The Dynamic Structure of Group Inequality

By Young Chul Kim M.A., Brown University, 2005 M.P.P., KDI School of Public Policy and Management, 2004 B.S., Seoul National University, 2000

Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Economics at Brown University

Providence, Rhode Island May 2009

c Copyright 2009 by Young Chul Kim °

This dissertation by Young Chul Kim is accepted in its present form by the Department of Economics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.

Date Glenn C. Loury, Director

Recommended to the Graduate Council

Date Oded Galor, Reader

Date Kenneth Y. Chay, Reader

Approved by the Graduate Council

Date Sheila Bonde Dean of the Graduate School

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VITA The author was born on May 3rd, 1976 in Seoul, the Republic of Korea. He received his Bachelor of Science Degree in Electronics from Seoul National University in Feb 2000. After spending two years in compulsory military service, he entered KDI School of Public Policy and Management in 2002 and received a Master of Public Policy Degree in Economics and Public Policy in 2004. He also achieved a Master Diploma in Economics from Seoul National University in 2004. While studying in Seoul, he worked as a junior researcher for the Korea Development Institute in 2003 and 2004. In 2004, he was selected as a Fulbright Fellow for the Ph.D. study in the United States. He received his Master of Arts Degree in Economics from Brown University in 2005. He participated in the Exchange Scholar Program and studied in the Department of Economics at Harvard University in 2005-2006. He received his Doctor of Philosophy Degree in Economics from Brown University in May 2009. He was honored to receive the Fulbright Grant for 2004-2006, Distinguished Fellowship and Award for Academic Excellence from KDI School of Public Policy and Management in 2002, a fellowship from the Woosan Scholarship Foundation for 2003 and 2004, a fellowship from the Doorae Scholarship Foundation for 2003 and 2004, a Brain Korea 21 Scholarship from the Ministry of Education of the Republic of Korea in 2003, and a Merit Dissertation Fellowship Award from the Graduate School at Brown University in 2008.

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PREFACE Socioeconomic disparities between social groups constitute a challenge in many countries around the world. The following are just a few examples: Caste in India, Chinese and Malays in Singapore and Malaysia, blacks and whites in the United States and South Africa, indigenous people and European descendants in the Americas and immigrants and non-immigrants in many Western countries. Even though social groups may educate their children within an identical educational system and work in the same market economy, the skill achievement ratios and wage levels are significantly different. The overt discrimination against some social groups (such as Afro-Americans in the United States during the Jim Crow period) played an important role in deepening the disparities. While the overt discrimination practices have diminished over the last decades in many societies, the persistent group disparities are not yet solved in the modern societies. Even though the cases are distinct from one another, two salient features of the issue are consistent throughout all the cases: divided social interactions between social groups and negative stereotypes (or stigma) imposed on disadvantaged groups. One’s economic success is significantly affected by the quality of his social network, both over the education period and over the working period. Also, one’s individual reputation in the market is strongly associated with the collective reputation of his identity group. The young members of the disadvantaged groups, who do not get enough support from their social network and are less favorably treated by employers in the market, have less incentive to invest in skill achievements. Thus, the inequality between social groups does not disappear over time. In the dissertation entitled The Dynamic Structure of Group Inequality, I focus on these two economic externalities that may serve as root causes of persistent group disparities: social network externality in Chapter 1 and group reputation externality in Chapter 2. Constructing the dynamic models, I suggest the concepts of Network Trap and Reputation Trap, in which a social group cannot improve its social network quality and collective reputation without any external interventions such as affirmative action. Extending the discussion in Chapter 2, in Chapter 3, I discuss social activities

v

such as passing and selective out-migration for the talented members of the disadvantaged groups to take in order to prevent discrimination in the market. In Chapter 1, titled Lifetime Network Externality and the Dynamics of Group Inequality, I explore the dynamic structure generated by social network externalities. I examine the interaction between education-period network externality (Loury 1977, 1981) and what I call Lifetime Network Externalities that are effective over the working period (Granovetter 1973). The former operates as a a historical force that restricts the improvement of a group’s skill level within its level of network quality: the better the network quality, the more young members of the group invest in skills; the worse the network quality, the fewer invest in skills. The latter operates as a mobilization force that can lead a social group to enhance (or shrink) skill investment by holding an optimistic (or pessimistic) view about the future network quality. If group members believe that the network quality will be better in the future, more young members invest in skills because the expected benefits of skill investment increase with their belief in the improved network effects. The optimistic belief is self fulfilled by higher skill investment and better network quality in the future. The historical force generates an agglomeration effect and the mobilization force generates a self-fulfilling effect. Understanding these two dynamic forces, I identify the Network Trap and explore egalitarian policies to mobilize the disadvantaged groups out of the trap. I also examine the macroeconomic effects of inequality. The social capital approach suggests the positive effect of equality on economic growth in later stages of economic development, and the positive effect of inequality in the early stage of economic development, consistent with Galor and Zeira (1993). In Chapter 2, titled Group Reputation and the Dynamics of Statistical Discrimination, which is coauthored with Glenn Loury, I examine the externality of group reputation and the collective action of reputation building. An individual’s reputation - the employer’s belief about his true productivity - is affected by the reputation of the group that he belongs to when precise information on an individual is not available. Employers looking to hire new employees with limited information about job candidates will have more incentive to hire a worker from a group with a reputation vi

for high productivity, and will have less incentive to hire a worker from a group with a reputation for low productivity. This is often referred to as the “statistical discrimination” practice (Arrow 1973; Coate and Loury 1993). This reputation externality generates an interesting aspect of group members’ behavior - collective action to build up (or destroy) the group’s reputation together. For example, once group members believe that they can improve the group’s reputation, a higher fraction of young members will invest in skills because the benefits they expect from skill achievement increase with their belief in employers’ more favorable hiring in the future. As this is repeated in the following generations, the group’s reputation improves over time through this self-fulfilling process. However, this collective action is not always feasible. If the group’s initial reputation is very poor, it is impossible to improve the reputation just by changing young members’ expectations about the future. Thus, it would be interesting to identify the deterministic ranges, in which the reputation is fixed at a high or low level, and the overlap range, in which the reputation can be built up, or destroyed, by the collective action of group members. The lower deterministic range is defined as the Reputation Trap. Developing this dynamic model, we can explain why a certain group is at a high reputation state and another group is at a low reputation state depending on their initial positions and available dynamic paths to stable equilibria. In Chapter 3, titled Group Reputation and the Endogenous Group Formation, which is also coauthored with Glenn Loury, I discuss the identity-switching behaviors of the most talented young members of a stereotyped population in a reputation trap, who have greater incentives to differentiate themselves from the stereotyped masses. I have presented three different identity-switching activities for the differentiation: passing, partial passing and elite culture development. Passing for an advantaged group would be the most efficient way for differentiation if the identity switching cost is not large. The most talented who succeed in passing can take advantage of the superior collective reputation of the group immediately. When passing is not available, the talented members of the stereotyped population may consider “partial” passing. They “pass” for a better-off subgroup with unique cultural traits in order to send signals of their higher productivity to employers (Loury vii

2002). The partial passing is a common activity among physically marked stereotyped people such as blacks in the United States. Finally, the most talented individuals may develop distinguished cultural indices that are not affordable to the less talented members of the stereotyped population (Fang 2001). The most talented members adopting the indices may form an elite cultural subgroup, whose members are distinguished from the rest in the population and will be preferentially treated by employers. In this dissertation, I have developed two different dynamic models to explain the emergence and the persistence of group disparities. I have a plan to integrate the two channels of group inequality evolution, network externality and reputation externality, in one dynamic system. The concept of Double Trap will be suggested, in which a social group is trapped both by the negative network influences and the reputational disadvantages. In this way, we may deepen our understanding of the dynamic structure of group inequality. We may discuss how integration between social groups may eliminate the negative stereotype imposed on the disadvantaged group (Chaudhuri and Sethi, 2008) and how the preferential treatment by employers may help the group out of the network trap via encouraging young members’ collective action to build up the group’s reputation. Noting the importance of social networks and collective reputation to one’s economic success and the evolution of group disparities, the lack of theoretical works along these lines is awaiting research in the field of economics.

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ACKNOWLEDGEMENTS I am deeply indebted to my thesis advisor Glenn Loury for his thoughtful guidance and continual encouragement since the early stage of my dissertation. When I switched my research topic to group inequality in the middle of my Ph.D. study, he willingly became my advisor and strongly expressed his confidence in my potential and talents to the faculty in my department. I am highly obliged to two readers, Oded Galor and Kenneth Chay, for their sincere supports and valuable comments. The dynamic models in my dissertation would not have been as much developed without Oded Galor’s continuous attention to my work. Kenneth Chay was a most valuable source for sharpening the empirical work in my thesis. He also provided me deep insights in understanding many issues related to inequality. It has been a privilege to have these three great economists on my committee and to have learned so much from each of them in the course of my last years at Brown. I would like to emphasize that I benefited greatly from discussions with Andrew Foster, Woojin Lee, Jung-Kyu Choi and Yeon-Koo Che, who provided me important suggestions for improvement. This dissertation was enriched by conversations with my best friends and colleagues, Toru Kitagawa, Norov Tumennasan, Sung Jae Jun, Daeho Kim, Youngsuk Lee and Hongwoo Kwon. Their encouragement and emotional support enabled me to go through the long trip of writing the dissertation. My special thanks must go to Rajiv Sethi, an economist at Columbia University. Since I met him at a conference, he has been the greatest supporter and a trusted advisor. His comments on my earlier work encouraged me enough to complete this dissertation. I am deeply grateful to my mentor Jean, who sacrificed enormous hours for me. Without her unconditional support in my hardship, I must have quit my study at Brown. I can never thank her enough. At the root of everything is my family - my parents Dong-Ho Kim and Young-Hee Kim and my brothers, Young-Sik and Chang-Hoon. They have always believed in me, and gave me the confidence that were necessary to reach this final line. Lastly, but most importantly, I thank my wife Sujin and my daughter Arin: Sujin for her love,

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patience and encouragement over the whole course of my Ph.D. study; and Arin, who was born just before the qualifying exam and had to wait for more than a month before her dad realized that he had become a father. They have been the most fun and joy in my life. I greatly acknowledge financial support from the J. William Fulbright Fellowship in my first two years at Brown and the Merit Dissertation Fellowship in my last year at Brown.

x

Contents Vita

iv

Preface

v

Acknowledgements

ix

1 Lifetime Network Externality and the Dynamics of Group Inequality

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Basic Structure of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

Dynamic Model with Network Externalities . . . . . . . . . . . . . . . . . . . . . . .

13

1.3.1

Education Period Network Externality and Evolution of Group Skill Level . .

13

1.3.2

Lifetime Network Externality and Evolution of Group Benefits of Investment

14

1.3.3

Dynamic System with Network Externalities . . . . . . . . . . . . . . . . . .

17

Homogeneous Group Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.4.1

Steady States and Economically Stable States . . . . . . . . . . . . . . . . . .

18

1.4.2

Overlap and Deterministic Ranges . . . . . . . . . . . . . . . . . . . . . . . .

22

1.4.3

Mobilization Force and Historical Force . . . . . . . . . . . . . . . . . . . . .

23

1.4.4

Size of Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Heterogeneous Group Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.5.1

Heterogeneous Economy with Total Segregation . . . . . . . . . . . . . . . . .

28

1.5.2

Heterogeneous Economy in General

32

1.4

1.5

xi

. . . . . . . . . . . . . . . . . . . . . . .

1.5.3

Social Consensus and Network Trap . . . . . . . . . . . . . . . . . . . . . . .

42

Egalitarian Policies in Network Trap . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

1.6.1

Integration Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

1.6.2

Affirmative Action Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

1.6.3

Policy Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Macroeconomic Effects of Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

1.7.1

Multiple Equilibria as Different Development Stages . . . . . . . . . . . . . .

50

1.7.2

Positive Effect of Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

1.8

Application: Regional Group Inequality in South Korea . . . . . . . . . . . . . . . .

54

1.9

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

1.10 Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

1.6

1.7

2 Group Reputation and the Dynamics of Statistical Discrimination

73

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

2.2

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

2.3

Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

2.4

Dynamic Reputation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

2.4.1

Group Reputation and Individual Reputation . . . . . . . . . . . . . . . . . .

82

2.4.2

Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

2.4.3

Simple Reputation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

2.4.4

Generalization of Simple Reputation Model . . . . . . . . . . . . . . . . . . .

93

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

2.5.1

US Racial Disparity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

2.5.2

Male-Female Disparity in Patriarchal Societies . . . . . . . . . . . . . . . . .

97

Monopolistic Principals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

2.5

2.6

2.6.1

Adjustment of Reputation Threshold . . . . . . . . . . . . . . . . . . . . . . . 100

2.6.2

Subsidy of Training Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.6.3

Improvement of Screening Process . . . . . . . . . . . . . . . . . . . . . . . . 102 xii

2.7

Egalitarian Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.7.1

Colorblind Hiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.7.2

Strict Quota System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.7.3

Asymmetric Training Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.8

Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

2.9

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.10 Appendix A: Market Learning Process . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.11 Appendix B: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3 Group Reputation and the Endogenous Group Formation

126

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.2

Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.3

3.4

3.2.1

Employers’ Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.2.2

Workers’ Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.3.1

Dynamics with Identity Switches Restricted . . . . . . . . . . . . . . . . . . . 136

3.3.2

Dynamics with Identity Switches from Type j to Type i . . . . . . . . . . . . 139

Endogenous Group Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.4.1

Group i Equilibrium Path with Skill Inflows from Group j . . . . . . . . . . . 147

3.4.2

Search For Final State Given Initial State (Πb0 , Πa0 ) . . . . . . . . . . . . . . . 149

3.4.3

Autonomous Emergence of an Elite Group among the Stereotyped . . . . . . 156

3.5

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.6

Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Reference

165

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List of Figures Figure 1.1

Steady States in the Homogeneous Economy .................................................

172

Figure 1.2

Equilibrium Paths in the Homogenous Economy ...........................................

173

Figure 1.3

Comparison of Three Distinct Cases ..............................................................

174

Figure 1.4

Economically Stable States with Total Segregation .......................................

175

Figure 1.5

Manifold Ranges and Overlaps with Total Segregation ..................................

176

Figure 1.6

Partial Steady States (si∗ , σ i∗ ) Given sj ........................................................

177

Figure 1.7

Global Steady States with η and β Given ......................................................

178

Figure 1.8

Steady States for Each Level of η (Given Small β 1 ) ......................................

179

Figure 1.9

s˙ 1t and σ˙ t1 Demarcation Surfaces .....................................................................

180

Figure 1.10

Stable States and Location of Stable Manifold ..............................................

181

Figure 1.11

Stable Manifolds and Folded Overlaps ...........................................................

182

Figure 1.12

Sable States and Manifold Ranges for Each Level of η (Given Small β 1 ) ......

183

Figure 1.13

Integration Effect: Economic State Move as η Declines .................................

184

Figure 1.14

Affirmative Action Policies .............................................................................

185

Figure 1.15

Equalization Policy Implementation ...............................................................

186

Figure 1.16

Macroeconomic Effects of Inequality ..............................................................

187

Figure 1.17

[Application] Urbanization and Regional Groups ...........................................

188

Figure 1.18

[Application] Evolution of Regional Group Disparity .....................................

189

Figure 1.19

[Application] College Advancement Rate .......................................................

190

xiv

Appendix Figure 1.1

Size of Overlap in a Simplified Economy .........................................

191

Appendix Figure 1.2

Manifold Ranges with No Lifetime Externalities .............................

192

Appendix Figure 1.3

Regional Group Disparity in S. Korea .............................................

193

Appendix Figure 1.4

Regional Voting in Presidential Elections ........................................

194

Figure 2.1

Multiple Steady States with Noisy Signals .....................................................

195

Figure 2.2

Unique Steady State with Better Signals .......................................................

196

Figure 2.3

Phase Space in Simple Model .........................................................................

197

Figure 2.4

Differential Equations in Modified Simple Model ...........................................

198

Figure 2.5

Phase Space in Modified Simple Model ..........................................................

199

Figure 2.6

Phase Diagram in Generalized Model ............................................................

200

Figure 2.7

Equilibrium Paths in Generalized Model .......................................................

201

Figure 2.8

Development of US Racial Disparity ..............................................................

202

Figure 2.9

Male-Female Disparity in Patriarchal Society ................................................

203

Figure 2.10

Strategy of Monopolistic Principals ...............................................................

204

Figure 2.11

Egalitarian Policies .........................................................................................

206

Appendix Figure 2.1

Market Learning Process .................................................................

208

Appendix Figure 2.2

Proof of Lemma 1 ............................................................................

209

Figure 3.1

Costs Distribution and Newborns Decision ......................................................

210

Figure 3.2

Dynamics with No Switches between Types ....................................................

211

Figure 3.3

Dynamics of Group i with Inflows from Group j .............................................

212

Figure 3.4

Dynamics of Group j with Outflows to Group i ..............................................

213

Figure 3.5

Group i Equilibrium Path with Inflows from Group j .....................................

214

Figure 3.6

State Evolution From Initial State (Πb0 , Πa0 ) ....................................................

215

Figure 3.7

Endogenous Group Formation .........................................................................

216

Figure 3.8

Autonomous Emergence of Elite Group ...........................................................

217

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Chapter 1

Lifetime Network Externality and the Dynamics of Group Inequality 1.1

Introduction

The acquisition of human capital occurs within a social context, and can be facilitated by access to the right social networks. I examine one mechanism by which such social network externalities affect the evolution of economic inequality between social groups.1 The interaction between network externalities during the education period and during the working period produces a unique dynamic structure for the evolution of group inequality. The education period network externalities operate as a historical force that restricts a group to be subject to the current network quality, while the working period (or lifetime) network externalities operate as a mobilization force that leads a group to enhance (or shrink) the skill investment activities by holding an optimistic (or pessimistic) view about the future network quality. In the model to follow I identify what I will call the Network Trap, in which the human capital development of a social group is trapped by the externality of social networks. Also, I examine possible egalitarian policies to mobilize a disadvantaged group out 1 Another approach to explaining group inequality explores the discrimination story: either taste-based discrimination (Becker 1957) or statistical discrimination through imperfect information (Arrow 1972, Phelps 1972 and Coate and Loury 1993). The work in this chapter focuses on the social network externality not because discrimination is a less important issue, but because the purpose of the work is to explore the dynamic structure of group inequality through the network externalities. A companion paper (Kim and Loury 2008), which is in Chapter 2, explores the dynamic structure of group inequality due to reputational externalities in the context of statistical discrimination.

1

2

of the trap and improve its skill investment activities. Considering that human capital is a prime engine of economic growth in the modern economy, I describe the macroeconomic effects of group inequality on economic development (Loury 1981, Galor and Zeria 1993). The model I create finds a positive effect of equality on the economic growth in most developmental stages. Unlike the previous literature, this conclusion is derived without imposing the standard assumption of imperfect credit markets. Therefore, the model implies a positive effect of equality even in an economy with no credit constraints: equality, a more equal distribution of social network capital in this study, has a positive effect on the economic development, by helping the disadvantaged groups to move out the network trap and enhance the skill investment activities, even in the society with public provision of schooling. Lifetime Network Externality Socioeconomic disparities between social groups constitute a challenge in many countries around the world. Even though social groups may educate their children within an identical educational system and work in the same market economy, their skill achievement ratios and wage levels can be significantly different. It is hard to conceive of a single root cause of inequality between groups since the manner in which social groups are formed is unique to each society. For instance, groups form along racial line in societies such as the Unites States, South Africa, New Zealand and Australia, but form along religious lines in Turkey, Iraq, Pakistan, Northern Ireland and Israel. While ethnicity is important in some countries such as Singapore, Indonesia, and the Balkan states, we often see caste-like social division in India and Gypsies in Europe. In many western countries, the population is divided into immigrants and non-immigrants, while population in the Americas is divided into indigenous peoples and European descendants.2 Even though these cases are distinct from one another, a salient feature of the issue is consistent throughout all the cases: divided social interactions between groups occurs over the whole lifetime. 2 Also, groups have formed along linguistic lines in nations such as Canada, Switzerland, and South Africa (AngloAfrican and Afrikaners). Region of family origin influences the social interactions in nations such as Spain, the United Kingdom, and South Korea (Youngnam, Honam).

3

The social network externality around the skill acquisition period and the consequent development bias has long been discussed since the pioneering work of Loury (1977). In his theory, a human being is socially situated in that familial and communal resources explicitly influence a person’s acquisition of human capital through various routes, including the constraints of training resources, of nutritional and medical provision, of after-school parenting, of peer effects, of role models, and even of the psychological processes that shape one’s outlook on life.3 A number of subsequent theoretical works discussed the development bias, emphasizing the network externalities over the skill acquisition period, including Akerlof (1997), Lundberg and Startz (1998) and Bowles, Loury and Sethi (2007).4 However, the theoretical work continues to confront some empirical evidence that it cannot fully embrace. Consider a few examples: 1. Over the industrialization process of South Korea in the 1970s and 1980s, the socioeconomic disparity between Youngnam and Honam regional groups increased significantly, even when the educational system was strictly based on the public provision of schooling, and when the familial and communal environment did not carry a big difference between two regional groups: both groups were in an early stage of development, poor and low skilled, and shared a similar cultural base. It is often argued that social connections and mentoring networks played a key role in the emergence of group disparity in South Korea (Ha 2007, Kim 2002). 2. In France, where the public school system is well established, the violence of second generation immigrant youth in 2005 caused nearly 9,000 cars to be torched and dozens of buildings damaged in a riot. Most of the rioters were unemployed youth who arguably suffered from social exclusion in French society, and from the lack of a job network. 3 His theory is supported by numerous empirical work, which includes the peer influence (Anderson 1990), community effects (Cutler and Glaeser 1997, Weinberg et al. 2004), racial network effect (Hoxby 2000, Hanusheck, Kain and Rivkin 2002) and academic peer effect (Kremer and Levy 2003, Zimmerman and Williams 2003). 4 Akerlof (1997) provides a theoretical argument, which states that concerns for status and conformity are the primary determinant of an individual’s educational attainment, child bearing, and law-breaking behavior. Lundberg and Startz (1998) argue that group disparities in earnings can persist indefinitely when the average level of human capital in a community affects the accumulation of human capital of the following generations. The recent work by Bowles, Loury and Sethi (2007) shows how group disparity can persist in a highly segregated society, and how it can disappear as integration is facilitated, in the presence of network externality over the skill acquisition period.

4

3. In his examination of the jobless black underclass in New York City, Waldinger (1996) concludes that black unemployment originates from the lack of access to the ethnic networks through which workers are recruited for jobs in construction and service industries.5 These examples illustrate the importance of social network externalities that operate beyond the education period – what I am calling the Lifetime Network Externalities. This effect has been emphasized in numerous empirical papers in the economics and sociology literature. The sociologist Granovetter (1975) has been one of the pioneers of this line of inquiry. His work sheds light on the role played by interpersonal relationships, such as friends and relatives, in channeling information about jobs and job applications. He and other researchers have found that approximately fifty percent of all workers employed found their jobs on the basis of recommendation and word-ofmouth (Granovetter 1973, Myers and Shultz 1951, Rees and Shultz 1970, Campbell and Marsden 1990).6 The role of ethnic networks in job search is emphasized in numerous empirical work such as immigrants in Australia (Mahuteau and Junankar 2008), Mexican immigrants in the US (Livingston 2006 and Munshi 2003) and migrants to urban centers in India (Banerjee 1981, 1983).7 The effects of social network go beyond just finding jobs. Friends and acquaintances of the same occupation may help workers to increase productivity and decrease the psychological stress of maintaining the occupation. Empirical papers show that a worker with richer social networks can be more efficient in 5 In the postwar era of New York, the manufacturing industries where the blacks occupied jobs moved out or eroded while the job opportunities in the service sector continued to grow with whites moving out of the sector. The immigrants who entered the low skilled service sector expanded their economic base through the ethnic networks, while the native blacks left behind jobless. Given employers’ preference for hiring through networks, information about job openings rarely penetrated outside the immigrant groups (Waldinger 1996). This empirical evidence brings a very different perspective from the spatial mismatch hypothesis (Kain 1968, Raphael 1998, Ross 1998), which insists that blacks in central cities lost jobs as employment moved to suburbs. The case in New York City reveals that blacks lost jobs even when whites moved out leaving jobs for minorities in the cities. 6 Other researchers concludes that, among many different job search methods, personal connection of friends or relatives is most widely used among unemployed youth in the US (Holzer 1987,1988, and Blau and Robins 1990), and in the UK (Gregg and Wadsworth 1996) and in Egypt (Assaad 1997, Wahba and Zenou 2005): Holzer (1988) finds 85.2% of jobseekers used friends/relatives ties, 79.6% used direct application without referral, 53.8% used state agency, and 57.8% used newspaper advertisement.In their study, the acceptance rate of job offers obtained through personal connection is highest (eg. about 82 percent in Holzer (1988)), implying that job offers through personal connection generally have higher wages or more appealing nonwage characteristics. 7 Observing the evidence, Montgomery (1991) constructs a theoretical model that explains why firms hiring through referral might earn higher profits and why workers who are well connected might fare better than poorly connected workers. Montgomery (1992) suggests another interesting model in which the widespread use of employee referrals, combined with a tendency to refer others within their own social network, might generate persistent inequality between groups of workers.

5

contacting business partners (clients and customers) and handling specific work troubles (Fafchamps and Minten 1999, Laband and Lentz 1995, 1999, Falk and Ichino 2005, Khwaja e al. 2008). The mentoring effects of the social network can help to increases job satisfaction, to minimize the turnover rate (Rockoff 2008, Castilla 2005, Cardoso and Winter-Ebmer 2007, Bilimoria et al. 2006), and to heighten the recognition of opportunities in the entrepreneurial process (Ozgen and Baron 2007). The empirical work suggests that the better the quality of one’s social network, the higher the benefits one can expect, and, consequently, the more incentive one has to invest in the acquisition of skills. Dynamic Structure of Group Inequality We conclude that both kinds of externalities – those operating during the education period and those at work over the course of a worker’s lifetime – affect a social group’s overall skill investment rate.8 As mentioned, in this chapter, I explore the dynamic structure of group inequality generated by the interaction between these two types of network externalities. These two effects operate via different channels. With the education period network externality, change in a group’s status tends to be subject to the “past”: by altering skill investment cost, the current stock of network human capital directly affects the investment rate in a newborn cohort. By contrast, with the lifetime network externality, change in a group’s status tends to be subject to the “future”: by altering the future benefits anticipated to accrue from skill acquisition, the expected success of one’s network influences skill investment in an entering cohort. This latter effect implies a unique feature of the dynamic structure: the possibility of workers acting together to improve, or deteriorate, the quality of a group’s social network. For instance, suppose that a group’s network quality is relatively poor, but that a newborn cohort happens to believe the quality of group’s network will be better in the future. If this belief leads more newborn group members to acquire skills, then the next newborn cohort will find the overall network quality 8 For example, when a group’s social network contains more highly skilled members, then more of its newborns will invest in skills – not only because they have lower costs over the skill acquisition period, but also because they expect greater benefits from a given skill investment to accrue over their lifetimes.

6

has improved because of the enhanced skill investment of the previous cohort. If the next newborn cohort, and the following cohorts, continue to hold the optimistic view of the future, they will keep the enhanced skill investment rate and the quality of group’s social network will improve over time thereby justifying the optimistic beliefs of earlier cohorts. However, suppose that the newborn cohort held a pessimistic view that the network quality will be even worse in the future. Fewer members of the newborn cohort will invest in the skill achievement because the expected benefits have declined. As the following cohorts continue to hold the pessimistic view, the network quality will be deteriorated over time. So, this pessimistic belief could also be self-fulfilling. However, collective action to influence such beliefs may not be feasible for all social groups with unequal network quality. The potential impact of altering beliefs is restricted by the strength of education period network externalities. That is, collective action through optimism or pessimism cannot play any role when the quality of network is too good or too bad.9 Therefore, the analysis of the dynamic structure of network externalities focuses on the identification of the following two ranges: (1) the network quality range mainly governed by the historical force of the education period network externality, and (2) the network quality range mainly governed by the mobilization force of the lifetime network externality. The former is defined as deterministic range, and the latter as overlap, as Krugman (1991) denotes in his argument for the relative importance of history and expectations. In the dynamic system developed in this chapter, there exists a unique equilibrium path in a deterministic range, and there are two equilibrium paths available in an overlap in which a group’s expectation toward the future determines the path to be taken.10 This insight is expanded 9 Suppose that a group’s network quality is very poor. The newborn group members may consider enhancing the skill investment rate by holding the expectation that the group’s network quality will be improved over time and their skill investment will be paid back in the future. However, they will realize very soon that the scenario would never occur in the real world: the following generations cannot invest enough due to the serious adverse effects of poor quality network externality over the education period, and, consequently, the network quality cannot be improved substantially even in the far future. This is the situation of “the past” that traps the disadvantaged group. The opposite scenario is plausible for the case of network quality that is too good. The newborn group members may consider lowering their skill investment rate by holding a pessimistic expectation toward the future. However, they will soon realize that the scenario would never occur because a sufficient number of following generations would continue to invest, due to the good quality network externality over the education period. 10 Adsera and Ray (1998) argue that overlap is generated only when agents can have an incentive to choose the option that offers less appealing benefits at the moment of decision. In the example of Krugman (1991) regarding industry specialization, overlap is generated because agents can have an incentive to choose the option that offers even loss at the moving moment, because its cost is lower than the cost of moving in the future. In my model, the

7

to the multi-group economy, defining notions of social consensus and folded overlap. In a folded overlap, two or more equilibrium paths can exist. The path to be taken is determined by the social consensus, which is a combination of groups’ expectations toward the future.11 An interesting feature of the multi-group economy is the existence of a network trap, where a social group maintains a high skill investment rate, and another group is trapped by the “past,” that is, the adverse effects of bad-quality education period network externality. To mobilize the disadvantaged group out of the trap, two egalitarian policies are examined: integration and affirmative action. If the disadvantaged group is a minority, the integration policy alone can save the group out of the trap. If it is not, integration may cause both groups to fall down to the lower investment rates, as discussed in Bowles et al. (2007). In this case, a combination of the two policy measures may help to solve the problem. Macroeconomic effects of Inequality Finally, I examine the macroeconomic effects of group inequality. Since human capital has been the prime engine for economic growth in the modern economy, aggregate skill investment activities can be directly interpreted as a stage of economic development. Thus far, most of the literature has discussed the topic under the assumption of an imperfect credit market (Loury 1981, Galor and Zeira 1993, Benabou 1996, Durlauf 1996).12 The work in this chapter shows the positive effects of equality on economic growth in most development stages, consistent with Galor and Zeira (1993), even without imposing the assumption of an imperfect credit market. When social network capital, the average human capital in one’s social network, is more equally distributed, more social groups can be encouraged to develop their skill investment ratios, moving out of the network trap incentive is originated by the nature of the overlapping generation structure. Since agents are given only one chance to choose their occupational type at the early stage of their lives, they choose a type that gives less appealing benefits at the moment of skill investment decision, expecting the average lifetime benefits accumulated in the future. 11 For example, suppose a two-group economy. In a folded overlap with four equilibrium paths available, the economy may evolve to the highest (lowest) level of development with both groups’ optimistic (pessimistic) expectations. If one group holds an optimistic expectation and the other holds a pessimistic expectation, the economy will evolve to a mediocre level of development with unequal distribution of wealth between two groups. By identifying folded overlaps and deterministic ranges, we can analyze the dynamic process of the group inequality evolution: conditions under which group disparities grow and conditions under which groups’ network qualities converge. 12 Bowles et al. (2007) have successfully modeled the intergenerational human capital externalities without imposing the imperfect credit market assumption.

