White-Black Test Gaps by Age, Single-Mother Households



% of Total Change

PSID-CDS WJ-R Reading Test


% of Total Change

PSID-CDS WJ-R Math Test




Time Inputs

Maternal Act. 8.190 62.98 3.233 24.60

Maternal Pas. -5.405 -41.56 -0.974 -7.41

Paternal Act. 0.094 0.72 2.291 17.43

Paternal Pas. 0.473 3.64 -0.198 -1.51

Incomes 7.531 57.92 5.294 40.27

School Inputs 1.474 11.34 0.615 4.68

Abilities

Child's IQ 3.628 27.90 2.540 19.33

Mother's Edu. 1.006 7.74 -0.493 -3.75

Mother's PC Score 6.090 46.84 2.613 19.88



Time Inputs

Maternal Act. -0.449 -3.26 -0.124 -0.80

Maternal Pas. -0.126 -0.92 -0.005 -0.03

Incomes -2.359 -17.11 -1.544 -10.01

School Inputs -0.097 -0.70 -0.285 -1.85

Abilities

Child's IQ 2.872 20.84 2.013 13.06

Mother's Edu. 1.014 7.36 1.177 7.64

Mother's PC Score 5.736 41.62 5.273 34.20

TABLE 3.7 (b): Decomposition of the White-Black Test Gaps, Active & Passive Time Separated

TABLE 3.7 (a): Decomposition of the White-Black Test Gaps, Active & Passive Time Separated

PSID-CDS WJ-R Applied Problem (Math) and Letter-Word Identification (Reading) Tests, Married Households



% of Total Change


% of Total Change

% of Total Change

PSID-CDS WJ-R Applied Problem (Math) and Letter-Word Identification (Reading) Tests, Single-Mother Households



% of Total Change





Time Inputs

Maternal

Levels 2.335 17.96 0.799 6.08

Returns 9.530 73.29 12.621 96.02

Paternal

Levels -0.463 -3.56 -0.095 -0.72

Returns 18.724 144.00 -1.475 -11.22

Incomes

Levels 7.969 61.29 3.217 24.47

Returns -1.828 -14.06 3.739 28.44

Abilities

Child's IQ

Levels 1.874 14.41 1.526 11.61

Returns 5.611 43.15 1.116 8.49

Mother's Edu.

Levels 2.626 20.19 0.620 4.72

Returns -14.542 -111.84 -2.961 -22.53

Mother's PC Score

Levels 2.239 17.22 1.590 12.09

Returns -6.523 -50.17 -9.496 -72.25

Total

Levels 16.160 124.28 7.853 59.74

Returns 0.290 2.23 7.990 60.79

TABLE 3.8 (a): Decomposition of the White-Black Test Gaps, Married Households




% of Total Change


% of Total Change




Time Inputs

Maternal

Levels -0.095 -0.69 -0.140 -0.91

Returns 20.866 151.38 2.571 16.67

Incomes

Levels 26.587 192.88 36.026 233.67

Returns -0.873 -6.33 -3.866 -25.08

Abilities

Child's IQ

Levels 1.159 8.41 1.153 7.48

Returns 24.132 175.07 1.337 8.67

Mother's Edu.

Levels 0.319 2.32 0.242 1.57

Returns 9.907 71.88 8.006 51.93

Mother's PC Score

Levels 4.302 31.21 2.094 13.58

Returns -13.345 -96.82 15.492 100.49

Total

Levels 30.545 221.59 38.815 251.77

Returns 11.158 80.95 9.936 64.450

Change, W-B

% of Total Change


% of Total Change

White Black

TABLE 3.8 (b): Decomposition of the White-Black Test Gaps, Single-Mother Households




Whi

teB

lack

Diff

.W

hite

B

lack

Diff

.W

hite

Bla

ckD

iff.

Whi

te

Bla

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iff.

Tot

al19

.939

16.5

573.

382

20.3

7416

.844

3.53

014

.243

12.0

312.

213

13.7

4913

.246

0.50

3(1

2.62

5)(1

1.05

8)(1

4.08

6)(1

2.72

8)(1

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(10.

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By

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r29

.952

22.8

177.

134

25.4

1620

.180

5.23

517

.393

16.8

530.

540

13.9

9413

.670

0.32

4(1

4.63

0)(1

1.67

9)(1

5.43

2)

(14.

525)

(12.

205)

(15.

979)

(1

1.88

2)(1

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6)

6 -8

yr

21.9

3118

.139

3.79

220

.708

16.8

893.

818

14.5

9912

.809

1.78

912

.109

13.5

18-1

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(9.9

97)

(10.

749)

(12.

590)

(10.

659)

(10.

047)

(9.5

99)

(9.7

22)

(9.6

97)

9 - 1

1 yr

19.8

8618

.075

1.81

118

.911

19.5

32-0

.620

14.4

3212

.200

2.23

212

.421

13.5

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.153

(12.

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(10.

732)

(12.

427)

(13.

697)

(1

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(11.

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(12.

378)

12 -

15 y

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13.9

662.

226

20.1

6614

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6.02

713

.617

12.0

631.

555

14.9

4613

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1.26

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0.91

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(14.

728)

(10.

350)

(10.

779)

(11.

149)

(1

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)>

15 y

r14

.693

11.4

233.

270

19.0

3014

.654

4.37

611

.807

7.45

64.

351

17.1

3111

.139

5.99

2(1

3.37

7)(1

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7)

(16.

531)

(15.

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(1

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)(1

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y G

ende

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19.2

1815

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3.90

320

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16.7

133.

832

15.2

5013

.228

2.02

213

.189

13.6

07-0

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(12.

566)

(1

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9)(1

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0)

(12.

491)

(11.

524)

(11.

043)

(1

1.78

3)(1

0.11

1)G

irl

20.6

0918

.876

1.73

320

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17.0

873.

126

13.3

079.

792

3.51

514

.270

12.5

711.

699

(12.

645)

(1

1.00

4)(1

3.47

7)(1

3.16

5)(1

0.62

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1.00

9)(1

2.90

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2.28

9)

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): D

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atis

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ouse

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s

Mat

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al T

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Act

ive

Pass

ive

Act

ive

Pass

ive


TABLE 3.9 (b): Descriptive Statistics of Parental Time, Single-Mother Households

White Black Diff. White Black Diff.

Total 16.325 16.985 -0.660 18.659 18.406 0.253

(10.616) (15.785) (11.532) (13.785)

By Children's Age

3 - 5 yr 24.978 33.078 -8.100 18.282 19.545 -1.263

(10.852) (17.898) (11.709) (11.203)

6 -8 yr 18.921 18.019 0.902 20.446 20.808 -0.363

(10.410) (10.299) (13.494) (15.772)

9 - 11 yr 16.135 21.484 -5.349 18.990 14.818 4.172

(9.002) (17.235) (10.353) (11.400)

12 - 15 yr 13.987 12.277 1.710 17.773 20.810 -3.036

(11.770) (15.838) (8.777) (14.316)

> 15 yr 9.226 7.766 1.459 16.722 16.305 0.418

(6.874) (10.690) (14.613) (12.389)

By Gender

Boy 15.487 13.946 1.541 19.683 20.567 -0.884

(9.473) (11.082) (11.255) (14.121)

Girl 17.693 20.364 -2.671 16.986 16.003 0.983

(12.144) (19.186) (11.769) (12.990)

Maternal Time

Active Passive


TotalDifference in actual parental time inputs

40.313 33.400 6.912 100 27.992 25.277 2.715 100


Incomes 2.563 37.08 -5.629 -207.31

Parent's Edu. Level -0.552 -7.99 1.357 49.96

School Inputs 0.407 5.89 0.126 4.65

Child's Noncog. -0.228 -3.30 0.014 0.50

Child's Ability 0.838 12.12 -1.156 -42.56

ACTIVEDifference in actual parental time inputs

19.939 16.557 3.382 100 14.243 12.031 2.212 100


Incomes -0.067 -1.99 -3.960 -179.01


School Inputs -0.074 -2.18 -0.128 -5.79

Child's Noncog. -0.111 -3.28 -0.165 -7.47

Child's Ability -1.049 -31.01 -1.557 -70.37

PASSIVEDifference in actual parental time inputs

20.374 16.844 3.530 100 13.749 13.246 0.503 100


Incomes 3.470 98.30 -0.988 -196.48


School Inputs 0.500 14.15 0.270 53.77

Child's Noncog. -0.183 -5.19 0.122 24.24

Child's Ability 1.333 37.75 -0.012 -2.48

TABLE 3.10 (b): Decomposition of the White-Black Parental Time Differences (Active, Passive Separated), Married Households

TABLE 3.10 (a): Decomposition of the White-Black Parental Time Differences, Married Households

Change, W-B

% of Total Change

Maternal Time Paternal Time


% of Total Change

White Black


Change, W-B

% of Total Change


% of Total Change

White Black


TotalDifference in actual parental time inputs

34.983 35.391 -0.407 100


Incomes 5.862 -1439.32

Parent's Edu. Level -1.014 249.07

School Inputs 0.788 -193.48

Child's Noncog. 0.933 -229.02

Child's Ability -4.162 1021.88

ACTIVEDifference in actual parental time inputs

16.325 16.985 -0.660 100


Incomes -7.854 1189.98

Parent's Edu. Level -1.577 238.97

School Inputs 0.750 -113.71

Child's Noncog. 1.344 -203.62

Child's Ability -5.056 766.12

PASSIVEDifference in actual parental time inputs

18.659 18.406 0.253 100


Incomes -3.381 -1336.30

Parent's Edu. Level 1.004 396.73

School Inputs -0.046 -18.27

Child's Noncog. -0.637 -251.74

Child's Ability 1.925 760.80

TABLE 3.10 (d): Decomposition of the White-Black Parental Time Differences (Active, Passive Separated), Single-Mother Households

TABLE 3.10 (c): Decomposition of the White-Black Parental Time Differences, Single-Mother Households

Maternal Time


% of Total Change


% of Total Change

Maternal Time


3.6. Appendix A: Tables


Description Unit

Reading The standardized scores of the Letter-Word Identification Test Standard Point

Math The standardized scores of the Applied Problems Test Standard Point

Time (t i ) Maternal Time (MT)The weighted weekly total hours mother spends with the child either actively or passively

Hour/Week

Paternal Time (FT)The weighted weekly total hours father spends with the child either actively or passively

Hour/Week

Income (I c ) Mother's Labor Income Mother's annual total labor income Thousand in 1997 Dollar

Father's Labor Income Father's annual total labor income Thousand in 1997 Dollar

Non-Labor Income Household 5-year average annual total asset income Thousand in 1997 Dollar

School (X sch ) Pupil-Teacher Ratio The county-level pupil-teacher ratio Ratio

Average Teacher Salaries The state-level average teacher salaries Thousand in 1997 Dollar

Child Characteristics

White Dummy (=1 if the targeted child is white) 0/1

Black Dummy (=1 if the targeted child is black) 0/1

Boy Dummy (=1 if the targeted child is a boy) 0/1

No. siblings Number of siblings living with the targeted child Number

Age (month) The targeted child's age Month

IQ The WISC Digit Span Test scores for short term memory Point

Noncognitive Indices

BPI The targeted child's problem behavior index Index

Health at birth (worse)Dummy (=1 if the targeted child health is worse than average at birth)

0/1

Birth weight (ounce) The targeted child's birth weight in counce Ounce

Mother Characteristics

Age (year) Mother's age Year

Education Mother's education level Year

PC Test Scores Mother's Passage Comprehension raw test scores Point

Drink Dummy (=1 if mother has drink problem) 0/1

Smoke Dummy (=1 if mother is smoking) 0/1

Father Characteristics

Age (year) Father's age Year

Education Father's education level Year

Local Variables

MSA Dummy (=1 if the household lives in metropolitan area) 0/1

Northeast Dummy (=1 if the household lives in Northeast) 0/1

Midwest Dummy (=1 if the household lives in Midwest) 0/1

South Dummy (=1 if the household lives in South) 0/1

West Dummy (=1 if the household lives in West, reference group) 0/1

Div_ind The state-level divorce legislation index Index

Unemployment Rate The state-level annual unemployment rate %

Percentage of service The state percentage of employment in service sector 100%

Hourly wage rate The state-level median hourly wage rate 1997 Dollar

TABLE A3.1: Variable Description

Test Scores

Inputs

Others



Reading 108.762 100.532 Child White 0.815 0.339

( 18.491 ) ( 20.198 ) ( 0.389 ) ( 0.474 )

Math 109.044 102.638 Black 0.068 0.506

( 16.519 ) ( 18.666 ) ( 0.252 ) ( 0.500 )

Boy 0.488 0.520

Maternal Time ( 0.500 ) ( 0.500 )

Total 39.247 34.743 No. siblings 1.536 1.378

( 18.586 ) ( 17.787 ) ( 1.035 ) ( 1.181 )

Active 19.316 16.647 Age (month) 128.864 131.866

( 12.522 ) ( 14.067 ) ( 44.874 ) ( 44.029 )

Passive 19.931 18.096 IQ 13.494 13.146

( 13.843 ) ( 13.081 ) ( 5.058 ) ( 5.161 )

Paternal Time BPI 7.601 9.267

Total 27.502 - ( 5.657 ) ( 6.402 )

( 16.707 ) - Health at birth (worse) 0.079 0.117

Active 13.728 - ( 0.270 ) ( 0.321 )

( 10.963 ) - Birth weight (ounce) 120.510 116.234

Passive 13.774 - ( 19.622 ) ( 22.748 )

( 12.247 ) - Mother Age (year) 38.958 36.977

Income ( 6.221 ) ( 7.772 )

Mother's Labor Income 17.321 15.627 Education 13.420 12.257

( 21.012 ) ( 14.314 ) ( 2.661 ) ( 2.137 )

Father's Labor Income 53.248 - PC Test Scores 33.101 29.519

( 80.940 ) - ( 4.654 ) ( 5.608 )

Non-Labor Income 170.282 32.251 Drink 0.387 0.251

( 878.372 ) ( 331.329 ) ( 0.487 ) ( 0.434 )

School Smoke 0.096 0.276

Pupil-Teacher Ratio 17.133 17.098 ( 0.295 ) ( 0.447 )

( 2.871 ) ( 2.693 ) Father Age (year) 41.258 -

Average Teacher Salaries 51.411 50.568 ( 6.750 ) -

( 7.864 ) ( 8.533 ) Education 13.547 -

( 2.828 ) -

MSA 0.753 0.739 Unemployment Rate 4.824 5.163

( 0.432 ) ( 0.439 ) ( 1.812 ) ( 1.770 )

Northeast 0.169 0.160 Percentage of service 0.172 0.170

( 0.375 ) ( 0.367 ) ( 0.016 ) ( 0.014 )

Midwest 0.269 0.208 Hourly wage rate 13.347 13.037

( 0.444 ) ( 0.406 ) ( 1.250 ) ( 1.406 )

South 0.262 0.458 Div_ind 1.286 -

( 0.440 ) ( 0.499 ) ( 0.781 ) ‐


● Both samples are weighted using PSID-CDS PCG/Child level weights.