8

where they had fallen. It is noteworthy that, contrary to the previous theoretical works (Galor and Moav 2004), equality of social network capital can enhance the process of economic development even in the society with a perfect public school system, or in well-developed countries where credit constraints are no longer binding for human capital investment. In addition, this social capital approach demonstrates the positive effects of inequality on economic growth in the early stage of economic development. In the early stage of development, the concentration of social network capital to selective groups may help the economy move out of the low skill investment steady state by giving opportunities for the groups to take the collective action needed to improve their skill investment ratios. Overall, the model is consistent with the empirical finding that income tends to be more equally distributed in developed countries than in less developed countries, a phenomenon many economists, including Kuznets (1955), tried to explain. By departing from the poor equal society with low skill investment ratios, the economy may move to a more developed stage with some groups in the network trap, where selective groups maintain high investment ratios while others continue the low skill investment activities. Egalitarian policies can move the economy towards the high investment equal society, in which all social groups participate in the high skill investment activities. As an application of the dynamic network model, I address the regional group inequality issue that emerged in South Korea during its industrialization process. Both Youngnam and Honam groups were in the low investment steady state after the Korean War (1950-53). The Youngnambased regime in the 1960s through the 1970s helped the regional group to be more successful in an initial state-led industrialization process and, consequently, to move into an overlap area, where the group was given an advantaged position to exercise collective action to increase the group’s human capital investment and build a better network quality. After rapid industrialization and urbanization in the 1970s and 1980s, Homan was identified as being in a network trap, where the group’s skill investment ratio was significantly less than that of the Youngnam group. As the between-group social interactions proceeded and the political power was transferred to Homan in the 1990s, younger

9

members of the Honam significantly enhanced their skill investment activities. Therefore, both the dominating position of Youngnam group in the early stage of economic development, and the more equal distribution of social network capital in the later stage of economic development, promoted the greater human capital investment of the economy and the faster economic growth. This chapter is organized into the following sections: Section 1.2 describes the basic structure of the model; Section 1.3 develops the dynamic model with network externalities and economic players’ forward-looking decision making; Section 1.4 provides an analysis on the homogeneous group economy; Section 1.5 provides an analysis on the multiple group economy; Section 1.6 examines the egalitarian policies to mobilize disadvantaged groups out of the network trap; Section 1.7 examines the macroeconomic effects of inequality; Section 1.8 presents an application of the dynamic model on the regional group disparity in South Korea; and Section 1.9 contains the conclusion.

1.2

Basic Structure of the Model

The analysis in this chapter focuses on the two-group economy because the most interesting features of dynamic structure associated with social interactions between groups are contained in the twogroup economy. The way to extend to arbitrary n-group economy is discussed in Section 1.7.1. The two social groups are denoted by group one and group two. Population shares are denoted by β 1 and β 2 respectively with β 1 + β 2 ≡ 1. Suppose that there are two types of occupations, skilled or white color jobs and unskilled or blue color jobs. Each agent decides whether to be a skilled worker or not at his early days of life. Once he becomes a skilled worker, he lives as a skilled worker until he dies. Otherwise, he lives as a unskilled worker until he dies. Let sit denote the fraction of skilled workers in group i ∈ {1, 2} at time t, which is called group i skill level at time t. The fraction of skilled workers in the overall population at time t is then s¯t ≡ β 1 s1t + β 2 s2t , which is a proxy of economic development as the economic growth is largely attributed to the human capital accumulations in the modern economy (Abramovitz 1993).

10 Let σti denote the fraction of skilled workers in the social network of an individual belonging to group i ∈ {1, 2} at time t, which is called group i network quality at time t. This depends on the levels of human capital in each of the two groups as well as the extent of segregation η: σti ≡ ηsit +(1−η)¯ st . When η=1, σti is equal to s¯t for any group i, indicating that there is no difference in the network quality across social groups. When η = 0, σti is equal to the skill level of group i (sit ), indicating the total segregation across groups. Note that, with this structure, the number of total contacts by group 1 members of group 2 members equals that by group 2 members of group 1 members: (1 − η)β 2 times population share of group 1 (β 1 ) equals (1 − η)β 1 times population share of group 2 (β 2 ). The σti is a convex combination of sit and sjt with their wights k i and 1 − k i ,

σti ≡ k i sit + (1 − k i )sjt , where k i = η + (1 − η)β i .

(1.1)

The k i represents the degree of influence from its own group skill level and 1 − k i represents that from the other group’s skill level. It is noteworthy that k i is an increasing function of the societal segregation level (η), and that of the population share of the group (β i ): as the society is more segregated, its own group skill level influence more on its network quality, and as the population size is bigger, the network quality is less affected by the other group’s skill level. Each newborn individual at time t makes a skill investment decision, comparing the cost of skill acquisition with the expected benefits of investment. The cost to achieve skill at time t depends on the innate ability a and the quality of social network at time t σti : C(a, σti ) is an increasing function in both arguments a and σti , which satisfies lima→0 C(a, σti ) = ∞ and lima→∞ C(a, σti ) = 0 for any σti ∈ [0, 1]. The cost includes both mental and physical costs that are incurred for the skill achievement. The lower one’s innate ability or the worse the quality of one’s social network, the more mentally stressful the skill acquisition process is or the more materials he must spend for the achievement. The expected benefits of investment to a newborn individual of group i born at time t, Πit ∈

11 (0, ∞), is the benefits of skill investment to be realized over the whole lifetime from time t until he dies, which is the difference between the expected lifetime benefits of being skilled (Bsi (t)) and that of being unskilled (Bui (t)): Πit ≡ Bsi (t) − Bui (t). I rule out the high and low skill complementarity for the reasons discussed below. Thus, both Bsi (t) and Bui (t) are functions of the sequence of the i i ∞ expected network quality from time t to infinite: {στi }∞ τ =t , implying that Πt ≡ Π({στ }τ =t ). Let us

call Πit group i benefits of investment at time t, because it depends on the group specific sequence of network quality. The benefits reflect both psychological and material benefits, about which we will discuss in section 1.3.2. With this setup, the education period network externality comes into C(a, σti ) and lifetime network externality comes into Π({στi }∞ τ =t ). A newborn individual of group i at time t will invest for the skill achievement only when C(a, σti ) is less than or equal to Π({στi }∞ τ =t ). Suppose that ability a ∈ (0, ∞) is distributed among newborn cohort in a S-shaped CDF function G(a): there exists a ˆ such that G00 (a) > 0, ∀a ∈ (0, a ˆ) and G00 (a) < 0, ∀a ∈ (ˆ a, ∞), implying that its PDF function is bell-shaped. Suppose that G(a) is identical for all groups, consistent with Loury’s (2002) axiom of anti-essentialism. We can find the threshold ability level for group i such that newborn individuals of group i whose innate ability is at least the threshold invest in the skill acquisition. Lemma 1. Given {σti }t→∞ , there exists a unique threshold ability level a ˜it that satisfies C(˜ ait , σti ) = Π({στi }∞ τ =t ). Proof. This is derived from the fact that C(a, σt ) is a decreasing function with respect to a that satisfies lima→0 C(a, σt ) = ∞ and lima→∞ C(a, σt ) = 0 for any σt ∈ [0, 1].  Let us define a function A which represents the unique threshold ability: a ˜it ≡ A(σti , Πit ). A is a decreasing function with respect to both arguments σti and Πit . Thus, the fraction of time t newborn individuals of group i who invest in skill, denoted by xit , is expressed by

xit = 1 − G(A(σti , Πit )).

(1.2)

12

In developing the basic structure of this model, I am indebted to Bowles et al. (2007) for suggesting the simplest way to encompass both the intergenerational network externality and the extent of segregation. The smart representation of the newborn cohort’s decision making helps to reflect the intergenerational network externality without imposing the assumption of credit market imperfections, which most previous theoretical work on the intergenerational dynamics relied on (Loury 1981, Banerjee and Newman 1993, Galor and Zeira 1993). The so called (η, β) structure used in the paper to represent the segregation level in a society enables the model to reflect the integration effects in the simplest and the most effective way.13 Bowles et al., assuming the presence of education period network externality, prove the instability of an equal society in a highly segregated economy combined with the complementarity of high- and low-skill workers, as well as the instability of an unequal society in a highly integrated economy. My main departure from their model is the replacement of the two-period overlapping generational structure with an infinite horizon structure. With this adjustment, we let the economic agents at the decision moment of skill investment to consider the benefits of skill investment accrued over the entire lifetime. Another departure is the assumption of exogenous wages: I rule out the high-low skill complementarity, which was an essential part of Bowles et al., in order to concentrate on the net effects of the two types of network externalities, education period and lifetime. The assumption of no complementarity can arguably be accepted for the correct description of the modern economy, where the human capital is considered the prime engine of economic development, due to skill-biased technologies and international capital flow (Goldin and Katz 2001, Galor and Moav 2004). The wage divergence between skilled and unskilled workers caused by trade openness and globalization does support the exogenous wages in the modern world (Wood 1994, Richardson 1995). One important departure is an introduction of the S-shaped G(a) function. With this specific functional form, which is arguably the best way to reflect the innate ability distribution among a newborn cohort, I can explicitly express the dynamic structure with network externalities in a heterogeneous group 13 The η indicates the level of segregation in a society, and β indicates the population share of a disadvantaged group. Chaudhuri and Sethi (2008) is another paper that adopts the structure.

13

economy, and explore the macroeconomic effects of inequality.

1.3

Dynamic Model with Network Externalities

In this section, I construct the dynamic system for the two-group economy, in which two types of network externalities, education period and lifetime, play a crucial role in explaining the evolution of group skill level sit and that of group benefits of investment Πit .

1.3.1

Education Period Network Externality and Evolution of Group Skill Level

I assume that a worker is subject to the “poisson death process” with parameter α: in a unit period, each individual faces α chances to die. We assume that the total population of each group is constant, implying that the α fraction of a group’s population is replaced by newborn group members in a unit period. Since xit is the fraction of newborn group members who invest in skill, the group i’s skill level sit evolves in a short time interval ∆t in the following way:

sit+∆t

 ≈ α ∆t ·

xt + xt+∆t 2

 + (1 − α∆t) · st .

By the rearrangement of the equation, we have

 i  sit+∆t − sit xt + xit+∆t ∆sit i ≡ ≈α − st . ∆t ∆t 2

Taking ∆t → 0, we have the evolution rule of group skill level,

s˙ it = α[xit − sit ].

(1.3)

14 There is a direct way to achieve the same result. We can define sit as sit ≡

Rt −∞

αxiτ e−α(t−τ ) dτ.

Taking a derivative with respect to time t, we have s˙ t = α[xt − st ]. The speed of group skill level change is determined by the difference of the skill level of newborn cohort and the skill level of “old” cohorts. If the fraction of newborn group members who invest in skill acquisition is greater than the fraction of skilled workers in the group population, the group skill level improves. Otherwise, it declines. There is no change in skill level, if the fraction of newborn members who invest in skill is equal to the group’s current skill level. Combining this with the determination rule of xit in equation (1.2), we have the evolution rule with education period network externality reflected,

s˙ it = α[1 − G(A(σti , Πit )) − sit ].

1.3.2

(1.4)

Lifetime Network Externality and Evolution of Group Benefits of Investment

As discussed earlier, we rule out the high and low skill complementarity. The expected benefits of skill investment depends on the expected quality of the network in the future and the exogenous wage levels for each type of worker. Let us assume the base level salary for a skilled worker (whitecolor worker) is ws and that for an unskilled worker (blue-color worker) is wu . A skilled worker obtains extra benefits from his social network, both psychological and material. The more skilled workers he has in his network, the more appropriate job position he may find for his specific skills. He may get more comfort and mentoring out of the informal network, and the cost for maintaining jobs may decline. Information flows along the synapses of the social network. A skilled worker can be more efficient in contracting his customers and handling specific work troubles with more skilled workers in his network. Let Ss (σti ) denote the extra benefits of having skilled workers in the social network of a skilled worker from group i. Even an unskilled worker may obtain more benefits from having more skilled workers in his network, but to a lesser degree than a skilled worker

15 would get. Let Su (σti ) denote the extra benefits of having skilled workers in the social network of an unskilled worker from group i. Both Ss (σti ) and Su (σti ) are increasing functions of σti . We assume ∂Ss ∂σti

that Sj (0) = 0, ∀j ∈ (s, u), and

>

∂Su , ∂σti

implying that a skilled worker would obtain higher

marginal benefits for having an additional skilled worker in his social network. In the same way, an unskilled worker would obtain more extra benefits from having more unskilled workers in his network. For example, a car mechanic would find a better car center that fits his speciality when he has more mechanics in his network. He would be more efficient in handling a specific mechanical problem when he can confer with more mechanics in his network. Since (1 − σti ) represents the fraction of unskilled workers in the social network of worker from group i, let Uu (1 − σti ) denote the extra benefits of having unskilled workers in the social network of an unskilled worker from group i. Even a skilled worker would obtain more benefits from having more unskilled workers in his network, but obviously to a lesser degree than an unskilled worker would get. In the previous example, at least, he would find a better car center to fix his broken car when he has more mechanics in his network. Let Us (1 − σti ) denote the extra benefits of having unskilled workers in the social network of a skilled worker from group i. Both Uu (1 − σti ) and Us (1 − σti ) are increasing functions of (1 − σti ). We assume that Uj (0) = 0, ∀j ∈ (s, u), and

∂Uu ∂(1−σti )

>

∂Us , ∂(1−σti )

implying that

an unskilled worker would obtain higher marginal benefits than a skilled worker from having an additional unskilled worker in his social network. Suppose a worker discounts the benefits realized in the future with a discounting factor ρ. We have assumed that each worker faces α chances to die in a unit period, under the poisson process. Accordingly, given the sequence of the expected network quality from time t to infinity, {στi }∞ τ =t , the expected lifetime benefits of being skilled (Bsi (t)) and that of being unskilled (Bui (t)) are

Bsi (t) Bui (t)

Z

∞

= Zt ∞ = t

[ws + Ss (στi ) + Us (1 − στi )]e−(ρ+α)(τ −t) dτ, [wu + Su (στi ) + Uu (1 − στi )]e−(ρ+α)(τ −t) dτ,

16

where ρ is a discounting factor and α is a poisson death rate. Since the expected benefits of investment to a group i individual born at time t is Πit ≡ Bsi (t) − Bui (t),

Πit =

Z

∞

[ws − wu + Sh (στi ) − Su (στi ) + Us (1 − στi ) − Uu (1 − στi )]e−(ρ+α)(τ −t) dτ.

t

¯ and Sh (σ i ) − Su (σ i ) + Us (1 − σ i ) − Uu (1 − σ i ) with f (σ i ), we have Replacing ws − wu with δ, τ τ τ τ t

Πit

Z =

∞

[δ¯ + f (στi )]e−(ρ+α)(τ −t) dτ,

(1.5)

t

where δ¯ indicates the base salary differential and, f (στi ) the difference between the extra benefits of being skilled and that of being unskilled at time τ . Let us call δ¯ + f (στi ) time τ net benefits of being skilled, which is an increasing function of σti because

∂Ss ∂σti

>

∂Su ∂σti

and

∂Uu ∂(1−σti )

>

∂Us . ∂(1−σti )

The more

skilled workers at time τ in a worker’s social network, the greater the net benefits of being skilled. I assume that δ¯ + f (0) > 0, which implies that the base salary differential is big enough that the net benefits of being skilled is always positive. Taking the derivative with respect to time t, we have the evolution rule of the group i benefits of investment Πit ,   δ¯ + f (σti ) i i ˙ . Πt = (ρ + α) Πt − ρ+α

(1.6)

The change of the lifetime benefits of investment evaluated at time t is determined by the difference between the current level of lifetime benefits of investment and the current level of net benefits of ¯  δ+f (σti ) being skilled. If the current level of net benefits of being skilled is greater than the current ρ+α lifetime benefits of investment (Πit ), at the next time t + ∆t, lifetime benefits of investment would be smaller than the current level: Πit+∆t < Πit . If they are equal to each other, there will be no change in the expected lifetime benefits of investment.

17

1.3.3

Dynamic System with Network Externalities

Thus far, we have examined how two variables, group skill level sit and group benefits of investment Πit , evolve over time. The first variable indicates the fraction of skilled workers in the population of group i. This is adjusted every minute by the fraction of skilled workers among the group i newborn cohort. Thus, it is a flow variable, which cannot make a sudden jump at a point of time. The second variable indicates the expected benefits of investment for a newborn individual of group i born at time t, which is determined by the sequence of the expected network quality in the future, {στ }∞ τ =t . Since it depends on the expected network qualities, it can make a sudden jump at any point of time by changing the expectation of {στ }∞ τ =t . Thus, it is a jumping variable. In the dynamic system that represents a two-group economy, there exist two flow variables, s1t and s2t , and two jumping variables, Π1t and Π2t . Using equations (1.4) and (1.6), we can construct the following dynamic system.14 Theorem 1 (Dynamic System). In a two-group economy, the dynamic system with two flow variables s1t and s2t and two jumping variables Π1t and Π2t is summarized by the following four-variable differential equations:

s˙t 1

  = α 1 − G(A(σt1 , Π1t )) − s1t

s˙t 2

  = α 1 − G(A(σt2 , Π2t )) − s2t   δ¯ + f (σt1 ) 1 = (ρ + α) Πt − ρ+α   ¯ δ + f (σt2 ) 2 = (ρ + α) Πt − , ρ+α

˙ 1t Π ˙ 2t Π 14 Refer

to Section 1.7.1 for the expansion of this dynamic system to the arbitrary n-group economy.

18

where

1.4

σt1

=

k 1 s1t + (1 − k 1 )s2t , with k 1 = η + (1 − η)β 1

σt2

=

k 2 s2t + (1 − k 2 )s1t , with k 2 = η + (1 − η)β 2 .

Homogeneous Group Economy

The dynamic system in Theorem 1 is defined in a four-dimensional space of (s1 , s2 , Π1 , Π2 ). It is hard to have a clear imaginary view of the dynamic structure through the phase diagram, the direction field, and the stationary points in this complex system. Even after succeeding in clarifying those technical aspects, the intuitive interpretation of the system is a more challenging task. Let us start with a simplest structure of the economy, in which two social groups are fully integrated becoming a homogeneous social group, or in which a social group is totally separated from all other social groups. In the middle of the analysis of this homogeneous group economy, we define the essential concepts to interpret the dynamic system with network externalities. In section 1.5, we will come back to the two-group economy.

1.4.1

Steady States and Economically Stable States

Suppose a homogeneous social group or social groups in a fully integrated society, in which st = σt . The skill level at time t directly represents the network quality at time t. The dynamic system is simply

s˙ t ˙t Π

= α[1 − G(A(st , Πt )) − st ]   δ¯ + f (st ) , = (ρ + α) Πt − ρ+α

(1.7)

19

and two demarcation loci (isoclines) of the time dependent variables are represented by

s˙ t = 0 Locus

: st = 1 − G(A(st , Πt ))

˙ t = 0 Locus Π

:

Πt =

(1.8)

δ¯ + f (st ) . ρ+α

(1.9)

˙ t , Πt is simply an increasing function of st , as denoted by a dotted curve In the demarcation locus for Π in Panel B of Figure 1.1. The demarcation locus for s˙ t is represented by (st , Πt ) that satisfies the following two equations that are associated with the threshold ability level for the skill achievement:

1 − G(˜ a)

st

=

a ˜

= A(st , Πt ).

The first one is denoted by the solid curve in Panel A of Figure 1.1 in (st , a ˜) domain, which is simply a S-shaped curve. The second one is denoted by the dotted curves for each level of Πt (iso-Π curves), in the same panel. As Πt increases, the corresponding iso-Π curve moves down. (The curves tend to be convex as the marginal impact of a network quality on the threshold ability level may decline as st increases.) The combinations of (st , Πt ) that satisfy the above two equations are represented by the solid curve in Panel B of the figure, which is the demarcation locus for s˙ t . Since we have achieved two demarcation loci, we can identify steady states in this system. ¯

0

(s ) Lemma 2. If there exist s0 and s00 , where s0 < s00 , that satisfy both s0 > 1 − G(A(s0 , δ+f ρ+α )) and ¯

00

(s ) s00 < 1 − G(A(s00 , δ+f ρ+α )), there are at least three steady states.

Proof. See the proof in the appendix.  Note that, as long as the the network externality in the skill acquisition period is strong enough ∂A ( ∂s is big enough), the dotted curves in Panel A (iso-Π curves) are tangent to the S-shaped curve t

(st = 1 − G(˜ a) curve) at two distinct points, (su , a ˜u ) with Πt = Πu and (sd , a ˜d ) with Πt = Πd , where Πu > Πd . In the specific case with two tangent points, the multiple steady state condition in

20

Lemma 2 is simply satisfied when Πu >

¯ (su ) δ+f ρ+α

and Πd <

¯ (sd ) δ+f ρ+α ,

as you can observe in Panel B of

Figure 1.1. Without loss of generality, we assume that there exist three steady states when the condition of Lemma 2 is satisfied.15 When the condition is not satisfied, it is most likely that there exists a unique steady state.16 For example, if the base salary differential δ¯ is too big or too small, the ˙ t = 0 locus will be placed too high or too low, and there is a unique intersection between the two Π loci. Whatever the initial position s0 is, the group state will move toward the unique steady state.17 That is, if the base salary for a skilled job is much greater (smaller) than that for an unskilled job, the group skill level st converges to a high (low) skill steady state, regardless of the initial network quality. This is certainly not an interesting case: the network externalities do not generate any difference in the final economic outcome. Therefore, we will focus on the case with three steady states in this research. Proposition 1. Without loss of generality, there is a range of the base salary differential δ¯ ∈ (δ¯l , δ¯h ), with which there exists three distinct steady states in a homogenous group economy. Proof. Without loss of generality, assume that the dynamic system has two distinct salary differ˙ t = 0 locus is tangent to the s˙ t = 0 locus in the (st , Πt ) plane. entials δ¯l and δ¯h , with which the Π Between the two levels, there will be at least three steady states. Without loss of generality, there are three steady states in the interval (δ¯l , δ¯h ).  Let us denote the three steady states by El (sl , Πl ), Em (sm , Πm ) and Eh (sh , Πh ), where sl < sm < sh . The final economic state (s, Π) will be on either one of them. In order to examine the conversing process to a steady state, we need a phase diagram with direction arrows (laws of motion), which are displayed in Figure 1.2. The characteristics of the steady states are summarized by the following lemma. 15 There

exist more than three steady states only for very peculiar functional forms of G, A or f . ˙ t = 0 locus is tangent to the s˙ t = 0 is possible that there are only two steady states, for example, when the Π locus. We ignore this case because it can occur with a measure zero probability. 17 The unique steady state is a saddle point, which is easily proven as the same way for the proof of Lemma 3. The economic state with an arbitrary s0 converges to the unique steady state following the saddle path. 16 It

21

Lemma 3. Among three steady states, El (sl , Πl ), Em (sm , Πm ) and Eh (sh , Πh ), El and Eh are saddle points and Em is a source. Proof. See the proof in Appendix.  We can identify the equilibrium path (saddle path) to each saddle point, El and Eh , as described in Figure 1.2. In the given example of the figure, the equilibrium paths spiral out of a source Em . Given an initial network quality s0 ∈ (0, 1), the newborn agents will calculate the expected benefits of investment Π0 . Based on the calculated Π0 , each agent with different innate ability (a) makes his own skill investment decision. In their calculation of Π0 , they will use the evolution rules of st and Πt , summarized in the formula (1.7). They understand that either Eh or El should be the final economic state. If they believe that Eh will be realized in the future, they will choose Π0 on the equilibrium path to Eh . As the following generations keep the same belief, the group state will be moving along the equilibrium path, and eventually arrive at Eh . If they believe that El will be realized in the future, they will choose Π0 on the equilibrium path to El . As the following generations keep the same belief, the group state will move toward El along the path. Since there is no equilibrium path that converges at Em , the newborn agents who understand the evolution rules of st and Πt will never choose Em as the final group state. Choosing Eh is called sharing an optimistic social consensus, while choosing El is called sharing a pessimistic social consensus. Although all three steady states are mathematically unstable, two saddle points, El and Eh , are “economically” stable in a sense that there exists a converging path to these states for any perturbation at the states: rational economic agents who understand the dynamic system can take the saddle path to lead the group back to the original state. However, the source Em is “economically” unstable, because any small perturbation from the state will lead the group to move away from it: rational economic agents will take either the optimistic path to Eh or the pessimistic path to El , because there is no converging path to Em . Definition 1 (Economically Stable States). A state (s0 , Π0 ) is an economically stable state if there

22 exists a converging phase path to the state for any s in the neighborhood of s0 . Otherwise, it is an economically unstable state. In the given economy with three steady states, El and Eh are economically stable states and Em is an economically unstable state.

1.4.2

Overlap and Deterministic Ranges

Let the network quality eo denote the lower bound of the optimistic path to Eh , and the network quality ep the upper bound of the pessimistic path to El . As shown in the example of Figure 1.2, there exists a unique optimistic path for an initial network quality in the interval (ep , 1): if the initial network quality is good enough, there is only one reasonable social consensus about the future, that is Eh , and the group state (st , Πt ) will move toward the high skill equilibrium (sh , Πh ) by self-fulfilling investment activities. Also, there exists a unique pessimistic path for an initial network quality in the interval (0, eo ): if the initial network quality is poor enough, there is only one reasonable social consensus about the future, that is El , and the group state (st , Πt ) moves toward the low skill equilibrium (sl , Πl ), by the self-fulfilling investment activities. However, if the initial network quality is mediocre in (eo , ep ), there exist multiple social consensuses about the future, El and Eh , that are available to the group. The final economic state depends on the social consensus that the newborn agents of the group choose. If they and the following generations are optimistic, Eh will be realized in the end. If they all are pessimistic, El will be realized in the end. Therefore, in the interval [eo , ep ], the social consensus determines the future, while the historical position determines the future outside the interval. We denote [e0 , ep ] as overlap as Krugman (1991) denotes. We denote the ranges (0, ep ) and (eo , 1) as deterministic ranges as the macroeconomic literature denotes. Proposition 2. In a homogeneous group economy with two economically stable states El and Eh , there exists a positive range of overlap, [eo , ep ], with eo < ep , where the social consensus about the future determines the final economic state among El and Eh .

23

Proof. See the proof in Appendix.  Corollary 1. In a homogeneous group economy with two economically stable states El and Eh , the deterministic range for Eh is (ep , 1), and the deterministic range for El is (0, eo ), where eo < ep . Proof. Since there exists a unique equilibrium path outside the overlap, both ranges (ep , 1) and (0, eo ) are deterministic ranges. Since ep is the upper bound of the pessimistic path toward El , (ep , 1) is a deterministic range for Eh . Since eo is the lower bound of the optimistic path toward Eh , (0, eo ) is a deterministic range for El . 

1.4.3

Mobilization Force and Historical Force

There are two kinds of network externalities, education period and lifetime, that affect the structure of the economy. In order to examine the direct effect from each network externality, we compare three distinct cases: 1) lifetime network externality only 2) education period network externality only and 3) both education and lifetime network externalities. Lifetime Network Externality Only: the Mobilization Force Panel A of Figure 1.3 describes the first case: there exists very negligible peer effects or parental effects together with perfect provision of public schooling or no credit constraints in the skill acquisition period. Since there is almost no education period network externality, the iso-Π curves described in Panel A of Figure 1.1 is almost flat:

∂A(st ,Πt ) ∂st

' 0, or a ˜ ≡ A(Πt ), ignoring the st

term. Therefore, the s˙ t = 0 locus will be like a S-shaped curve that satisfies st = 1 − G(A(Πt )). As you can observe in Panel A of Figure 1.3, the overlap may to cover the whole range of network quality [0, 1].18 In this case, the historical position of initial network quality s0 does not provide any constraint in the determination of the final economic state. The final economic state entirely depends on the social consensus, regardless of s0 . (With the bigger discounting factor ρ, the overlap 18 In this case, another limit set exists, which is a loop located between the optimistic path and the pessimistic path. If economic agents believe that the network quality will fluctuate forever, the group state will be on this loop. I rule out this unique case in my study.

24 may not cover the whole range of network quality [0, 1].19 Even in this case, the overlap range will be much greater than that in Panel C of Figure 1.3.) In other words, skill investment activities of newborn agents tend to be subject to the “future”: the expected benefits of skill acquisition that accrues over the lifetime. Suppose the initial network quality is s0 ∈ (sl , sh ). If the group members believe that the final state is Eh instead of El , the future benefits anticipated to accrue from skill acquisition Πop 0 are ¯

(s0 ) greater than the current level of network benefits ( δ+f ρ+α ), and more newborns invest in skills.

The group’s network quality improves over time and the group state converges to (sh , Πh ) along the optimistic path, as displayed in Panel A of Figure 1.1. If they believe that the final state is El instead of Eh , the future benefits anticipated to accrue from skill acquisition Πpe 0 is smaller than the ¯

(s0 ) current level of network benefits ( δ+f ρ+α ), and less newborns invest in skills. The group’s network

quality deteriorates and the group state converges to (sl , Πl ). Therefore, the social consensus toward the future determines the future. Group members can work together to improve the quality of the group’s social network by sharing the optimistic social consensus, or to deteriorate the quality of the network by sharing the pessimistic social consensus. This is what I call the mobilization force of network externalities. Education Period Network Externality Only: the Historical Force Panel B of Figure 1.3 describes the opposite regime, where there is no network externality over the course of a worker’s lifetime. The benefits of skill acquisition is just the wage differential δ¯ at each period, and the consequent lifetime benefits are

δ¯ ρ+α .

In this case, the expectation toward the

future does not play any role because the benefits of skill acquisition is fixed. The skill investment activities of newborns are subject to the “past”: the cost level to achieve the skill. If the initial network quality is good, the cost for the skill achievement is low. Consequently, a large fraction of newborns invest in skill. Then, the network quality in the next period is even better because of the 19 Note that, the bigger ρ, the bigger Π ˙ t and the longer the vertical direction arrows in the phase diagram. As the slopes of the saddle paths are steeper, the overlap may not cover the whole range of network quality [0, 1], and is limited between [sl , sh ].

25

enhanced skill investment rate in the previous period. Thus, even more newborns invest in skill in the next period. The network quality improves over time. If the initial network quality is poor, a small fraction of newborns invest in skill. The next period network quality is even worse, and even less newborns invest in skills. The network quality deteriorates over time. The two examples are displayed in the Panel. If the initial network quality is bad, below sm , the network quality converges to the low skill equilibrium El . If it is good, above sm , the network quality converges to the high skill equilibrium Eh . Therefore, the final economic state entirely depends on the history, an initial network quality of the group. This is what I call the historical force of network externalities. Both Lifetime and Education Period Network Externalities: Two Forces Combined Panel C of the figure display how the two forces of network externalities are interwoven in the dynamic structure of network quality evolution. The mobilization force of lifetime network externalities is constrained by the historical force of education period network externalities. The overlap is the network quality range mainly governed by the mobilization force, while the deterministic ranges are the network quality ranges mainly governed by the historical force. In the overlap, the group can be mobilized toward the high skill steady state Eh by sharing the optimistic view together, or toward the low skill steady state El by sharing the pessimistic view together. Outside the overlap, it is the initial historical position that determines the final state. If it is high (low) enough, the group status moves toward the high (low) skill steady state.

1.4.4

Size of Overlap

In the previous sections, the importance of overlap has been emphasized. Whether the initial position is in the overlap or outside the overlap determines whether the group members can work together to change the future by sharing optimism (or pessimism). Outside the overlap, the future is determined through a mechanical tatonnement process. The bigger size of overlap indicates the

26

dynamic structure that is more governed by the mobilization force or the power of collective action. It is worthwhile to check how the overlap size is determined, because this is an indication of the relative power of the mobilization force and the historical force. In order to make a comparative statics analysis, let us simplify the given model using the linear functional forms of the cost function C(a, st ) and the benefits function f (st ):

C(a, st )

=

ψ(a) − k1 st

(1.10)

f (st )

=

q0 + q1 st ,

(1.11)

where k1 represents the influence of education period network externality, and q1 the influence of lifetime network externality. We further assume that the innate ability equals across the population: a≡a ¯. The newborn agents with the unique innate ability a ¯ decide to invest when the benefits is greater than the cost: if Πt > ψ(¯ a) − k1 st , xt = 1 and s˙ = α[1 − st ], and if Πt < ψ(¯ a) − k1 st , xt = 0 and s˙ = α[0 − st ]. Therefore, the slanting part of the s˙ = 0 locus in Appendix Figure 1.1 is Πt = ψ(¯ a) − k1 st , which is a demarcation line that sharply divides the evolution rule of s˙ t . The demarcation locus for Πt is Πt =

¯ 0 +q1 st δ+q . ρ+α

In this simple system without hurting the essential

structure of the economy, we can explicitly find the optimistic and pessimistic paths and the relevant size of overlap. Lemma 4. In the simple homogenous economy with equations (1.10) and (1.11) and the unique innate ability level of a ¯, the optimistic equilibrium path (sop , Πop ) above two demarcation loci and the pessimistic equilibrium path (spe , Πpe ) below two demarcation loci are

Πop

=

Πpe

=

Proof. See the proof in Appendix. 

q1 sop + ρ + 2α q1 spe + ρ + 2α

(δ¯ + q0 )(ρ + 2α) + q1 α (ρ + α)(ρ + 2α) ¯ δ + q0 . ρ+α

27

This linear equilibrium paths are described in Appendix Figure 1.1 with the corresponding demarcation loci. Using the calculated equilibrium paths and the slanting part of the s˙ = 0 locus ¯ (Π = ψ(¯ a) − k1 s), we can obtain the overlap size L:

L=

α . (α + ρ)(1 + (k1 /q1 )(ρ + 2α))

(1.12)

Using the outcomes, we have the following results that have deep economic implications. Proposition 3. In a simple economy defined in Lemma 4, the bigger the relative influence of lifetime network externality over education period network externality (the bigger q1 /k1 ), the larger the size of overlap. The less the economic agents discount the future (the smaller ρ), the larger the size of overlap. Proof. The first derivatives of equation (1.12) give the results.  The first result implies that, when the lifetime network externality is relatively more influential, the mobilization force of network externalities is stronger compared to their historical force; collective action facilitated by the formation of social consensus can play a bigger role in the determination of the final economic state. The lower discounting factor means the greater forward-looking decision making of economic agents, which implies the expectation toward the future can play a bigger role in the determination of the final outcome. This fact is reflected in the second result of the proposition.

1.5

Heterogeneous Group Economy

Now we come back to the two group economy summarized in the four variable dynamic system of Theorem 1. We assume that the conditions in Lemmas 2 and 3 are applied to this economy, so that there exist three steady states at skill levels sl , sm and sh in the fully integrated economy, or in each group’s economy with social interactions fully separated between two groups. Note that there will be no group disparity in the long run if there exists a unique steady state: whatever the initial skill

28 composition (s10 , s20 ) is, the economy state (s1t , s2t ) converges to the unique steady state as time goes by. Therefore, there will no issue for the persistent group disparity through the channel of network externalities in this case, and this is certainly not an interest in this study.

1.5.1

Heterogeneous Economy with Total Segregation

Let us start with the simplest case of the two group economy: fully separated social interactions between two groups (η = 1), which can help us to have an intuition about the four dimension dynamic structure of two group economy. The structure of this special case can be directly inferred from the properties of the homogeneous economy because there are no interactions between the two. Using the same definition of economically stable states in the homogeneous economy (Definition 1), a steady state (s10 , s20 , Π10 , Π20 ) is called an economically stable state if there exists a converging phase path to the state in the neighborhood of (s10 , s20 ). Obviously, there are nine steady states in this economy: Qij (si , sj , Πi , Πj ) for i ∈ {l, m, h} and j ∈ {l, m, h}. Among them, the following four are economically stable states: Qll (sl , sl , Πl , Πl ), Qlh (sl , sh , Πl , Πh ) Qhl (sh , sl , Πh , Πl ) and Qhh (sh , sh , Πh , Πh ). Those are depicted with dark circles in (s1 , s2 ) domain in Figure 1.4. The other five economically unstable states are depicted with gray circles in the domain. Two separated dynamic structures are displayed beside the domain in the figure. In an economically unstable state, any arbitrary shock to the position may lead the economic state (s1t , s2t , Π1t , Π2t ) to move away from it, while the economic state can come back to the original steady state after any small shock in an economically stable state. Since there are four economically stable states, a society with an initial skill composition (s10 , s20 ) will move to either of them in the long run. Once a social consensus about the future is constructed in the society, the economic state (s1t , s2t , Π1t , Π2t ) will move to the future state of social consensus following a unique converging path to the state. Let us check available social consensuses and corresponding equilibrium paths for each initial skill composition (s10 , s20 ) in this totally segregated economy.

29

1.5.1.1

Stable Manifolds and Manifold Ranges

First, regardless of the position of s20 , group 1 with an initial skill level s10 can move toward either the skill level sl or sh following the same rule in the homogeneous economy. The optimistic path to sh is available to the group with an initial skill position s10 ∈ [eo , 1] and the pessimistic path is available to the group with an initial skill position s20 ∈ [0, ep ]. Those available equilibrium paths are described in Panel A of Figure 1.5: the available optimistic path in pink and the pessimistic path in blue. In the interval (eo , ep ), which is an overlap, both optimistic and pessimistic paths are available. The same is true for group 2, as displayed in Panel B of the figure. Therefore, when both s10 and s20 are greater or equal to eo , the equilibrium path to Qhh (sh , sh , Πh , Πh ) is available to the society. The unique converging path to Qhh is a combination of two optimistic equilibrium paths, (s1t , Π1t )op for group 1 and (s2t , Π2t )op for group 2. In the same way, the equilibrium path to Qll (sl , sl , Πl , Πl ) is available when both s10 and s20 are smaller or equal to ep . The unique converging path to Qll is the combination of two pessimistic paths, (s1t , Π1t )pe and (s2t , Π2t )pe . The set of initial positions (s10 , s20 ) where the converging path to Qhh is available is called Manifold Range for Qhh , and colored in darker green in Panel C of Figure 1.3. The set of initial positions where the converging path to Qll is available is called Manifold Range for Qll , and colored in lighter green in the same panel. In the same way, we define the Manifold Ranges for Qhl and Qlh . The manifold range for Qhl is the set of (s10 , s20 )s with s1 ≥ eo and s2 ≤ ep , and is described in lighter orange in Panel D. The Manifold Range for Qlh is the set of (s10 , s20 )s with s1 ≤ ep and s2 ≥ eo , and is described in darker orange in the panel. In geometry, a collection of points on all converging paths to a limit set Q is defined as a stable manifold to the limit set Q.20 Using the concept, we define the stable manifold to an economically stable state Qij . Definition 2 (Stable Manifold SMij ). Stable manifold SMij is a collection of (s10 , s20 , Π10 , Π20 )s that 20 A limit set in geometry is the state a dynamic system reaches after an infinite amount of time has passed, by either going forward or backward in time.

30

converge to an economically stable state Qij in the dynamic system defined in Theorem 1:

SMij ≡ {(s10 , s20 , Π10 , Π20 ) ∈ R4 |(s1t , s2t , Π1t , Π2t )|(s10 ,s20 ,Π10 ,Π20 ) → Qij }.

The manifold range to Qij is redefined using the concept of stable manifold, which is just a projection of the stable manifold to Qij to the (s1 , s2 ) plane. Definition 3 (Manifold Range Mij ). Manifold range Mij is a collection of (s10 , s20 )s in SMij :

Mij ≡ {(s10 , s20 ) ∈ [0, 1]2 |(s1t , s2t , Π1t , Π2t )|(s10 ,s20 ,Π10 ,Π20 ) → Qij }.

1.5.1.2

Folded Overlaps and Deterministic Ranges

All manifold ranges are put together in Panel E of Figure 1.5. There are nine distinct areas: all manifold ranges are folded in the center square, two manifold ranges are folded in the rectangles surrounding the center square, and a unique manifold range exists in each of the corner. In the center square, where four manifold ranges are folded, four social consensuses about the future are available to the members in the society: the consensus of Qhh , that of Qlh , that of Qhl and that of Qll . Depending on the constructed social consensus, the economic state will move toward one of four economically stable states along a unique converging path. The social consensus is a combination of the expectation to group 1’s final state and that of group 2’s final state, as summarized in the following table. Group 2 \ Group 1

Pessimistic Expectation

Optimistic Expectation

Optimistic Expectation

Qlh

Qhh

Pessimistic Expectation

Qll

Qhl

If the initial skill composition (s10 , s20 ) is in one of four rectangles, where two social consensuses are available, the expectation to one group’s final state is critical in the determination of the social consensus about the future. For example, in the rectangle placed in the top middle, two manifold ranges

31

are folded, Mlh and Mhh , and two social consensus about the future are available: the consensus of Qlh and that of Qhh . Group 2 will move toward sh regardless of social consensus, because only the optimistic equilibrium path is available to the group: s2t → sh . Group 1’s expectation toward the future is important in the determination of social consensus and in the consequent equilibrium path. If group 1 holds an optimistic expectation toward the future, the economic state (s1t , s2t , Π1t , Π2t ) will converge to Qhh . Otherwise, it will converge to Qlh . If an initial skill composition (s10 , s20 ) is in one of four corner areas, the economic state will converge to the unique economically stable state (Qij ) in the area. Social consensus about the future is fixed as Qij among the rational economic agents. Thus, the expectation toward the future does not play any critical role in the determination of the final state, but the location of the initial skill composition, so called history, determines the final state. To clarify the distinct areas determined by manifold ranges, I define a folded overlap where multiple manifold ranges are folded, and a deterministic range which is covered by a unique manifold range. Definition 4 (N-Folded Overlap). N-folded overlap of M 1 , M 2 , ... M n is an overlapped area of those n manifold ranges, M 1 , M 2 , ... M n . Definition 5 (Deterministic Range). Deterministic range for Qij is the part of the manifold range for Qij (Mij ) that does not belong to any folded overlaps. Therefore, the characteristics of the two group economy with social interactions fully segregated between groups are summarized in the following way: Proposition 4. In the two group economy with total segregation, the (s1 , s2 ) domain is composed of one four-folded overlap, four two-folded overlaps and four deterministic ranges. If the initial skill composition (s10 , s20 ) is in the four-folded overlap, four final economic states are available, depending on the social consensus about the future. If it is in a two-folded overlap, one group’s expectation toward the future determines the final economic state among two possible destinations. If it is in a deterministic range, the economic state converges to the unique economically stable state belonging

32

to the range.

1.5.2

Heterogeneous Economy in General

Now we turn to the two group economy with an arbitrary segregation level η. Concepts of stable manifolds, manifold ranges, folded overlaps and deterministic ranges, defined in the previous section, will be useful in the following analysis of the complex dynamic structure with arbitrary η.

1.5.2.1

Identifying Steady States

In order to proceed with the analysis of the dynamic system in Theorem 1, we first need to identify ˙ 2t = 0. Let (s1∗∗ , s2∗∗ , σ 1∗∗ , σ 2∗∗ , Π1∗∗ , Π2∗∗ ) denote ˙ 1t = Π the steady states that satisfy s˙ 1t = s˙ 2t = Π a steady state, where two sets, (s1 , s2 ) and (σ 1 , σ 2 ), are bijective with parameters η, β 1 and β 2 . First, let us identify “partial” steady states (si∗ , σ i∗ , Πi∗ )|sj which are (sit , σti , Πit )s that satisfies ˙ it = 0 and σt = k i sit + k j sjt , given sjt . The following three equations characterize the set both s˙ it = Π of partial steady states (si∗ , σ i∗ , Πi∗ )|sj :

s˙ i = 0 Condition

: si∗ = 1 − G(A(σ i∗ , Πi∗ ))

˙ i = 0 Condition Π

:

Πi∗ =

δ¯ + f (σ i∗ ) ρ+α

(1.14)

Clearing Condition

:

σ i∗ = k i si∗ + (1 − k i )sj .

(1.15)

(1.13)

¯ (σ ) ˜ i∗ ) ≡ A(σ i∗ , δ+f By equation (1.14), A(σ i∗ , Πi∗ ) is a function of σ i∗ . Let us denote A(σ ρ+α ). Panel i∗

˜ i∗ ), which is a decreasing function and steeper than iso-Π curves A of Figure 1.6 describes A(σ ˜ i∗ ) A(σ, Π) at each point (σ i∗ , Πi∗ ).21 In the panel, I place si∗ = 1 − G(˜ a) locus together with A(σ ˜ i∗ ) locus must pass through si∗ = 1 − G(˜ locus, sharing the x-axis together. Note that A(σ a) locus at three points of si∗ (σ i∗ ), sl , sm and sh , because we have assumed three steady states (El , Em and Eh ) in a homogeneous economy. In Panel B, using two curves in Panel A, we can denote the set of ˜ 21 ∂ A (= ∂A ∂σ σ i∗ ∂σ (σ i∗ ,Πi∗ )





+

∂A | ∂Π (σ i∗ ,Πi∗ )

·

∂Π ) ∂σ σ i∗



<

∂A . ∂σ (σ i∗ ,Πi∗ )



33 ˜ i∗ )) (si∗ , σ i∗ )s that satisfy both equations (1.13) and (1.14), which is represented by si∗ = 1 − G(A(σ curve in Panel B.22 This curve must pass through three symmetric points, (sl , sl ), (sm , sm ) and (sh , sh ). Finally, given sj , (si∗ , σ i∗ ) must satisfy the clearing condition in equation (1.15), which is represented by the slashed dotted line in the panel. Thus, for given sj , three points colored in blue ˜ i∗ )) indicate the corresponding partial steady states, which are intersections of the si∗ = 1 − G(A(σ curve and the σ i∗ = k i si∗ + (1 − k i )sj line. The second step is to collect all partial steady states (s1∗ , σ 1∗ , Π1∗ )|s2 , and (s2∗ , σ 2∗ , Π2∗ )|s1 , in order to identify (global) steady states (s1∗∗ , s2∗∗ , σ 1∗∗ , σ 2∗∗ , Π1∗∗ , Π2∗∗ ). Panel A of Figure 1.7 indicates the former and Panel B of the figure indicates the latter. In the top figure of Panel A, the slashed lines with different levels of s2 help to identify (s1∗ , σ 1∗ ) for each level of s2 . Note that the slope of the slashed line is k 1 . Consequently, all partial steady states are denoted by s1∗ (s2 ) locus in the bottom figure. In the same way, in the top figure of Panel B, the slashed lines with different levels of s1 help to identify (s2∗ , σ 2∗ ) for each level of s1 . The slope of the slashed line is

1 k2 .

All

partial steady states are denoted by s2∗ (s1 ) locus in the bottom figure of the panel. As we overlap the two partial steady state curves, s1∗ (s2 ) and s2∗ (s1 ), finally we can identify the (global) steady states in Panel C. Note that each partial steady state locus is characterized with, using equations (1.13), (1.14) and (1.15),

˜ i si∗ + (1 − k i )sj )), ∀sj ∈ [0, 1]. si∗ (sj ) Locus : si∗ = 1 − G(A(k

(1.16)

The following proposition characterizes the (global) steady states. Proposition 5. The (global) steady states (s1∗∗ , s2∗∗ ) are a set of (s1 , s2 )s that satisfy the following 22 Refer

to two examples (σa , sa ) and (σb , sb ) illustrated in Panels A and B.

34

two equations:

s1∗∗

=

˜ 1 s1∗∗ + (1 − k 1 )s2∗∗ )) 1 − G(A(k

s2∗∗

=

˜ 2 s2∗∗ + (1 − k 2 )s1∗∗ )), 1 − G(A(k

(1.17)

¯ (σ) ˜ where A(σ) ≡ A(σ, δ+f ρ+α ).

1.5.2.2

Characteristics of Steady States

Let us denote four regions in the (s1 , s2 ) plane by Regions 1, 2, 3 and 4 in clockwise order, which are divided by one vertical line (s1 = sm ) and one horizontal line (s2 = sm ), and the left and top region is denoted by region 1, as displayed in Panel C of Figure 1.7. In order to analyze the number of steady states in each region, I impose the following assumption without hurting the essential structure of the model. ˜ Assumption 1 (Smoothing Condition). The function G(A(σ)) has one point of inflection as the ability distribution G(a) has one point of inflection. There must be at least one inflection point between σ = sl and σ = sh , because we have assumed three steady states in a homogeneous economy. As graphics in Figure 1.7 manifest, the ˜ economic structure cannot embed more than two steady states when the function G(A(σ)) has no inflection point. The assumption imposes the uniqueness of the inflection point. This assumption is achieved when the curvature of the S-shaped G(a) function is strong enough that the curvature ˜ ˜ of the function A(σ) does not distort the overall S-shape of G(A(σ)). This assumption helps the ˜ model to be more tractable than the case of G(A(σ)) with multiple inflection points. We call this ˜ assumption smoothing condition, because we rule out unnecessary local fluctuations of G(A(σ)) with 0 ˜ imposing this assumption. The assumption implies that there exists σ ˆ ∈ (sl , sh ) such that G(A(σ))

is decreasing in [0, σ ˆ ] and increasing in [ˆ σ , 1]. Let us define a function Dj (si∗ ) as the unique sj given si∗ on the si∗ (sj ) locus , which is Dj (si∗ ) ≡

˜−1 G−1 (1−si∗ )−ki si∗ A , 1−ki

according to formula (1.16). Then,

35

we have the following useful results. Lemma 5. Under Assumption 1, the si∗ (sj ) locus with η < 1 has one point of inflection, when the ˜ σ )), and locus is defined over the range of sj ∈ (−∞, ∞): Dj (si∗ ) is concave with si∗ < 1 − G(A(ˆ ˜ σ )). convex with si∗ < 1 − G(A(ˆ Proof. See the proof in Appendix.  The partial steady state loci are composed of a concave part and a convex part, regardless of η and β 1 . Lemma 6. As η declines, the partial steady state locus si∗ (sj ) tends to be flatter: the distance ∗

|Dj (si∗ ) − si | shrinks as η declines. The partial steady state locus si∗ (sj ) with the bigger β i is ∗

steeper than that with the smaller β i : The bigger β i , the larger the distance |Dj (si∗ ) − si | and the j i∗ )−si∗ ) steeper the slope ∂(D (s . ∂si∗ Proof. See the proof in Appendix.  The Dj (si∗ ) curve gets closer to the diagonal as η declines, which implies the si∗ (sj ) locus tends to be flatter as η declines. The second one implies that, the bigger the size of the group, the more distant the curve Dj (si∗ ) is from the diagonal and the steeper the curve is. Suppose group 1 is the minority and group 2 is the majority. Figure 1.8 shows the two partial steady state loci, D2 (s1∗ ) and D1 (s2∗ ), for each segregation level η. Note that the locus for group 2 is less sensitive to the change of η because the population size of the group is bigger than group 1 (β 1 < β 2 ). From the above lemmas, we have the following results in terms of the number of steady states. Proposition 6. The total number of steady states decreases from nine to three as η declines from one to zero, regardless of the population composition (β 1 , β 2 ). Regardless of η and (β 1 , β 2 ), there exist three symmetric steady states, (sl , sl ), (sm , sm ) and (sh , sh ). All other steady states are asymmetric. Proof. See the proof in Appendix.  The network qualities (or skill levels) of two groups at a steady state vary depending on the location of each steady state:

36 Corollary 2. Regardless of η < 1 and (β 1 , β 2 ), there exists a unique steady state (sh , sh ) in Region 2, and (sl , sl ) in Region 4. Regardless of η < 1 and (β 1 , β 2 ), any steady state (s1∗∗ , s2∗∗ , σ 1∗∗ , σ 2∗∗ ) in Region 1 satisfy sl < σ 1∗∗ (s1∗∗ ) < sm and sm < σ 2∗∗ (s2∗∗ ) < sh , and any steady state in Region 3 satisfy sm < σ 1∗∗ (s1∗∗ ) < sh and sl < σ 2∗∗ (s2∗∗ ) < sm . Proof. See the proof in Appendix.  Therefore, we can conclude that two groups are equally better off if the economy state is on any steady state in Regions 2 and 4. Group 2 is better off than group 1 on any steady state in Region 1 and group 1 is better off than group 2 on any steady state in Region 3. In order to analyze the welfare implications of the model, we need to define the Pareto dominant (or inferior) steady state. Definition 6 (Pareto Dominance). A steady state (s1 , s2 ) is a strictly Pareto dominant steady state if both s1 > s10 and s2 > s20 are satisfied for any other steady state (s10 , s20 ). Definition 7 (Pareto Inferiority). A steady state (s1 , s2 ) is a strictly Pareto inferior steady state if both s1 < s10 and s2 < s20 are satisfied for any other steady state (s10 , s20 ). With the definitions, we have the following result. Corollary 3. Regardless of η < 1 and β 1 , (sh , sh ) is a strictly Pareto dominant steady state and (sl , sl ) is a strictly Pareto inferior steady state. Proof. Because |Dj (si∗ ) − si∗ | is monotonically decreasing as η declines (Lemma 6), both groups’ skill levels become less than sh at any steady state with η < 1, except the fixed steady state (sh , sh ). Also, both groups’ skill levels become greater than sl at any steady state with η < 1, except the fixed steady state (sl , sl ).  We adopt the following notation rule for the deeper analysis in the following sections. When η = 1, each steady state (si , sj ) is denoted by Qij for i, j ∈ {l, m, h}. As η declines, each steady state is denoted by its original notation at η = 1. Thus, when we have less than nine steady states, we can identify each steady state by following its original name in the economy with η = 1. Note that the locations of the following three steady

37

states do not change with varying η or β: Qll , Qmm and Qhh . Other steady states continuously move as η or β changes. Particulary, the locations of Qlh and Qhl are denoted by the following rule. The skill levels of group 1 and group 2 at Qlh are denoted by (s0l , s0h ) and the skill levels of group 1 and group 2 at Qhl are denoted by (s00h , s00l ). When η < 1, the followings should hold by corollary 2: s0l > sl , s0h < sh , s00h < sh and s00l > sl .

1.5.2.3

Demarcation Surfaces

˙ 1t , Π ˙ 2t ) is (0, 0, 0, 0) at each steady state identified in the previous section. The vector field (s˙ 1t , s˙ 2t , Π At any other state (s1t , s2t , Π1t , Π2t ), the vector field is identified by the state’s location in a four dimensional Euclidian space E4 . It is hard to observe the moving direction at each state because the four dimensional space is not visible. Fortunately, the dynamic system in Theorem 1 implies each component of the vector field can be displayed in a three dimensional Euclidian space E3 , either in the (s1 , s2 , Π1 ) coordinates or in the (s1 , s2 , Π1 ) coordinates. Therefore, we can identify the demarcation manifolds in each three dimensional Euclidian space, which turn out to be two dimensional manifolds, surfaces:

s˙ 1t = 0 Surface

:

1 − G(A(σ 1 , Π1 )) = s1

(1.18)

s˙ 2t = 0 Surface

:

1 − G(A(σ 2 , Π2 )) = s2

(1.19)

˙ 1t = 0 Surface Π

:

˙ 2t = 0 Surface Π

:

δ¯ + f (σ 1 ) ρ+α ¯ + f (σ 2 ) δ Π2 = . ρ+α Π1 =

(1.20) (1.21)

In the space above the s˙ it = 0 surface, sit increases over time and, in the space below the surface, it ˙ it = 0 surface, Πit increases over time and, in the space decreases over time. In the space above the Π below the surface, it decreases over time. Therefore, when an initial steady state (s10 , s20 , Π10 , Π20 ) is identified to be in the above (below) s˙ 1t = 0 surface in the (s1 , s2 , Π1 ) coordinates, s˙ 10 is positive ˙ 1t = 0 surface in the (negative). When the initial state is identified to be in the above (below) Π

38 ˙ 1 is positive (negative). The same is true for s˙ 2 and Π ˙ 2 in the (s1 , s2 , Π2 ) (s1 , s2 , Π1 ) coordinates, Π 0 0 0 coordinates. ˙ 1t = 0 surface together in one (s1 , s2 , Π1 ) Figure 1.9 illustrates the s˙ 1t = 0 surface and the Π coordinates. The second pictures in the five panels show the sliced segments of the two surfaces ˙ 1 = 0 surface is captured with Π1 = for each level of s2 . The Π t

¯ (σ 1 ) δ+f ρ+α

each panel. The sliced segment of the surface in each panel is Π1 =

in the second picture of

¯ (σ 1 ) δ+f ρ+α

with σ 1 ranging over

[(1 − k 1 )s2 , k 1 + (1 − k 1 )s2 ]. For example, in panel A with s2 = 1, σ 1 ranges over [1 − k 1 , 1]. The segment of the s˙ 1 = 0 surface should satisfy the following two conditions with s2 given:

a ˜ 1 − G(˜ a)

= A(σ 1 , Π1 ) = s1

Points that satisfy the first condition are depicted as dotted gray curves for different levels of Π1 , named by iso-Π curves, in the first picture of each panel. Points that satisfy the second condition are depicted as a solid curve with the range of σ 1 in [(1 − k 1 )s2 , k 1 + (1 − k 1 )s2 ] in the same picture. The points that satisfy both conditions together constitute the segment of the surface given s2 . This is displayed in the second picture of each panel as a solid curve. Adding up all these segments for each ˙ 1 = 0 surface and the s˙ 1 = 0 surface. Note that the intersections s2 in [0, 1], we can construct the Π t t of two segments given s2 are the partial steady states (s1∗ , σ 1∗ , Π1∗ )|s2 .

1.5.2.4

Economically Stable States and Stable Manifolds

˙ 1t , Π ˙ 2t ), we are ready to identify Since we have identified steady states and the vector field (s˙ 1t , s˙ 2t , Π the economically stable states. Theorem 2. Regardless of η and (β 1 , β 2 ), steady states Qll , Qhh , Qlh and Qhl are economically stable states and all others are economically unstable states. Proof. See the proof in Appendix. 