TABLE A3.2: Descriptive Statistics

Test Scores Others

Inputs



Dep. Variable

(1)a(2) (3) (4) (1)a

(2) (3) (4)

Maternal Time

Total Time 1.020 - 0.623 0.338 0.581 - 0.493 0.116

(0.147)*** - (0.186)*** (0.186)* (0.135)*** - (0.134)*** (0.151)

Active Time - 2.421 - - - 0.956 - -

- (0.510)*** - - - (0.367)*** - -

Passive Time - -1.531 - - - -0.276 - -

- (0.746)** - - - (0.542) - -

Paternal Time

Total Time -0.055 - 0.570 -0.170 0.230 - -0.093 -0.035

(0.230) - (0.204)*** (0.210) (0.213) - (0.171) (0.171)

Active Time - 0.042 - - - 1.036 - -

- (0.608) - - - (0.463)** - -

Passive Time - 0.941 - - - -0.394 - -

- (0.546)* - - - (0.437) - -

Mother's Labor Income -0.192 -0.834 -0.032 -0.140 0.340 -0.203 0.194 0.054

(0.229) (0.390)** (0.250) (0.132) (0.196)* (0.265) (0.188) (0.107)

Father's Labor Income 0.010 -0.080 0.072 0.216 0.117 0.035 0.092 0.048

(0.045) (0.076) (0.045) (0.088)** (0.039)*** (0.051) (0.033)*** (0.071)

Non-Labor Income -0.003 0.033 -0.012 0.000 -0.005 0.019 -0.003 0.013

(0.006) (0.013)** (0.007)* (0.021) (0.005) (0.009)** (0.005) (0.017)

Pupil-Teacher Ratio -0.700 -0.819 -0.683 -1.792 -0.610 -0.547 -0.470 0.047

(0.275)** (0.423)* (0.352)* (0.437)*** (0.250)** (0.296)* (0.250)* (0.352)

Average Teacher Salaries 0.114 0.579 -0.054 -0.124 -0.087 0.230 -0.119 -0.072

(0.095) (0.192)*** (0.114) (0.138) (0.084) (0.135)* (0.081) (0.111)

Mother's Education 1.339 1.046 1.619 2.731 -0.408 -0.513 0.418 0.645

(0.643)** (0.905) (0.796)** (0.779)*** (0.526) (0.530) (0.635) (0.629)

Mother's PC Test Scores 0.184 1.140 0.191 0.419 -0.014 0.489 -0.035 0.298

(0.161) (0.327)*** (0.219) (0.180)** (0.147) (0.239)** (0.155) (0.145)**

IQ 2.277 2.970 1.981 1.534 1.439 2.080 1.338 1.249

(0.207)*** (0.415)*** (0.255)*** (0.225)*** (0.187)*** (0.292)*** (0.181)*** (0.182)***

Boy -1.686 -0.089 -1.744 3.884 2.039 4.979 -2.028

(1.285) (2.694) (1.635) (1.171)*** (1.927) (1.148)*** (1.920)

BPI 0.078 0.207 0.005 -0.671 -0.299 -0.203 -0.306 -0.163

(0.120) (0.198) (0.159) (0.142)*** (0.109)*** (0.136) (0.112)*** (0.114)

Health at birth -1.068 0.589 -1.694 4.506 -2.529 -1.722 -0.924 -0.220

(2.277) (3.662) (3.167) (3.386) (2.069) (2.514) (2.229) (2.725)

Birth weight 0.021 0.003 0.092 -0.123 -0.005 0.015 -0.017 0.008

(0.031) (0.055) (0.044)** (0.037)*** (0.029) (0.038) (0.031) (0.030)

No. of Obs 1692 1692 1161 404 1692 1692 1161 404

F-Stat 21.08 8.45 1403.62 25.43 22.46 14.53 2917.02 8.58


TABLE A3.3: Estimation Results, Production Function, Married Households

PSID-CDS WJ-R LW TEST PSID-CDS WJ-R AP TEST

▪ a(1) = Baseline Model, (2) = Different Time Measures, (3) = White Sample, (4) = Black Sample.

▪ All samples are weighted using PSID-CDS PCG/Child level weights.



Dep. Variable

(1)a(2) (3) (4) (1)a

(2) (3) (4)

Maternal Time

Total Time 0.501 - 0.822 0.232 0.529 - 0.417 0.344

(0.168)*** - (0.344)** (0.126)* (0.149)*** - (0.265) (0.127)***

Active Time - 0.680 - - - 0.684 - -

- (0.181)*** - - - (0.156)*** - -

Passive Time - -0.500 - - - -0.285 - -

- (0.251)** - - - (0.216) - -

Mother's Labor Income -0.334 -0.641 -0.319 -0.438 -0.181 -0.434 -0.051 -0.190

(0.240) (0.257)** (0.257) (0.265)* (0.215) (0.220)** (0.201) (0.268)

Non-Labor Income 0.023 0.011 0.000 0.373 0.018 0.008 -0.009 0.472

(0.012)** (0.012) (0.009) (0.181)** (0.011)* (0.011) (0.007) (0.183)***

Pupil-Teacher Ratio 0.123 0.071 -0.324 -0.538 -0.126 -0.169 -1.025 -0.652

(0.325) (0.335) (0.841) (0.407) (0.291) (0.286) (0.657) (0.412)

Average Teacher Salaries -0.207 -0.160 -0.655 -0.014 -0.138 -0.098 -0.144 0.121

(0.096)** (0.099) (0.260)** (0.097) (0.086) (0.085) (0.203) (0.098)


(0.673) (0.715)** (1.597) (0.492) (0.602)** (0.612)*** (1.248) (0.498)


(0.217)*** (0.225)*** (0.555) (0.250)*** (0.194)*** (0.193)*** (0.434)** (0.253)

IQ 2.001 2.048 2.740 0.826 1.390 1.435 0.928 0.822

(0.230)*** (0.238)*** (0.427)*** (0.245)*** (0.205)*** (0.204)*** (0.334)*** (0.247)***

Boy 1.067 4.574 3.596 0.456 2.914 5.789 6.054 0.845

(1.640) (1.833)** (4.017) (1.911) (1.467)** (1.570)*** (3.144)* (1.932)

BPI -0.115 -0.189 -0.120 -0.132 -0.242 -0.304 -0.286 -0.196

(0.129) (0.134) (0.329) -0.159 (0.115)** (0.115)*** (0.258) (0.161)

Health at birth -7.968 -9.760 -9.305 -9.153 -8.131 -9.650 -6.976 -7.355

(2.876)*** (3.004)*** (5.590)* (3.318)*** (2.571)*** (2.571)*** (4.374) (3.355)**

Birth weight -0.038 -0.066 0.019 -0.062 0.095 0.072 0.065 0.113

(0.036) (0.037)* (0.088) (0.037)* (0.032)*** (0.032)** (0.069) (0.038)***

No. of Obs 763 763 128 600 763 763 128 600

F-Stat 15.07 14.56 289.98 3.86 18.82 19.28 492.13 1167.12



▪ a(1) = Baseline Model, (2) = Different Time Measures, (3) = White Sample, (4) = Black Sample.

TABLE A3.4: Estimation Results, Production Function, Single-Mother Households

PSID-CDS WJ-R LW TEST PSID-CDS WJ-R AP TEST



Dep. Variable

(1)a(2) (3) (1) (2) (3) (1) (2) (3)

Mother's Labor Income -0.020 -0.131 -0.088 0.079 0.084 0.315 1.040 -0.175 0.780

(0.225) (0.319) (0.168) (0.153) (0.157) (0.200) (0.856) (1.022) (0.491)

Father's Labor Income 0.018 -0.006 -0.001 -0.172 -0.111 -0.124 - - -

(0.044) (0.055) (0.120) (0.091)* (0.059)* (0.207) - - -

Non-Labor Income 0.012 0.020 0.011 0.004 0.002 0.031 0.009 0.020 0.474

(0.007)* (0.010)** (0.029) (0.008) (0.007) (0.039) (0.018) (0.018) (0.436)

Pupil-Teacher Ratio 0.064 -0.039 -0.347 0.404 0.317 -0.028 0.374 1.993 -0.166

(0.300) (0.470) (0.627) (0.366) (0.376) (0.816) (0.447) (1.165)* (0.641)

Average Teacher Salaries 0.176 0.304 0.293 0.077 0.168 0.221 0.362 0.005 0.274

(0.106)* (0.157) (0.222) (0.108) (0.115) (0.303) (0.167)** (0.370) (0.259)


Black -0.520 - - -3.213 - - 9.764 - -

(3.770) - - (3.684) - - (6.630) - -

White 4.385 - - 5.166 - - 7.929 - -

(2.338)* - - (2.468)** - - (5.960) - -

Boy -2.406 -3.252 -2.215 2.312 1.896 6.654 0.368 -3.429 -6.238

(1.079)** (1.698)* (2.339) (1.127)** (1.282) (3.390)** (2.158) (5.058) (3.300)*

No. of Siblings -0.516 -2.770 1.723 1.068 -1.109 -9.491 0.195 0.025 -2.887

(1.482) (2.934) (4.009) (1.835) (2.223) (4.926)* (1.923) (4.845) (3.789)

Age (month) -0.194 -0.258 0.065 0.194 0.066 0.330 -0.022 0.129 0.140

(0.066) (0.120)** (0.148) (0.070)*** (0.086) -0.212 (0.128) (0.227) (0.204)

Age2 (month) 0.000 0.000 -0.001 -0.001 0.000 -0.002 0.000 -0.001 -0.001

(0.0002) (0.0004) (0.0006) (0.0002)*** (0.0003) (0.0009)** (0.0005) (0.0007) (0.0007)

IQ -0.114 0.031 0.212 -0.107 -0.223 -0.070 -0.301 0.224 -0.727

(0.203) (0.326) (0.245) (0.215) (0.223) (0.340)* (0.234) (0.430) (0.393)*

BPI -0.275 -0.214 -0.224 0.012 0.021 -0.489 0.145 0.047 0.507

(0.096)*** (0.152) (0.177) (0.108) (0.113) (0.257) (0.163) (0.296) (0.260)*

Health at birth -0.137 -0.519 3.049 0.908 0.641 1.843 9.760 2.612 3.168

(2.040) (3.264) (3.446) (2.119) (2.349) (4.510) (4.819)** (5.978) (4.952)

Birth weight -0.028 -0.065 0.024 0.001 -0.043 0.112 0.024 -0.032 0.006

(0.027) (0.043) (0.050) (0.031) (0.034) (0.060)* (0.045) (0.076) (0.053)


Mother's Education -0.574 -0.999 0.152 - - - -1.716 -1.202 0.777

(0.610) (1.004) (0.756) - - - (2.020) (3.123) (0.781)

Mother's Age 0.047 0.055 -0.716 - - - -0.059 0.543 -0.339

(0.086) (0.142) (0.333)** - - - (0.122) (0.741) (0.223)

Mother's PC Test Scores 0.183 0.388 -0.073 -0.192 -0.282 -0.202 -0.603 -0.407 -0.932

(0.149) (0.247) (0.194) (0.154) (0.180) (0.240) (0.411) (0.703) (0.452)**

Father's Education - - - 1.411 1.291 -0.243 - - -

- - - (0.655)** (0.598)** (0.784) - - -

Father's Age - - - -0.024 -0.005 0.103 - - -

- - - (0.090) (0.097) (0.240) - - -

No. of Obs

F-Stat

TABLE A3.5: Estimation Results, Inputs Demand Function

Married Households

Maternal Time

Single-Mother Households


▪ a(1) = Baseline Model, (2) = White Sample, (3) = Black Sample.