39

Those economically stable states are illustrated in Figure 1.10 with dark circles. Since we do not impose functional forms of G, A and f , we do not have the explicit form of steady states. Consequently, we are not able to calculate eigenvalues for those steady states. However, it is possible to identify eigenvalues at three symmetric steady states because they are fixed points. We can show that there are two positive and two negative eigenvalues at steady states Qll and Qhh . Lemma 7. At the economically stable states Qll and Qhh , the solutions of the following equation constitute eigenvalues, among which two are positive and two are negative:

[λ2 − Rλ + H] · [λ2 − (ηR + (1 − η)ρ)λ + ηH − (1 − η)α(α + ρ)] = 0,

(1.22)

where R = [−GA0σ + ρ]Qii and H = [−α(α + ρ)(G0 A0σ + 1) − αG0 A0Π f 0 ]Qii . Proof. See the proof in Appendix.  Knowing this result, without loss of generality, we assume that two positive eigenvalues and two negative eigenvalues are at all economically stable states, because the dynamic structure of those economically stable states are locally identical. As far as the four eigenvalues are distinct, we can calculate the exact equilibrium path to each economically stable state in the neighborhood of the state. Corollary 4. Suppose four eigenvalues are distinct. The unique equilibrium path, given an initial skill composition (s1 (t0 ), s2 (t0 )) in the neighborhood of an economically stable state Qij (s10 , s20 , Π10 , Π20 ), is





 s1t 



 1

10



 10

s (t0 ) − s   s                s2   −1  2 Λ1 (t−t0 ) 20  W11  s (t0 ) − s    t  W11 e  s20   =  +         Π1     −1  1 W21 eΛ1 (t−t0 ) W11  t s (t0 ) − s10  Π10              2 2 20 Πt s (t0 ) − s Π20

(1.23)

, where Λ1 is a diagonal matrix containing two negative eigenvalues and B is a 4 × 4 matrix whose

40

column vectors are eigenvectors corresponding to two negative  eigenvalues  and two positive eigenvalW11 ues, which is composed of four 2 × 2 matrices Wij s, B =   W21

W12  .  W22

Proof. See the proof in Appendix.  The set of these equilibrium paths are stable manifold in the neighborhood of the state. The stable manifold theorem helps us to understand the shape of the stable manifold. Corollary 5. By the virtue of the Stable Manifold Theorem, the stable manifold to an economically stable state Qij (s10 , s20 , Π10 , Π20 ) is two dimensional and is tangent to the stable subspace ES of the linearized differential system at Qij . Suppose four eigenvalues are distinct. The stable subspace ES is represented by the following two planes:  ES :



Π1t 



 s1 −1  t

  = [W21 ][W11 ]   Π2t

10



 10

− s  Π   −  ,     Π20 s2t − s20

(1.24)

where the first plane is determined in the (s1 , s2 , Π1 ) coordinates and the second one in the (s1 , s2 , Π2 ) coordinates. Proof. The stable manifold theorem manifests that if the linearized dynamic system at an economically stable state Qij has two eigenvalues with negative real parts and two eigenvalues with positive real parts, a stable manifold SMij is two dimensional, which is tangent to the stable subspace ES of the linearized differential system at Qij . In Lemma 7, we have presented the characteristics of eigenvalues at any economically stable state. The stable subspace ES is derived from Corollary 4.  Therefore, we can infer that the stable manifold, which is two dimensional in a four dimensional Euclidian space E4 , is the combination of two surfaces, one which is identified in the (s1 , s2 , Π1 ) coordinates, and the other is identified in the (s1 , s2 , Π2 ) coordinates. The part of the stable manifold to the stable state Qlh is illustrated with blue and red curves in Figure 1.10. In the sliced phase spaces in the figure, we can check the relative position of the stable manifold with respect to demarcation

41 ˙ i = 0 surface). In Panel B surfaces of the time dependent variables (the s˙ i = 0 surface and the Π of Figure 11, I display the full picture of the stable manifold with the blue and red curves, which is a combination of two surfaces. The blue surface indicates Π1 on the stable manifold given (s1 , s2 ). The red surface indicates Π2 on the stable manifold given (s1 , s2 ). The manifold range (Mlh ), which is a projection of stable manifold SMlh to the (s1 , s2 ) plane, is depicted in orange color in Panel A of the figure, together with the blue and red curves. Panel C of the figure displays manifold ranges of all economically stable states and their overlapped areas. In the middle, we have a fourfolded overlap, and there are two-folded overlaps surrounding that. We can observe tiny areas of three-folded overlaps. The number of stable states decreases from four to two as η declines. The manifold ranges and their overlapped areas are depicted for each level of η in Figure 1.12. Two manifold ranges Mlh and Mhl disappear at some level of η, as the two stable states disappear. Two manifold ranges Mll and Mhh tend to expand as η declines, while the other two manifold ranges Mlh and Mhl tend to shrink as it declines. All manifold ranges are greater when the lifetime network externalities are more influential, as the size of overlap is greater with the stronger lifetime network externalities in a homogeneous economy (Proposition 3). Proposition 7. As η declines, the manifold ranges Mll and Mhh tend to expand, while the manifold ranges Mlh and Mhl tend to shrink. All manifold ranges tend to expand with the stronger lifetime network externalities (greater f 0 (σ)). Proof. The first argument is obvious when we look at the case with no lifetime externalities f (σ)0 = 0. In this case, manifold ranges are not overlapped at all, as illustrated in Appendix Figure 1.2. Each manifold range is a basin of attraction for an attractor (economically stable state). The basins are separated by separatrices that are connecting saddle points (economically unstable states). As economically unstable states get closer to the diagonal with the declined η (Lemma 6), the basins for economically unstable states should be shrinking. This analysis for the special case of f 0 (σ) = 0

42

is directly applied to the general case, because the only difference is bigger manifold ranges with greater lifetime externalities (positive f 0 (σ)). The second argument is true because the size of overlap in a homogeneous economy is analogous to the folded overlaps in a heterogeneous economy. When f 0 (σ) = 0, there is no overlap in a homogeneous economy, and no folded overlap in a heterogeneous economy. With greater f 0 (σ), they get larger, as Propostion 3 proves. 

1.5.3

Social Consensus and Network Trap

In the given economy, a maximum of four economically stable states exist. In Qhh (Qll ), both groups’ skill levels and network qualities are sh (sl ). In Qlh , group 1’s skill level and network quality are below sm , while group 2’s skill level and network quality are above sm , according to Corollary 2. In Qhl , we have the opposite result. Social consensus plays a critical role in the determination of the final destination of the economy. If optimism prevails and newborn members believe in the better network quality of the future, they are encouraged to invest more, expecting the higher benefits of investment to accrue over their lifetimes. If pessimism prevails and newborn members believe that network quality of the future gets worse, they are discouraged to invest, due to the declined expected benefits of investment. The social consensus about the future should be one of the above four stable states, because any other state is not stable and thus cannot be the final destination. In the analysis of the model, I propose that members in a society can agree to a social consensus within a reasonably short time period. Suppose the society is located in a four-folded overlap. The society faces four possible destinations. The final destination is determined by the belief of members in the society. If both group members are optimistic about the future, Qhh will be realized. If both are pessimistic about the future, Qll will be realized. If newborn members of group 1 are pessimistic about the group’s network quality and newborn members of group 2 are optimistic about the group’s network quality, the social consensus about the future will be formed as Qlh . The economic state (s1t , s2t ) will move toward the final destination Qlh gradually. In the same way, if the social consensus is formed with

43

group 1’s optimism and group 2’s pessimism, the state will move toward Qhl . Thus, the society can be mobilized to any stable state depending on the chosen social consensus: the mobilization force of lifetime network externalities strongly influence in this economy. Suppose the society is located in a two-folded overlap: for example, an overlap of Mhh and Mlh, in which group 2’s network quality is relatively better than group 1’s network quality. Newborns of group 2 will keep the optimistic view toward the group’s future because the following generations will maintain the high investment rate, due to the good quality of network externalities over the education period. Thus, there are two social consensuses available to the members in the society: Qhh and Qlh. . If newborn members of group 1 share the optimistic view toward the group’s future, the social consensus will be formed as Qhh instead of Qlh . The skill investment rate of group 1 will be enhanced significantly and the economic state (s1t , s2t ) will move toward Qhh . If they are pessimistic, the skill investment rate of group 1 will deteriorate over time and the economic state will move toward Qlh . Thus, group 1’s expectation toward the future determines the social consensus about the future, and strongly affects the skill investment pattern of group 1 newborns. The mobilization force of the lifetime network externalities influences group’1 future, while the historical force of the education period externalities influences group 2’s skill investment. When an initial economic state is in a deterministic range, “rational” group members understand that there is only one possible future. The social consensus will be quickly formed, and the economic state will move toward the stable state gradually. Among four deterministic ranges, two of them lead the economic state (s1t , s2t ) to an asymmetric steady state. If an initial economic state is in one of those ranges, “rational” newborns of one group will share a pessimistic view toward the group’s future, while “rational” newborns of the other group will share an optimistic view toward the group’s future. Envision a society in a deterministic range for Qlh . The current network quality of group 1 is so poor that there is no way to recover the skill investment rate of the group, while that of group 2 is so good that newborn members can maintain the high skill investment rate, benefiting from the good quality education period network externalities. Thus, the historical force of education period

44

network externalities determines the future. In many societies around the world, the economic state of the society is at an asymmetric stable state belonging to a deterministic range: for example, either Qlh or Qhl in a two group economy. At this state, a disadvantaged group cannot be mobilized to improve its skill investment rate, because the network externalities over the skill acquisition period have a strong adverse effect on the cost of skill achievement. This is a case where a social group is trapped by the network externalities. Definition 8 (Network Trap). Qlh (Qhl ) is called a network trap of group 1 (group 2) if it belongs to a deterministic range.

1.6

Egalitarian Policies in Network Trap

In this section, we discuss the egalitarian policies in a society placed in a network trap. Suppose that the economic state (s1 , s2 ) is at Qlh in a deterministic range, as Panel B of Figure 1.12 illustrates. Group 1 is disadvantaged in the social network structure. The skill disparity between two groups will persist indefinitely, without any governmental intervention or a structural change of the economy. Since Qhh is a Pareto dominant state, the government has an incentive to pursue egalitarian policies to mobilize the society toward the equality of his skills. We analyze two kinds of egalitarian policies, the integration between groups and the implementation of affirmative actions such as quota and training subsidies. Then, we will discuss the way to implement an effective policy for the different sizes of the disadvantaged group. In the end, we emphasize the importance of good leadership of the disadvantaged group in the mobilization out of the trap: encouragement of optimism and fostering of within-group cooperation.23 23 This is what civic leaders, civic organizations, and religious groups can contribute for a more egalitarian society. In US history, many civil rights activists and organizations including Rev. Martin Luther King Jr. and the National Urban League contributed to the improvement of the black community.

45

1.6.1

Integration Effect

Imagine a society in a network trap of group 1, Qlh . First, note that the asymmetric stable state disappears as integration is facilitated. The threshold level of the segregation level ηˆ depends on the population size of the disadvantaged group: ηˆ ≡ ηˆ(β 1 ). The following Lemma summarizes the shape of the function ηˆ(β 1 ). ˆ and strictly increasing in Lemma 8. There exists βˆ such that ηˆ(β 1 ) is strictly decreasing in (0, β) ˆ 1). (β, Note that, as integration proceeds, either Qlh and Qmh are merged together or Qlh and Qlm are merged together, before Qlh disappears. Therefore, before the steady state Qlh disappears, the state must get into either Mhh or Mll or both, because the manifold range Mhh (Mll ) always covers the unstable state Qmh (Qlm ) as far as η 6= 1. With this fact integrated with the above Lemma, we have the full picture of the integration effect, which is summarized in Figure 1.13. In the diagram, η ∗ (β 1 ) ˆ indicates the threshold level of η for Qlh ’s entering Mhh . Note that η ∗ (β 1 ) > ηˆ(β 1 ) with β 1 ∈ (0, β). η ∗∗ (β 1 ) indicates the threshold level of η for Qlh ’s entering Mll . Note that η ∗∗ (β 1 ) > ηˆ(β 1 ) with ˆ 1). Let us denote β ∗ with which η ∗ (β 1 ) = ηˆ(β 1 ), and β ∗∗ with which η ∗∗ (β 1 ) = ηˆ(β 1 ), where β ∈ (β, β ∗ > β ∗∗ , as displayed in the figure. As far as both groups are optimistic rather then pessimistic, the integration can lead the society to the pareto dominant Qhh when β 1 is in (0, β ∗ ), because Qlh will move into the manifold range Mhh before its disappearance. However, if β 1 > β ∗ , the integration will lead to the manifold range Mll and the society may fall down to Qll as Qlh disappears. Proposition 8. Suppose members in the society are optimistic rather than pessimistic. As integration proceeds, the economic state moves to the pareto dominant state Qhh with β 1 ∈ (0, β ∗ ), and it moves to the pareto inferior state Qll with β 1 ∈ (β ∗ , 1). However, if members in the society are pessimistic rather optimistic, they may tend to choose Qll rather than Qhh for their social consensus about the future, when the economic state is in the overlap of Mll and Mhh . They would tend to stay at Qlh rather than moving toward Qhh , when the

46

equilibrium path to Qhh is feasible. In this pessimistic society, integration will lead both groups to fall down to Qll as far as β 1 > β ∗∗ , as Figure 1.13 illustrates. Corollary 6. Suppose members in the society are pessimistic rather than optimistic. As integration proceeds, the economic state moves to the pareto dominant state Qhh with β 1 ∈ (0, β ∗∗ ), and it moves to the pareto inferior state Qll with β 1 ∈ (β ∗∗ , 1). In either situation, integration has an adverse effect for the welfare improvement when the population size of the disadvantaged group is relatively big. In this case, we should consider other policy tools together with the integration for the effective implementation of egalitarian policies.

1.6.2

Affirmative Action Policies

We consider two types of affirmative action policies: training subsidies and quota system. First, consider the training subsidy policy. Government can impose some taxes on the advantaged group (or skilled workers in general) and transfer the resources to the disadvantaged group (or unskilled workers in general) for the purpose of the enhanced skill investments. This policy targets decreasing the education cost for the disadvantaged group, while increasing the cost for the advantaged group. For simplicity, suppose that the policy is implemented in a way that all members of the disadvantaged group experience a certain amount of cost decrease for skill acquisition and those of the advantaged group experience a certain amount of cost increase for its acquisition. This effect is well reflected ˙ 1t = 0 surface and the shifted-down Π ˙ 2t = 0 surface, because the cost increase by the shifted-up Π (decrease) has the exactly same impact with the increased (decreased) benefits of investment on each individual’s decision making process, which is a simple cost-benefits comparison in the model. ˙ 1t = 0 surface leads the shift-down of the D2 (s1∗ ) If then, as Figure 1.9 shows, the shift-up of the Π ˙ 2t = 0 surface leads the shift-up of the D2 (s1∗ ) curve (or the curve, and the shift-down of the Π shift-to-right in (s1 , s2 ) plane). This impact is summarized in Panel A of Figure 14. As the curves shift enough, the steady state Qlh , which was a network trap of group 1, moves into the manifold

47

range of Qhh . Thus, by overturning the social consensus from Qlh to Qhh , the economic state can move toward a high skill symmetric state. Quota policy places some group 1 members, who are unskilled, into skilled job positions that otherwise would go to skilled members of group 2. Suppose the current economic state is Qlh (s0l , s0h ). The skilled job positions are fixed as s¯ = β 1 s01 + β 2 s0h in this economy. The government intervenes to mitigate the skilled job disparity between two groups, |s2 − s1 |. The higher fraction of group 1 take the skilled job positions, and the lesser fraction of group 2 takes the skilled job positions under the constraint of s¯ = β 1 s01 + β 2 s0h . If this external intervention can lead the economic state (s1 , s2 ) into the manifold range Mhh , the society will start to move toward the high skill symmetric state. More of group 1 newborns will be motivated to invest in skill acquisition, by sharing the optimistic view about the future. This process is displayed in Panel B of Figure 1.14. With the imposition of affirmative actions, members of group 2 may suffer temporarily, but the group state will improve in the end: both the group’s skill level and the network quality will approach sh , which is greater than s0h . The effectiveness of affirmative action is restricted by the size of the disadvantaged group. As it increases in size, the stronger action is required to mobilize the disadvantaged group out of the trap. If it is too big, there might be no way to implement effective affirmative policies to make society equal.

1.6.3

Policy Implementation

As discussed earlier, the population size of the disadvantaged group is critical for the effective implementation of the egalitarian policies. If it is small enough, any one type of policy may solve the problem (Proposition 8). This is depicted in the Panel As of Figure 1.15. The state X in Panel A1 of the figure indicates the original economic state, which is a network trap of group 1. As integration proceeds, X moves to X’ in Panel A2, which belongs to the manifold range Mhh . With the overturn of social consensus, the economic state starts to move toward Qhh . If the integration policy is difficult to implement due to the rigid division of social interactions, other affirmative

48 action policies can handle the problem. A quota way is illustrated in Panel A1: the state (s1 , s2 ) is relocated to Z 0 in Mhh by the quota implementation, and moves toward Qhh as social consensus overturns from Qlh to Qhh . (The training subsidy strategy is illustrated in Panel A of Figure 1.14.) However, if the population size of group 1 is too big, there is be no way for any type of policy to improve the situation. This is depicted in the Panel Bs of Figure 1.15. The integration alone will lead the original economic state X in Panel B1 to the pareto inferior state X 00 in Panel B3. A quota way alone cannot solve the problem as illustrated in Panel B1: the straight line constraint (¯ s = β 1 s01 +β 2 s0h ) does not go through the manifold range Mhh . The same is true for training subsidy policy: if the disadvantaged group is too big, per capita training subsidies would be very small and ˙ 1 surface does not shift would not significantly change the dynamic structure of the model (eg. the Π up enough). It is important to know that the mixed policies can be effective to mobilize the society to the pareto dominant state Qhh , even when the majority group is disadvantaged. The mixed policy of integration and quota system is depicted in Panel B2 of Figure 1.15. With the integration between groups, the group state will moves closer to the center: X in Panel B1 to X 0 in Panel B2. Then, the straight line from the position X may pass through the manifold range Mhh . Thus, this mixed policy helps the majority group to move out of the trap. A bundle of three policy methods integration, quota and training subsidy - might be more effective in the implementation. In the implementation of egalitarian policies, one important factor is the social consensus. Even though an effective policy leads the economic state to the overlap of Mhh and Mlh , the society may stay at Qlh consistently if the pessimism prevails in the society and newborns are not motivated to improve their skill investment rates. Therefore, the effective policy should come with the overturn of social consensus. Forward looking decision making and the optimistic view toward the future are crucial parts of the effective policy implementation. Another factor that can improve the effectiveness of policy implementation is the fostering of within-group cooperation. Even though the social interaction between two social groups cannot be proceeded significantly, it can be easier to improve the quality of social interactions among the disadvantaged group members. If the quality of

49 relationship is improved, the lifetime benefits of investment increase: the slope f (σ 1 ) gets steeper. We can check that the manifold range Mhh expands with the steeper f (σ 1 ) in the given model. The fostering of within-group cooperation can help the group move out the trap with the expanded folded overlap, when the proper egalitarian policies are activated together. These two factors, the optimism in the society and the within-group cooperation, can be facilitated by non-governmental institutions such as civic groups and religious institutions, and by civic leaders who can motivate and integrate the disadvantaged group members.

1.7

Macroeconomic Effects of Inequality

Human capital has been the prime engine of economic growth in the modern economy (Goldin and Katz 2001, Abramovitz and David 2000). The accumulation of intangible capital contributed to growth significantly, replacing the importance of physical capital accumulation in the early stage of the Industrial Revolution (Galor and Moav 2004). Because inequality is greatly associated with overall human capital achievement, it is natural to think about the macroeconomic effects of inequality (Benabu 1996). Along this line, Loury (1981) shows the positive effect of egalitarian policies on overall economic activities, under the intergenerational transfer of skill achievement. Galor and Zeira (1993) show the positive effect of equality on economic development, identifying the multiple equilibria with the assumption of indivisible human capital investment. The credit market constraint is the underlying force of the intergenerational mobility restriction in these studies. Unlike the previous literature, I suggest the positive effect of equality on economic development without imposing the assumption of credit market imperfection. Even in an economy with a perfect credit market, the social network externalities still restrict the skill achievements of the disadvantaged groups. With more equal distribution of social capital across social groups, the society can encourage more newborns from disadvantaged groups to invest in skill acquisition, and reach a more developed stage of an economy.

50

Of course, social network externality is a broad concept that can include accessibility to physical resources. For example, children of a rich community can afford the higher tuition for private schools and tend to have a better quality of schooling. It is easier for college graduates of the rich community to obtain the seed money for starting a business than those of the poor community. Therefore, in a society with credit constraints, the social network externalities will be stronger, both during education period and over the lifetime. However, the conclusion of the model sharply contrasts to the prediction of the previous literature, which suggests an equal society in the matured economy where credit constraint does not bind for the skill investment (Galor and Moav 2004), or an equal society with the centralized provision of training (perfect public school system) (Loury 1981). Even in the sufficiently developed economy with no binding of credit constraint or in the society with public provision of schooling, the social network externalities over the skill acquisition period (such as peer effects, parental effects, role models, and medical and nutritional provision) and over the working period (such as mentoring, job search and business connections ) still influence the incentives for skill acquisition and work as a major force of the intergenerational mobility constraint. Therefore, unlike the conclusions of the previous studies, equality, namely more equal distribution of social network capital in this study, will have a positive effect on the economic development even in the matured economy or in the society with a perfect public school system.

1.7.1

Multiple Equilibria as Different Development Stages

In the developed model, we have four economically stable states, two symmetric ones and two asymmetric ones. The two symmetric states, Qhh and Qll , indicate the most developed stage and the least developed stage: s¯ = sh and s¯ = sl for each. Two asymmetric states indicate the mediocre levels of development with group 2’s better off (Qlh ) and group 1’s better off (Qhl ), which are defined as network traps in the model: s¯ = β 1 s0l + β 2 s0h and s¯ = β 1 s00h + β 2 s00l for each. When the economy is trapped in either Qlh or Qhl , the egalitarian policies discussed in the previous section can help society to be mobilized to the most developed stage of Qhh : integration or affirmative actions such

51

as training subsidies and quotas can be the tool to motivate more newborns of disadvantaged groups to invest in skills. The structural change of the educational system or redistribution policy of income can help the group to move out of the trap by mitigating the adverse effects of poor education period network externalities, such as better public education system or more progressive tax system: with this structural change, the folded overlap may expand covering the unequal steady state. Panel A depicts the threshold level of economic development by the red line, above which egalitarian policies can promote the economic growth helping the economic state move into the manifold range Mhh . Below the threshold level, the policies may not be effective in the promotion of growth. The development stages can be more than four in a multi-group economy. If the number of social groups is n, the maximum number of economically stable state is 2n . Each of them can serve as a development stage of an economy. The case of three group economy is displayed in Panel C of Figure 1.16. Maximum eight development stages are identified in a three dimensional Euclidian space with the coordinates (s1 , s2 , s3 ), in which h (l) indicates the skill level of a group above (below) the medium skill level sm . For example, (h, h, l) indicates group 1 and group 2 achieve the higher skill level while group 3 is left behind with the low skill acquisition rate. The following proposition analogous to Theorem 1 summarizes n group economy, denoting the integration level between group i and group j by ηij and the average skill level of the two groups by s¯ij (t) ≡

β i sit +β j sjt β i +β j :

Proposition 9 (N-Group Economy). In a n-group economy, the dynamic system with n flow variables (s1t , · · ·, snt ) and n jumping variables (Π1t , · · ·, Πnt ) is summarized by the following 2n-variable differential equations: 

 i

s˙t   ˙ it Π

= =

α[1 − G(A(σti , Πit )) − sit ]  i h ¯ (σ i )  δ+f i (ρ + α) Πt − ρ+αt

,

(1.25)

i∈{1,2,···,n}

where σti = (ηi1 , ηi2 , · · ·, ηin ) · (¯ si1 (t), s¯i2 (t), · · ·, s¯in (t)), with ηij = ηji and

Pn

k=1

ηik = 1.

In this expanded (η, β) structure, the quality of social network of group i (σti ) is an inner product

52

of two vectors, a vector of between-group integration levels and a vector of between-group average skill levels. ηij = 1 indicates the perfect integration between group i and group j and their perfect segregation from all other social groups. Then, the quality of the social network of group i is equal to the average skill level between two groups: σti = s¯ij (t). ηij = 0 indicates zero contacts between the two social groups. Then, the quality of the social network of group i is not affected by the group j’s skill level:

∂σti ∂sj (t)

= 0. In this dynamic system, there are maximum 3n steady states including

maximum 2n economically stable states. The stable manifold to each economically stable state is an n dimensional manifold defined in 2n dimensional Euclidian space E2n . The manifold range of an economically stable state is a projection of n dimensional stable manifold to n dimensional Euclidian space En with the coordinates (s1 , s2 , · · ·, sn ). Using the same notation rule defined in Notation 1.5.2.2, we have the following implication for this economy. Corollary 7. In an economy with n social groups, there are maximum 2n distinct development stages. Qhh···h is the most developed stage and a pareto dominant steady state, and Qll···l is the least developed stage and a pareto inferior steady state. Proof. The proof is analogous to the proofs for the two group economy at Corollary 3 and Theorem 2.  This implies countries in the world might be in different development stages due to the different social network structure. In order to understand how equalitarian policies can promote economic development, check the following simple example: Suppose the initial economic state is (h, l, l) in a three group economy. Group 1 is a sufficiently big group and those three social groups are fully segregated. Suppose the initial state is in a deterministic range. Thus, the unequal state persists. If the integration between group 1 and group 2 is facilitated, the economic state will move into the folded overlap area of Mhll and Mhhl . As members of group 2 are motivated to increase the skill investment rate, the economy will move toward the state (h, h, l). If the integration is between group 1 and group 3, the economic state will move toward (h, h, h), which is the most developed

53

stage of the economy. Thus, an egalitarian policy, integration, will help the economy grow. The case is roughly illustrated in Panel C of Figure 1.16.

1.7.2

Positive Effect of Inequality

In most development stages, the egalitarian policies might facilitate the economic growth. However, if the economy is in its early stage of development, the effect is obscure. As Panel A of Figure 1.16 describes, there is no way to enter Mhh if the economic state is positioned below the threshold level of economic development depicted by the red line. Instead, there can be a positive effect of inequality, consistent with Galor and Tsiddon (1997). Suppose the initial economic state is at Qll in a simple two group economy. Also, suppose the government has resources to invest for human capital development in the society, which might be borrowed from abroad or gained from selling natural resources. Panel B of Figure 1.16 illustrates the resource constraint for human capital development. As far as two social groups are separated significantly, the unequal distribution of development resources can be the best strategy for growth, because the unequal distribution may lead the skill composition into the manifold range Mlh or Mhl , while the equal distribution is more likely to lead it into a deterministic range of Qll . That is, when the resources are limited in the early stage of development, the concentration of social capital to some selective groups can promote the groups to enhance their skill investment rates significantly because they expect the increased return on skill achievement through the network externality channel. This might explain the concentration of education facilities in selective cities in many developing countries, rather than the equal distribution all over the countryside. The current group inequality that exists in many less developed countries can be a byproduct of an initial economic development promotion. This positive effect of inequality in the early stage of development along with the positive effect of equality in the later stage of development is consistent with the empirical findings that income tends to be more equally distributed in developed countries than less developed countries and the early stage of economic development often comes along with the growing inequality.

54

1.8

Application: Regional Group Inequality in South Korea

In this section, I present a historical example of between-group disparity - regional group disparity in South Korea. The example displays how an initially advantaged group enhances its skill acquisition activities by holding an optimistic view about the group’s network quality, and reinforces its dominant position. Most social interactions in Korean society had occurred within each region (Youngnam, Honam, Chungcheong, Kangwon, etc.) before the rapid urbanization in the last decades, as displayed in Figure 1.17. Even after the urbanization, which caused a huge population to migrate to South Korea’s main metropolis, Seoul, over the industrialization process, the regional based social interactions have been the strongest in the social interactions among Seoul migrants through hometown gatherings, high school alumni, or extended family reunions. Two regional groups, Youngnam and Honam, are most distinguished due to their rivalry size and geographical separation by the Taebaek Mountains that separate the peninsula.24 In the 1950s after the Korean war (1950-53), there was a negligible difference between these two regional groups: both were poor and low skilled, as indicated by point A in the skill composition map of Panel A of Figure 1.18. Over the next decades, the between-group disparity has grown significantly: for example, among leaders in contemporary Korean society, 43.35% were born in Youngnam and 21.88% were born in Honam.25 As Appendix Figure 1.3 demonstrates, members of the Youngnam are much more represented in most professions than those of the Honam. The following explanations present the process of the group inequality evolution in the early stage of economic development and the diminishing between-group inequality in the later stage of economic development. Emergence of Initial Group Disparity in the 1960s and 1970s In the 1960s and 70s, the industrialization was strongly pushed forward by President Park’s administration, whose regional origin was Youngnam. Ministerial officials from his native province, 24 According to the 1949 Census, Youngnam constituted 31.43%, Honam 25.24%, Metropolitan Area (Seoul and Gyunggi) 20.69%, Chungcheong 15.73%, Kangwon 5.65% and Cheju 1.26% of the total population. 25 Source: Chosun Daily Leaders’ Database in 2002 (www.dbchosun.com); Eui-Young Yu (2003). Note that these are calculated excluding Seoul born leaders because Seoul natives (about 5% of the population) were exceptionally more successful than migrants from the outskirts.

55 Youngnam, were favored for the stability of the military regime (Ha 2007).26 Youngnam-created companies and businessmen took advantage of the social connections to the administration, while the rivalry social group Honam, which was least connected to the administration, was most disadvantaged. In the early 80s, about half of the largest conglomerates were Youngnam-based, while only ten percent of them were Honam-based.27 More Youngnam-born workers were hired by big companies and promoted to the manager level using the social ties and connections. The Youngnam-dominating circumstance led Honam group to be against the Park’s political party denoted by “Industrial party”, and Youngnam group to be strongly supportive for the party as reflected in the presidential elections since 1971 in Appendix Figure 1.4. The disparity emerged under the Park’s regime is described in Panel B of Figure 1.18: denoting the skill levels of Honam and Youngnam groups by sh and sy respectively, the state (sh , sy ) moved from a low-skilled equal state A to a unequal state A0 in an overlap range of Mll and Mlh . Enhanced Human Capital Investment of the Youngnam Group since the Mid 1970s Even after the assassination of President Park in 1979, the Youngnam based military regime continued until the early 1990s. Youngnam-created business conglomerates were successful in the global market. Members of the Youngnam maintained the optimistic view about the future that the network quality of the group persistently improves over time. As the dynamic model of this chapter predicts, they enhanced human capital investment expecting the higher returns accrued over their lifetimes. The college advancement rate in Figure 1.19 well reflects the enhanced skill investment activities of young members of the Youngnam: since the late 70s, their college advancement rate started to be significantly higher than other regional groups. It maintained 7 to 13 percent higher rate than the national average in the 80s and 90s. More importantly, the higher college advancement rate continued under the strictly equal provision of schooling in those days: since the late 70s, all 26 Among ministerial officials between 1962 and 1984, excluding Seoul or North Korea born officials, 48.76% were born in Youngnam while only 16.25% were born in Honam; Among CEOs of major banks, 52.63% were born in Youngnam and 5.26% were born in Honam. (Hankook Daily 1/27/1989) 27 Among the founders of the largest fifty conglomerates in 1985, 22 were born in Youngnam, four in Honam, twelve in Seoul and Gyunggi, five in North Korea and seven in other regions (MH Kim 1991). The politically connected firms are favored by lenders in the developing countries (Khwaja and Mian 2005).

56

private secondary schools were merged into the public school system and the salary of the teachers became identical across all secondary schools in the country.28 The evolution of skill composition (sh , sy ) over the period is described in Panel C of Figure 1.18. The members of the Youngnam significantly improved skill investment rates with an optimistic view about the future, while those of the Honam continued the lower skill investment activities with a pessimistic view. Enhanced Human Capital Investment of the Honam Group since the Early 1990s The situation started to change in the early 1990s. The first democratic regime took place in the 1992 presidential election and a Honam-born candidate was elected as the President for the first time in the next election of 1997.29 Social integration between two regional groups proceeded over time. The power transfer from Youngnam to Honam and the progressed social interaction between two regional groups helped young members of the Honam to hold the optimistic view that the network quality of the group will improve over time. They enhanced the skill investment activities expecting the higher lifetime returns on the investment. Figure 1.19 demonstrates the highest level of college advancement rate of Honam since 1994. The enhanced skill investment activities of Honam is described in Panel D of Figure 1.18: the skill composition (sh , sy ) was placed in an overlap range of Mlh and Mhh in the early 1990s due to the integration effect and the power transfer to Honam. As members of the Honam hold an optimistic view about the future, the skill composition (sh , sy ) started to move toward the high-skilled equal state (sh , sh ).30 The dominating position of the Youngnam regional group helped the increased human capital investment of the Korean economy in the 70s and 80s. Noting that the human capital accumulation is a driving force of the economic development, the dominating position of a selective group provided a 28 Even more, over the same period, any type of private tutoring was prohibited by law. Also, students were randomly assigned to the schools in most cities. 29 As Appendix Figure 1.4 displays, a fraction of the original democratic party was merged into the industrial party in 1992, which had been led by President Park in the 70s. The democratic leader YS Kim, the candidate of the “new” industrial party, competed against another democratic leader DJ Kim, the candidate of the democratic party, in the 1992 presidential election. 30 The underclass of Seoul continue lower skill investment activities. It is plausible that they are trapped in the network structure due to the urban poverty problem and the consequent low quality of education period network externalities. This might persist in the future. It is noteworthy that a considerable percentage of the Seoul underclass are migrants from Honam who moved in the 70s and 80s.