Dep. Variable

Active Passive Active Passive

Mother's Labor Income 0.163 -0.210 0.243 -0.134 1.290 -0.503

(0.139) (0.176) (0.092)*** (0.103) (1.044) (0.745)

Father's Labor Income 0.041 0.025 -0.053 -0.068

(0.029) (0.035) (0.051) (0.060)

Non-Labor Income -0.003 0.009 -0.005 0.003 0.019 -0.012

(0.005) (0.006) (0.005) (0.005) (0.022) (0.016)

Pupil-Teacher Ratio 0.065 -0.087 0.135 0.164 0.318 0.120

(0.192) (0.262) (0.216) (0.248) (0.547) (0.384)

Average Teacher Salaries -0.028 0.207 -0.047 0.124 0.383 -0.161

(0.067) (0.090)** (0.064) (0.073)* (0.204)* (0.144)


Black -3.701 4.109 -5.401 2.209 11.183 -3.713

(2.471) (3.086) (2.289)** (2.387) (8.095) (5.748)

White 1.938 3.096 2.492 2.827 8.258 -2.442

(1.523) (2.008) (1.507)* (1.634)* (7.281) (5.167)

Boy -1.858 -0.173 2.413 0.185 -2.954 3.469

(0.730)** (0.890) (0.706) (0.719) (2.632) (1.851)*

No. of Siblings 0.111 0.802 0.952 0.328 -0.263 0.702

(0.870) (1.450) (1.037) (1.321) (2.331) (1.668)

Age (month) -0.164 -0.159 0.123 -0.052 -0.072 0.090

(0.039)*** (0.059)*** (0.042)*** (0.048) (0.155) (0.111)

Age2 (month) 0.000 0.000 -0.001 0.000 0.000 0.000

(0.0001)** (0.0002)** (0.0001)*** (0.0002) (0.0006) (0.0004)

IQ -0.254 0.049 -0.309 0.091 -0.151 -0.292

(0.130)* (0.167) (0.129)** (0.143) (0.286) (0.201)

BPI -0.207 -0.090 -0.040 0.022 0.186 -0.050

(0.065)*** (0.079) (0.066) (0.070) (0.199) (0.140)

Health at birth -0.486 -0.368 0.269 0.013 10.715 -2.389

(1.371) (1.681) (1.322) (1.367) (5.892)* (4.173)

Birth weight -0.014 -0.021 -0.019 0.014 0.058 -0.052

(0.019) (0.023) (0.019) (0.020) (0.055) (0.039)


Mother's Education -0.196 -0.127 - - -2.669 1.698

(0.370) (0.441) - - (2.470) (1.759)

Mother's Age -0.003 0.079 - - 0.106 -0.331

(0.050) (0.080) - - (0.149) (0.106)***

Mother's PC Test Scores -0.138 0.238 -0.221 -0.023 -0.781 0.376

(0.099) (0.127)* (0.095)** (0.101) (0.503) (0.356)

Father's Education - - 0.710 0.583 - -

- - (0.357)** (0.420) - -

Father's Age - - -0.047 0.084 - -

- - (0.053) (0.060) - -

No. of Obs

F-Stat

TABLE A3.6: Estimation Results, Inputs Demand Function, Different Time Measures

Married Households Single-M Households


Passive




Maternal Time Paternal TimeActive

Chapter 4

Bargaining Power Shift and

Children’s Cognitive Achievement

(ABSTRACT)

This chapter studies the impact of bargaining power shift on children’s cognitive achieve-

ment. A global bargaining power change in favor of the females may be beneficial to the

children who live with both parents, if mothers are more concerned about children’s out-

comes. The effects on children in single-mother households, however, are not always positive

when the tradeoff between parental time and labor income is taken into account. We extend

the collective model with household production to the general equilibrium framework and

show that children’s welfare may deteriorate when female market equilibrium wage increases

resulting from decreased labor supply by the married women. Policies that intend to protect

females usually shift the bargaining power within a household towards married women, but

they may have unintended effects on the optimal resource allocations for the single mothers.

4.1. Introduction

An extensive body of previous literature has supported the argument that different household

members do not behave the same towards intra-household resource allocations from either

106


theoretical or empirical perspective1. It is widely believed that a relative bargaining position

change among different household members has significant impacts on the decision process of

household resource allocations. For instance, in a standard husband-wife household, a power

shift in favor of wife encourages her to enjoy more leisure and household non-labor income.

The effects, however, have the externality that changes the market equilibrium prices after

we take the demand side into consideration. Gersbach and Haller (2009) have discussed the

price effects based on a collective model in a general equilibrium framework. In this paper,

we examine the wage effects on children’s cognitive development and show that because of

the price effect children living with single mothers may not benefit from the bargaining power

shifts in favor of female.

In this chapter, we first develop a collective model with household production. Without con-

sidering the market prices fluctuation, we show that under certain assumptions both parents

will spend more time with children when mothers have more bargaining power, and they

are more sensitive to the share of residual incomes compared with fathers. Labor demand is

considered to capture the wage effects on household optimal choices. An increase of female

bargaining power may suppress the labor supply of married women, and increase equilibrium

wage rate. Single mothers probably choose to work more but compensate their children with

goods inputs after observing a higher wage. If children’s cognitive achievement depends on

both parental time and goods investment, the wage effects may lower the welfare of children

living in single-mother households.

Since mothers are believed to be more concerned about children’s outcomes compared with

fathers, a bargaining power shift in favor of females usually is thought to be beneficial to

children’s development. Mothers with larger bargaining powers are likely to spend more time

and monetary inputs on children and therefore, have children with higher quality. However,

when the power shift is large, market labor supply can be affected and reach a new equilib-

1Lundberg and Pollak (1994) has a very good summary on the theories. Most empirical work, involving

income pooling test, labor supply, child-specific expenditures and etc., is based on a series of papers assuming

the results are Pareto optimal, such as Chiappori (1992), Chiappori (1997), Chiappori, Fortin and Lacroix

(2002), and Thomas, Contreras and Frankenberg (1997)


rium female wage rate. A price effect can alter the optimal decisions of the single mothers.

If child’s outcomes depend on both time and monetary inputs, the resulting tradeoff be-

tween maternal time and goods expenditure does not necessarily guarantee a higher level of

outcomes for children who live with their mothers alone. In other words, policies aiming to

improve children’s welfare by increasing all the female’s social position could be ineffective,

or even worse, hinder the development of children from single-mother families.

The study provides the implications for future empirical research on early childhood devel-

opment. A series of current empirical literature on household resource allocations starts to

rely on the exogenous distribution factors to help identify the structure, for example Bobonis

(2009), Chiappori, Fortin and Lacroix (2002), and You and Davis (2010). The papers, how-

ever, are under the partial equilibrium framework and thereby only employ the distribution

factors as instruments for married households. We show that because the market equilibrium

wage rates are affected by the global bargaining power change single-mother families can be

affected through the channel of wages. For example, studies about the effects of maternal

time inputs on children who live with single mothers usually evolve an endogeneity issue

caused by the potential correlation between female’s wage rate and unobserved ability. Our

model implies that, except for some specific policy reform, exogenous distribution factors

can be instruments, too, to help improve the efficiency of the estimates. Moreover, as long

as the data are available, the model is a great starting point for a structural analysis to

include the demand side of female labor market.

The chapter is organized as follows. Section 4.2 develops a collective model with household

production under a partial equilibrium framework. In section 4.3, we extend the model

to include the labor demand and discuss the effects of a global bargaining power shift on

optimal home inputs choices. Section 4.4 concludes.


4.2. Collective Model with Household Production:

A Partial Equilibrium Case

In this section, we extend the collective labor supply model, allowing for the household pro-

duction, and especially assume that children’s cognitive achievement, A, is “produced” by

parental time and goods inputs. We show that, under certain general assumptions of child’s

cognitive production technology, changes of the bargaining power within a household have

predictable impacts on home inputs. The model can also help generate testable conditions

about household resource allocation decisions, and serve as the foundation for the empirical

analyses.

4.2.1 Basic Setting

We develop a standard collective model with child’s cognitive achievement A, which is “pro-

duced” by two types of home inputs, time and money. For simplicity, the household fertility

choice is not taken into account in the model by assuming that there is only one child in

each household. The household consists of two individuals (i = p,m)2 with distinct utility

functions which depend on individual’s private leisure li, consumption Ci, and child’s cogni-

tive achievement A, determined by maternal time tm, paternal time tp and monetary inputs

X3. Therefore, in contrast with the standard collective model, we allow for the existence of

non-marketable household production, in particular, the child’s cognitive production in the

model. Moreover, the decision process on the intra-household resource allocation leads to

Pareto-efficient outcomes.

The price of goods, either for the private or the public consumption, is normalized to one,

for our future results are independent of the variations in the price of consumption goods.

Wages are denoted by wp and wm, respectively, and the household non-labor income is I. To

2We use “p” for father and “m” for mother to avoid the ambiguity such as “m” for male or mother.3Here we assume the monetary input to children’s cognitive achievement is a composite good with nor-

malized price 1.


achieve identification of the model, we assume the presence of some continuous distribution

factors z. According to the literature, these factors are defined as variables that affect the

observed behaviors through the impacts on the household members’ respective bargaining

positions alone without any influence on either family member’s utility. Some examples of

the distribution factors in the previous literature are local sex ratio [Chiappori (2002)], non-

labor income [Thomas (1990)], wealth at marriage [Thomas et al (1997)], and etc.

As is standard, the choice variables lp, lm, tp, tm, X, and C(= Cp + Cm) are assumed to be

continuous and observed, while the distribution of consumption within the couple may not.

We also assume that all the choice variables are functions of wp, wm, I, and z, and twice

continuously differentiable. The level of child’s cognitive achievement A thereby is also a

function of wp, wm, I, and z, and twice continuously differentiable. The bundle (wp, wm, I, z)

varies within a compact subset K of R3+ × R. Individual i’s preferences on leisure, private

consumption, and child’s cognitive achievement are assumed to be differentiable, strictly

increasing, and strictly convex, and therefore, the individual utility Ui is represented by a

twice continuously differentiable function and strictly concave on (li, Ci, A). The household

production technology A(·) is assumed to be twice continuously differentiable and concave

on (tp, tm, X).

4.2.2 Decision Process

Following the standard literature on the collective model, the Pareto-efficient intra-household

resource allocations are determined by an optimization program P1:

max{li,ti,Ci,X}i=p,m

H(lp, lm, tp, tm, C,X;λ)

= λUp(lp, Cp, A(tp, tm, X)) + (1− λ)Um(lm, Cm, A(tp, tm, X)) (P )

s.t. Cp + Cm +X + wp(lp + tp) + wm(lm + tm) = wp + wm + I.

Here, λ ∈ [0, 1] is the Pareto weight between two household members, mother and father in

our model. Based on the definition of the distribution factors, the Pareto weight, λ, depends

on wp, wm, I, and z, and the household utility function H itself becomes price-dependent.


It is the key distinction between a collective household model and a traditional unitary one,

where price and non-labor income do not affect the household utility directly. Notice that

since Ui is assumed to be strictly concave, H is also strictly concave, and in addition, we

assume that the Pareto weight λ is continuously differentiable in (wp, wm, I, z).

The household’s observed behaviors are the solutions to the optimization of a household

utility function H, which itself varies with wage rates, non-labor income, and distribution

factors. However, even with the household production, H still exhibits partial separabil-

ity properties for the household labor supply and private consumptions, in the sense that

the household’s marginal rates of substitution between individual i’s leisure and private con-

sumption is λ-independent. The trade-off between individual i’s leisure and consumption will

not be affected as the bargaining power position changes within the household. Technically,

the first order conditions of (P ) imply that

λ∂Up∂C

= (1− λ)∂Um∂C

, (4.1)

and by the envelope theorem, we have:

∂H

∂lp= λ

∂Up∂lp

,∂H

∂lm= λ

∂Um∂lm

,∂H

∂C= (1− λ)

∂Um∂C

. (4.2)

Therefore,∂H/∂li∂H/∂Ci

=∂Ui/∂li∂Ui/∂Ci

. (4.3)

The condition (4.3) is crucial for us to re-state our model as a two-stage process. It implies

that individual i’s margining rates of substitution between leisure and private consumption

only depends on his or her own optimal choices of li and Ci, as well as the level of child’s

cognitive achievement A produced by the optimal home inputs, but has no direct relevance

to the spouse’s choices of leisure, private consumption, and the Pareto weight. The program

(P ) thereby is equivalent to a two-stage decision process. Both members agree on the home

inputs (tp, tm, X) to their child’s cognitive achievement production and a particular sharing

rule towards the residual non-labor income in the first stage, and then choose their own


private consumption and leisure freely in the second stage to maximize individual’s utility,

subject to the optimal level of child’s cognitive achievement and the budget constraint stem-

ming from the first stage.

Using backward induction, we can solve the program by a two-step procedure: 1) maximize

each individual’s utility to obtain the optimal private consumption and leisure, conditional

on child’s cognitive achievement A and a sharing rule; and 2) determine the optimal leisure

and private consumptions from the second stage. Therefore, because of the condition (4.3),

the equilibrium to the program (P ) is equivalent to the Subgame Perfect Nash Equilibrium

to the following re-constructed two-stage game.

First, define ρi as the mutual consentient sharing rule. In the literature, ρi is also called the

conditional sharing rule in the sense that it is the residual non-labor income after parents

decide on the home inputs to child’s cognitive development. ρi can be positive or negative,

but the summation of ρp and ρm equals to the household total non-labor income left for

private consumption after determining the home inputs. That is,

ρi(wp, wm, I, z) = wi × li(wp, wm, I, z) + Ci(wp, wm, I, z)− wi,

and therefore,

ρp(wp, wm, I, z) + ρm(wp, wm, I, z)

= I −X(wp, wm, I, z)− wp × tp(wp, wm, I, z)− wm × tm(wp, wm, I, z).

Given the level of child’s cognitive achievement A and ρi, the second-stage problem for

individual i becomes:

max{li,Ci}

Ui(li, Ci, A) (P1)

s.t. wi × li + Ci = wi + ρi(wp, wm, I, z).

Each individual chooses the private consumption and leisure freely in this stage to maximize

his or her own utility under a budget constraint depending on the conditional sharing rule

and the wage rate. Note that since the optimal choices of Ci and li clearly depends on A and


ρi, the Pareto-efficient solutions to the program (P ) would only occur when the home inputs

are chosen optimally in the first stage and there exists an efficient conditional sharing rule ρi.

Based on the program (P1), the solutions li and Ci (i = 1, 2) are functions of (wi, ρi, A).

Hence, we can define a conditional indirect utility function Vi(wi, ρi, A) as the value of the

program (P1) for each individual i. Moreover, since it is a one-to-one correspondence between

the individual i’s direct and indirect utility function, the first stage problem for the household

now can be written as the following optimization program:

max{ρp,ρm,A}

λVp(wp, ρp, A(tp, tm, X)) + (1− λ)Vm(wm, ρ2, A(tp, tm, X)) (P2)

s.t. ρ1 + ρ2 = I −X − wp × tp − wm × tm.

The solutions to the program are the Pareto-efficient outcomes for the collective model with

child’s cognitive production.

4.2.3 Determination of Home Inputs

There are two methods to obtain the optimal home inputs (tp, tm, X) for a typical two-

member household: a directly way based on the program (P ), and an “indirectly” way

based on the program (P2). Both solve for optimal home inputs (t?p, t?m, X

?) as functions of

(wp, wm, I, z). If the existence of interior solutions is assumed, at the equilibrium, we have:

∂A

∂tp= wp

∂A

∂X, (4.4)

and∂A

∂tm= wm

∂A

∂X. (4.5)

Therefore, when wi > wj for i 6= j, at the optimal point, the marginal productivity of in-

dividual i’s time input towards child’s cognitive production is larger than that of individual j.

Technically, we could have the corner solutions to the program (P2), but they only occur

when the home inputs are perfect substitutes and have constant marginal substitution rates,


such as MRStp,tm = wpwm

, since the iso-quant of our well behaved cognitive production func-

tion is continuous and convex to the origin. However, in this case, individual i, who has

higher wages rate (i.e., higher time cost), would invest zero time inputs for child’s cognitive

production. It is somewhat unreasonable because paternal time, maternal time and the par-

ents’ monetary inputs usually are non-homogenous, and therefore, in the following section,

we simply assume that there exist interior solutions for all home inputs. Empirically, this

can be tested when appropriate data are available.

The first order conditions of the program (P ) imply that:

(∂Up/∂A

∂Up/∂Cp+

∂Um/∂A

∂Um/∂Cm)∂A

∂X= 1, (4.6)

which takes the standard Bowen-Lindahl-Samuelson form. Equivalently, we can obtain the

same form from the program (P2), which has following first order conditions:

λ∂Vp∂ρp

= (1− λ)∂Vm∂ρm

= [λ∂Vp∂A

+ (1− λ)∂Vm∂A

]∂A

∂X. (4.7)

Thus,

(∂Vp/∂A

∂Vp/∂ρp+

∂Vm/∂A

∂Vm/∂ρm)∂A

∂X= 1. (4.8)

Define the ratio

MWPi =∂Vi/∂A

∂Vi/∂ρi

as i’s marginal willingness to pay (MWPi) for children’s cognitive achievement A. Together

with the conditions (4.4) and (4.5), we show that the product of each individual’s marginal

willingness to pay for A and the marginal productivity of one home input must add up to the

price of that home input, or even simply that the summation of each individual’s marginal

willingness to pay for one home input must equal to the price of it.

4.2.4 Comparative Statics

One key motivation for our developing a collective model with household production is

that the distribution of intra-household income and changes of such distribution impact the


household resource allocation decisions. The directions of the effects, however, are ambigu-

ous when the household production exists. Child’s cognitive production technology thereby

plays a key role. We show hereinafter that a welfare analysis can still be conducted if we

impose some general form to the child’s cognitive production function. In other words, it is

possible to answer the question how changes of the bargaining power within the household

- that is, changes of the Pareto weight λ - affect the home inputs on child cognitive achieve-

ment when the household production function has some specific functional forms.

For simplicity, let us denote ρ ≡ ρp and thereby ρm = I −X − wptp − wmtm − ρ. The first

order conditions for the program (P2) then become:

λ∂Vp∂ρp

= (1− λ)∂Vm∂ρm

, (4.9)

(MWPp +MWPm)∂A

∂X= 1, (4.10)

∂A

∂tp= wp

∂A

∂X, (4.11)

∂A

∂tm= wm

∂A

∂X. (4.12)

Taking the total differentiation of the conditions above and applying the implicit function

theorem, we can obtain the following:

∂ρ

∂λ=

1

D(∂Vp∂ρp

+∂Vm∂ρm

)(−Γ1) (4.13)

∂X

∂λ=

1

D(∂Vp∂ρp

+∂Vm∂ρm

)(Γ2) (4.14)

∂tp∂λ

=1

D(∂Vp∂ρp

+∂Vm∂ρm

)(−Γ3) (4.15)

∂tm∂λ

=1

D(∂Vp∂ρp

+∂Vm∂ρm

)(Γ4), (4.16)


where

−Γ1 = −{(Π1AX + Π2AX,X)[(Atp,tp − wpAX,tp)(Atm,tm − wmAX,tm)

−(Atp,tm − wmAX,tp)(Atp,tm − wpAX,tm)]

+(Π1Atp + Π2AX,tp)[(Atp,tm − wpAX,tm)(AX,tm − wmAX,X)

−(AX,tp − wpAX,X)(Atm,tm − wmAX,tm)]

+(Π1Atm + Π2AX,tm)[(AX,tp − wpAX,X)(Atp,tm − wmAX,tp)

−(Atp,tp − wpAX,tp)(AX,tm − wmAX,X)]}

Γ2 = AX(∂MWPp∂ρ1

− ∂MWPm∂ρ2

)[(Atp,tp − wpAX,tp)(Atm,tm − wmAX,tm)

−(Atp,tm − wmAX,tp)(Atp,tm − wpAX,tm)]

−Γ3 = AX(∂MWPp∂ρ1

− ∂MWPm∂ρ2

)[(Atp,tm − wpAX,tm)(AX,tm − wmAX,X)

−(AX,tp − wpAX,X)(Atm,tm − wmAX,tm)]

Γ4 = AX(∂MWPp∂ρ1

− ∂MWPm∂ρ2

)[(AX,tp − wpAX,X)(Atp,tm − wmAX,tp)

−(AX,tm − wmAX,X)(Atp,tp − wpAX,tp)],

and Π1 = ∂MWPp∂A

AX + ∂MWPm∂A

AX − ∂MWPm∂ρm

, Π2 = MWPp +MWPm.

Given the standard assumptions that the conditional sharing rule ρ and child’s cognitive

achievement A are normal, MWPi is increasing in ρi and decreasing in A. Then Π1 < 0 and

Π2 > 0. Without loss of generality, we may further assume that the difference between two

MWP ’s is positive, i.e.,

DMWP ≡ ∂MWPm∂ρm

− ∂MWPp∂ρp

> 0.

It implies that mother’s MWPm is more sensitive to the changes in her share than that of

the father’s. Then, the signs for the equations (4.13)-(4.16) will be totally determined by

the functional form of the child’s cognitive production technology A(·).

In theory, shapes of the isoquant for A(·) show the trade-off between two home inputs,

which determines the signs of (4.13)-(4.16). When the household production is allowed, a

marginal change in an individual i’s Pareto weight can induce opposite-direction movements


towards the home inputs and thereof exaggerate or dampen the effects of λ to child’s cog-

nitive achievement. Therefore, certain welfare questions such as how the bargaining power

changes within a household affect the child’s well-being may only be answered empirically.

It is possible that policies, aiming to boost parents’ goods inputs by increasing mother’s

bargaining power within a household, may potentially “crowd out” parental time inputs and

eventually lower the cognitive achievement of children.

4.2.5 Examples

A General CES Production Function: assume that the cognitive production technology

A(·) takes the general form of a CES production function, i.e.,

A = F [α1/δXδ−1δ + β1/δ(tp)

δ−1δ + γ1/δ(tm)

δ−1δ ]

δδ−1 ,

then we have:

• AX,ti > 0, i = p,m;

• AX,X < 0 and Ati,ti < 0, i = p,m;

• Atp,tm − wmAX,tp = Atp,tm − wpAX,tm = 0.

Therefore, ∂ρ∂λ

, ∂K∂λ

, ∂T1

∂λ, and ∂T 2

∂λshare the same sign under the assumption that DMWP > 0.

It implies that, when individual 1’s marginal willingness to pay for child’s cognitive achieve-

ment A is more sensitive to the change of his/her share than that of individual 2, an increase

of his/her bargaining power λ will make both individuals contribute more time and monetary

inputs to their child.

A Separable Production Function: assume that the production technology A(·) is sep-

arable between the time and monetary inputs. That is, parents’ monetary inputs does not

affect the trade-off between maternal time and paternal time inputs and have no impact on

the marginal effects of parental time inputs. Then we have:


• AX,ti = 0, i = p,m;

• AX,X < 0 and Ati,ti < 0, i = p,m;

• Atp,tpAtm,tm − A2tp,tm ≥ 0.

In this case, a change of bargaining power position in favor of child’s mother, for example,

can increase both parents’ time inputs but have no impact on the goods inputs to child’s

cognitive development as long as mother’s marginal willingness to pay for child’s cognitive

achievement is more sensitive to changes in her share than that for her male partner.

4.2.6 Identification Issue

The identification of the whole structure certainly becomes more complicated when we con-

sider the household production in a collective model. Blundell et al (2005) paper has proved

that the functions

ψi(X, tp, tm) =∂uX/∂ti∂uX/∂X

, (i = 1, 2)

are identifiable when the determinant D(wp,wm,I)(X, tp, tm) do not vanish over a convex set. In

their model, parents care for their child’s utility level uX , which is assumed to be “produced”

by parental time. Since the utility uX is unobservable, even with the time use data, people

should choose an increasing transform of uX to identify the functions ψi (i = 1, 2) and the

whole structure. In our model, however, we assume that parents actually “produce” child’s

cognitive achievement A, which can be measured and well observed by the researchers.

The household production can be fully identified since both the inputs and the output are

observed. All the results proved in Blundell et al (2005) are automatically applied in our

case. Another identification issue, however, arises for the empirical work when people’s

heterogeneity cannot be fully captured. The estimation based on the structural model can

be very complicated or even hard to achieve if the data are not satisfactory.


4.3. Collective Model with Household Production:

A General Equilibrium Case

4.3.1 Basic Setting

In this section, we extend the model to a simple general equilibrium framework. Consider a

society with two types of households, a standard two-parent family and a single parent family.

Each person in the society has an individual preference, Ui (i = p,m), with a Cobb-Douglas

functional form denoted by:

Ui = αi lnCi + βi ln li + (1− αi − βi) lnA. (4.17)

The Pareto weight to a two-member household is λ for the member “p” and (1− λ) for the

member “m”. It simply equals to one when there is only one person in the household.

Assume that there are M two-member households, N1 single families for the female, and N2

for the male4. The female labor supply and male labor supply are not perfect substitutable,

and therefore, we simply assume the production function for the demand side has the form:

f(Lps, Lms) = δ1 ln(Lps + 1) + δ2 ln(Lms + 1), (4.18)

where Lps is the total male labor and Lms is the total female labor.

4In the real world, there are families, either multi-person or single-person, that have no children, but

in our model, we do not consider the case for two reasons. First, children’s cognitive production can be

seen as a typical household production, and therefore, in general our results can be extended to the case

with household production which are almost unavoidable. Second, even in 2008, about 18% U.S. women

end their childbearing years without having a child, and more than half of them are well-educated, high-

income group based on the Current Population Census data. Our analyses thereby still bear the important

policy implications for children coming from low-income families, especially from low-income single-mother

households.


4.3.2 Children’s Cognitive Production Depends on Goods Only

First we assume that the children’s cognitive achievement A is produced by goods inputs X

only, and for simplicity, let lnA = lnX. Then our typical two-member household problem

becomes:

max{λUp + (1− λ)Um}

s.t. Cp + Cm +X = wp(1− lp) + wm(1− lm) + I.

The problem for the single person household is:

max{Ui}

s.t. Ci +X = wi(1− li) + I .

By assuming the existence of interior solutions, we can solve the problem for two types of

households and obtain the optimal choices for the type 1 household as:

C?p = λαp(wp + wm + I),

C?m = (1− λ)αm(wp + wm + I),

l?p =λβpwp

(wp + wm + I),

l?m =(1− λ)βm

wm(wp + wm + I),

X? = [λ(1− αp − βp) + (1− λ)(1− αm − βm)](wp + wm + I)

and for the type 2 household as:

C?′

p = αp(wp + I),

C?′

m = αm(wm + I),

l?′

p =βpwp

(wp + I),

l?′

m =βmwm

(wm + I).

X?′

m = (1− αm − βm)(wm + I)

Therefore, the total male labor supply equals:

Lps = M(1− l?p) +N1(1− l?′

p )

= M [1− λβpwp

(wp + wm + I)] +N1[1−βpwp

(wp + I)],


and the total female labor supply is:

Lms = M(1− l?m) +N2(1− l?′

m)

= M [1− (1− λ)βmwm

(wp + wm + I)] +N2[1−βmwm

(wm + I)].

The production sector efficiency requires that the marginal productivity equals to the wage

rate. It therefore implies that:

wp =δ1

Lps + 1,

wm =δ2

Lms + 1.

Thus,

M [wp − λβp(wp + wm + I)] +N1[wp − βp(wp + I)] + wp = δ1, (4.19)

M [wm − (1− λ)βm(wp + wm + I)] +N2[wm − βm(wm + I)] + wm = δ2. (4.20)

By the implicit theorem, the equation (4.19)-(4.20) give us the following results:

∂wp∂λ

=1

D·M(wp + wm + I)βp[(M +N2)(1− βm) + 1]

∂wm∂λ

= − 1

D·M(wp + wm + I)βm[(M +N1)(1− βp) + 1],

where

D = [M(1−λβp)+N1(1−βp)+1][M(1−(1−λ)βm)+N2(1−βm)+1]−M2λ(1−λ)βpβm > 0.