57

positive effect on economic growth, which is consistent with what the given dynamic model suggests for the early stage of economic development. The power transfer from one group to another group and the more equally distributed social network capital helped another social group Honam to improve its skill investment activities significantly. This promoted the further economic growth by the improved human capital investment in the economy. Thus, as the model predicts, the equality has a positive effect on economic growth in the later stage of economic development.

1.9

Conclusion

I explore the dynamic structure of group inequality evolution through the channel of social network externalities. The interaction of two kinds of network externalities, those operating during the skill acquisition period and those at work over the course of a worker’s life, provides a unique dynamic picture with folded overlaps and deterministic ranges. The former are the skill composition ranges in which the mobilization force of lifetime network externality is most influential, and the latter are ranges in which the historical force of education period network externality is most influential. Unequal stable states in deterministic ranges are defined as network traps, in which a disadvantaged social group cannot improve its skill investment rate without a governmental intervention or a structural change of the economy. Egalitarian policies to mobilize the group out of the trap are examined. Any type of egalitarian policy, integration, quota or training subsidies, can be effective in an economy with a minority disadvantaged group. If the disadvantaged group is the majority, one policy alone cannot solve the problem, but a combination of different policies may mobilize the group to change its skill investment activities. The dynamic model in this chapter identifies multiple steady states of groups’ skill levels that can be interpreted as different development stages, considering that the economic growth is driven by the accumulation of human capital in the modern economy. The positive effect of egalitarian policies on the economic development is discussed. When social network capital, the quality of social

58

network, is more equally distributed between social groups, more disadvantaged group members are motivated to invest in their skills with the increased return on skill acquisition to accrue over their lifetime and, thus, the economy grows. However, if the economy is in early stage of development, the unequal distribution of social capital could be better for the economic growth to take off, because at least selective groups are motivated to develop their skills even under the strong adverse effects of poor quality network externalities over the skill acquisition period. This implies that unequal skill distribution between social groups in many less developed countries might be the byproduct of an initial economic development promotion. It is noteworthy that the macroeconomic effects of equality/inequality have been examined even without imposing the standard assumption of an imperfect credit market. Therefore, unlike the previous studies (Loury 1981, Galor and Zeira 1993), the result implies a positive effect of equality even in an economy where a credit constraint is not binding for the skill achievement, or where public provision of education is well cultivated. The theoretical framework in this chapter is unique in terms of its dynamic structure with multiple overlaps. The folded overlaps and social consensus in the model are innovative ideas to deal with distinct social groups with different economic statuses and expectations toward the future. The concepts can be applied to other research areas dealing with heterogeneous economic groups, such as trade between nations and games between teams. Also, the rational expectation framework, combined with an overlapping generation structure, provides a unique way to analyze the intergenerational social mobility. A similar method is applied to my companion paper (Kim and Loury 2008), which is summarized in Chapter 2, for the analysis of the evolution of group reputation. Future research related to intergenerational social mobility may adopt this method. Noting the importance of social networks to one’s economic success, the lack of theoretical works along this line is awaiting research in the field of economics. The theoretical framework suggested in this paper could be a good building block for more research on the social networks and social capital.

59

1.10

Appendix: Proofs

Proof of Lemma 2. ¯

(sx ) 0 00 0 0 Let us define a function sy (sx ): sy ≡ 1 − G(A(sx , δ+f ρ+α )). Then, s and s satisfy s > sy (s ) and ¯

(0) s00 < sy (s00 ), according to the given condition. Because A(0, δ+f ρ+α ) < ∞, sy (0) > 0, which implies ¯

(1) that at least one steady state exists in (0, s0 ). Because A(1, δ+f ρ+α ) > 0, sy (1) < 1, which implies

that at least one steady state exists in (s00 , 1). By the continuity of sy (sx ), there must be at least one steady state in (s0 , s00 ). QED.

Proof of Lemma 3. Using an implicit function theorem, we have the following result from equation (1.8) for any state on the s˙ = 0 locus: ∂Π 1 + G0 A0s =− . ∂s (s=0 G0 A0Π (s=0 ˙ locus ) ˙ locus )

(1.26)

˙ = 0 locus: From equation (1.9), we have the following result for any state on the Π

f 0 ∂Π = . ˙ ˙ ∂s (Π=0 locus ) ρ + α (Π=0 locus )

(1.27)

From the demarcation loci described in Panel B of Figure 1.1, we know that the slope at the s˙ = 0 ˙ = 0 locus at the steady states El and Eh , and the slope at the locus is greater than that at the Π ˙ = 0 at the steady state Em : s˙ = 0 locus is smaller than that at the Π

1 + G0 A0s G0 A0Π 1 + G0 A0s − G0 A0Π −

> <

f0 ρ+α f0 ρ+α

(at El or Eh ).

(1.28)

(at Em ).

(1.29)

60 ¯ is Given the dynamic system in equation (1.7), its linearization around a steady state (¯ s, Π)

s˙ t

¯ = α[−G0 A0s − 1](st − s¯) + α[−G0 A0Π ](Πt − Π)

˙t Π

=

¯ −f 0 (st − s¯) + (ρ + α)(Πt − Π).

Therefore, the Jacobian matrix JE evaluated at a steady state is 



−αG0 A0s − α JE ≡   −f 0

−αG0 A0Π    ρ+α

.

¯ (¯ s,Π)

Consequently, its transpose is trJE = −αG0 A0s + ρ and the determinant is |JE | = −α(ρ + α)(G0 A0s + 1) − αf 0 G0 A0Π . Since trJE is positive, every steady state is unstable. |JE | is negative at El and Eh because of equation (1.28), which implies that those are saddle points. |JE | is positive at Em because of equation (1.29), which implies that Em is a source, either an unstable node or an unstable focus. QED.

Proof of Proposition 2. ˙ = 0 locus between sm Suppose eo ≥ sm . This means that the saddle path to Eh intersects the Π and sh . Let us denote the intersection point by C(sc , Πc ), where sm ≤ sc < sh . Because it is on the ˙ = 0 locus, Πc | ˙ Π Π=0 locus =

¯ (sc ) δ+f ρ+α .

Because it is on the saddle path to Eh ,

Z

c

Π |saddle path =

∞

[δ¯ + f (sτ )]e−(ρ+α)(τ −t) dτ,

t

where st = sc , sτ > sc , ∀τ > t, and limτ →∞ sτ = sh . Therefore, we have

c

Π |saddle path

Z =

∞

[δ¯ + f (sc ) + (f (sτ ) − f (sc ))]e−(ρ+λ)(τ −t) dτ Z ∞ δ¯ + f (sc ) + [f (sτ ) − f (sc )]e−(ρ+λ)(τ −t) dτ. ρ+α t t

=

(1.30)

61 c Then, Πc |saddle path > Πc |Π=0 ˙ locus because f (sτ )−f (s ) > 0, ∀τ > t. This contradicts the assumption

that there exists an intersection of the locus and the saddle path between sm and sh . Therefore, eo < sm . In the same way, we can prove that ep > sm . Thus, a positive range of overlap [eo , ep ] exists, where eo < ep . Since two saddle paths in the overlap exist (one path to Eh , and the other path to El ), the social consensus determines the one to be taken. QED.

Proof of Proposition 3. [Optimistic Path] Above the two demarcation loci, the dynamic system is determined by

s˙ t

= α(1 − st )

˙t Π

=

(ρ + α)Πt − q1 st − δ¯ − q0 .

(1.31)

¯

¯ op is (1, δ+q0 +q1 ). In this dynamic system, two eigenvalues are −α and α+ρ and the steady state (¯ s, Π) ρ+α Then, we have the explicit functions of st and Πt that satisfy the saddle path condition limτ →∞ (st , Πt ) = ¯ op : (¯ s, Π)

st

= C ∗ e−αt +

Πt

=

C∗

δ¯ + q0 + q1 ρ+α

ρ + 2α −αt e + 1, q1

(1.32)

where C ∗ depends on the initial condition (s0 , Π0 ) on the saddle path. Thus, we have the saddle ¯

op 0 +q1 path that converges to Eh (1, δ+q = ρ+α ): Π

q1 op ρ+2α s

+

¯ 0 )(ρ+2α)+q1 α (δ+q . (ρ+α)(ρ+2α)

[Pessimistic Path] Under the two demarcation loci, the dynamic system is determined by

s˙ t

= −αst

˙t Π

=

(ρ + α)Πt − q1 st − δ¯ − q0 .

(1.33)

62 ¯ 0 ¯ pe is (0, δ+q In this dynamic system, two eigenvalues are −α and α + ρ and the steady state (¯ s, Π) ρ+α ).

Then, we have the explicit functions of st and Πt that satisfy the saddle path condition limτ →∞ (st , Πt ) = ¯ pe : (¯ s, Π)

st

= C ∗ e−αt +

Πt

= C∗

δ¯ + q0 ρ+α

ρ + 2α −αt e , q1

(1.34)

where C ∗ depends on the initial condition (s0 , Π0 ) on the saddle path. Thus, we have the saddle ¯

pe 0 = path that converges to El (0, δ+q ρ+α ): Π

q1 pe ρ+2α s

+

¯ 0 δ+q ρ+α .

QED.

Proof of Lemma 5. By the implicit function theorem imposed at equation (1.16), we have the following first order derivative:   1 1 d sj i = −k . ˜ i∗ ))0 d si∗ 1 − k i −(G(A(σ

(1.35)

0 ˜ By assumption 1, there exists σ ˆ such that G(A(σ)) is decreasing in [0, σ ˆ ] and increasing in [ˆ σ , 1].

As equation (1.16) implies, σ i∗ is monotonically increasing with si∗ . Therefore,

d sj d si∗

is decreasing

˜ σ )), and increasing where si∗ < 1 − G(A(ˆ ˜ σ )). Equivalently, Dj (si∗ ) is concave where si∗ < 1 − G(A(ˆ ˜ σ )), and convex where si∗ < 1 − G(A(ˆ ˜ σ )). QED. where si∗ < 1 − G(A(ˆ

Proof of Lemma 6. ˜−1 −1 (1−si∗ )−si∗ Note that |Dj (si∗ ) − si∗ | = A G 1−k = i

|σ i∗ −si∗ | 1−ki ,

˜ i∗ )). |σ i∗ − si∗ | because si∗ = 1 − G(A(σ

is fixed as Panel B of Figure 1.6 describes. The first derivative gives

∂|D j (si∗ )−si∗ | ∂η

=

|σ i∗ −si∗ | (1−β i )(1−η)2 .

Thus, |Dj (si∗ ) − si∗ | shrinks as η increases. Also, it becomes larger with the bigger β i because

63 ∂|D j (si∗ )−si∗ | ∂β i

=

|σ i∗ −si∗ | (1−β i )2 (1−η) .

j i∗ )−si∗ ) Finally, let us denote the slope ∂(D (s by Q: ∂si∗

  1 1 i , because of equation (1.35). Q = − k − 1 i ˜ i∗ ))0 1 − k −(G(A(σ 1 1 . − 1 = ˜ i∗ ))0 1 − k i −(G(A(σ

Then, the first derivative with respect to β i is

∂Q ∂β i

=

1 (1−β i )2 (1−η)

(1.36)

1 −(G(A(σ ˜ i∗ ))0 − 1 . Thus, the slope

is steeper with the bigger β i . QED.

Proof of Proposition 6. The total number of steady states is nine with η = 1, as discussed in section 1.5.1. By the dynamic system in a homogeneous economy summarized in (1.7), the three states, (sl , sl ), (sm , sm ) and (sh , sh ), are steady states in a heterogeneous economy regardless of η and β 1 . For example, in the ˙1 = Π ˙ 1 = 0 in the dynamic case of (sh , sh ), σ 1 = σ 2 = s1 = s2 = sh . They satisfy s˙ 1 = s˙ 2 = Π system summarized in Theorem 1. First of all, I claim that there are no symmetric steady states other than those three. Suppose that a symmetric steady state (ˆ s, sˆ) exists, which is not one of the three. Since σ 1 = σ 2 = s1 = s2 = sˆ, this implies

s˙t 1

=

s˙t 2

  α 1 − G(A(ˆ s, Π2t )) − sˆ = 0   δ¯ + f (ˆ s) 1 = (ρ + α) Πt − =0 ρ+α   δ¯ + f (ˆ s) = (ρ + α) Π2t − = 0. ρ+α

˙ 1t Π ˙ 2t Π

  α 1 − G(A(ˆ s, Π1t )) − sˆ = 0

=

This contradicts that there are only three skill levels (sl , sm , sh ) that satisfy formula (1.7). Therefore, there are only three steady states regardless of η and β 1 . Secondly, let us prove that the total number of steady states is three with η = 0. This is true when there are no asymmetric steady states with η = 0. Suppose an asymmetric steady state

64 (ˆ s1 , sˆ2 ) exists, where sˆ1 6= sˆ2 . Since two groups are fully integrated, σ 1 = σ 2 = s¯. Since it is a (global) steady state, it should be a partial steady state. By equations (1.13) and (1.14), si∗ is uniquely determined by σ i∗ , which implies that sˆ1 =ˆ s2 when σ 1 = σ 2 . This contradicts that it is an asymmetric steady state. Therefore, there is no asymmetric steady state when η = 0. Since there are only three symmetric steady states, the number of steady states is three when two groups are fully integrated. The total number of steady states monotonically decreases from nine to three as η declines, because |Dj (si∗ ) − si∗ | is monotonically decreasing as η declines (Lemma 6) and there is a unique inflection point in the partial steady state loci (D2 (s1∗ ) and D1 (s2∗ )) (Lemma 5). This implies the number of steady states decreases from three to zero as η declines, in Regions 1 and 3, and there is always a unique steady state in Regions 2 and 4. QED.

Proof of Corollary 2. The uniqueness of the steady states in Regions 2 and 4 is already proven in the proof of Proposition 6. Let us prove that all steady states satisfy σ 1∗∗ < sm and σ 2∗∗ > sm in Region 1. The distance |Dj (si∗ ) − si∗ | is monotonically decreasing as η declines (Lemma 6), which means the partial steady state loci move closer to the diagonal as η declines. This implies that the following should hold: s1∗∗ < sm and s2∗∗ > sm at any steady state (s1∗∗ , s2∗∗ ) with η < 1. In Region 1, the partial steady state locus s1∗ (s2 ) is below the sm = k 1 s1∗ + (1 − k 1 )s2 line, because (s1∗ , σ 1∗ ) must satisfy ˜ 1∗ )) from equations (1.13) and (1.14), and, due to its monotonicity, σ 1∗ < sm s1∗ = 1 − G(A(σ when s1∗ < sm . By the same reasoning, the partial steady state locus s2∗ (s1 ) is above the sm = k 2 s2∗ + (1 − k 2 )s1 line, because σ 2∗ > sm when s2∗ > sm . Therefore, σ 1∗∗ is smaller than sm because any steady state (s1∗∗ , s2∗∗ ) in Region 1 must be below the sm = k 1 s1∗ + (1 − k 1 )s2 line. σ 2∗∗ is greater than sm because the steady state must be above the sm = k 2 s2∗ + (1 − k 2 )s1 line. Now, let us prove that σ 1∗∗ > sl and σ 2∗∗ < sh . Since the distance |Dj (si∗ )−si∗ | is monotonically decreasing as η declines (Lemma 6), any steady state in Region 1 should satisfy the following two

65 conditions with η < 1: sl < s1∗∗ < sm and sm < s2∗∗ < sh . This implies that sl < σ 1∗∗ < sh and sl < σ 2∗∗ < sh . Therefore, we can conclude that sl < σ 1∗∗ (s1∗∗ ) < sm and sm < σ 2∗∗ (s2∗∗ ) < sh for any steady state in Region 1 with η < 1 given. In the same way, we can prove that all steady states satisfy sm < σ 1∗∗ (s1∗∗ ) < sh and sl < σ 2∗∗ (s2∗∗ ) < sm in Region 3. QED.

Proof of Theorem 2. Let us check the local stability at one steady state Qhh . We have the following Jacobian matrix at the steady state Qhh (sh , sh , Πh , Πh ): 

 0

JQhh

A0σ (η

0

1

A0σ (1

2

− η)β ] + (1 − η)β ) − 1] α[−G α[−G    α[−G0 A0σ (η + (1 − η)β 2 ) − 1] α[−G0 A0σ (1 − η)β 1 ]  =   −fσ0 (η + (1 − η)β 1 ) −fσ0 (1 − η)β 2    −fσ0 (η + (1 − η)β 2 ) −fσ0 (1 − η)β 1



0

α[−G

A0Π ]

0 ρ+α 0

0

   0 0  α[−G AΠ ]    0    ρ+α Qhh



J11 Let us denote JQhh −λI using 2×2 matrices Jij s: JQhh −λI =   J21

J12  . We need to calculate the  J22

−1 determinant of JQhh −λI in order to find eigenvalues. Note that |JQhh −λI| ≡ |J22 |·|J11 −J12 J22 J21 |. −1 Let us denote the second term by J 0 : J 0 ≡ J11 − J12 J22 J21 . Using the explicit forms of Jij s, J 0 is



  0

J0

α[−G = J11 −   0 

A0Π ]

 −1

 (ρ + α − λ) ·   0 0 0 α[−G AΠ ]  0

−fσ0 (1 − η)β 2  −fσ0 (η + (1 − η)β 1 ) . ·   0 1 0 2 −fσ (1 − η)β −fσ (η + (1 − η)β )

0 (ρ + α − λ)−1

  

(1.37)

.

66

Thus, its determinant is  η + (1 − η)β 1 |J 0 | = J11 − αξ   (1 − η)β 1 =

 0 0 0 (1 − η)β   , where ξ = G AΠ fσ .  ρ+α−λ η + (1 − η)β 2 2

[λ − α(−G0 A0σ η − 1) + αξη]· [λ − α(−G0 A0σ − 1) + αξ].

(1.38)

The result is achieved with a bit messy calculation. Therefore, we have the determinant of J − λI:

|JQhh − λI| = |J22 | · [λ − α(−G0 A0σ η − 1) + αξη] · [λ − α(−G0 A0σ − 1) + αξ] =

[λ2 − λ(−αG0 A0σ + ρ) − α(α + ρ)(G0 A0σ + 1) − αG0 A0Π fσ0 ]Qhh ·[λ2 − λ(−αG0 A0σ η + ρ) − α(α + ρ)(G0 A0σ η + 1) − αG0 A0Π fσ0 η]Qhh . (1.39)

Taking |JQhh − λI| = 0, we can obtain four eigenvalues at the steady state. First, note that [−α(α+ρ)(G0 A0σ +1)−αG0 A0Π fσ0 ]Qhh < 0 by equation (1.28). Thus, the first term of the determinant has one positive and one negative eigenvalue. That is, the local stability condition at Ehh in a homogeneous economy implies one negative and one positive eigenvalue in a heterogeneous economy at Qhh . Also, we have [−α(α + ρ)(G0 A0σ η + 1) − αG0 A0Π fσ0 η]Qhh < 0 because −α(α + ρ)(G0 A0σ η + 1) − αG0 A0Π fσ0 η = η(−α(α + ρ)(G0 A0σ + 1) − αG0 A0Π fσ0 ) − α(α + ρ)(1 − η). Therefore, there are two positive eigenvalues and two negative eigenvalues. There exists a unique equilibrium path if the number of jumping variables equals the number of eigenvalues with a positive real part (Buiter, 1984). Since we have two jumping variables, Π1t and Π2t , we know the existence of the unique equilibrium path in the neighborhood of (sh , sh ). Therefore, Qhh is an economically stable state. The four steady states Qll , Qhh , Qlh and Qhl are identical in terms of their local dynamic structures, as manifested by local demarcation surfaces at those states. We can conclude that those four steady states are economically stable states. Without loss of generality, we can infer that two eigenvalues with a positive real part and two with a negative

67

real part exist at those states. All other steady states, Qlm , Qmh , Qml , Qhm and Qmm , are economically unstable steady states. For example, check the local stability of Qmm . Using equation (1.39), we have the determinant JQmm − λI:

|JQmm − λI| =

[λ2 − λ(−αG0 A0σ + ρ) − α(α + ρ)(G0 A0σ + 1) − αG0 A0Π fσ0 ]Qmm ·[λ2 − λ(−αG0 A0σ η + ρ) − α(α + ρ)(G0 A0σ η + 1) − αG0 A0Π fσ0 η]Qmm (1.40)

We know that [−α(α + ρ)(G0 A0σ + 1) − αG0 A0Π fσ0 ]Qmm > 0, by equation (1.29), and −αG0 A0σ + ρ > 0 because of A0σ < 0. Thus, the first term of the determinant implies two eigenvalues with positive real parts. The second term implies at least one eigenvalue with positive real part because −αG0 A0σ η + ρ > 0. Therefore, at least three eigenvalues have positive real parts. Since we have only two jumping variables, we cannot always find a unique equilibrium path in the neighborhood of (sm , sm ). Thus, Qmm is an economically unstable state. Now check other states. Since all other four are identical in terms of their dynamic structures, we need to check only one of them: Qmh . When η = 1, there must be three eigenvalues with positive real parts and one negative eigenvalue, because group 1 is at an economically unstable state Em and group 2 is at an economically stable state Eh in the separated dynamic structures of two groups. Thus, Qmh is an economically unstable state since the number of positive eigenvalues exceeds the number of jumping variables: except s1 = sm , there is no converging path to the state in the neighborhood of (sm , sh ). We cannot explicitly calculate the signs of eigenvalues with η < 1. However, the qualitative approach using demarcation surfaces identified in section 1.5.2.3 helps us to conclude that it cannot be an economically stable state for any η, because we can easily find at least one point (s1 , s2 ) in the neighborhood of (s0m , s0h ), in which a converging equilibrium path to Qmh (s0m , s0h ) does not exist. QED.

Proof of Lemma 7.

68 The given determinant equation is obtained in the proof of Theorem 2. Both R and ηR + (1 − η)ρ are positive because A0σ < 1. Both H and ηH − (1 − η)α(α + ρ)] are negative at economically stable states, Qll and Qhh , because of El and Eh are economically stable states in a homogeneous economy and satisfy condition (1.28). QED.

Proof of Corollary 4. At an economically stable state Qij , the linearized dynamic system is expressed with the Jacobian matrix JQij : 





 s˜˙ (t)  J11 =     ˙ ˜ Π(t) J21  J11 where JQij =   J21





J12   s˜(t)   ,   ˜ J22 Π(t)



 s1t



(1.41)



 Π1t

10

10

−Π  J12  −s  ˜ , in which Jij is 2 × 2 ma and Π(t)  , s˜(t) =  =      20 20 2 2 Πt − Π J22 st − s

trix. Let us define the expectation operator E with I(t), which is the information set conditioning expectations formed at time t: for any vector x, Et x(τ ) ≡ E(x(τ )|I(t)). This means that Et x(τ ) is the expected value x at time τ given the information set at time t. Let us define x(t) ˙   ) as x(t) ˙ ≡ limu→t x(u)−x(t) . Then, Et x(τ ˙ ) = E limu→τ x(u)−x(τ I(t) . Taking the expectation u−t u−τ operator at both sides of the above equation, we have 

  ˙  Et s˜(t)  J11  =    ˙ ˜ Et Π(t) J21





J12   Et s˜(t)   .   ˜ J22 Et Π(t)

(1.42)

Note that there are two positive eigenvalues and two negative eigenvalues at an economically stable state according to Theorem 2. Since they are distinct by assumption, there are four linearly independent eigenvectors. Then, we have Jordan form with a diagonal matrix Λ:

JQij = BΛB −1 ,

(1.43)

69 

 Λ1 in which Λ =   0

0  with Λ1 (Λ2 ) containing two negative (positive) eigenvalues, and the column  Λ2

vectors Let us partition B and B −1 into four 2×2 matrices:  eigenvectors.   of B are the  corresponding W11 B=  W21  

W12  V11  and B −1 =    V12 W22  

V21   . Let us define two dimensional vectors p˜(t) and q˜(t) as  V22

 s˜(t)  p˜(t) . Then, using the Jordan form, we have the following result.  = B −1       ˜ Π(t) q˜(t)

       ˙ ˙ Et p˜(t) Et p˜(t)  Et s˜(t)   Et s˜(t)  .  = Λ ⇒  = BΛB −1           ˙ ˜ ˜ ˙ Et q˜(t) Et q˜(t) Et Π(t) Et Π(t) 

(1.44)

Therefore, we have Et q˜˙(t) = Λ2 Et q˜(t). This is true for any time τ ≥ t: Eτ q˜˙(τ ) = Λ2 Eτ q˜(τ ). Taking the expectation operator Et at both sides, we have Et Eτ q˜˙(τ ) = Λ2 Et Eτ q˜(τ ). Note that, for any random vectors u,v and w, E(E(u|v, w)|w) = E(u|w). Since I(τ ) ⊇ I(t), we have the consequent result, Et q˜˙(τ ) = Λ2 Et q˜(τ ).

(1.45)

This means that a forward looking individual’s expectation for time τ variation of q˜(τ ), given information set I(t), follow the above equation. An forward looking individual can expect q˜(τ ) to be Et q˜(τ ) = eΛ2 τ K,

∀τ ≥ t,

(1.46)

in which K is a two dimensional arbitrary constant. The forward looking “rational” individuals who know that q˜(τ ) should not explode over time will adjust their jumping variables (Π1t , Π2t ) in order to make Et q˜(∞) 6= ∞, which implies K = 0. This is a typical transversality condition. Therefore, we have Et q˜(τ ) = 0. This should be true for all τ ≥ t. We have

˜ Et q˜(t) = 0 ⇒ V21 s˜(t) + V22 Π(t) = 0,

(1.47)

70 ˜ because q(t) = V21 s˜(t) + V22 Π(t) and, for any vector x(t), Et (x(t)) = x(t). Therefore, we have

˜ = J11 s˜(t) + J12 Π(t)

s˜˙ (t)

(∵ equation (1.41))

=

−1 (J11 − J12 V22 V21 )˜ s(t)

=

−1 (J11 + J12 W21 W11 )˜ s(t)

=

−1 (J11 W11 + J12 W21 )W11 s˜(t)

−1 = W11 Λ1 W11 s˜(t)

(∵ equation (1.47)) (∵ BB −1 = I)

(∵ JQij B = BΛ).

Therefore, we know how the skill composition evolves around an economically stable state given (s10 , s20 ):

s˜(t)

−1

= eW11 Λ1 W11

(t−t0 )

s˜(t0 )

−1 = W11 eΛ1 (t−t0 ) W11 s˜(t0 ).

(1.48)

The corresponding benefits of investments are

˜ Π(t)

−1 = W21 W11 s˜(t)

(∵ equation (1.47) and BB −1 = I)

−1 = W21 eΛ1 (t−t0 ) W11 s˜(t0 ).





s1t

10

 Π1t

(1.49) 

10

−s  −Π  ˜  and Π(t)  ,we have the unique equilibrium path given Applying s˜(t) =  =     2 20 2 20 st − s Πt − Π (s10 , s20 ) in the neighborhood Qij : 



 s1t 



 1

10



 10

s (t0 ) − s   s                s2   −1  2 Λ1 (t−t0 ) 20  W11  s (t0 ) − s    t  W11 e  s20   =   +  .        Π1     −1  1 W21 eΛ1 (t−t0 ) W11 s (t0 ) − s10  Π10   t             2 2 20 s (t0 ) − s Π20 Πt

71

QED.

Proof of Lemma 8. Note that, as integration proceeds, either Qlh and Qmh are merged together or Qlh and Qlm are merged together before Qlh disappears. First, envision a threshold segregation level for a sufficiently small β 10 : ηˆ(β 10 ). With the threshold level, the D2 (s1 ∗) curve will be tangent to the D1 (s2∗ ) curve and Qlh will be merged with Qmh , as Panel C of Figure 1.12 illustrates approximately. Now, let us increase β 10 to β 10 +  holding η = ηˆ(β 10 ). With this increase, D1 (s2∗ ) moves away from a diagonal because of the increased β 1 and D2 (s1∗ ) curve moves closer to the the diagonal because of the increased β 2 , according to Lemma 6. Thus, two steady states, Qlh and Qmh , get more distant from each other. In order to merge them and to make D2 (s1 ∗) curve tangent to the D1 (s2 ∗) curve, the lower segregation level is required. Therefore, ηˆ(β 10 ) > ηˆ(β 10 + ), which implies ηˆ(β 1 ) is a strictly deceasing function with the lower level of β 1 . Now, imagine a threshold segregation level for a sufficiently great β 100 : ηˆ(β 100 ). With this threshold level, the D1 (s2 ∗) curve will be tangent to the D2 (s1∗ ) curve and Qlh will be merged with Qlm , as Panel B-2 of Figure 1.15 illustrates approximately. Now, let us decrease β 100 to β 100 −  holding η = ηˆ(β 100 ). With this decrease, D2 (s1∗ ) moves away from a diagonal because of the increased β 2 and the D1 (s2∗ ) curve moves closer to the the diagonal because of the decreased β 1 , according to Lemma 6. Thus, two steady states, Qlh and Qlm , get more distant from each other. In order to merge them and to make the D1 (s2 ∗) curve tangent to the D2 (s1 ∗) curve, the lower segregation level is required. Therefore, ηˆ(β 10 ) > ηˆ(β 10 − ), which implies ηˆ(β 1 ) is a strictly increasing function with the higher level of β 1 . ˆ with which all three steady states, Qlh , Qmh Finally, imagine a group 1 population size of β, ˆ With an increase of β 1 to βˆ + , and Qlm , are merged together at some level of segregation: ηˆ(β). D1 (s2∗ ) moves away from a diagonal and the D2 (s1∗ ) curve moves close to the the diagonal, which means only one steady state Qmh survives and the two others disappear. This implies the threshold

72 ˆ < ηˆ(βˆ + ). With a decrease of β 1 to βˆ − , level of segregation should be higher with βˆ + : ηˆ(β) D2 (s1∗ ) moves away from a diagonal and the D1 (s2∗ ) curve moves closer to the the diagonal, which means only one steady state Qlm survives and the two others disappear. This implies the threshold ˆ < ηˆ(βˆ − ). Therefore, ηˆ(β) ˆ is a local minima. level of segregation should be higher with βˆ − : ηˆ(β) ˆ Qlh and Qmh are merged at the threshold segregation level (before Therefore, with β 1 ∈ (0, β), their disappearance), and the threshold level ηˆ(β 1 ) is a strictly decreasing function of β 1 . With ˆ 1), Qlh and Qlm are merged at the threshold segregation level, and the threshold level β 1 ∈ (β, ηˆ(β 1 ) is a strictly increasing function of β 1 . QED.