Therefore, ∂wp∂λ

> 0 and ∂wm∂λ

< 0. Furthermore, since we can derive the following:

dl?mdλ

=∂l?m∂wm

∂wm∂λ

+∂l?m∂wp

∂wp∂λ

+∂l?m∂λ

=(1− λ)βm − 1

w2m

∂wm∂λ

+(1− λ)βm

wm

∂wp∂λ− βm(wm + wp + I)

wm, (4.21)

dl?′m

dλ= −βmI

w2m

∂wm∂λ

, (4.22)

the equation (4.22) implies that the household bargaining power shift could impact the op-

timal labor supply of single mothers through the channel of wages. A global power change


in favor of the females (in our model, a decrease in λ) lowers the optimal leisure for the

single mothers because dl?′m

dλ> 0 and thereby implies an increase of the labor supply. For the

married women, the global change of bargaining power reduces the total effect of λ. In a

partial equilibrium case, only the last term in equation (4.21) exists with a negative sign,

but now because both male and female wages are affected by the bargaining power shift, the

general equilibrium effects captured by the first two term offset the partial equilibrium effects.

The intuitive explanation is that a global bargaining power change in favor of the females

increases the leisure for the married women and thereby provides a downward pressure to

the total female labor supply. The labor supply between men and women is not perfectly

substitutable so the market equilibrium wage rate for the female may increase and offer

incentives to the single mothers to sell more time in the market with no compensation from

their partners.

The total effect of mother’s bargaining power increase to the child’s cognitive achievement is

positive in the single-mother families under the the Cobb-Douglas functional form assump-

tion. The impact in the married household, however, can be ambiguous even though we

assume mothers are more sensitive to child’s outcomes than fathers (i.e., (1 − αm − βm) >

(1 − αp − βp)). The general equilibrium effects through two wage rates depend on each

gender’s sensitivity to the leisure and their labor force participation.

We present some numerical examples in Table 4.1(a) - Table 4.1(c) to further illustrate the

effects of bargaining power shift on the optimal household resource allocations, and also plot

the comparative statistics of the case in Table 4.1(a) in Figure 4.1 against λ. For simplicity,

we assume that: 1) N1

M= 0.1 and N2

M= 0.2 which are a close approximation of the U.S. current

family structure; 2) δ1 = 0.6 and δ2 = 0.4 due to higher average wage rate for the males; and

3)αp = αm = βm = 0.3 and βp = 0.4 in order to guarantee that (1−αm−βm) > (1−αp−βp).Moreover, we employ three different levels of non-labor income I = 0, 0.05, and 0.15 to see

5In order to keep the magnitude consistent, I belongs to [0, 0.1], but because of the properties implied by

the Cobb-Douglas functional form, single-person household time allocations behave differently when I = 0


the changes of the optimal resources allocations for 2 household types as the bargaining

power shifts.

In our example, a change in the bargaining power has significant effects on the market equi-

librium wage rates, though the effects decrease as household non-labor income decreases.

When father’s bargaining power λ goes from 0 to 1, market equilibrium wage rate for the

male increases about 30% and for the female decreases more than 40%. Moreover, for the

specific functional forms, female’s market wage rate are more sensitive to the bargaining

power change than the male’s. For example, when we set I = 0.1, female’s market wage rate

declines about 50.13% as their married males shift from completely no bargaining power to

absolute family authority. If only consider a little switch in the household bargaining status,

for instance λ shifts from 0.45 to 0.55, the effects on the market equilibrium wage rates are

around 3.26% to 4.06% which are still sizable.

In general, labor supply changes a lot for married people when λ only increases from 0.45 to

0.55, but the general equilibrium effects caused by market price effects are much smaller and

tend to offset the pure impacts from a partial equilibrium case. In our example, the ratio

between two effects is about one-tenth. For example, when I = 0.1, married fathers decrease

their labor supply by around 11.20% but married mothers have to work more by about

11.36% in total when household bargaining power shifts slightly in favor of the males. If we

do not consider the market price effects, however, the changes of the labor supply in married

households are more than 12%. In single-parent families, labor supply changes around 0.8%

when I = 0.1, and the impacts disappear completely when non-labor income goes to zero.

Two points are worth noting: 1) the magnitude of the effects are function-form dependent;

and 2) later when we take parental time into consideration, the effects become larger for

the same parameters. Therefore, the price effects on the labor supply for the single-person

families deserve some further attention in the future empirical studies.

For our numerical example, an increase in λ (indicating a shift of bargaining power in favor

and therefore, we test two extreme cases I = 0 and I = 1 and one mediocre case I = 0.05


of the male!) hurts the cognitive achievement of children in married households because of

a decrease in the monetary inputs. The monetary input X in a typical married household

may decrease by around 2.2% in total when λ increases from 0.45 to 0.55, but less than one-

tenth of the changes result from the market equilibrium price effects. Children’s cognitive

achievement for the single-parent households, however, is affected more in magnitude by the

general equilibrium effects. As the bargaining power increases in favor of the male, children

living with single-fathers may acquire more monetary inputs by around 3.06% but those

living with single-mothers enjoy less inputs by around 2.84%. It may imply that certain

policies targeting those single-mother children need to pay much attention to the market

price effects induced by a global bargaining power shift, such as a consistent status change

in favor of the female.

4.3.3 Children’s Cognitive Production Depends on Goods and

Time

In this part, we consider a more complicated case where child’s cognitive achievement A is

produced by both parental time (tp, tm) and goods inputs (X). Assume that for children

living with both parents the cognitive production function is lnA = π1 lnX+π2 ln tp+π3 ln tm

where πj (j = 1, 2, 3) is the elasticity of each home input to children’s achievement A and

π1 + π2 + π3 = 1. Moreover, the production function for children who live with single parent

is assumed to be lnA = (1 − πi) lnX + πi ln ti (i = t, p). Hence, in the married household,


the optimal choices are:

C?p = λαp(wp + wm + I),

C?m = (1− λ)αm(wp + wm + I),

l?p =λβpwp

(wp + wm + I),

l?m =(1− λ)βm

wm(wp + wm + I),

t?p =π2wp

[λ(1− αp − βp) + (1− λ)(1− αm − βm)](wp + wm + I),

t?m =π3wm

[λ(1− αp − βp) + (1− λ)(1− αm − βm)](wp + wm + I),

X? = π1[λ(1− αp − βp) + (1− λ)(1− αm − βm)](wp + wm + I).

In the single-person household, the analytical solutions are:

C?′

p = αp(wp + I)

l?′

p =βpwp

(wp + I)

t?′

p =πp(1− αp − βp)

wp(wp + I)

X?′

p = (1− πp)(1− αp − βp)(wp + I)

C?′

m = αm(wm + I)

l?′

m =βmwm

(wm + I)

t?′

m =πm(1− αm − βm)

wm(wm + I)

X?′

m = (1− πm)(1− αm − βm)(wm + I)

Under the condition that market is clearing, we can obtain the following equations:

wp[M(1− l?p − t?p) +N1(w − l?′

p − t?′

p )] + wp = δ1 (4.23)

wm[M(1− l?m − t?m) +N2(w − l?′

m − t?′

m)] + wm = δ2 (4.24)

The absolute values of ∂wp∂λ

(∂wm∂λ

) is smaller (larger) than the absolute values in the first

case because of the existence of parental time input. And meanwhile, for the single-mother


households, we can derive:

dt?′m

dλ= −πm(1− αm − βm)I

w2m

∂wm∂λ

, (4.25)

dX?′m

dλ= (1− πm)(1− αm − βm)

∂wm∂λ

. (4.26)

Therefore, single mothers may lower the maternal time inputs but compensate their chil-

dren with monetary investment when there is a global bargaining power shift in favor of the

females. Since the total impact of bargaining power change on children’s cognitive develop-

ment depends on the production technology, i.e. πi (i = p,m), it is possible that the goods

inputs cannot offset the negative effects of less maternal company. Children’s total welfare,

thereby, can be undermined. Later, our numerical analyses demonstrate the negative effects

of an increase in the female bargaining power on children’s cognitive achievement. We also

show that the effects can be significant if marginal impacts of parental time are relatively

larger than those of goods inputs.

Our numerical results are displayed in Table 4.2(a) to Table 4.2(f) and we plot the compar-

ative statistics of the choice variables against λ in Figure 4.2(a) to Figure 4.2(f). We keep

the same values for the parameters that have been specified in the previous case. Mean-

while, because real parental time data show that, in married households, the paternal time

input is consistently smaller than the maternal time input, combined with the first order

conditions, it implies that π2 ≤ π3. We therefore set π2 = 0.3 and π3 = 0.4. Moreover, we

implement 4 different combinations of πp and πm for the single-parent families, which are

(πp = 0.6, πm = 0.8), (πp = 0.5, πm = 0.9), (πp = πm = 0.7), and (πp = πm = 0.9), to see the

behaviors of children’s cognitive achievement A.

The changes in the market equilibrium wage rate for the males are slightly smaller than

the comparable scenario in the previous example without considering the difference between

leisure and parental time, but much more significant for the female’s wage rate. For exam-

ple, in one scenario I = 0.1 (and πp = 0.6, πm = 0.8p), the female’s wage rate decreases

more than 76.56% as fathers’ bargaining power increases from 0 to 1, or 5.72% as λ goes

from 0.45 to 0.55 (in the previous case, they are about 50.13% and 4.06%, respectively). In


other words, when we assume children’s cognitive production requires parents’ time inputs,

a global shift of bargaining power within multi-person households may have greater impacts

on the female market equilibrium wage. The male’s market wage rate, however, is much less

sensitive to the bargaining power change6.

For married households, a bargaining power shift in favor of the female increases father’s

labor supply and parental time input, but decreases mother’s labor supply and maternal

time input no matter what values we choose for πp and πm. Under our model specification,

the numerical solutions may imply that in a two-parents household the parental time inputs

behave like the market labor supply. They react to the bargaining power change in the same

direction. Those married mothers tend to enjoy more pure leisure by letting their spouses

work more both domestically and on the market when they have relatively larger bargaining

power than fathers do within a household. For our most general case, i.e., I = 0.1, πp = 0.6,

and πm = 0.8, as λ increases from 0.45 to 0.55, father’s labor supply (time input) decreases

by around 13.25% (6.83%) and mother’s labor supply (time input) increases by about 21.39%

(2.39%) in total, where the general equilibrium effects caused by the wage rates changes are

still less than one-tenth. Therefore, it may safely conclude that the effects on market labor

supply are more substantial than those on the parental time. The monetary input X de-

creases about 3.18% totally in this case such that even though, as fathers’ bargaining power

shifts from 0.45 to 0.55, mothers spend more time with their children, children’s cognitive

achievement is still lowered by about 1.94%. The total effects on A slightly increases as

household non-labor income decreases7 and almost irresponsible to the choices of πp and πm.

For the single-person families, the price effects on the market labor supply are more sizable

than those in the case in 4.3.2. For example, when I = 0.1, πp = 0.6, and πm = 0.8,

single-mother’s labor supply decreases 6.75% (it is only 0.87% in the previous case), and

the maternal time increases by about 1.35% as λ goes from 0.45 to 0.55. Unlike the choices

made by married people, single person’s optimal labor market supply changes in an opposite

6In order to check whether it is mainly because of the parameter choice of N1, we actually try another

two cases, N1

M = N2

M = 0.2 and N1

M = 0.4 > N2

M = 0.2. Our results are robust.7When we set I = 0, it lowers by about 2.02%.


direction to their parental time input choices. Furthermore, single parents indeed substi-

tute the parental time input with the goods inputs except when no non-labor income exists

in the family. Since children’s cognitive achievement A only depends on one parent’s time

input and the goods investment, when A is very sensitive to the parental time input, such

time-good tradeoff can impair the total welfare of children. In our general case, children’s

cognitive achievement decreases by about 3.29% for those living with single mothers if mar-

ried men’s household bargaining power shifts from an absolute authority to the completely

none power. The negative effects caused by a bargaining shift in favor of the female are

related with household non-labor income as well as the marginal impacts of parental time

inputs. As long as the parental time inputs have very large effects on children’s cognitive

achievement, the negative price influence caused by general equilibrium can be sizable. For

an interesting case where we set πp = πm = 0.9, an increase of married fathers’ bargaining

power even hurts the children living with single fathers.

4.4. Conclusion Remarks

We study the impact of bargaining power shift on children’s cognitive achievement in this

chapter. A global bargaining power change in favor of the females improves children’s wel-

fare in married households, but does not necessarily benefit children who live with single

mothers. After we extend the collective model with household production to the general

equilibrium framework, market equilibrium wage rates are no longer exogenous and become

dependent on the bargaining power. We show that it is possible that children’s welfare may

deteriorate when female market equilibrium wage increases as married women work less and

single parents substitute their time inputs to goods investment. Policies aiming to protect

females usually shift the bargaining power within a household towards married women, but

they may have unintended effects on the optimal resource allocations for the single mothers.

For future work, an empirical application for the single-mother case would be very interest-

ing. The effects of certain distribution factors on wage rates can have indirect impacts on


children’s outcomes. This may be important to evaluate the effectiveness of the policy.

4.5. Tables


wp

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1180

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1938

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660.

5930

0.11

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1521

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1141

0.10

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650.

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0.91

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3321

0.26

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0205

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4695

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1219

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1219

0.11

270.

150.

3131

0.27

130.

8689

0.35

660.

2635

0.03

080.

1745

0.47

230.

5894

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390.

1485

0.12

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1114

0.20

0.31

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2667

0.82

830.

3822

0.26

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0412

0.16

480.

4749

0.58

750.

1260

0.14

670.

1260

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250.

3266

0.26

210.

7891

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880.

2583

0.05

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1550

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5856

0.12

800.

1449

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0.30

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4365

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0.10

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350.

3403

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7148

0.46

550.

2530

0.07

280.

1352

0.48

250.

5814

0.13

210.

1412

0.13

210.

1059

0.40

0.34

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2482

0.67

950.

4957

0.25

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0835

0.12

520.

4848

0.57

910.

1342

0.13

930.

1342

0.10

450.

450.

3542

0.24

350.

6454

0.52

730.

2477

0.09

420.

1151

0.48

710.

5768

0.13

620.

1374

0.13

620.

1031

0.50

0.36

120.

2388

0.61

240.

5603

0.24

500.

1050

0.10

500.

4892

0.57

440.

1383

0.13

550.

1383

0.10

160.

550.

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0.23

400.

5804

0.59

490.

2423

0.11

590.

0948

0.49

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5718

0.14

050.

1336

0.14

050.

1002

0.60

0.37

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0.54

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1268

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4934

0.56

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1426

0.13

170.

1426

0.09

880.

650.

3824

0.22

440.

5194

0.66

930.

2368

0.13

780.

0742

0.49

540.

5663

0.14

470.

1298

0.14

470.

0973

0.70

0.38

960.

2195

0.49

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7093

0.23

400.

1489

0.06

380.

4973

0.56

340.

1469

0.12

780.

1469

0.09

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750.

3968

0.21

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4621

0.75

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2312

0.16

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0534

0.49

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5602

0.14

900.

1259

0.14

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0944

0.80

0.40

410.

2097

0.43

470.

7958

0.22

840.

1713

0.04

280.

5010

0.55

700.

1512

0.12

390.

1512

0.09

290.

850.

4114

0.20

480.

4081

0.84

260.

2256

0.18

260.

0322

0.50

280.

5535

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340.

1219

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0914

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1998

0.38

230.

8921

0.22

280.

1940

0.02

160.

5045

0.54

980.

1556

0.11

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1556

0.08

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950.

4262

0.19

480.

3572

0.94

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2199

0.20

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0.50

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5460

0.15

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1179

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0884

1.00

0.43

370.

1897

0.33

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0000

0.21

700.

2170

0.00

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5078

0.54

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1601

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8227

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0.09

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0.09

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0.72

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0.07

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0.23

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3763

0.20

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0.18

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0.08

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0.06

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850.

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0.19

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0.86

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0.14

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0.11

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0797

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0.90

0.39

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1952

0.46

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9094

0.18

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6000

0.70

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0.07

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0.05

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4387

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0.12

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0.53

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0.10

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0.10

0.30

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0.91

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3713

0.24

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0.16

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5343

0.64

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1063

0.12

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1063

0.09

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150.

3104

0.26

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0.39

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0.15

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0.12

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0.09

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0.25

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5521

0.69

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0.12

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0675

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6318

0.12

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0.12

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0.70

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2155

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7304

0.21

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1355

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0.63

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0.03

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0.20

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6258

0.13

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1008

0.13

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0756

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1975

0.42

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9006

0.20

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1766

0.01

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5508

0.62

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1370

0.09

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1370

0.07

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4133

0.19

290.

3966

0.94

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2001

0.18

700.

0098

0.55

160.

6222

0.13

900.

0972

0.13

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0729

1.00

0.42

010.

1883

0.37

311.

0000

0.19

750.

1975

0.00

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5524

0.62

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1410

0.09

530.

1410

0.07

15

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wp wm hp hm tp tm X A Cp Cm

0.00 0.3456 0.3872 0.7108 0.0106 0.2892 0.3441 0.0999 0.2254 0.0000 0.2498

0.05 0.3531 0.3786 0.6738 0.0268 0.2791 0.3471 0.0986 0.2229 0.0125 0.2370

0.10 0.3605 0.3700 0.6383 0.0438 0.2695 0.3502 0.0972 0.2204 0.0249 0.2242

0.15 0.3680 0.3614 0.6044 0.0614 0.2603 0.3534 0.0958 0.2180 0.0373 0.2115

0.20 0.3754 0.3529 0.5719 0.0799 0.2515 0.3568 0.0944 0.2156 0.0497 0.1988

0.25 0.3827 0.3444 0.5408 0.0993 0.2431 0.3603 0.0930 0.2133 0.0620 0.1861

0.30 0.3901 0.3359 0.5109 0.1196 0.2350 0.3640 0.0917 0.2111 0.0743 0.1735

0.35 0.3975 0.3274 0.4822 0.1409 0.2272 0.3678 0.0903 0.2089 0.0866 0.1608

0.40 0.4048 0.3189 0.4546 0.1632 0.2198 0.3719 0.0890 0.2068 0.0988 0.1483

0.45 0.4121 0.3105 0.4281 0.1867 0.2126 0.3762 0.0876 0.2047 0.1111 0.1357

0.50 0.4194 0.3021 0.4026 0.2114 0.2057 0.3807 0.0863 0.2027 0.1232 0.1232

0.55 0.4266 0.2937 0.3780 0.2375 0.1990 0.3854 0.0849 0.2008 0.1354 0.1107

0.60 0.4339 0.2854 0.3542 0.2651 0.1926 0.3904 0.0836 0.1989 0.1475 0.0983

0.65 0.4411 0.2770 0.3314 0.2942 0.1864 0.3957 0.0822 0.1971 0.1595 0.0859

0.70 0.4483 0.2687 0.3093 0.3250 0.1804 0.4013 0.0809 0.1953 0.1716 0.0735

0.75 0.4555 0.2604 0.2880 0.3577 0.1746 0.4073 0.0796 0.1936 0.1836 0.0612

0.80 0.4627 0.2522 0.2674 0.3925 0.1691 0.4136 0.0782 0.1919 0.1956 0.0489

0.85 0.4698 0.2439 0.2474 0.4295 0.1637 0.4204 0.0769 0.1903 0.2075 0.0366

0.90 0.4769 0.2357 0.2282 0.4690 0.1585 0.4275 0.0756 0.1887 0.2194 0.0244

0.95 0.4841 0.2275 0.2095 0.5113 0.1534 0.4352 0.0743 0.1873 0.2313 0.0122

1.00 0.4911 0.2193 0.1914 0.5565 0.1485 0.4435 0.0729 0.1858 0.2431 0.0000

wp wm hp' hm' tp' tm' Xp' Xm' Ap' Am' Cp' Cm'

0.00 0.3456 0.3872 0.2522 0.2199 0.2321 0.4026 0.0535 0.0390 0.1290 0.2524 0.1337 0.1462

0.05 0.3531 0.3786 0.2557 0.2162 0.2310 0.4045 0.0544 0.0383 0.1295 0.2524 0.1359 0.1436

0.10 0.3605 0.3700 0.2591 0.2124 0.2299 0.4065 0.0553 0.0376 0.1300 0.2525 0.1382 0.1410

0.15 0.3680 0.3614 0.2624 0.2085 0.2289 0.4085 0.0562 0.0369 0.1305 0.2526 0.1404 0.1384

0.20 0.3754 0.3529 0.2655 0.2043 0.2280 0.4107 0.0570 0.0362 0.1310 0.2527 0.1426 0.1359

0.25 0.3827 0.3444 0.2685 0.2000 0.2270 0.4129 0.0579 0.0355 0.1315 0.2528 0.1448 0.1333

0.30 0.3901 0.3359 0.2713 0.1954 0.2261 0.4153 0.0588 0.0349 0.1320 0.2530 0.1470 0.1308

0.35 0.3975 0.3274 0.2741 0.1906 0.2253 0.4177 0.0597 0.0342 0.1324 0.2532 0.1492 0.1282

0.40 0.4048 0.3189 0.2767 0.1856 0.2245 0.4203 0.0606 0.0335 0.1329 0.2535 0.1514 0.1257

0.45 0.4121 0.3105 0.2793 0.1803 0.2237 0.4231 0.0614 0.0328 0.1334 0.2537 0.1536 0.1232

0.50 0.4194 0.3021 0.2817 0.1748 0.2229 0.4259 0.0623 0.0322 0.1339 0.2541 0.1558 0.1206

0.55 0.4266 0.2937 0.2841 0.1689 0.2222 0.4289 0.0632 0.0315 0.1344 0.2544 0.1580 0.1181

0.60 0.4339 0.2854 0.2863 0.1627 0.2215 0.4321 0.0641 0.0308 0.1349 0.2549 0.1602 0.1156

0.65 0.4411 0.2770 0.2885 0.1562 0.2208 0.4355 0.0649 0.0302 0.1353 0.2553 0.1623 0.1131

0.70 0.4483 0.2687 0.2906 0.1493 0.2202 0.4391 0.0658 0.0295 0.1358 0.2559 0.1645 0.1106

0.75 0.4555 0.2604 0.2927 0.1419 0.2195 0.4429 0.0667 0.0288 0.1363 0.2565 0.1667 0.1081

0.80 0.4627 0.2522 0.2946 0.1341 0.2189 0.4469 0.0675 0.0282 0.1367 0.2571 0.1688 0.1056

0.85 0.4698 0.2439 0.2965 0.1258 0.2183 0.4512 0.0684 0.0275 0.1372 0.2579 0.1709 0.1032

0.90 0.4769 0.2357 0.2984 0.1169 0.2177 0.4558 0.0692 0.0269 0.1377 0.2587 0.1731 0.1007

0.95 0.4841 0.2275 0.3002 0.1075 0.2172 0.4607 0.0701 0.0262 0.1382 0.2596 0.1752 0.0982

1.00 0.4911 0.2193 0.3019 0.0973 0.2166 0.4659 0.0709 0.0255 0.1386 0.2607 0.1773 0.0958

Type 1 HH: Married HH

Father's Bargaining Power: λ

Market Wage Rates

TABLE 4.2(a): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp = 0.6, & πm = 0.8)


Market Wage Rates

Type 2 HH: Single-Person HH



0.00 0.3336 0.3437 0.7564 0.0935 0.2436 0.3153 0.0813 0.1943 0.0000 0.2032

0.05 0.3397 0.3367 0.7242 0.1101 0.2360 0.3174 0.0802 0.1922 0.0101 0.1928

0.10 0.3458 0.3298 0.6933 0.1273 0.2286 0.3196 0.0790 0.1901 0.0203 0.1824

0.15 0.3518 0.3228 0.6635 0.1452 0.2215 0.3218 0.0779 0.1880 0.0304 0.1720

0.20 0.3578 0.3158 0.6348 0.1639 0.2146 0.3242 0.0768 0.1860 0.0404 0.1617

0.25 0.3638 0.3089 0.6071 0.1833 0.2080 0.3267 0.0757 0.1840 0.0505 0.1514

0.30 0.3698 0.3020 0.5804 0.2036 0.2016 0.3292 0.0746 0.1820 0.0605 0.1411

0.35 0.3758 0.2951 0.5546 0.2248 0.1955 0.3319 0.0735 0.1801 0.0704 0.1308

0.40 0.3818 0.2882 0.5297 0.2469 0.1895 0.3347 0.0724 0.1783 0.0804 0.1206

0.45 0.3877 0.2814 0.5056 0.2700 0.1838 0.3377 0.0713 0.1764 0.0903 0.1104

0.50 0.3936 0.2745 0.4823 0.2942 0.1782 0.3407 0.0702 0.1746 0.1002 0.1002

0.55 0.3995 0.2677 0.4597 0.3196 0.1729 0.3439 0.0691 0.1729 0.1101 0.0901

0.60 0.4054 0.2609 0.4379 0.3462 0.1676 0.3473 0.0680 0.1711 0.1199 0.0800

0.65 0.4113 0.2541 0.4167 0.3742 0.1626 0.3509 0.0669 0.1694 0.1298 0.0699

0.70 0.4172 0.2474 0.3963 0.4036 0.1577 0.3546 0.0658 0.1678 0.1396 0.0598

0.75 0.4230 0.2406 0.3764 0.4346 0.1530 0.3585 0.0647 0.1661 0.1493 0.0498

0.80 0.4288 0.2339 0.3571 0.4673 0.1484 0.3627 0.0636 0.1645 0.1591 0.0398

0.85 0.4347 0.2272 0.3384 0.5019 0.1439 0.3670 0.0625 0.1630 0.1688 0.0298

0.90 0.4405 0.2205 0.3202 0.5384 0.1396 0.3717 0.0615 0.1615 0.1785 0.0198

0.95 0.4462 0.2139 0.3025 0.5771 0.1353 0.3766 0.0604 0.1600 0.1881 0.0099

1.00 0.4520 0.2072 0.2854 0.6182 0.1313 0.3818 0.0593 0.1585 0.1978 0.0000


0.00 0.3336 0.3437 0.4200 0.3800 0.1800 0.3200 0.0400 0.0275 0.0987 0.1959 0.1001 0.1031

0.05 0.3397 0.3367 0.4200 0.3800 0.1800 0.3200 0.0408 0.0269 0.0994 0.1951 0.1019 0.1010

0.10 0.3458 0.3298 0.4200 0.3800 0.1800 0.3200 0.0415 0.0264 0.1001 0.1943 0.1037 0.0989

0.15 0.3518 0.3228 0.4200 0.3800 0.1800 0.3200 0.0422 0.0258 0.1008 0.1934 0.1055 0.0968

0.20 0.3578 0.3158 0.4200 0.3800 0.1800 0.3200 0.0429 0.0253 0.1015 0.1926 0.1073 0.0948

0.25 0.3638 0.3089 0.4200 0.3800 0.1800 0.3200 0.0437 0.0247 0.1021 0.1917 0.1092 0.0927

0.30 0.3698 0.3020 0.4200 0.3800 0.1800 0.3200 0.0444 0.0242 0.1028 0.1909 0.1109 0.0906

0.35 0.3758 0.2951 0.4200 0.3800 0.1800 0.3200 0.0451 0.0236 0.1035 0.1900 0.1127 0.0885