Chapter 2

Group Reputation and the Dynamics of Statistical Discrimination Joint work with Glenn C. Loury.

2.1

Introduction

Previous literature on statistical discrimination explained stereotypes based on the existence of multiple equilibria, in which principals have different self-confirming beliefs about different social groups (Arrow, 1973, Coate and Loury, 1993). However, the literature has not provided an account of where the principals’ prior beliefs come from, nor an account of which particular groups should be expected to have an advantage when unequal group stereotypes become confirmed in equilibrium (Moro and Norman, 2004, Chaudhuri and Sethi, 2008). Moreover, the static models dominating the literature cannot be used to understand the dynamic paths that lead to each equilibrium. In this chapter, we (I and Glenn Loury) develop a dynamic version of statistical discrimination in which economic players’ forward-looking behaviors determine the dynamic path to each equilibrium. With the paths identified, the self-confirming belief is explained by the consequence of the historical

73

74

development of the overall quality of each group. The developed dynamic model can provide conditions to reach each equilibrium point and to switch from one equilibrium point to another. Consequently, we can identify groups to be advantaged or disadvantaged, based on their initial historical positions. By understanding the dynamic mechanism, we can provide a richer analysis of egalitarian policies than static models can, by reflecting on the forward-looking decision making of principals and agents. We start by distinguishing group reputation from individual reputation. Group reputation is defined as the average characteristics of the group members, which is shared by principals. Individual reputation is defined as the probability that an individual is qualified for a certain task, given his group identity and his personal records, which is assessed by the principals who hire him. The essential point is that an individual’s reputation is influenced by the reputation of the group to which he belongs, when his personal records are insufficient to clarify his qualification for the task. The more insufficient the records are, the more the principals rely on the average characteristics of the group in their assignment decision. Therefore, given the same personal records, an individual with a good group reputation is treated more favorably than one with a bad group reputation. This implies that an individual’s decision for skill investment to be qualified for a task is affected by others’ skill investment in the same identity group; each individual makes his investment decision by considering the expected group reputation in the future, which is determined by other group members’ skill investment now and in the future. If more of them are expected to invest, he has more incentive to invest in the qualification for the task because the expected payoff will be greater. This externality of group reputation implies the possibility of collective action to improve or worsen group reputation, which is simply a self-fulfilling process: if each group member believes that other group members will invest, the expected payoff is high and it is likely that more members will invest, but if each of them doubts that others will invest, the expected payoff is low and it is likely that few will invest. In this chapter, we identify the multiple steady states, as most statistical discrimination models

75

do. Then, we will check how these dynamic aspects of group reputation help to explain the dynamic paths to reach each steady state. For a concrete analysis, we adopted a basic set-up of job assignment models introduced by Coate and Loury (1993). There are two jobs, task one and task zero, and task one is the more rewarding and demanding job. Principals determine who will be assigned to task one. Given the bell-shaped distribution of investment cost among the population, we identify three steady states. The dynamic system engaged with the externality of group reputation proves that two are saddle points and one is an unstable source. The dynamic path that leads to each saddle point is easily traced in the phase diagram. By having two equilibrium paths to two saddle points, high and low reputation steady states, we define the overlap of the two paths. Within the overlap, either the good or bad reputation steady state can be reached, which means that if a group shares an optimistic view about the future, the high reputation state is gradually realized in the future, while the bad reputation state is realized if the group shares a pessimistic view toward the future. Outside the overlap, the historical position, an initial reputation level, determines the final reputation state; the group with an initial reputation above the overlap range converges to the high reputation steady state, while the group with initial reputation below it converges to the low reputation state. There is an important point regarding the overlap: the condition to recover the good reputation level from the bad steady state is determined by the range of overlap. If the overlap covers the low reputation steady state, the group trapped in the low reputation is given a chance to improve the quality of the group by sharing the optimism together and can ultimately reach the high reputation steady state. If it does not cover the low reputation steady state, the group trapped in the state cannot escape it unless there is a policy intervention. By using this dynamic structure of group reputation, we explain the persistent racial inequality in the United States. When the overt discrimination in the past results in a very low reputation of the black group, the group will improve its reputation over time as the practice of overt discrimination disappears. However, the reputation of the group may improve only up to the low reputation steady

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state and stay persistently there because of the non-existence of a path to the high reputation steady state. The white group, which is initially better positioned than the black group, is advantaged by being given the path to the high reputation level. The model can be applied to other inequality contexts, including the male-female disparity in a patriarchical society, where parents give sons more opportunities than to daughters to develop their potential. (In turn, grown sons are obliged to support their aged parents.) Using the model, we show that the female group is more likely to be trapped in the low reputation steady state, compared to the male group. The high reputation steady state is pareto dominant to the low state. Principals may have an incentive to help the disadvantaged group in the reputation trap to improve its skill investment rate, so that principals can increase their profits. We distinguish monopolistic principals from competitive principals (Loury 2002). Competitive principals cannot change the status of the disadvantaged group because the size of each principal is relatively insignificant and one’s actions cannot affect the overall behavior of numerous agents. However, monopolistic principals, which are defined as a very small number of principals in the economy or principals well coordinated by a mediator such as government, are able to change the structure of the economy and affect the behavior of the disadvantaged group. We investigate three possible strategies that principals may consider: (1) applying a favorable hiring standard, (2) subsidizing the training cost, and (3) improving the screening process. Each may incur some cost to principals. Principals, if they are well coordinated, may decide which one is to be chosen by comparing the costs and increased profits of the strategies. However, even monopolistic principals may not take action if they are too myopic to anticipate the far future. Furthermore, actions of monopolistic principals may not change the behavior of the disadvantaged group if their “promise” to continue the actions is not considered credible by workers within the group. Therefore, even in the monopolistic situation, the farsightedness of principals and the credibility of their actions are pre-conditions for the market to voluntarily cure the misery of the disadvantaged group. Suppose that the market fails to cure the between-group inequality, due to any of the reasons

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mentioned above. Then, we expect that the government implements equalizing policies, which would be one or a combination of the following: colorblind hiring enforcement, quota systems, and subsidies for training costs. We examine the conditions that are required for each policy to be effective. The dynamic reputation model developed in this chapter can be applied to other subjects in which collective reputation matters, such as brand management (Tirole 1996), crime and racial stereotype (O’Flaherty and Sethi 2004), and institutional (e.g. college) reputation. This chapter is organized into the following sections. Section 2.2 explains the related literature. Section 2.3 describes the motivation of this research. Section 2.4 develops the dynamic reputation model. Section 2.5 shows the applications of the model. Section 2.6 describes the strategies of monopolistic principals. Section 2.7 examines the egalitarian policies. Section 2.8 presents further discussions. Section 2.9 contains the conclusion.

2.2

Literature

This work was inspired by an insight of Jean Tirole (1996), who examined the persistent corrupt behaviors of group members. He derived the existence of multiple stereotypes from “history dependence” rather than from self-confirming prior belief, which statistical discrimination literature had been based on since the seminal work of Arrow (1973). A member’s past behavior is imperfectly observed by principals. Thus, principals use collective reputation as well as the member’s imperfect track record in the determination of hiring. Poor collective behavior in the past may make the current good behavior a low-yield individual investment and thus generates poor collective behavior in the future. Tirole concludes that a negative stereotype, once developed, can be long lasting: a onetime, non-recurrent shock due to the behavior of a group can prevent the group from ever returning to a satisfactory state, even long after the people affected by the original shock have died. That is, bad (or good) collective reputation steady state originates from poor (or good) behavior in the past, not from the employers’ prior beliefs. In addition, the paper points out the dynamic aspects

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of reputation building: a group with a negative stereotype can collectively work to re-establish trust by enduring incurred loss until the group’s reputation returns progressively to the good-reputation level. Tirole’s game-theoretical approach, however, has several limitations. His work ignores the importance of group expectations about the future: over some range of initial reputation, either a good reputation steady state or a bad reputation steady state can be a final destination of the group, depending on the shared beliefs among group members about the future. For example, under some circumstances, even a group with a good reputation may fall to the bad reputation steady state if pessimism prevails among group members. This coordination issue is not addressed properly in his work. Secondly, his work fails to provide the critical level of an initial group reputation that determines the existence of a recovery path, as well as the time spent to reach each steady state from an initial reputation level. Moreover, it is hard to observe how parameters affect the dynamic structure of group reputation. With these shortcomings, his model cannot be used to evaluate various policy methods targeting the elimination of negative stereotypes of a disadvantaged group. Third, his model is not applicable to the analysis of complex real-world issues with heterogeneous agents or continuous “signal” distribution. Our model overcomes the shortcomings mentioned above and enables us to examine the complex issues regarding heterogeneity by graphical display of equilibrium paths and phase diagrams. We are indebted to Krugman’s insight about the interpretation of two equilibrium paths leading to two steady states (Krugman 1991). Within the overlap, which is the area where both equilibrium paths are available to the agents, either steady state can be reached by the agents’ expectations about the future. Outside the overlap, a unique equilibrium path exists: the agents have no choice but to take the path ending in a designated steady state. Therefore, within the overlap, expectation determines the final state, while, outside the overlap, the final state is determined by history, the initial position. Being inspired by Krugman’s work, I develops a dynamic model of social mobility with network externality in Chapter 1, Kim(2008), in which Krugman’s history versus expectation

79

structure is combined with the overlapping generation model of Bowles, Loury and Sethi (2007). Adsera and Ray (1998) provide an important argument that overlap is generated only when agents have an incentive to choose the option that offers less appealing benefits at the decision moment. In Krugman’s example, overlap is generated because agents can have an incentive to choose the option that even offers loss at the moving moment, due to the moving cost being lower than the cost of moving in the future. In my model in Chapter 1, the incentive is originated by the structure of the overlapping generation model. Since agents are given only one chance to choose their occupational type at the early stage of their lives, they choose a type that offers less appealing benefits at the decision moment, expecting accumulated benefits in the future. This forward-looking decision making generates the overlap. The work in this chapter adopts a dynamic framework similar to that in Chapter 1, Kim (2008), in which overlap is generated by the forward-looking behaviors of agents in an overlapping generational model.

2.3

Motivation

In this section, we identify the multiple steady states in a job assignment model introduced by Coate and Loury (1993) and argue the limitation of static statistical discrimination models. Those who want to go to the dynamic reputation model directly may skip this section. Imagine a large number of identical employers and a larger population of workers. Each employer will be randomly matched with many workers from this population. Employers assign each worker to one of two jobs, called task one and task zero. Task one is a more demanding and rewarding assignment: workers get the gross benefit w if assigned to task one. All workers prefer to be assigned to task one, whether or not they are qualified for the task. Employers gain a net return Xq if they assign a qualified worker to task one and suffer a net loss Xu if they assign an unqualified worker to task one. Define ρ ≡ Xq /Xu to be the ratio of net gain to loss. A worker’s gross returns and an employer’s net return from an assignment to task zero are normalized to zero.

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Employers are unable to observe whether a worker is qualified for task one. Employers observe ¯ The distribution of θ depends on whether each worker’s group identity and a noisy signal θ ∈ [0, θ]. or not a worker is qualified. The signal might be the result of a test, an interview, or some form of onthe-job monitoring. The signal is distributed for a qualified worker as fq (θ), and for an unqualified worker as fu (θ), as displayed in Panel A of Figure 1. Define ψ(θ) ≡ fu (θ)/fq (θ), to be the likelihood ¯ which implies Fq (θ) ≤ Fu (θ) for all θ. ratio at θ. We assume that ψ(θ) is nonincreasing on [0, θ], Employers’ assignment policies will be characterized by the choice of hiring standard s for each group, such that only those workers with a signal observed to exceed the standard are assigned to the more demanding task. Given the proportion of qualified workers Πi among group i population, employers assign a group i worker who “emits” signal θ to task one position if the expected payoff, Xq · P rob[qualified|θ] − Xu · P rob[unqualified|θ], is nonnegative. Using Bayes’ rule, the posterior probability that he is qualified is

Πi fq (θ) Πi fq (θ)+(1−Πi )fu (θ) .

Therefore, the hiring standard s is a function

of Πi : 

¯ s (Π ) ≡ min θ ∈ [0, θ]|ψ(θ) ≤ ∗

i

ρΠi 1 − Πi

 ,

(2.1)

where s∗ (Πi ) is a nonincreasing function of Πi . Note that s∗ (0) ≤ θ¯ and s∗ (1) = 0. We now turn to a worker’s investment decision. Workers are qualified only if they made some ex ante investment. The cost of becoming qualified varies among workers and is distributed as CDF G(c) in (0, ∞). We assume that G(0) > 0 and G(W ) < 1, which implies that there is a fraction of the workers who will invest for very tiny expected benefits of investment, and there is a fraction of workers who will not invest even for the highest possible benefits W . If the assignment standard is s, the probability of assignment is 1 − Fq (s) when qualified, and 1 − Fu (s) when unqualified. A worker with investment cost c invests if and only if the net return of being qualified is greater or equal to the net return of being unqualified; invest if and only if W [Fu (s) − Fq (s)] ≥ c. Thus, ˜ among all workers facing the standard s, the proportion that becomes qualified is G(β(s)), denoting ˜ ˜ ˜ θ)) ¯ > 0. β(s) ≡ W [Fu (s) − Fq (s)]. Note that G(β(0)) = G(β(

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Figure 1 describes the multiple steady states given two noisy signals fu (θ) and fq (θ). Checking ˜ the boundary conditions of s∗ (Πi ) and G(β(s)), it is obvious that at least one steady state exists. It is most likely that there are three steady states if the number of steady states is not unique. Proposition 10 (Multiple Steady States). Assume that ψ(θ) is continuous and strictly decreasing ¯ and G(c) is continuous and satisfies G(0) > 0. If there are s1 and s2 in [0, θ] ¯ for which on [0, θ], ˜ 1 )) > G(β(s

ψ(s1 ) ρ+ψ(s1 ) ,

˜ 2 )) < G(β(s

ψ(s2 ) ρ+ψ(s2 )

and s1 < s2 , then at least three steady states exist.

For the same parameters and G(c) function, if signal functions fu and fq are more informative, there tends to be a unique steady state. This point is described in Figure 2: the multiplicity is originated from the imperfect information about the workers’ true characteristics. Note that the ˜ domain as well. In this domain, the dotted curve represents the steady states are identified in (Π, β) expected benefits of investment that is determined by employers’ hiring standard s∗ (Πi ). The S shaped solid curve indicates the proportion of workers who will invest given the benefits of investment ˜ β. In the previous statistical discrimination literature, those steady states are explained by selfconfirming prior beliefs: employers’ beliefs about the likelihood of a group’s members being qualified will determine the hiring standard for the group, and the standard will determine the fraction of each group who become qualified. When workers from one group (B’s, say) are believed less likely to be qualified, the belief for group B will be self conformed at the lower steady state, while workers from the other group (W’s, say) are believed more likely to be qualified, the belief for group W will be self confirmed at the higher steady state, as displayed in Figure 1. This is a situation of discriminatory behavior by employers and persistent skill disparity between two groups. However, the static model does not provide an explanation for where the employers’ prior beliefs come from, and why employers start to have different beliefs about different social groups. Also, it cannot explain the case that the initial employers’ belief is not at one of those steady states. The belief will be updated over time and may converge to one of the steady states. The model does not

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provide the evolution path from an initial state that is not a steady state. Also, one group stuck in one steady state may move to another steady state under some circumstances. The model cannot analyze the condition that enables the switch from one steady state to another. Above all, it ignores the forward looking behavior of group members. In the static model, it assumes that workers react to the employers’ prior belief without accounting for expected payoff in the future. This myopic assumption limits the dynamic analysis of the model. In sum, the static model does not provide any explanation except the possible scenario on each steady state. Nothing can be discussed for the states other than the steady states. For these reasons, its analysis on the policy implication is restricted around the steady states and limited by employers’ prior beliefs, for which the model does not provide an account. In this chapter, we try to overcome the shortcomings of the static models of statistical discrimination by introducing the fully dynamic framework with the economic agents’ forward looking decision making reflected.

2.4 2.4.1

Dynamic Reputation Model Group Reputation and Individual Reputation

Instead of relying on the employers’ prior beliefs, we propose that employers use the objective information about the overall quality of each group in their decision to set up the hiring standard applied to a group. The overall quality is the proportion of qualified workers Πi in the market in the given job assignment model. The objective information for the overall quality is directly computed from the following formula: F i (θ) = Πi Fq (θ) + (1 − Πi )Fu (θ). F i (θ) represents the fraction of group i workers who emit a signal below θ, which is easily observed by employers who are matched with a large population of each group. Assuming that Fq (θ) and Fu (θ) are common knowledge, each employer can obtain the information about the proportion of qualified workers among group i

83 members in the market using the aggregate information F i (θ),

Πi =

Fu (θ) − F i (θ) , ∀θ. Fu (θ) − Fq (θ)

(2.2)

Let us call the quality of group Πi , which is shared among employers, group reputation. Facing a job candidate of group identity i and signal θ, an employer will try to calculate the probability that he is qualified, so that he can make a decision of whether to assign him to task one. Let us call it an individual reputation of group identity i and signal θ, and denote it by R(i, θ):

R(i, θ) =

Πi fq (θ) . Πi fq (θ) + (1 − Πi )fu (θ)

(2.3)

Direct observation is that an individual reputation R(i, θ) is an increasing function of group reputation Πi :

∂R(i,θ) ∂Πi

> 0. The higher the expected individual reputation, the more incentive each

individual has to make skill investment. Consequently, an individual’s skill investment is affected by the expected group reputation in the future. Each individual will consider others’ investment decisions now and in the future, in his current decision of skill investment. This externality of group reputation contains the possibility of collective action to build up better group reputation together, or to drag down the good group reputation to the worse reputation state. In the following section, we will examine these dynamic aspects of group reputation, and will try to find the answers to the questions raised for the static models in the earlier section.

2.4.2

Dynamic System

In the dynamic model, we assume that each worker makes skill investment at the early stage of his life and then works for the rest of his life. He is subject to a “Poisson death process” with parameter λ (Kim 2008, Tirole 1996): in a unit period, each individual faces a probability of death λ. We assume that the total population of each group is constant. Therefore, in a unit period, a

84

fraction λ of workers are replaced by newborn agents. Suppose that each individual discounts future payoffs at the rate δ, and employers discount future payoffs at the rate r. Suppose that a worker is randomly matched with employers every period, which implies that he will go through the regular screening process every period. In the appendix, we will loosen this assumption by introducing a market learning process, in which the true characteristic of each worker is more likely to be revealed as he spends more time in the market. The expected extra benefit to being qualified at time τ (βτ ) is ω[Fu (sτ ) − Fq (sτ )], where ω is the wage rate at task 1, which satisfies W ≡

R∞ t

ωe−(δ+λ)(τ −t) dτ to be consistent with the wage level

in the static model. Employers gain a net return xq from the correct assignment and incur a net loss xu from incorrect assignment, where Xq ≡

R∞ t

xq e−r(τ −t) dτ and Xu ≡

R∞ t

xu e−r(τ −t) dτ to be

consistent with the profit level in the static model. Note that ρ ≡ Xq /Xu = xq /xu . The expected lifetime benefits of investment for workers born at time t is

R∞ t

βτ e−(δ+λ)(τ −t) dτ. For convenience,

we denote as Vt the “normalized” lifetime benefits of investment:

Z Vt = (δ + λ)

∞

βτ e−(δ+λ)(τ −t) dτ.

(2.4)

t

Taking a derivative with respect to t, we can describe how Vt evolves over time,

V˙t = (δ + λ)[Vt − βt ].

(2.5)

Let φt denote the fraction of workers born at time t who invest and become qualified. Since λ of the total population is replaced with newborn agents in a unit period, Πt evolves in short time interval ∆t in the following way,

 Πt+∆t ≈ λ∆t ·

φt + φt+∆t 2

 + (1 − λ∆t) · Πt .

(2.6)

85

By the rearrangement of this equation, we have

  ∆Πt Πt+∆t − Πt φt + φt+∆t ≡ ≈λ − Πt . ∆t ∆t 2

Taking ∆t → 0, we can express how Πt evolves over time,

˙ t = λ[φt − Πt ]. Π

Note that there is a direct way to achieve the same result. We can define Πt as Πt ≡

(2.7)

Rt −∞

λφτ e−λ(t−τ ) dτ ,

˙ t = λ[φt − Πt ]. and taking a derivative with respect to t, we have Π Theorem 3 (Dynamic System). The dynamic system with a flow variable Πt and a jumping variable Vt is summarized by the following two-variable differential equations:

˙t Π

=

λ[φt − Πt ]

V˙t

=

(δ + λ)[Vt − βt ],

(2.8)

with demarcation loci of

˙ t = 0 Locus : Π

Πt = φt

V˙t = 0 Locus : Vt = βt .

(2.9)

We can interpret the theorem as follows: the difference between the investment rate of the newborn cohort and the overall investment rate of group i workers determines the speed of group reputation change. The change in accrued benefits of investment at time t is determined by the difference between the accrued benefits of investment at time t and the time t level of the benefits of being qualified. Note that there is no change in group reputation if the fraction of the newborn cohorts who invest is exactly the same as the level of group reputation, and there is no change in

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lifetime benefits of investment if the current benefits of being qualified is exactly equal to the level of lifetime benefits of investment.

2.4.3

Simple Reputation Model

In order to understand the dynamic system correctly, we will start with the simplest functional forms that do not hurt the essential structure of the economy: fu (θ) is uniformly distributed in [0, θu ] and ¯ where θq < θu . The population of each group is constituted fq (θ) is uniformly distributed in [θq , θ], of three types of agents: Πl fraction of workers whose investment cost is very small and close to zero, 1 − Πh fraction of workers whose investment cost is very high and beyond the highest possible benefit from investment

ω δ+λ ,

and Πh − Πl fraction of workers whose investment cost is intermediate

and fixed as cm . Then, cost distribution G(c) is Πl for c ∈ (, cm ), and Πh for c ∈ (cm , ω/(δ + λ)). In this case, employers will set the hiring standard as either θu or θq . If the signal is below θq , the worker must be unqualified, and, if the signal is above θu , the worker must be qualified. If the signal is between θq and θu , the signal is unable to tell the true characteristic of the worker. Let us denote the probability that, if a worker does invest, his test outcome proves that he is qualified by Pq (=

1−θu 1−θq )

and the probability that, if a worker does not invest, his test outcome proves that he is

unqualified by Pu (=

θq θu ).

Assumption 2 (Imperfect Information). A qualified worker’s signal is less informative, compared to an unqualified worker’s signal. This is, the payoff uncertainty is greater for qualified workers ¯ compared to for unqualified workers: Pq < Pu , and equivalently, θq + θu > θ. From this assumption, we propose that non-qualification of workers is easily detected by employers. However, qualification of workers is relatively hard for employers to confirm. This is an essential part of the imperfect information in the labor market. If the investment of workers can be easily confirmed, workers do not have to worry that their chance to be assigned to a good job is affected by their group’s reputation.

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Employers must make a decision on whether or not to give the benefit of the doubt (BOD) if the signal is unclear. If they give BOD to a group, the hiring standard for the group is θu , but, if not, the hiring standard for the group is θq . Employers’ decision to give BOD is determined by the sign of expected payoff, xq · P rob[qualified|θ] − xu · P rob[unqualified|θ], for θq < θ < θu . Using Bayes’ rule, the posterior probability that the worker with group identity i and an unclear signal (θq < θ < θu ) is qualified is

Πi (1−Pq ) Πi (1−Pq )+(1−Πi )(1−Pu ) .

Thus, we can find the threshold level Π∗ , above

which employers give BOD and below which they do not give BOD, where Π∗ ≡ ρ=

xq xu .

1−Pu ρ(1−Pq )+1−Pu

Note that the threshold level can be obtained using equation (2.1) as well: Π∗ ≡

with

1−θq ρθu +1−θq ,

which is identical to the above. If agents with unclear signals are assigned to task one, that is, BOD is given, the extra benefit βτ to being qualified is ωPu , because βτ = ω[Fu (θq ) − Fq (θq )] = ωθq /θu . If agents with unclear signals are not assigned, that is, BOD is not given, the extra benefit to being qualified is ωPq , because βτ = ω[Fu (θu ) − Fq (θu )] = ω(1 − θu )/(1 − θq ). Therefore, at time t the extra benefit to being qualified is summarized by

βt (Πt ) =

    ωPu

for Πt ∈ [Π∗ , 1]

   ωPq

for Πt ∈ [0, Π∗ ).

(2.10)

Given the cost distribution G(c) among the newborn cohort, the fraction of newborn agents who become qualified is  φt = G

Vt δ+λ

 .

(2.11)

Using βt and φt , we can draw demarcation loci as displayed in Panel A of Figure 3. In the left (right) side of βt locus, the movement of V is westward (eastward). Above (below) φt locus, the movement of Π is southward (northward). As far as Π∗ is between Πh and Πl and (δ + λ)cm is between ωPq and ωPu , there will be multiple steady states, which are denoted as Qh (ωPu , Πh ), Qm ((δ + λ)cm , Π∗ ) and Ql (ωPq , Πl ) in the Panel. Note that the middle one Qm ((δ + λ)cm , Π∗ ) is a “conditional” steady state: that is, it becomes steady state only when φt = Π∗ for Vt = (δ + λ)cm

88 and βt = (δ + λ)cm for Πt = Π∗ .1 In the following sections, we will assume that Π∗ ∈ (Πl , Πh ) and (δ + λ)cm ∈ (ωPq , ωPu ). Otherwise, there is a unique steady state and nothing to be discussed because there will be no reputation disparity between social groups. In Panel B of Figure 3, we display the equilibrium path that leads to each steady state, Qh and Ql . In the next sections, we will provide concrete explanations about this dynamic structure and the economic meanings of equilibrium paths.

2.4.3.1

Properties of Simple Reputation Model

In order to have a deeper analysis of the dynamic model, we will focus on the gray box in Panel B of Figure 3, within which meaningful dynamic structure is constructed, by the adjustment of the scaling of Vt . Let us define vt as a linear transformation of Vt such as Vt = ωPq + ω(Pu − Pq )vt . Then, as vt ranges over [0, 1], Vt ranges over [ωPq , ωPu ], which is the entire range of the gray box. Let us denote by ξt the indicator of giving BOD: ξt = 1 if Πt > Π∗ and ξt = 0 if Πt < Π∗ . Since βt = wPu · ξt + wPq · (1 − ξt ), applying to equation (2.4), we have

Z Vt = ωPq + ω(Pu − Pq )(δ + λ)

∞

ξτ e−(δ+λ)(τ −t) dτ

t

Thus, vt simply indicates the normalized lifetime BOD:

Z vt = (δ + λ)

∞

ξτ e−(δ+λ)(τ −t) dτ.

t 1 The first condition implies that the fraction of the newborn worker who invest is Π∗ so that there is no change in the overall group reputation. The second condition implies the principals’ mixed strategy assigns only a fraction (δ+λ)c −wP v ∗ (= w(P m−P ) q ) of workers who emit an unclear signal to the task 1 position. Note that the latter is not consistent u

q

with the hiring standard rule defined in equation (2.1), in which s∗ (Π∗ ) = θq and βt = ωPu .

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The dynamic system in this modified model with a flow variable Πt and a jumping variable vt is

˙t Π

= λ[φt − Πt ]

v˙t

=

(δ + λ)[vt − ξt ],

with demarcation loci of

˙ t = 0 Locus Π

:

v˙t = 0 Locus

: vt = ξ t .

Πt = φt

The critical level of Vt , (δ + λ)cm , is denoted in this (vt , Πt ) domain as

v∗ ≡

(δ + λ)cm − wPq . w(Pu − Pq )

The differential equations in each region are divided by two lines vt = v ∗ and Πt = Π∗ and are displayed in Figure 4, named by regions I, II, III and IV, going counterclockwise. The corresponding steady states are Qh (1, Πh ), Qm (v ∗ , Π∗ ) and Ql (0, Πl ). In regions I and II, principals give BOD, which they do not give BOD in other regions. In regions II and III, only a fraction Pl of the newborn cohort invests, while a fraction Ph of the newborn cohort invest in regions I and IV. Definition 9 (Economically Stable State). A state (V 0 , Π0 ) is an economically stable state if there exists an equilibrium path that converges to the state for any Π in the neighborhood of Π0 . This means that a state is defined as “economically stable” if when nearby to the state, economic agents can find a reasonable equilibrium path that converges to it, even though the state itself is mathematically unstable: it is a saddle point in general (Kim 2008). Lemma 9 (Spiraling Out Paths). In the simple reputation model, the state Qm (v ∗ , Π∗ ) is unstable, and the phase paths around it spiral out.

90

Proof. See the proof in the appendix.  Lemma 10 (Curvature of Paths). In the simple reputation model, the equilibrium paths are concave on the right hand side of the v = v ∗ line, and convex on the left hand side of the v = v ∗ line. Proof. See the proof in the appendix. 

Using direction arrows in Panel A of Figure 3, we can easily identify the equilibrium paths to the steady states Qh and Ql , which are vertical straight lines nearby the states. Lemma 1 tells us that the paths spiral out around the state Qm . Proposition 11 (Dual Economically Stable States). In the simple reputation model, there exist two economically stable states, Qh (1, Πh ) and Ql (0, Πl ). Because there are two economically stable states, group members with group reputation Π0 ∈ [0, 1] may rationally conjecture that the final state should be either the high reputation state Qh or the reputation state Ql . Let us suppose that group members can make a consensus about the future state all together. Suppose that, once the consensus is built up, it can be passed to the next generations. For example, group members with its group reputation around Π∗ may guess that the final state would be Πh . Then, by rational reasoning, they will find the optimal path that leads to the high reputation state. Based on the optimal path and the expected payoff, a newborn cohort will make an investment decision. Generations following will make an investment decision based on the same optimal path leading to Qh , as far as the optimistic consensus is passed to the next generations. By this self-confirming process, the group will gradually approach the state Qh . However, if the group shares the pessimistic view toward the future and the pessimistic consensus is passed to the next generations, the reputation of the group may gradually fall down to the low reputation level Πl . Since either Qh or Ql is realized in the future for any given initial reputation level, it is worth checking which is superior to the other.

91

Proposition 12 (Pareto Dominance). In the simple reputation model, Qh is strictly Pareto dominant to Ql ; all economic agents, including employers and workers with different investment costs, are better off when the group state (vt , Πt ) stays at Qh than at Ql . Proof. See the proof in the appendix. 