0.40 0.3818 0.2882 0.4200 0.3800 0.1800 0.3200 0.0458 0.0231 0.1041 0.1891 0.1145 0.0865

0.45 0.3877 0.2814 0.4200 0.3800 0.1800 0.3200 0.0465 0.0225 0.1048 0.1882 0.1163 0.0844

0.50 0.3936 0.2745 0.4200 0.3800 0.1800 0.3200 0.0472 0.0220 0.1054 0.1873 0.1181 0.0824

0.55 0.3995 0.2677 0.4200 0.3800 0.1800 0.3200 0.0479 0.0214 0.1060 0.1863 0.1199 0.0803

0.60 0.4054 0.2609 0.4200 0.3800 0.1800 0.3200 0.0487 0.0209 0.1067 0.1854 0.1216 0.0783

0.65 0.4113 0.2541 0.4200 0.3800 0.1800 0.3200 0.0494 0.0203 0.1073 0.1844 0.1234 0.0762

0.70 0.4172 0.2474 0.4200 0.3800 0.1800 0.3200 0.0501 0.0198 0.1079 0.1834 0.1252 0.0742

0.75 0.4230 0.2406 0.4200 0.3800 0.1800 0.3200 0.0508 0.0193 0.1085 0.1824 0.1269 0.0722

0.80 0.4288 0.2339 0.4200 0.3800 0.1800 0.3200 0.0515 0.0187 0.1091 0.1814 0.1287 0.0702

0.85 0.4347 0.2272 0.4200 0.3800 0.1800 0.3200 0.0522 0.0182 0.1097 0.1803 0.1304 0.0682

0.90 0.4405 0.2205 0.4200 0.3800 0.1800 0.3200 0.0529 0.0176 0.1103 0.1792 0.1321 0.0662

0.95 0.4462 0.2139 0.4200 0.3800 0.1800 0.3200 0.0535 0.0171 0.1108 0.1781 0.1339 0.0642

1.00 0.4520 0.2072 0.4200 0.3800 0.1800 0.3200 0.0542 0.0166 0.1114 0.1770 0.1356 0.0622

TABLE 4.2(b): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0, πp = 0.6, & πm = 0.8)


Market Wage Rates Type 1 HH: Married HH


Market Wage Rates Type 2 HH: Single-Person HH



0.00 0.3396 0.3655 0.7332 0.0496 0.2668 0.3306 0.0906 0.2102 0.0000 0.2265

0.05 0.3464 0.3577 0.6985 0.0660 0.2580 0.3331 0.0894 0.2079 0.0113 0.2149

0.10 0.3531 0.3499 0.6652 0.0831 0.2495 0.3358 0.0881 0.2056 0.0226 0.2033

0.15 0.3599 0.3421 0.6333 0.1010 0.2413 0.3385 0.0869 0.2034 0.0338 0.1918

0.20 0.3666 0.3344 0.6026 0.1196 0.2335 0.3414 0.0856 0.2012 0.0451 0.1802

0.25 0.3733 0.3266 0.5731 0.1390 0.2260 0.3444 0.0844 0.1990 0.0562 0.1687

0.30 0.3800 0.3189 0.5447 0.1594 0.2188 0.3475 0.0831 0.1969 0.0674 0.1573

0.35 0.3866 0.3112 0.5174 0.1806 0.2118 0.3508 0.0819 0.1949 0.0785 0.1458

0.40 0.3933 0.3036 0.4910 0.2029 0.2051 0.3543 0.0807 0.1929 0.0896 0.1344

0.45 0.3999 0.2959 0.4657 0.2263 0.1986 0.3579 0.0794 0.1909 0.1007 0.1231

0.50 0.4065 0.2883 0.4412 0.2509 0.1924 0.3617 0.0782 0.1890 0.1117 0.1117

0.55 0.4131 0.2807 0.4175 0.2767 0.1864 0.3656 0.0770 0.1872 0.1227 0.1004

0.60 0.4197 0.2731 0.3946 0.3038 0.1805 0.3698 0.0758 0.1854 0.1337 0.0891

0.65 0.4262 0.2656 0.3726 0.3325 0.1749 0.3743 0.0746 0.1836 0.1447 0.0779

0.70 0.4327 0.2581 0.3512 0.3627 0.1695 0.3789 0.0733 0.1819 0.1556 0.0667

0.75 0.4393 0.2505 0.3305 0.3947 0.1642 0.3839 0.0721 0.1802 0.1665 0.0555

0.80 0.4458 0.2430 0.3105 0.4285 0.1591 0.3891 0.0709 0.1786 0.1773 0.0443

0.85 0.4522 0.2356 0.2911 0.4644 0.1542 0.3946 0.0697 0.1770 0.1881 0.0332

0.90 0.4587 0.2281 0.2724 0.5026 0.1494 0.4005 0.0685 0.1754 0.1989 0.0221

0.95 0.4652 0.2207 0.2541 0.5432 0.1447 0.4068 0.0673 0.1739 0.2097 0.0110

1.00 0.4716 0.2133 0.2365 0.5865 0.1402 0.4135 0.0661 0.1725 0.2205 0.0000


0.00 0.3396 0.3655 0.3346 0.2952 0.2065 0.3638 0.0468 0.0332 0.1140 0.2254 0.1169 0.1246

0.05 0.3464 0.3577 0.3363 0.2933 0.2060 0.3647 0.0476 0.0326 0.1146 0.2250 0.1189 0.1223

0.10 0.3531 0.3499 0.3379 0.2914 0.2055 0.3657 0.0484 0.0320 0.1152 0.2247 0.1209 0.1200

0.15 0.3599 0.3421 0.3394 0.2894 0.2050 0.3668 0.0492 0.0314 0.1158 0.2243 0.1230 0.1176

0.20 0.3666 0.3344 0.3409 0.2873 0.2046 0.3679 0.0500 0.0307 0.1164 0.2239 0.1250 0.1153

0.25 0.3733 0.3266 0.3423 0.2851 0.2041 0.3690 0.0508 0.0301 0.1170 0.2236 0.1270 0.1130

0.30 0.3800 0.3189 0.3437 0.2828 0.2037 0.3702 0.0516 0.0295 0.1176 0.2232 0.1290 0.1107

0.35 0.3866 0.3112 0.3450 0.2804 0.2033 0.3714 0.0524 0.0289 0.1182 0.2229 0.1310 0.1084

0.40 0.3933 0.3036 0.3463 0.2779 0.2029 0.3727 0.0532 0.0283 0.1188 0.2225 0.1330 0.1061

0.45 0.3999 0.2959 0.3475 0.2753 0.2025 0.3741 0.0540 0.0277 0.1193 0.2222 0.1350 0.1038

0.50 0.4065 0.2883 0.3487 0.2725 0.2021 0.3755 0.0548 0.0271 0.1199 0.2219 0.1369 0.1015

0.55 0.4131 0.2807 0.3498 0.2696 0.2018 0.3770 0.0556 0.0265 0.1205 0.2216 0.1389 0.0992

0.60 0.4197 0.2731 0.3509 0.2665 0.2014 0.3786 0.0564 0.0259 0.1210 0.2213 0.1409 0.0969

0.65 0.4262 0.2656 0.3520 0.2633 0.2011 0.3802 0.0571 0.0252 0.1216 0.2211 0.1429 0.0947

0.70 0.4327 0.2581 0.3530 0.2599 0.2008 0.3820 0.0579 0.0246 0.1221 0.2208 0.1448 0.0924

0.75 0.4393 0.2505 0.3540 0.2563 0.2005 0.3839 0.0587 0.0240 0.1227 0.2206 0.1468 0.0902

0.80 0.4458 0.2430 0.3549 0.2524 0.2002 0.3858 0.0595 0.0234 0.1232 0.2204 0.1487 0.0879

0.85 0.4522 0.2356 0.3559 0.2484 0.1999 0.3879 0.0603 0.0228 0.1237 0.2202 0.1507 0.0857

0.90 0.4587 0.2281 0.3568 0.2441 0.1996 0.3901 0.0610 0.0222 0.1243 0.2200 0.1526 0.0834

0.95 0.4652 0.2207 0.3577 0.2395 0.1993 0.3925 0.0618 0.0217 0.1248 0.2199 0.1545 0.0812

1.00 0.4716 0.2133 0.3585 0.2346 0.1991 0.3950 0.0626 0.0211 0.1253 0.2198 0.1565 0.0790

TABLE 4.2(c): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.05, πp = 0.6, & πm = 0.8)







0.00 0.3451 0.3899 0.7097 0.0148 0.2903 0.3427 0.1002 0.2255 0.0000 0.2505

0.05 0.3526 0.3812 0.6725 0.0310 0.2802 0.3456 0.0988 0.2229 0.0125 0.2376

0.10 0.3600 0.3725 0.6369 0.0479 0.2706 0.3486 0.0974 0.2204 0.0250 0.2248

0.15 0.3674 0.3639 0.6029 0.0656 0.2613 0.3518 0.0960 0.2180 0.0374 0.2120

0.20 0.3748 0.3553 0.5704 0.0841 0.2525 0.3552 0.0946 0.2156 0.0498 0.1992

0.25 0.3822 0.3467 0.5392 0.1034 0.2440 0.3586 0.0933 0.2133 0.0622 0.1865

0.30 0.3896 0.3381 0.5092 0.1236 0.2358 0.3623 0.0919 0.2110 0.0745 0.1738

0.35 0.3969 0.3296 0.4805 0.1449 0.2280 0.3661 0.0905 0.2089 0.0868 0.1612

0.40 0.4042 0.3211 0.4528 0.1671 0.2205 0.3702 0.0891 0.2067 0.0990 0.1486

0.45 0.4115 0.3126 0.4263 0.1906 0.2133 0.3744 0.0878 0.2046 0.1113 0.1360

0.50 0.4188 0.3041 0.4007 0.2153 0.2063 0.3788 0.0864 0.2026 0.1234 0.1234

0.55 0.4261 0.2957 0.3761 0.2413 0.1996 0.3835 0.0851 0.2007 0.1356 0.1109

0.60 0.4333 0.2873 0.3524 0.2687 0.1932 0.3885 0.0837 0.1988 0.1477 0.0985

0.65 0.4405 0.2789 0.3295 0.2977 0.1869 0.3937 0.0824 0.1969 0.1598 0.0860

0.70 0.4477 0.2705 0.3074 0.3285 0.1809 0.3993 0.0810 0.1951 0.1718 0.0736

0.75 0.4549 0.2622 0.2860 0.3611 0.1751 0.4052 0.0797 0.1934 0.1838 0.0613

0.80 0.4621 0.2538 0.2654 0.3957 0.1695 0.4114 0.0783 0.1917 0.1958 0.0490

0.85 0.4692 0.2455 0.2455 0.4326 0.1641 0.4181 0.0770 0.1901 0.2078 0.0367

0.90 0.4763 0.2373 0.2262 0.4720 0.1589 0.4252 0.0757 0.1885 0.2197 0.0244

0.95 0.4834 0.2290 0.2076 0.5140 0.1538 0.4328 0.0743 0.1870 0.2315 0.0122

1.00 0.4905 0.2208 0.1895 0.5591 0.1489 0.4409 0.0730 0.1856 0.2434 0.0000


0.00 0.3451 0.3899 0.2906 0.1707 0.1935 0.4523 0.0668 0.0196 0.1137 0.3305 0.1335 0.1470

0.05 0.3526 0.3812 0.2940 0.1668 0.1925 0.4544 0.0679 0.0192 0.1143 0.3313 0.1358 0.1444

0.10 0.3600 0.3725 0.2972 0.1628 0.1917 0.4566 0.0690 0.0189 0.1150 0.3321 0.1380 0.1418

0.15 0.3674 0.3639 0.3003 0.1586 0.1908 0.4589 0.0701 0.0186 0.1157 0.3330 0.1402 0.1392

0.20 0.3748 0.3553 0.3033 0.1542 0.1900 0.4613 0.0712 0.0182 0.1163 0.3339 0.1425 0.1366

0.25 0.3822 0.3467 0.3061 0.1496 0.1892 0.4638 0.0723 0.0179 0.1170 0.3349 0.1447 0.1340

0.30 0.3896 0.3381 0.3088 0.1448 0.1885 0.4665 0.0734 0.0175 0.1177 0.3360 0.1469 0.1314

0.35 0.3969 0.3296 0.3114 0.1397 0.1878 0.4692 0.0745 0.0172 0.1183 0.3371 0.1491 0.1289

0.40 0.4042 0.3211 0.3139 0.1344 0.1871 0.4721 0.0756 0.0168 0.1190 0.3383 0.1513 0.1263

0.45 0.4115 0.3126 0.3164 0.1289 0.1864 0.4752 0.0767 0.0165 0.1196 0.3396 0.1535 0.1238

0.50 0.4188 0.3041 0.3187 0.1230 0.1858 0.4784 0.0778 0.0162 0.1203 0.3409 0.1556 0.1212

0.55 0.4261 0.2957 0.3209 0.1168 0.1852 0.4818 0.0789 0.0158 0.1209 0.3424 0.1578 0.1187

0.60 0.4333 0.2873 0.3231 0.1102 0.1846 0.4853 0.0800 0.0155 0.1215 0.3439 0.1600 0.1162

0.65 0.4405 0.2789 0.3252 0.1033 0.1840 0.4891 0.0811 0.0152 0.1222 0.3455 0.1622 0.1137

0.70 0.4477 0.2705 0.3272 0.0960 0.1835 0.4931 0.0822 0.0148 0.1228 0.3473 0.1643 0.1112

0.75 0.4549 0.2622 0.3291 0.0882 0.1830 0.4973 0.0832 0.0145 0.1234 0.3492 0.1665 0.1086

0.80 0.4621 0.2538 0.3310 0.0800 0.1825 0.5018 0.0843 0.0142 0.1240 0.3512 0.1686 0.1062

0.85 0.4692 0.2455 0.3328 0.0712 0.1820 0.5066 0.0854 0.0138 0.1246 0.3534 0.1708 0.1037

0.90 0.4763 0.2373 0.3345 0.0618 0.1815 0.5117 0.0864 0.0135 0.1253 0.3557 0.1729 0.1012

0.95 0.4834 0.2290 0.3362 0.0518 0.1810 0.5172 0.0875 0.0132 0.1259 0.3583 0.1750 0.0987

1.00 0.4905 0.2208 0.3379 0.0411 0.1806 0.5230 0.0886 0.0128 0.1265 0.3610 0.1771 0.0962

TABLE 4.2(d): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp = 0.5, & πm = 0.9)







0.00 0.3461 0.3845 0.7120 0.0063 0.2880 0.3456 0.0997 0.2253 0.0000 0.2492

0.05 0.3536 0.3760 0.6751 0.0226 0.2780 0.3486 0.0983 0.2228 0.0124 0.2364

0.10 0.3610 0.3675 0.6397 0.0395 0.2685 0.3517 0.0969 0.2203 0.0249 0.2237

0.15 0.3685 0.3590 0.6059 0.0573 0.2594 0.3550 0.0956 0.2179 0.0372 0.2110

0.20 0.3759 0.3505 0.5735 0.0758 0.2506 0.3584 0.0942 0.2156 0.0496 0.1983

0.25 0.3833 0.3421 0.5424 0.0952 0.2423 0.3619 0.0928 0.2133 0.0619 0.1857

0.30 0.3906 0.3336 0.5126 0.1155 0.2342 0.3656 0.0915 0.2111 0.0742 0.1731

0.35 0.3980 0.3252 0.4839 0.1369 0.2265 0.3696 0.0901 0.2090 0.0864 0.1605

0.40 0.4053 0.3168 0.4564 0.1593 0.2191 0.3737 0.0888 0.2069 0.0987 0.1480

0.45 0.4126 0.3085 0.4299 0.1828 0.2119 0.3780 0.0874 0.2048 0.1108 0.1355

0.50 0.4199 0.3001 0.4044 0.2076 0.2050 0.3825 0.0861 0.2028 0.1230 0.1230

0.55 0.4272 0.2918 0.3798 0.2338 0.1984 0.3873 0.0848 0.2009 0.1351 0.1106

0.60 0.4344 0.2835 0.3561 0.2614 0.1920 0.3924 0.0834 0.1990 0.1472 0.0982

0.65 0.4417 0.2752 0.3333 0.2906 0.1859 0.3977 0.0821 0.1972 0.1593 0.0858

0.70 0.4489 0.2670 0.3112 0.3215 0.1799 0.4034 0.0808 0.1954 0.1713 0.0734

0.75 0.4561 0.2587 0.2899 0.3544 0.1742 0.4094 0.0794 0.1937 0.1833 0.0611

0.80 0.4633 0.2505 0.2693 0.3893 0.1686 0.4158 0.0781 0.1921 0.1953 0.0488

0.85 0.4704 0.2423 0.2494 0.4264 0.1633 0.4226 0.0768 0.1905 0.2072 0.0366

0.90 0.4776 0.2341 0.2301 0.4661 0.1581 0.4299 0.0755 0.1890 0.2192 0.0244

0.95 0.4847 0.2260 0.2114 0.5085 0.1530 0.4377 0.0742 0.1875 0.2310 0.0122

1.00 0.4918 0.2178 0.1933 0.5540 0.1482 0.4460 0.0729 0.1861 0.2429 0.0000


0.00 0.3461 0.3845 0.2138 0.2692 0.2707 0.3528 0.0402 0.0581 0.1527 0.2054 0.1338 0.1454

0.05 0.3536 0.3760 0.2175 0.2657 0.2694 0.3545 0.0408 0.0571 0.1529 0.2050 0.1361 0.1428

0.10 0.3610 0.3675 0.2210 0.2622 0.2682 0.3562 0.0415 0.0561 0.1532 0.2046 0.1383 0.1402

0.15 0.3685 0.3590 0.2245 0.2584 0.2670 0.3580 0.0422 0.0551 0.1535 0.2042 0.1405 0.1377

0.20 0.3759 0.3505 0.2277 0.2545 0.2659 0.3599 0.0428 0.0541 0.1537 0.2038 0.1428 0.1352

0.25 0.3833 0.3421 0.2308 0.2504 0.2648 0.3619 0.0435 0.0530 0.1540 0.2034 0.1450 0.1326

0.30 0.3906 0.3336 0.2338 0.2462 0.2638 0.3639 0.0442 0.0520 0.1543 0.2030 0.1472 0.1301

0.35 0.3980 0.3252 0.2367 0.2417 0.2628 0.3661 0.0448 0.0510 0.1546 0.2027 0.1494 0.1276

0.40 0.4053 0.3168 0.2395 0.2369 0.2618 0.3684 0.0455 0.0500 0.1549 0.2024 0.1516 0.1250

0.45 0.4126 0.3085 0.2422 0.2320 0.2609 0.3708 0.0461 0.0490 0.1551 0.2021 0.1538 0.1225

0.50 0.4199 0.3001 0.2447 0.2267 0.2600 0.3733 0.0468 0.0480 0.1554 0.2018 0.1560 0.1200

0.55 0.4272 0.2918 0.2472 0.2212 0.2592 0.3760 0.0474 0.0470 0.1557 0.2015 0.1582 0.1175

0.60 0.4344 0.2835 0.2496 0.2154 0.2583 0.3788 0.0481 0.0460 0.1560 0.2013 0.1603 0.1150

0.65 0.4417 0.2752 0.2519 0.2093 0.2575 0.3817 0.0488 0.0450 0.1563 0.2010 0.1625 0.1126

0.70 0.4489 0.2670 0.2541 0.2027 0.2568 0.3849 0.0494 0.0440 0.1566 0.2008 0.1647 0.1101

0.75 0.4561 0.2587 0.2563 0.1958 0.2560 0.3882 0.0500 0.0430 0.1569 0.2007 0.1668 0.1076

0.80 0.4633 0.2505 0.2583 0.1885 0.2553 0.3918 0.0507 0.0421 0.1572 0.2006 0.1690 0.1051

0.85 0.4704 0.2423 0.2603 0.1806 0.2546 0.3956 0.0513 0.0411 0.1575 0.2005 0.1711 0.1027

0.90 0.4776 0.2341 0.2623 0.1723 0.2540 0.3996 0.0520 0.0401 0.1578 0.2005 0.1733 0.1002

0.95 0.4847 0.2260 0.2642 0.1633 0.2533 0.4039 0.0526 0.0391 0.1581 0.2005 0.1754 0.0978

1.00 0.4918 0.2178 0.2660 0.1537 0.2527 0.4085 0.0533 0.0381 0.1584 0.2006 0.1775 0.0953

TABLE 4.2(e): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp = πm = 0.7)







0.00 0.3479 0.3904 0.7109 0.0122 0.2891 0.3436 0.1006 0.2257 0.0000 0.2515

0.05 0.3555 0.3817 0.6738 0.0284 0.2791 0.3465 0.0992 0.2231 0.0126 0.2386

0.10 0.3630 0.3730 0.6384 0.0452 0.2695 0.3496 0.0978 0.2207 0.0251 0.2257

0.15 0.3705 0.3643 0.6046 0.0628 0.2602 0.3529 0.0964 0.2182 0.0376 0.2129

0.20 0.3780 0.3557 0.5721 0.0812 0.2514 0.3563 0.0950 0.2159 0.0500 0.2001

0.25 0.3854 0.3471 0.5410 0.1005 0.2430 0.3598 0.0937 0.2136 0.0624 0.1873

0.30 0.3929 0.3385 0.5112 0.1207 0.2349 0.3635 0.0923 0.2113 0.0748 0.1746

0.35 0.4003 0.3299 0.4826 0.1418 0.2271 0.3674 0.0909 0.2092 0.0872 0.1619

0.40 0.4077 0.3213 0.4550 0.1641 0.2196 0.3715 0.0895 0.2071 0.0995 0.1492

0.45 0.4151 0.3128 0.4286 0.1875 0.2124 0.3758 0.0882 0.2050 0.1118 0.1366

0.50 0.4225 0.3043 0.4031 0.2121 0.2055 0.3804 0.0868 0.2030 0.1240 0.1240

0.55 0.4298 0.2958 0.3786 0.2381 0.1988 0.3851 0.0855 0.2011 0.1362 0.1115

0.60 0.4371 0.2874 0.3549 0.2655 0.1924 0.3902 0.0841 0.1992 0.1484 0.0989

0.65 0.4444 0.2790 0.3321 0.2946 0.1862 0.3955 0.0828 0.1973 0.1606 0.0865

0.70 0.4517 0.2706 0.3101 0.3253 0.1802 0.4012 0.0814 0.1956 0.1727 0.0740

0.75 0.4590 0.2622 0.2888 0.3579 0.1744 0.4072 0.0801 0.1938 0.1848 0.0616

0.80 0.4662 0.2538 0.2683 0.3926 0.1689 0.4136 0.0787 0.1922 0.1968 0.0492

0.85 0.4734 0.2455 0.2484 0.4295 0.1635 0.4203 0.0774 0.1906 0.2088 0.0369

0.90 0.4807 0.2372 0.2292 0.4689 0.1582 0.4276 0.0761 0.1890 0.2208 0.0245

0.95 0.4878 0.2289 0.2106 0.5111 0.1532 0.4354 0.0747 0.1876 0.2328 0.0123

1.00 0.4950 0.2206 0.1926 0.5563 0.1483 0.4437 0.0734 0.1862 0.2447 0.0000


0.00 0.3479 0.3904 0.1374 0.1709 0.3476 0.4522 0.0134 0.0196 0.2511 0.3304 0.1344 0.1471

0.05 0.3555 0.3817 0.1415 0.1671 0.3460 0.4543 0.0137 0.0193 0.2504 0.3312 0.1366 0.1445

0.10 0.3630 0.3730 0.1454 0.1630 0.3444 0.4565 0.0139 0.0189 0.2498 0.3321 0.1389 0.1419

0.15 0.3705 0.3643 0.1492 0.1588 0.3429 0.4588 0.0141 0.0186 0.2492 0.3329 0.1411 0.1393

0.20 0.3780 0.3557 0.1527 0.1544 0.3414 0.4612 0.0143 0.0182 0.2487 0.3339 0.1434 0.1367

0.25 0.3854 0.3471 0.1562 0.1498 0.3400 0.4637 0.0146 0.0179 0.2481 0.3349 0.1456 0.1341

0.30 0.3929 0.3385 0.1595 0.1450 0.3387 0.4664 0.0148 0.0175 0.2477 0.3359 0.1479 0.1315

0.35 0.4003 0.3299 0.1626 0.1399 0.3374 0.4691 0.0150 0.0172 0.2472 0.3371 0.1501 0.1290

0.40 0.4077 0.3213 0.1657 0.1346 0.3362 0.4720 0.0152 0.0169 0.2467 0.3383 0.1523 0.1264

0.45 0.4151 0.3128 0.1686 0.1290 0.3350 0.4751 0.0155 0.0165 0.2463 0.3395 0.1545 0.1238

0.50 0.4225 0.3043 0.1714 0.1231 0.3339 0.4783 0.0157 0.0162 0.2459 0.3409 0.1567 0.1213

0.55 0.4298 0.2958 0.1741 0.1169 0.3328 0.4817 0.0159 0.0158 0.2455 0.3423 0.1589 0.1188

0.60 0.4371 0.2874 0.1767 0.1104 0.3318 0.4853 0.0161 0.0155 0.2452 0.3439 0.1611 0.1162

0.65 0.4444 0.2790 0.1792 0.1034 0.3308 0.4890 0.0163 0.0152 0.2448 0.3455 0.1633 0.1137

0.70 0.4517 0.2706 0.1817 0.0961 0.3298 0.4931 0.0166 0.0148 0.2445 0.3473 0.1655 0.1112

0.75 0.4590 0.2622 0.1840 0.0883 0.3288 0.4973 0.0168 0.0145 0.2442 0.3492 0.1677 0.1087

0.80 0.4662 0.2538 0.1863 0.0800 0.3279 0.5018 0.0170 0.0142 0.2439 0.3512 0.1699 0.1061

0.85 0.4734 0.2455 0.1885 0.0711 0.3270 0.5067 0.0172 0.0138 0.2436 0.3534 0.1720 0.1036

0.90 0.4807 0.2372 0.1906 0.0617 0.3262 0.5118 0.0174 0.0135 0.2433 0.3558 0.1742 0.1011

0.95 0.4878 0.2289 0.1927 0.0516 0.3253 0.5173 0.0176 0.0132 0.2431 0.3583 0.1764 0.0987

1.00 0.4950 0.2206 0.1946 0.0408 0.3245 0.5232 0.0179 0.0128 0.2428 0.3611 0.1785 0.0962

TABLE 4.2(f): Numerical Solutions for the Example in 4.3.3 (Non-labor income I = 0.1, πp = πm = 0.9)






4.6. Figures

Figure 4.1: Numerical Solutions for 4.3.2, I = 0.1

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Figure 4.2(a): Numerical Solutions for 4.3.3, I = 0.1, πp = 0.6, and πm = 0.8

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Figure 4.2(b): Numerical Solutions for 4.3.3, I = 0, πp = 0.6, and πm = 0.8

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Figure 4.2(c): Numerical Solutions for 4.3.3, I = 0.05, πp = 0.6, and πm = 0.8

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Figure 4.2(d): Numerical Solutions for 4.3.3, I = 0.1, πp = 0.5, and πm = 0.9

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Figure 4.2(e): Numerical Solutions for 4.3.3, I = 0.1 and πp = πm = 0.7

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