Thus, the high reputation state Qh is socially more desirable than the low reputation state Ql . It is noteworthy that all types are better off at the high reputation state; even high investment cost individuals who will never invest for the job qualification are better off in this state. One interesting point is that employers are better off when group reputation is good than bad.

2.4.3.2

Interpretation of Simple Reputation Model

Denote the lower boundary of the equilibrium path to Qh as π o , and the upper boundary of the equilibrium path to Ql as π p . Denote the initial reputation level of group i as Πi0 . At any initial reputation level Πi0 ∈ [π o , 1], group i can converge to the high reputation state Πh by sharing an optimistic view of the future among group members. At any initial reputation level Πi0 ∈ [0, π p ], group i can converge to the low reputation state Πl by sharing a pessimistic view of the future among group members. Thus, we call the equilibrium path to Qh the “optimistic path”, and the equilibrium path to Ql the “pessimistic path”. In the given simple reputation model, the optimistic path passes through (1, Π∗ ). The overall shape of the optimistic path is determined by how much the concave curve in region IV is bent. If the path passes through between Πl and Π∗ at v = v ∗ , the path changes its direction entering region III, and π o becomes greater than Πl . Otherwise, the path maintains its direction entering the region and π o becomes zero. The pessimistic path passes through (0, Π∗ ). The overall shape of the path is determined by how much the convex curve in region II is bent. If the path passes through between Πh and Π∗ at v = v ∗ , the path changes its direction entering region I, and π p becomes smaller than Πh . Otherwise, the paths maintains its direction entering the region and π p becomes one.

92 Definition 10 (Overlap). The range of group reputation level [π o , π p ] is called “overlap”; if the initial group reputation Πi0 is within the overlap, group i can converge either to the high reputation state Qh by sharing an optimistic view among group members, or to the low reputation state Ql by sharing a pessimistic view among them. Note that, with an initial reputation level Πi0 ∈ (πp , 1], group i “must” converge to the high reputation state Qh because the optimistic path is the only reasonable path. With an initial reputation level Πi0 ∈ [0, πo ), group i “must” converge to the low reputation state Ql because the pessimistic path is the only reasonable path. Therefore, those ranges, (πp , 1] and [0, πo ), are respectively called a deterministic range for Qh and a deterministic range for Ql . Definition 11 (Reputation Trap). The low reputation state Ql is called a“reputation trap” if Πl belongs to the deterministic range for Ql , [0, πo ), namely below the overlap [πo , πp ]. Thus, if a group is in the reputation trap, there is no way to recover its reputation without a change in the dynamic structure. Lemma 11 (Separation). In the simple reputation model, if π o > Πl and π p < Πh , that is, the two economically stable states Qh and Ql are “separate” from each other; a group in either state cannot move to the other state by changing its expectations:

max{− ln(1 − v∗), − ln(v ∗ )} − 1 δ > . λ ln(Πh − Πl ) − ln(Π∗ − Πl )

Proof. Using π o and π p listed below, the condition is directly obtained. 

In order to analyze the properties of overlap, let us suppose δ is big enough that two economically stable states are “separate” from each other. Then, the lower boundary of the optimistic path and

93

the upper boundary of the pessimistic path are

λ

πo

=

Πh + (Π∗ − Πh )v ∗− δ+λ ,

πp

=

Πl + (Π∗ − Πl )(1 − v ∗ )− δ+λ .

λ

The size of overlap L(≡ π p − π o ) is directly computed as

λ

λ

L = Πl − Πh + (Π∗ − Πl )(1 − v ∗ )− δ+λ + (Πh − Π∗ )v ∗− δ+λ .

Therefore, we have the following properties of overlap size:

∂L ∂δ

< 0 and

∂L ∂λ

(2.12)

> 0. This implies that,

the more weight that workers place on future payoffs (lower δ), or the faster generations are replaced by newborns (higher λ), the size of the overlap tends to be bigger (bigger L), which means that the expectation toward the future plays a greater role in the determination of the final economic outcome. Also, the overlap shifts up with higher level of investment cost:

∂π p ∂cm

> 0 and

∂π o ∂cm

> 0.

This implies that the higher (lower) the investment cost, the more likely that a group converges to the low (high) reputation state Ql (Qh ). Proposition 13 (Properties of Overlap). Under Lemma 11, the size of overlap tends to be bigger with the smaller δ, and the larger λ: expectation toward the future tends to plays a bigger role if workers discount the future payoff less, and if generations are replaced faster. The range of overlap tends to shift up with bigger cm : social groups are more likely to converge to the low reputation state Ql when investment cost is bigger.

2.4.4

Generalization of Simple Reputation Model

Now let us come back to the static statistical reputation model of Coate and Loury (1993) introduced in the section on motivation. Given the noisy signals fu (θ) and fq (θ) displayed in Panel A of Figure 1, we have identified three steady states in Panel B of the same figure. However, in the static

94

model, we could not answer questions about the dynamic paths that lead to those steady states, and conditions under which a group can switch from one state to the other. Using the developed dynamic model, we will answer those questions. As Theorem 3 says, the demarcation loci are φt and βt . φt is G



Vt δ+λ



, as noted in equation

(2.11). βt is ω[Fu (st ) − Fq (st )] and st is a function of Πt , as noted in equation (2.1). Those two demarcation loci are displayed in Panel A of Figure 6 with direction arrows. In Panel B of the same Figure, we can identify three steady states out of the demarcation loci. Note that those steady states are exactly the same as the steady states in the static model displayed in Figure 1.2 Let us denote the steady states as Qh (Vh , Πh ), Qm (Vm , Πm ) and Ql (Vl , Πl ). Lemma 12 (Saddle Points). Among three steady states, Qh , Qm and Ql , Qh and Ql are saddle points and Qm is a source. Proof. See the proof in the appendix. 

We might wonder whether the equilibrium paths around Qm spiral out or not. The following lemma shows that it depends on the relative size of δ and λ. Lemma 13 (Spiraling Out). There exists a critical level of (δ/λ)∗ below which equilibrium paths spiral out in the neighborhood of Qm , where (δ/λ)∗ satisfies 1 +

λ δ


λ δ

=



1 4(φ0t βt0 −1) (Vm ,Πm ) .

Proof. See the proof in the appendix. 

This implies that the less workers discount the future payoffs, the more likely that the equilibrium paths will spiral out around Qm . Theorem 4 (Dual Economically Stable States). Under Lemma 12, there exist two economically stable states, and equilibrium paths to those states overlap for a certain range of Π. ˜ to the β(≡ w[Fu (s) − Fq (s)]) in Panel B of Figure 1, βt is scaled down as much as (δ + λ) because R ˜ in the same figure, φt is scaled down as much as (δ + λ) in the w ≡ t∞ ωe−(δ+λ)(τ −t) dτ . Compared to G(β) horizontal direction. Therefore, two demarcation loci are identical to the “scaled down” reaction curves in the static model. 2 Compared

95

Proof. Two states are saddles points, and consequently economically stable states. Using the phase diagram in Figure 6, the existence of overlap is directly proven. 

In Panel A of Figure 7, we display the optimistic path to Qh and the pessimistic path to Ql . In Panel B of the same figure, we identify the overlap of the two equilibrium paths and the reputation trap. Note Ql becomes a reputation trap when Πl is located below the overlap, as discussed earlier. Once a group is in the trap, the group cannot move out of the trap unless there is a structural change in the labor market. If the overlap is between Πl and Πh , then a group in either economically stable state cannot move to the other state even through the collective action by the group. Also, if an economically stable state is covered by the overlap, a group in the state can move to the other economically stable state.

2.5

Applications

In this section, we try to explain some real world issues using the developed reputation model.

2.5.1

US Racial Disparity

Over the Jim Crow period and until the civil right movement in the 1960s, African-Americans were discriminated against in an overt manner in the US labor market. This discrimination decreased significantly over the last decades. However, we still observe the persistent black-white disparity of skill achievement. The advocates of the black group insist that they are discriminated against continuously. The dynamic reputation model explains one possible origin of the persistent disparity, and the continuing “statistical” discrimination practice in the market, which uses the “group reputation” under the imperfect information about the job candidates. When overt discrimination in America results in a very low ratio of qualified workers among blacks (very low ΠB 0 ), the quality of the group will improve over time after the disappearance of

96

taste-based discriminatory practice. However, as Figure 9 displays, the group reputation or the quality of the group may improve only up to the low reputation state Πl , which is a reputation trap under some circumstances. If the group is in the trap and is continuously disadvantaged by the market’s “statistical” discriminatory practice, the group may stay permanently in the state. The collective action of building up the better group reputation cannot work in this situation, because rational agents know that other group members will not invest for the change of group reputation when their skill achievement is not paid back enough in the future due to the low group reputation. Some might wonder why the white group is advantaged with the group’s higher reputation over the same time frame. The initial group reputation of the white group should be much higher than that of the black group. As Figure 9 displays, if the initial group reputation is higher and belongs to the overlap, the group can take the optimistic path that leads to the high reputation state Qh by sharing the optimistic view toward the future together. They will invest more than the black group in skill achievement, because the expected benefits of investment are greater due to the market’s favor for the higher group reputation. The white group is advantaged with the market’s “statistical” discriminatory practice, while the black group is disadvantaged with that. Note that we have assumed that the underlying characteristics of the two groups are identical. Thus, the disparity between groups originates solely from the reputation role embedded in the market structure. One important implication of the dynamic model is that the reputation gap between two groups can even grow over time by the agglomeration effects of collective reputation. This should be an absurd argument to the people who believe that the elimination of market discrimination of taste shrinks group disparity. The dynamic model claims that it does not need to be true all the time: depending on the initial reputation levels of groups, the gap between two groups can grow or shrink over time.

97

2.5.2

Male-Female Disparity in Patriarchal Societies

In many parts of the world, the female group is discriminated against in the labor market. Even when it is not overt discrimination, we can observe that employers are reluctant to hire female workers for the good job positions. When the overall quality of female job applicants in the market is worse than that of the male applicants, employers can lower the risk of the wrong assignment by hiring more male workers. Suppose that an employer who has two applicants, male and female, with identical records. He may prefer to hire a male applicant with the consideration of the higher likelihood that he is qualified for the job. In this section, we argue that the male-female disparity tends to grow or persist in the society with the patriarchal family structure. In the patriarchal society, parents give sons more chances to develop their potential than daughters. In turn, grown sons are obliged to support their aged parents. Grown daughters are often treated as belonging to their husbands’ families after getting married. This cultural aspect leads parents to have less incentive to invest in daughters’ skill achievement. Especially when parents have limited resources but raise many children, they distribute their education resources unequally between sons and daughters. This unequal treatment not only causes the lower skill investment by parents for the female group, but also brings out the pessimistic skill-achievement behavior of female workers because the market may “statistically” discriminate for the lower qualification ratio of the female group. The fundamental difference between male and female groups is represented by the different skill investment rates (φt ) for the same level of benefits of investment (Vt ), as displayed in Panel A of Figure 9. Given a certain level of benefits of skill investment, a higher percentage of the male group may invest than the female group due to the influence of the patriarchal family structure: φft emale < φmale . This difference creates different dynamics of group reputation for the two groups, t as displayed in Panel B of the same figure. The lower investment rate of the female group can be interpreted as the higher investment cost for the group. As noted in Proposition 13, the overlap

98

tends to shift up for the higher investment cost. The different locations of overlap in male and female reputation maps may cause the different reactions of male and female groups even for the same level of initial group reputation. Suppose that the initial group reputations are identical for both groups as Π(t0 ). Suppose the initial reputation level is located within the overlap in the male’s reputation map, and below the overlap in the female’s reputation map, as indicated in Panel B. Then, the female group is positioned in the deterministic range for the low reputation state Πfl emale , which means that the female group faces only one choice that it can take, a pessimistic path to the reputation trap Qfl emale . Since the male group is positioned in the overlap, the group can take either the optimistic path or the pessimistic path. Even when the group chooses the pessimistic path by sharing the pessimistic view to the future, they will be better off because Πmale > Πfl emale . If they choose the optimistic path, l the gap between the two groups may grow significantly due to the agglomeration effects of collective reputation: Πmale is much greater than Πfl emale . The discouraged female group at Qfl emale invests h very little for the skill achievement compared to the male group. This size of disparity cannot be explained solely by the parents’ unequal treatment, but should be explained together with the pessimism within the female group and the lower level of expected benefits of investment by the anticipated “statistical” discrimination behavior of employers in the workplace. Suppose that even the initial group reputations are different between two groups: Πf0 emale < Πmale , possibly due to the overt discrimination against female workers in the past. If then, it is 0 more obvious that the female group tends to end up with the reputation trap Qfl emale and the male group tends to end up with the high reputation state Qmale . The dynamic reputation model can h give an account for why we can observe the prevailing pessimistic behaviors among female students in the patriarchal society, even when overt discrimination against the female workers does not exist in the market.

99

2.6

Monopolistic Principals

In Proposition 12, we have shown that Qh is pareto dominant to Ql in the simple reputation model: employers can make bigger profits when a social group is at the high reputation state Qh than at the low reputation state Ql .3 Suppose a group B is at the reputation trap Ql , as described in Figure 5. Since employers prefer the group’s staying at Qh to its staying at Ql , they might help the group to move out of the trap and improve the group’s qualification ratio. However, this never happens in the competitive situation, which is defined as the market condition in which there are numerous employers and the size of each employer is relatively insignificant. In this situation, one’s action does not affect the overall behavior of group members. Each employer just accepts the market structure, and determines whether to give BOD based on the group reputation of B. Now suppose that they are in the monopolistic situation, which is the market condition in which there are a small number of employers, or employers are well coordinated by a mediator (eg. government). Let us call employers in the monopolistic situation “monopolistic principals”, as defined in Loury (2002). Monopolistic principals can change the market structure and affect the behavior of the group in the reputation trap. When the group is stuck in the reputation trap, the principals make profits as much as

Z

∞

YQl =

Pq Πl xq · e−r(τ −t0 ) dτ =

t0

Pq Π l x q . r

If their action can make the expected profit greater than YQl , they will take the action and help the group to move out of the trap. In this section, we will examine the following three strategies that they can take: 1) adjustment of reputation threshold (favorable treatment), 2) subsidy of training cost and 3) improvement of screening process. Note that the farsightedness of principals and the credibility of their actions are required for the effective implementation of each strategy. If principals are myopic, they will not be able to 3 Compare employers’ profits for a group at Q and that at Q for a unit period: Π x − (1 − Π )(1 − P )x > u u h l h q h Πl Pq xq , given Πl < Π∗ < Πh .

100

implement the long-term policy that gradually improves the qualification ratio of group B workers. Also, if principals’ actions are not credible, group B members may not change the conjecture about the expected benefits of investment, so that their skill investment rate will not be improved even with the employers’ action. Finally, we assume that the group in the reputation trap will move to the high reputation state Qh as soon as the optimistic path to the state is available to the group.

2.6.1

Adjustment of Reputation Threshold

Monopolistic principals have an incentive to lower the reputation threshold if the policy can increase profits by helping the disadvantaged group to move out of the reputation trap. In order to help the group to move toward the high reputation state Qh , the reputation threshold for BOD needs to be λ

lowered from Π∗ to Π∗0 , where Π∗0 = Πh − (Πh − Πl )v ∗ δ+λ , as displayed in Panel A of Figure 10. The state of group B will move along the following points in the panel as the group members share the optimistic view toward the future: Ql →jump a → b → c → Qh . Principals may suffer in the interval (b, c) by placing agents with unclear signals on task one, because the expected payoff of the assignment is negative with Πt < Π∗ . Let us call the corresponding reputation level [Π∗0 , Π∗ ] loss area. If they take this action, the total profits accrued over time will be

∗0

Z

tb

Pq Π τ x q · e

Y (Π ) = t0

−r(τ −t0 )

Z

∞

dτ +

[Πτ xq − (1 − Pu )(1 − Πτ )xu ] · e−r(τ −t0 ) dτ,

(2.13)

tb

∗

v −λ(τ −t0 ) . where tb = − ln δ+λ + t0 and Πτ = Πh − (Πh − Πl ) · e

Proposition 14. In the simple reputation model, well-coordinated monopolistic principals have an incentive to lower the reputation threshold for BOD of group B from Π∗ to Π∗0 if and only if Y (Π∗0 ) ≥ YQl .

101

2.6.2

Subsidy of Training Cost

Monopolistic principals have an incentive to support the training cost of group B members if the policy can increase profits by helping the group move out of the trap. If the training cost is subsidized enough by principals, the optimistic path can be available to the group in the trap, as displayed in Panel B of Figure 10. Then, the state of group B will move along the following points in the panel as the group members share the optimistic view toward the future: Ql →jump a → b → c → Qh .   δ+λ ∗ λ h −Π The training cost for the group should be lowered enough so that v ∗0 = Π . Since v ∗ = Πh −Πl 0

(δ+λ)cm −wPu w(Pu −Pq ) ,

the required size of subsidy (S ) is

0

S (≡ cm −

c0m )

(v ∗ − v ∗0 )w(Pu − Pq ) , with v ∗0 = = δ+λ



Πh − Π∗ Πh − Πl

 δ+λ λ .

The training subsidy should be implemented for the interval (a, b). Assuming the size of subsidy is constant as the group state moves from the point a to b, the total cost that the principals incur will be ∗0

Z

tb

0

S λ·e

T C(v ) =

−r(τ −t0 )

t0

where tb =

ln v ∗ δ+λ

+

Z

tc

1 λ

 1 − e−r(tb −t0 ) , dτ = S λ r 0



Πh −Πl · ln Π ∗ + t0 . The total revenue that the principals benefit from this strategy h −Π

is

Y (v ∗0 ) =

Pq Πτ xq · e−r(τ −t0 ) dτ +

t0

where tc =

1 λ

Z

∞

[Πτ xq − (1 − Pu )(1 − Πτ )xu ] · e−r(τ −t0 ) dτ,

(2.14)

tc

Πh −Πl −λ(τ −t0 ) · ln Π . ∗ + t0 and Πτ = Πh − (Πh − Πl ) · e h −Π

Proposition 15. In the simple reputation model, well-coordinated monopolistic principals have an incentive to subsidize the training cost of group B members as much as S 0 for the period tb − t0 if and only if Y (v ∗0 ) − T C(v ∗0 ) ≥ YQl .

102

2.6.3

Improvement of Screening Process

Monopolistic principals have an incentive to improve the screening process for group B members if the policy can increase profits by helping the group move out of the trap. In the given reputation model, only a fraction Pq of qualified workers are assigned to task one job for sure. A fraction 1 − Pq of them are assigned to task zero when employers do not give BOD to the group, even though they have invested in their early days of life in qualification for task one. If principals improve the assignment accuracy enough, the optimistic path can be available to the group in the trap, because the skill achievement will be a more valuable investment to the group than before. Suppose that group B is placed on the reputation trap Ql in Figure 3 and Pu is fixed. If assignment accuracy Pq improves up to Pq0 in Panel C of Figure 10, the optimistic path to Qh becomes available to the group in the trap. The group will move gradually along the following points in the panel, as they start to share the optimistic view toward the future: Ql (→ Q0l ) →jump a → b → Qh . Using the dynamic system and the demarcation loci in Theorem 3, we can calculate the critical level Pq0 :

Pq0

(δ + λ)cm − ZwPu = , with Z ≡ w(1 − Z)



Πh − Π∗ Πh − Πl

 δ+λ λ .

Suppose that principals should spend the amount k for a percentage improvement of Pq that applies for the screening process of group B members. Then, the total cost T C(Pq0 ) for the improveP0

ment of Pq to Pq0 becomes T C(Pq0 ) = 100k ln Pqq .4 The total revenue that the principals benefit from this strategy is

Y (Pq0 ) =

Z

tb

t0

where tb =

1 λ

Pq0 Πτ xq · e−r(τ −t0 ) dτ +

Z

∞

[Πτ xq − (1 − Pu )(1 − Πτ )xu ] · e−r(τ −t0 ) dτ,

(2.15)

tc

Πh −Πl −λ(τ −t0 ) · ln Π . ∗ + t0 and Πτ = Πh − (Πh − Πl ) · e h −Π

Proposition 16. In the simple reputation model, well-coordinated monopolistic principals have 4 By

definition of the cost for assignment improvement,

dc dP/P ·100

= k. Then, we have

R TC 0

dc =

R Pq0 Pq

100 Pk dPq . q

103 an incentive to improve the screening process for group B members from Pq to Pq0 if and only if Y (Pq0 ) − T C(Pq0 ) ≥ YQl .

2.7

Egalitarian Policies

Suppose there are two groups B and W in the economy with population ratios γ and 1 − γ; B is a disadvantaged group in the reputation trap Ql , while group W is at the high reputation state Qh in Figure 5. Suppose that the government goal is to achieve an egalitarian society with both groups at the high reputation state Qh , which is pareto dominant to the low reputation state Ql . If monopolistic principals act to “rescue” group B in the trap under conditions discussed in the previous section, the government goal is achieved by the market itself without government intervention. If needed, the government may work just as a mediator for coordination among employers. However, suppose that the coordination cost across employers is too high or, even though the cost is low, no strategy discussed earlier is profitable for employers. The condition of the principals’ farsightedness or credibility of their action may not be fulfilled for the market to cure the group disparity autonomously. Then the plight of group B will persist, and government intervention is necessary for the achievement of the egalitarian society. In this section, we will examine the following three policies that the government may implement for the elimination of the group disparity: 1) colorblind hiring, 2) strict quota system, and 3) asymmetric training cost.

2.7.1

Colorblind Hiring

Suppose a government policy that strictly enforces employers to hire workers without screening their group identity, which is called “colorblind hiring enforcement”. If the policy is effectively ˜ in their decision to give BOD to implemented, employers will use the average qualification ratio Π ˜ = γΠB + (1 − γ)ΠW . When Π ˜ ≥ Π∗ , employers give an agent with unclear signal θ ∈ [θq , θu ]: Π BOD and place any agents with unclear signal on task one job. The initial levels of ΠB and ΠW

104

are Πl and Πh for each. Therefore, employers give BOD to both groups right after the introduction of the policy as far as γ ≤ satisfies γ ≤

Πh −Π∗ Πh −Πl .

Πh −Π∗ Πh −Πl .

Suppose group B is a minority group and its population ratio

Then, BOD will be given to the group continuously as long as its reputation is

˜ ≥ Π∗ will be satisfied with ΠB at least Πl , because Π t ≥ Πl . Therefore, the expected lifetime BOD evaluated at time zero (ξ0 ) is one to group B members, which means that the optimal path to Qh is available to group B right away after the introduction of the policy, as displayed in Panel A of Figure 11. The state of group B will gradually move to Qh along the following points, as they share the optimistic view toward the future: Ql →jump a → b → c → Qh . In the figure, the phase diagrams are developed using implied reputation thresholds Π∗W and B Π∗B : given ΠW t , group B achieves BOD when Π ≥

W achieves BOD when ΠW ≥

Π∗ −γΠB t (= 1−γ

Π∗ −(1−γ)ΠW t γ

(= Π∗B (t)), and, given ΠB t , group

Π∗W (t)), under the colorblind hiring rule. Note that

W ∗ Π∗B (0) ≤ Πl (≡ ΠB 0 ) and ΠW (0) ≤ Πh (≡ Π0 ) if γ ≤

Πh −Π∗ Πh −Πl .

(Moreover, Π∗B (0) < Πl if γ <

Πh −Π∗ Πh −Πl ,

which implies that Ql becomes “economically” unstable in the group B’s reputation map and group B must move out of the state immediately.) Therefore, the optimistic path becomes available to group B right away, and the optimistic path for group W is maintained after the enforcement. The reputation level of group B will increase as the group shares the optimistic view taking the optimistic path to Qh , and that of group W will stay at Qh , as displayed in the figure. (Note that the implied reputation threshold Π∗W (t) is lowered over time and approaches

Π∗ −γΠh 1−γ (<

Π∗ ) as ΠB t increases

gradually up to Πh , while the implied reputation threshold Π∗B (t) is constant because ΠW t is fixed as Πh .) The condition of γ ≤

Πh −Π∗ Πh −Πl

is not a very demanding requirement: for example, if Π∗ is in

the middle of Πh and Πl , it requires the population ratio of group B to be less than fifty percent. Therefore, this policy looks very attractive if it can be implemented effectively in the real world. The critical issue regarding enforcement is that employers can easily deceive the enforcement authority due to the inevitable asymmetric information between employers and the authority about the characteristics of job candidates. Employers always have an incentive to deviate from the colorblind

105

hiring rule by using the group identity in their decision of assignment to make bigger profits, when they can defend their hiring decisions using the job candidates’ information, or the qualification list for the task, that is not available to the enforcement authority. Therefore, the policy is by nature deceitful in the sense that it is hard to enforce in the real world. Proposition 17. With the enforcement of a colorblind hiring policy, group B in the reputation trap moves out of the trap toward the high reputation state Qh , where group W is placed, as far as the population ratio of the disadvantaged group is small enough that γ ≤

Πh −Π∗ Πh −Πl .

However, the effective

enforcement of the policy is highly questioned due to the asymmetric information between employers and the enforcement authority.

2.7.2

Strict Quota System

Suppose that the government imposes a strict quota system that requires employers to meet the identical assignment ratio (ρ) of task one positions between groups B and W: ρW = ρB . The initial assignment ratios before the quota enforcement are ρ¯W ≡ Πh + (1 − Πh )(1 − Pu ) for group W, and ρ¯B ≡ Πl Pq for group B. Then, employers’ reaction to this policy is summarized as the following lemma. Lemma 14 (placement standard). Suppose that a population ratio of group W is big enough that 1−γ >

xu Πh (xq +xu ) .

If then, employers’ best response to the strict quota enforcement is to use ρ¯W as

a standard assignment ratio, regardless of ΠB t ∈ [Πl , Πh ]. Proof. See the proof in the appendix. 

We will examine whether the enforcement of a strict quota can lead the disadvantaged group to move out of the reputation trap Ql . Under the strict quota, all qualified workers of group B are assigned to task one jobs, and χt fraction of unqualified workers are assigned to task one jobs, where B χt (ΠB t ) > 1 − Pu : given Πt ∈ [0, Πh ), we have χt = 1 −

Pu (1−Πh ) , 1−ΠB t

B using ΠB ¯W . t + (1 − Πt )χt = ρ

106

Remember that there are three types of agents: always investing agents, always non-investing agents, and agents whose investment cost is cm . The mediocre cost group invest when the lifetime benefits of investment (βinv ) is greater than the lifetime benefits of non-investment (βnon ). The lifetime benefits of investment is fixed as βinv =

ω δ+λ

− cm , because all qualified workers of group B

are given BOD under the quota system. The interesting point is that the lifetime benefits of noninvestment depend on the collective action of the agents. If their collective action is not to invest 0 0 (ΠB t = Πl , ∀t), the lifetime benefits of non-investment (βnon ) is βnon =

ωχ ¯ δ+λ ,

with χ ¯ =1−

Pu (1−Πh ) 1−Πl

00 00 = ) is βnon . If their collective action is to invest, the lifetime benefits of non-investment (βnon

R∞ t0

ωχτ · e−(δ+λ)(τ −t0 ) dτ, where χτ = 1 −

Pu (1−Πh ) 1−ΠB τ

−λ(τ −t0 ) and ΠB . τ = Πh − (Πh − Πl ) · e

0 0 00 : all . We must handle the following three distinct cases: 1) βinv > βnon < βnon Note that βnon

agents with the investment cost cm invest and ΠB t converges to the high reputation level Πh . 2) 00 : all agents with the investment cost cm do not invest and ΠB βinv < βnon t stays at the low reputation 0 00 : the belief of agents with the investment cost cm determines < βinv < βnon level Πl , and 3) βnon

the final outcome. If they believe that they will all invest, then the benefits of non-investment is 00 < βinv ), and they will invest. If they believe that they will all less than that of investment (βnon 0 > βinv ), and not invest, then the benefits of non-investment is greater than that of investment (βnon

they will not invest. Denote the initial reputation gap between two groups by GR : GR ≡ Πh − Πl . Also, define G∗R ∗ 0 ∗∗ 00 and G∗∗ R as follows: GR ≡ {GR |βinv = βnon } and GR ≡ {GR |βinv = βnon }. Those three distinct ∗ cases are displayed in Panel B of Figure 11 with G∗R and G∗∗ R . We can have an explicit form of GR , h i wPu G∗R = (1 − Πh ) (δ+λ)c − 1 , but cannot find the explicit from of G∗∗ R . However, at least, we can m

confirm the existence of G∗∗ R using the following lemma. Lemma 15 (Existence of G∗R and G∗∗ R ). As long as v∗ =

(δ+λ)cm −ωPq ω(Pu −Pq ) ,

Pq Pu

> 1−

Πh (1+ρ0 )(1−v ∗ ) ,

∗ ∗∗ both G∗R and G∗∗ R exist in (0, Πh ) and GR < GR .

Proof. See the proof in the appendix. 

where ρ0 =

λ δ+λ

and

107

Therefore, we can conclude that if the initial reputation gap between two groups is too big, the benefits of non-investment are relatively big under the strict quota, and the disadvantaged group continues to keep the low skill investment behavior. The quota may help to alleviate the income difference between two groups, but does not eliminate the skill difference or the negative stereotype of group B, which is called a patronizing situation. If the initial gap is not so big and moderate, the belief of the group plays a crucial role in the determination of the skill investment rate. If they have an optimistic view that others will invest, they have more incentive to invest and the group reputation improves gradually with the enforcement of the quota. Otherwise, it may lead to a patronizing situation. If the initial gap is relatively small, the quota can effectively eliminate the negative stereotype of group B. The following proposition summarizes. Proposition 18. Under Lemmas 14 and 15, if the initial reputation gap between groups B and W is smaller than G∗R , the strict quota can eliminate the negative stereotype of group B and improves its group reputation, and if it is greater than G∗∗ R , the quota is patronizing and does not help to eliminate the negative stereotype of group B. If it is between G∗R and G∗∗ R , the effectiveness of the strict quota depends on the belief of group B members: the reputation of group B improves with the group’s optimism, but it does not improve with its pessimism.

2.7.3

Asymmetric Training Cost

One possible way to improve the skill disparity between two groups is to transfer the part of training resources of group W to group B. If the government imposes this policy, the training cost for group W members will increase while that for group B members will decrease. Denote the new level of investment cost for group W members whose investment cost was cm by cW m > cm , and that for group B members whose investment cost was cm by cB m < cm . Then, the transposed training cost v ∗ (≡

(δ+λ)cm −ωPq ω(Pu −Pq ) )

where v ∗ (i) ≡

in (v, Π) domain is expressed as v ∗ (W ) for group W, and v ∗ (B) for group B,

(δ+λ)cim −ωPq ω(Pu −Pq )

for i ∈ {W, B}. The reputation map for each group is compared in Panel

108 C of Figure 11 with those new levels of v ∗ (i). As you can observe in the reputation map for group   δ+λ ∗ λ h −Π B, the optimistic path becomes available to group B members when v ∗ (B) is Π .5 Πh −Πl The decrease of the investment cost for group B members (∆cm (B) ≡ cm −cB m ) will be ∆cm (B) = (v ∗ −v ∗ (B))ω(Pu −Pq ) . δ+λ

Then, the increase of the investment cost for group W members (∆cm (W ) ≡

cW m − cm ) must satisfy the clearing condition: γ∆cm (B) = (1 − γ)∆cm (W ), which implies that cW m =

γ(v ∗ −v ∗ (B))ω(Pu −Pq ) (1−γ)(δ+λ)

+ cm and, consequently, v ∗ (W ) =

γ(v ∗ −v ∗ (B)) 1−γ

+ v∗ .

We need to impose one restriction on v ∗ (W ): v ∗ (W ) ≤ 1. If v ∗ (W ) > 1, the high reputation state Qh becomes “economically” unstable immediately after the introduction of this policy, as you can check with the reputation map for group W in the panel: the state of group W may start to fall down to the low reputation state Ql . Note that v ∗ (W ) ≤ 1 if and only if γ ≤ long as the population ratio of group B is smaller than or equal to

1−v ∗ 1−v ∗ (B) ,

1−v ∗ 1−v ∗ (B) .

Therefore, as

the transfer of training

resources of group W to group B can effectively help group B to move out of the reputation trap and achieve the equal reputation level to group W, without disturbing the high reputation state of group W. The policy makers may want to minimize the period to impose this transfer policy, because the policy may cause political resistance from group W members who suffer from the increased training cost. In the given simple model, we can calculate the minimum period of policy enforcement, which is the movement interval between a and b in Panel C of Figure 11:

Ttransf er =

λ 1 Πh − Πl ln , where π o = Πh − (Πh − Π∗ )v ∗− δ+λ . λ Πh − π o

After this period, even without the policy enforcement, group B can move toward the high reputation state Qh , taking the available optimistic path and sharing the optimistic view to the future. Proposition 19. The policy of the training resource transfer from group W to group B can eliminate the negative stereotype of group B, without disturbing the high reputation state of group W, as long as 5 This

level is identical to v ∗0 in section 2.6.2.

109

the population ratio of group B is small enough that γ <

1−v ∗ 1−v ∗ (B) ,

where v ∗ (B) =

policy should be imposed at least for the following period of time: Ttransf er =

1 λ



Πh −Π∗ Πh −Πl

 δ+λ λ

. The

Πh −Πl o ln Π = o , where π h −π

λ

Πh − (Πh − Π∗ )v ∗ − δ+λ .

2.8

Further Discussion

In the previous section, we discussed the effectiveness of each egalitarian policy. Each of them may confront the political resistance from economic players who are negatively affected by the policy enforcement. The quota system may face resistance from employers who are forced to employ unqualified workers of group B to sensitive job positions. The resource transfer from group W members to group B members may bring the claim of adverse discrimination against group W, who suffers from the increased training cost to achieve the job qualification. The least political resistance may be from the enforcement of a colorblind hiring policy, often called Equal Opportunity (EO), which prohibits employers from using the group identity in the screening process. Economic players may easily agree to adopt the policy, because they share the view that the discrimination based on the immutable group identity is politically wrong. In this sense, among three possible policy choices, the colorblind policy could be the most attractive among the public, and get the widespread support as a policy tool for the egalitarian society. However, as we already discussed in the section of colorblind hiring, this policy is by nature deceitful. It is hard to be enforced in the real world. Employers who are by nature profit-maximizing agents have an incentive to deviate from the blind rule in the screening process: using the group identity is a very useful tool for employers to lower the risk of hiring unqualified workers to the sensitive jobs. Members of the disadvantaged group, who failed to get the job due to the “statistical” discrimination, may claim the violation of EO policy against employers. The enforcement authority may join to dissolve the dispute. However, the employers would win most of the disputes because they have a position superior to that of the enforcement authority, in terms of information about the list of

110

job qualifications and the characteristics of job candidates. This inevitable asymmetric information between employers and enforcement authority presents an obstacle to the effective implementation of the policy. Therefore, the government which is seeking a way to achieve the egalitarian society should consider the adoption of other policy tools than the EO. Either quota system or asymmetric training cost through the transfer would be a useful tool, if it can be adopted as a government policy overcoming the political resistance from the losing parties. However, a strict quota system has its limitation to be patronizing without the elimination of the negative stereotype, when the initial reputation gap between two groups is too big and the assignment of unqualified workers reinforces an incentive for the disadvantaged group not to invest. In this situation, a mixture of the two policies may give an answer: first, adopt the training cost subsidy through the transfer and decrease the reputation gap to the sufficient level, and secondly, stop the training cost subsidy and adopt the quota system with the lower reputation gap. In such a way, policy makers may avoid the possibility of a patronizing outcome, even in the worst situation of a high reputation gap. Also, a mixed policy would be beneficial in terms of the degree of political resistance: it will be easier for the government to persuade the public, since employers and advantaged group members may share the burden out of the mixed policy. Note that so far we have not discussed fully the spiraling out equilibrium paths. As denoted in Panel A of Figure 7, there often exist multiple points of lifetime benefits of investment Vt that are available to a group for a given level of initial group reputation. In the first graph, the group with a certain level of initial reputation may choose either point a or point b (or others if available) on the optimistic path to the high reputation state Qh . What would make the difference between choosing a as an expected Vt or choosing b? The answer is related to the expectation about the length of time to arrive at Qh . Choosing point b means that the group believes that the high reputation level Πh will be realized as soon as it can. This is a case of strong optimism. Choosing point a means that the group believes that the level Πh may take longer to come. If they believe in that way, the

111

benefits of investment would be lowered and less of newborn cohorts will have an incentive to invest, causing the group reputation level to drop for a while, even when they have an optimistic view that the group will arrive at Qh someday. Therefore, this is a case of weaker optimism. In principal, the weaker the optimism that a group possesses, the more time it may take to arrive at Qh and the more likely that the group reputation fluctuates over time. In the same way, we can interpret the cases for group pessimism. There often exist multiple points that are available to the group with the pessimistic view toward the future. The point c indicates the case of strong pessimism that the miserable future Ql may come very soon. With this view, the expected benefits of investment would be very low and, consequently, a small percentage of the newborn cohort may invest. Thus, the group reputation will decline rapidly to the low reputation level Πl . However, suppose that they believe that the state Ql may arrive someday, but it may take much longer to come. If then, the expected benefits of investment would be higher, and, consequently, more of the newborn cohort may invest. We may observe the increase of group reputation for a while, even when the group shares the pessimistic view to the future. Point d represents this case, namely weaker pessimism. In the developed reputation model, we have simplified the labor market by the assumption that each worker is randomly assigned to an employer every period and each of them gets through the regular screening process repeatedly. In this assumption, the true characteristic of each worker is never revealed in the market, no matter how long he spends in the workplace. In order to correct this point, we will add an additional assumption about the market learning process in which, the more time a worker spends in the workplace, the more likely the market learns his true characteristic. Once a market learns the true characteristic of a worker, he will not get through the regular screening process anymore. Instead, he is assigned according to his qualification. We use the poisson process to represent the random arrival of market learning for a worker’s true characteristic. This additional development is summarized in Appendix A. The critical difference from the original model is that the demarcation locus of V˙ t = 0 shifts to the right and the equilibrium levels of group reputation,

112

Πh and Πl , shift up. Finally, the reputation model has some empirical implications. First, the model can be directly applied to the issue of heterogeneous “tipping points” of white flight in the US housing market. Card et al. (2007) discuss this issue and conclude that the different white attitudes toward minority groups, the “racist” preference, explain the different tipping points across cities in the US. However, they do not explain the origin of the different white attitudes across cities, and the expected price change in the housing market is not reflected in their examination. The developed group reputation model provides a different perspective to the issue and suggests an empirical meaningful research agenda to overcome the limit of the previous tipping point literature. White residents may use the overall quality of the move-in minority group in their calculation of the expected housing price in the future, which means that they decide whether to flight out or not considering the collective reputation of the move-in group. (The different white attitudes mentioned above may simply reflect the different collective reputation of the move-in minority group.) If we can collect data on the quality of the move-in group, such as crime rate or educational achievement at each period of time for each city, we might be able to give an explanation for the heterogeneous tipping points across cities and periods in the US. Secondly, the dynamic model predicts the alleviation of male-female disparity after the collapse of the patriarchal family structure, as discussed in the Applications part. For example, the improvement of skill investment rate (φt ) for women is likely to help the optimistic path to be available to the female group, which was stuck in a reputation trap under the patriarchal structure. Once the optimistic path becomes available, the female group may recover its reputation or overall quality of workers over time by collective action. In many countries, including South Korea and China, the patriarchal family structure started to disappear with government intervention such as the one-child policy in China. An empirical economist may check whether the overall quality of female workers started to improve with the introduction of government policy leading to the collapse of patriarchal family structure.

113

2.9

Conclusion

In this chapter, we develop the dynamic version of statistical discrimination (Coate and Loury 1993). We have shown the importance of both the historical position and the expectation toward the future for the determination of the final group reputation, which is the overall qualification ratio of a group. By identifying two stable states of high and low reputations and dynamic paths leading to them, we have defined an overlap in which both optimistic and pessimistic paths are available to a group, and determined the conditions under which the low reputation state is a reputation trap, in which a group cannot move out of the trap unless the market structure is adjusted. We have argued how a black group in a white-dominant society, or the female group in a patriarchal society, is positioned in the reputation trap based on the initial level of group reputation and the existence of the optimistic path at the level. We have determined that a high reputation state is pareto dominant to the low reputation state in a simple reputation model. Principals can make bigger profits when a social group is at a higher reputation state. By distinguishing monopolistic principals from competitive principals, we have examined the strategy of profit-maximizing monopolistic principals to change the market structure and help the disadvantaged group escape the reputation trap. We have also examined affirmative action policies by using the dynamic model: enforcement of color-blind hiring, a quota system, and asymmetric training subsidy. By arguing that color-blind hiring policy is efficient but can be deceitful due to limitations in its enforcement, we suggest a combination of two other policies to be implemented for the achievement of an egalitarian society, with which government can avoid the possibility of patronization and minimize political resistance by the negatively affected parties. This dynamic reputation model is unique for explaining the collective reputation and the corresponding collective action to change the reputation. The model can be adjusted to examine other subjects concerned with collective reputation. Racial reputation for crime can be examined as O’Flaherty and Sethi (2004) do in a static model. Racial reputation for crime affects the reaction

114

of victims and, in turn, affects the behavior of criminals. Collective action can be discussed for the change of racial representation for crime. Brand is another topic that involves collective reputation. Enterprises may be concerned with how to build up a valuable brand that represents heterogeneous products of the company. Similar work is done in Tirole (1996) in a game-theoretical manner. Institutional reputation such as college reputation may be an interesting subject of study, because the overall quality of alumni determines the collective reputation, and the reputation affects the quality of entering students and their willingness to pay the tuition. By identifying the multiple equilibria and dynamic paths, we can discuss the strategies for building the reputation of an institution.

115

2.10

Appendix A: Market Learning Process

In the reputation model developed in this chapter, we assume that the true characteristics of each worker is not fully revealed in the market, even after he spends a long time in the workplace. To lose this assumption, we introduce the “market learning” process, in which the true characteristic of each worker is revealed and confirmed in the market under the Poisson process with parameter η: in a unit period, a worker faces average η chances to reveal his true characteristic in the market. Suppose that, once his true characteristic is revealed, he does not go through the regular screening process anymore, where signal θ and identity i determine the chances to be assigned to task 1. Instead, after the revelation, he is always assigned to task one if he is a qualified worker, and to task zero if he is not. Note that over T periods of time, the probability that his true characteristic is not revealed is e−ηT , and the probability that it is revealed is 1−e−ηT under the Poisson process of market learning. A worker with investment cost c invests only when the expected lifetime payoff of investment is greater than that of non-investment:

Lifetime Payoff =

   R∞    t e−η(τ −t) βq (Πτ ) + (1 − e−η(τ −t) )w e−(δ+λ)(τ −t) dτ − c, (Investment.)    R    ∞ e−η(τ −t) βu (Πτ ) + (1 − e−η(τ −t) )0 e−(δ+λ)(τ −t) dτ. t

(Non-investment.)

Therefore, a worker with investment cost c invests only when, noting βτ = βq (Πτ ) − βu (Πτ ),

Z

∞

βτ e−(δ+λ+η)(τ −t) dτ +

t

ηw > c. (δ + λ)(δ + λ + η)

Using this, we have a normalized lifetime benefits of investment Vt ,

Z Vt = (δ + λ) t

∞

βτ e−(δ+λ+η)(τ −t) dτ +

ηw . δ+λ+η

116

Taking the derivative, we have the evolution of lifetime benefits of investment,

 ˙ Vt = (δ + λ + η) Vt −

 δ+λ ηw βt − . δ+λ+η δ+λ+η

˙ t is the same to the original model, dynamic system with the consideration of market Since Π learning becomes,

˙t Π V˙t

= λ[φt − Πt ] =

 (δ + λ + η) Vt −

 ηw δ+λ βt − , δ+λ+η δ+λ+η

with demarcation loci of

˙ t = 0 Locus Π

:

V˙t = 0 Locus

: Vt =

Πt = φt ηw δ+λ βt + . δ+λ+η δ+λ+η

The dynamic paths with positive η are described in Appendix Figure 1. The revelation of workers’ true characteristics does not make a big difference to the original reputation model. The only difference is that the demarcation locus of V˙ t = 0, denoted by V˜t (Πt ), shifts to the right with the market learning consideration. The more the true characteristic of a worker is likely to be revealed, the more the demarcation locus shifts to the right:

∂ V˜t (Πt ) ∂η

> 0, which implies that two equilibrium

levels of group reputations (Πh , Πl ) and their corresponding benefits of investment (Vh , Vl ) increase with the higher degree of market learning. If you apply this to the simplified model with uniform distributions of fu (θ) and fq (θ), we can evaluate how overlap changes with the introduction of market learning:

∂πp ∂η

< 0, which means

that the deterministic range for Qh expands with the higher degree of market learning. (Refer to Appendix B for the proof.) The change is described in Panel B in the same figure. Therefore, a

117

group is more likely to converge to the high reputation state Qh when the market learns workers’ true characteristics faster.6 Note that there exists a degree of market learning η ∗ above which the lower reputation state Ql is not economically stable: Ql is not stable with η >

(δ+λ)((δ+λ)cm −wPq ) (= w−(δ+λ)cm

η ∗ ). (Refer to Appendix

B for the proof.) This implies that all social groups will converge to the high reputation state Qh , regardless of their initial group reputations, when the degree of market leaning is high enough. Therefore, if the market can learn the true characteristics of workers very quickly, there will be no group disparity caused by the difference of initial group reputations or statistical discrimination practice in the market. 6 Note

that the sign of

∂πo ∂η

is not clear in the simplified reputation model.

118

2.11

Appendix B: Proofs

Proof of Lemma 9. Suppose a starting point a on the Π = Π∗ line nearby the state (v ∗ , Π∗ ): a(va , Π∗ ). The initial state at a moves counterclockwise according to the direction arrows depicted in Panel A of Appendix Figure 2. Suppose the path starting from a passes across the v = v ∗ line at b(v ∗ , Πb ), the Π = Π∗ line at c(vc , Π∗ ), the v = v ∗ line at d(v ∗ , Πd ) and the Π = Π∗ line at a0 (va0 , Π∗ ), as described in the same panel. The first-order differential system in each region is described in Figure 4. In region I, the slope of the phase path is represented by

v˙ ˙ Π

(δ+λ)(vt −1) λ(Πh −Πt ) .

=

Thus, we can find the relationship

between va and Πb , v∗

Z

va

dv = (δ + λ)(v − 1)

Z

Πb

Π∗

ds =⇒ λ(Πh − Π)



1 − va 1 − v∗

Also, in region II, the slope of the phase path is represented by

λ  δ+λ

v˙ ˙ Π

=

=

Πh − Πb . Πh − Π∗

(δ+λ)(vt −1) λ(Πl −Πt ) .

(2.16)

Thus, we can find

the relationship between vc and Πb ,

Z

vc

v∗

dv = (δ + λ)(v − 1)

Z

Π∗

Πb

ds =⇒ λ(Πl − Πs )



1 − vc 1 − v∗

λ  δ+λ

=

Πb − Πl . Π∗ − Πl

(2.17)

From (2.16) and (2.17), we can derive the relationship between va and vc , taking Πb out. The following formula summarizes the result, denoting

λ δ+λ

as ρ0 :

(1 − va )ρ0 (Πh − Π∗ ) + (1 − vc )ρ0 (Π∗ − Πl ) = (1 − v ∗ )ρ0 (Πh − Πl ).

(2.18)

Therefore, va is a function of vc in vc ∈ [0, v ∗ ]: va (vc ). Note that va (v ∗ ) = v ∗ . In the same way, out of regions III and IV, we can derive the relationship between va0 and vc , taking Πd out. In region III, the slope of the phase path is represented by

v˙ ˙ Π

=

(δ+λ)vt λ(Πl −Πt ) .

Thus, we can find the relationship

119

between vc and Πd , v∗

Z

vc

Z

dv = (δ + λ)v

Πd

Π∗

λ  v  δ+λ dΠ Πd − Πl c =⇒ = ∗ . ∗ λ(Πl − Π) v Π − Πl

Also, in region IV, the slope of the phase path is represented by

v˙ ˙ Π

(2.19)

=

(δ+λ)vt λ(Πh −Πt ) .

Thus, we can find

=

Πh − Πd . Πh − Π∗

(2.20)

the relationship between va0 and Πd ,

Z

va 0

v∗

dv = (δ + λ)v

Z

Π∗

Πd

dΠ =⇒ λ(Πh − Π)



va0 v∗

λ  δ+λ

From (2.19) and (2.20), we have the following formula that indicates the relationship between va0 and vc , denoting

λ δ+λ

as ρ0 :

0

vaρ0 0 (Πh − Π∗ ) + vcρ 0 (Π∗ − Πl ) = v ∗ρ (Πh − Πl ).

(2.21)

Therefore, va0 is a function of vc in vc ∈ [0, v ∗ ]: va0 (vc ). Note that va0 (v ∗ ) = v ∗ . To prove Lemma 1, we need the following two conditions to be satisfied, as depicted in Panel B of Appendix Figure 2: - Condition 1: Two curves va (vc ) and va0 (vc ) are tangent at (v ∗ , v ∗ ): - Condition 2: va (vc ) is concave and va0 (vc ) is convex in [0, v ∗ ]:

d2 va dvc2

dva dvc (v ∗ ,v ∗ )



< 0 and

d2 va0 dvc2

=

dva0 dvc (v ∗ ,v ∗ )



> 0, ∀vc ∈ [0, v ∗ ].

Proof of Condition 1. From equation (2.18), let us define the function F, F = (1−va )ρ0 (Πh −Π∗ )+ (1 − vc )ρ0 (Π∗ − Πl ) − (1 − v ∗ )ρ0 (Πh − Πl ) = 0. By the implicit function theorem, Fva dva + Fvc dvc = 0. Therefore, we have 0

(1 − vc )ρ −1 Π∗ − Πl dva =− · . dvc (1 − va )ρ0 −1 Πh − Π∗ This gives the slope of the curve va (vc ) at (v ∗ , v ∗ ):



dva dvc

(2.22) ∗

(v ∗ ,v ∗ )

Π −Πl = −Π ∗ . From equation (2.21), let h −Π 0

us define the function G, G = vaρ0 0 (Πh − Π∗ ) + vcρ 0 (Π∗ − Πl ) − v ∗ρ (Πh − Πl ) = 0. By the implicit

120

function theorem, Gva0 dva0 + Gvc dvc = 0. Therefore, we have 0

dva0 v ρ −1 Π∗ − Πl . = − cρ0 −1 · dvc Πh − Π∗ va 0 This gives the slope of the curve va0 (vc ) at (v ∗ , v ∗ ):

dva0 dvc



(2.23)

∗

(v ∗ ,v ∗ )

Π −Πl = −Π ∗ . Therefore, condition 1 h −Π

is satisfied. Proof of Condition 2. From equations (2.18) and (2.22),

dva dvc

can be expressed in terms of vc ,

0

dva (1 − vc )ρ −1 Π∗ − Πl =− · 0 1 . ρ −1 dvc (Πh − Π∗ ) ρ0 [−(1 − vc )ρ0 (Π∗ − Πl ) + (1 − v ∗ )ρ0 (Πh − Πl )] ρ0 Taking the second derivative with respect to vc , the following result follows: From (2.21) and (2.23),

dva0 dvc

d 2 va dvc2

< 0 as far as δ > 0.

can be expressed in terms of vc , 0

dva0 vcρ −1 Π∗ − Πl =− · 0 1 . ρ −1 dvc (Πh − Π∗ ) ρ0 [−vcρ 0 (Π∗ − Πl ) + v ∗ρ0 (Πh − Πl )] ρ0 Taking the second derivative with respect to vc , the following result follows:

d 2 va 0 dvc2

> 0 as far as

δ > 0. Therefore, condition 2 is satisfied. The results imply that when starting in a neighborhood around (s∗ , v ∗ ), the equilibrium path spirals out. Therefore, the state (s∗ , v ∗ ) is unstable because any tiny perturbation to the state will make the state (st , vt ) move away from it. (However, note that

d 2 va dvc2

=

d2 va0 dvc2

= 0 if δ = 0. This im-

plies that the equilibrium paths are cyclical, like a vortex, if the time discount rate(δ) is zero.) QED.

Proof of Lemma 10. The slope of the equilibrium path in region IV is

˙ Π v˙ IV

=

λ(Πh −Πt ) (δ+λ)vt .

The first derivative of the slope

is d dvt

"

˙ Π v˙ IV

# =−

λ(Πh − Πt ) λΠh dΠt − · < 0. (δ + λ)vt2 (δ + λ)vt dvt

121

Therefore, in region IV, the equilibrium paths are concave. The slope of the equilibrium path in region I is

˙ Π v˙ IV

=

λ(Πh −Πt ) (δ+λ)(vt −1) .

The first derivative of the

slope is d dvt

"

# ˙ λ(Πh − Πt ) λΠh Π dΠt − · < 0. =− 2 v˙ I (δ + λ)(vt − 1) (δ + λ)(vt − 1) dvt

Therefore, in region I, the equilibrium paths are concave. In sum, at the righthand side of the vt = v ∗ line, equilibrium paths are concave. In the same way, we can prove that equilibrium paths are convex at the lefthand side of the vt = v ∗ line. QED.

Proof of Proposition 12. First, let us check expected benefits at Qh and Ql for workers with different investment cost levels: 1. Agents with Low Investment Cost They invest at either Qh or Ql . At Qh , the size of normalized lifetime benefits is ω. At Ql , it is ωPq . 2. Agents with High Investment Cost They do not invest at either Qh or Ql . At Qh , the size of normalized lifetime benefits is ω(1 − Pu ). At Ql , it is zero. 3. Agents with Medium Level of Investment Cost They invest at Qh , but do not invest at Ql . At Qh , the size of normalized lifetime benefits is ω − (δ + λ)cm , which is positive in the given dynamic structure. At Ql , it is zero. Second, let us check the profits of principals:

1. Profits at Qh : Principals give the benefit of doubt to agents with unclear signals. Thus, at point of time τ , the expected profits from the workers with low investment cost is Πl xq , the expected profit from the workers with high investment cost is (1 − Πh )(1 − Pu )(−xu ), and

122 the expected profit from the medium cost workers is (Πh − Πl )xq because they invest at Qh . Therefore, the total profit at time τ is Yτ (Qh ) = −(1 − Πh )(1 − Pu )xu + Πh xq . 2. Profits at Ql : Principals do not give the benefit of doubt. Thus, at point of time τ , the expected profit from the workers with low investment cost is Πl Pq xq , the expected profit from the workers with high investment cost or medium investment cost is zero, because they do not invest and principals do not give BOD. Therefore, the total profit at time τ is Yτ (Ql ) = Πl Pq xq . The given condition Πl < Π∗ < Πh implies that Yτ (Qh ) > Yτ (Ql ). Thus, all workers with different investment cost and principals are better off at Qh than at Ql . QED.

Proof of Lemma 12. ¯ is Given the dynamic system in equations (3), its linearization around a steady state (¯ v , Π)

V˙ t

=

¯ − (δ + λ)βt0 (Π)(Π ¯ ¯ ¯ (δ + λ)(V¯ − βt (Π)) t − Π) + (δ + λ)(Vt − V )

˙t Π

=

¯ − λ(Πt − Π) ¯ + λφ0t (V¯ )(Vt − V¯ ). λ(φt (V¯ ) − Π)

¯ is a steady state, it is Since (V¯ , Π)

V˙ t

=

¯ t + (δ + λ)[−V¯ + βt0 (Π) ¯ Π] ¯ (δ + λ)Vt − (δ + λ)βt0 (Π)Π

˙t Π

¯ = λφ0t (V¯ )Vt − λΠt + λ[−φ0t (V¯ )V¯ + Π].

Therefore, the Jacobian matrix JE evaluated at a steady state is  δ + λ JE ≡   λφ0t

 −(δ +

λ)βt0 

−λ

 

.

¯ (V¯ ,Π)

Consequently, its transpose is trJE = δ and the determinant is |JE | = λ(δ + λ)[βt0 φ0t − 1]. Since trJE  −1 ∂βt ∂φt is positive, every steady state is unstable. Note that |JE | is negative if and only if ∂Π < ∂Vt t

123 ¯ This is true for the two steady states Qh and Ql as easily confirmed in Panel B of Figure at (V¯ , Π). 6. Therefore, the characteristic roots for Qh and Ql are one positive and one negative. Therefore, those are saddle points. In the same way, we can confirm that |JE | is positive at Qm , which means two positive characteristics. Therefore, Qm is a source, either unstable node or unstable focus. QED.

Proof of Lemma 13. Note that the characteristic roots are, based on the proof of Lemma 12,

r1 , r2 =

δ±

p

δ 2 − 4λ(δ + λ)[βt0 φ0t − 1] . 2

Since we already shown that Qm is a source in the same lemma, if both r1 and r2 are imaginary numbers, then the trajectories around Qm spiral out. Therefore, if δ 2 − 4λ(δ + λ)[βt0 φ0t − 1](≡ D) is negative, they spiral out. The condition is summarized as follows:

  λ λ 1 . D < 0 ⇐⇒ 1 + > δ δ 4(φ0t βt0 − 1) (Vm ,Πm )

QED.

Proof of Lemma 14. When ρ¯W is a assignment ratio standard, the cut-off signal for group W is θq for ΠW t = Πh , while the cut-off signal for group B is less than θq for any ΠB t < Πh . Suppose that employers make a small increase in the assignment ratio. Then, the cut-off signal for group W will be a little above θq , while that for group B will be still less than θq . Therefore, we can compare the loss and gain from this

124

small increase of the assignment ratio:     Loss from group W workers :

(1 − γ)[xq Πh − xu (1 − Πh )]

   Gain from group B workers :

γxu .

(2.24)

Therefore, the loss is greater than the gain, given that 1 − γ >

xu Πh (xq +xu ) .

Secondly, suppose that

employers make a small decrease in the assignment ratio. Then, the cut-off signal for group W will be a little below θq , while that for group B will be still less than θq . If so, employers will suffer from a net loss for both group W and group B workers:     Loss from group W workers :

(1 − γ)xu

   Loss from group B workers :

γxu .

(2.25)

Therefore, the change causes a loss in profits. In sum, the assignment ratio ρ¯W is the locally optimal choice for employers, given 1 − γ >

xu Πh (xq +xu ) .

It is easy to show that this local optimal ρ¯W is in

fact a global optimal choice to employers using a similar method. QED.

Proof of Lemma 15. 00 |GR =0 = Since βnon

ω(1−Pu ) δ+λ ,

00 |GR =0 < βinv (≡ we have βnon

ω−(δ+λ)cm ). δ+λ

Therefore, the existence of

00 00 G∗∗ R is assured when βnon |GR =Πh > βinv , considering that βnon is an increasing function of GR . Note

that

00 βnon |GR =Πh

= = = >

∞

 Pu (1 − Πh ) ω 1− e−(δ+λ)(τ −t0 ) dτ 1 − Πh + Πh e−λ(τ −t0 ) t0  Z ∞  Pu Πh e−λ(τ −t0 ) ω 1 − Pu + e−(δ+λ)(τ −t0 ) dτ 1 − Πh + Πh e−λ(τ −t0 ) t0 Z ∞ ω(1 − Pu ) Pu Π h + ω· · e−(δ+2λ)(τ −t0 ) dτ −λ(τ −t0 ) δ+λ 1 − Π + Π e h h t0 ω(1 − Pu ) ωPu Πh + (∵ e−λ(τ −t0 ) ≤ 1). δ+λ δ + 2λ

Z



125 00 Therefore, when the following holds, we have βnon |GR =Πh > βinv :

ω(1 − Pu ) ωPu Πh ω − (δ + λ)cm Pq Πh + > ⇐⇒ >1− , 0 δ+λ δ + 2λ δ+λ Pu (1 + ρ )(1 − v ∗ ) where ρ0 =

λ δ+λ

and v ∗ =

(δ+λ)cm −ωPq ω(Pu −Pq ) .

In sum, if the above condition holds, G∗∗ R exists. Since

0 00 0 00 βnon = βnon at GR = 0 and βnon > βnon , ∀GR ∈ (0, Πh ), G∗R exists as well, when G∗∗ R exists in

(0, Πh ). QED.

Proof of Appendix A. First, we prove that δ+λ δ+λ+η ω(Pu

∂πp ∂η

< 0. If we transpose the dynamic system with Vt =

δ+λ δ+λ+η ωPq

+

ηw δ+λ+η

+

− Pq )vt , we can use the following result in section 2.4.3.2 directly from the simple λ

reputation model developed in this chapter: π p = Πl + (Π∗ − Πl )(1 − v ∗0 )− δ+λ+η , where v ∗0 = (δ+λ+η)(δ+λ)cm −(δ+λ)ωPq −ηω . (δ+λ)(ωPu −ωPq )

(Note that the differential equations with the transpose are identical

to those in Figure 4, except using (δ + λ + η) instead of using (δ + λ).) The partial derivative with respect to η gives

∂πp ∂η

< 0.

Secondly, we prove that there exists η ∗ above which Ql is not economically stable, where η∗ = is

(δ+λ)((δ+λ)cm −wPq ) . w−(δ+λ)cm

δ+λ δ+λ+η ωPq

+

ηω δ+λ+η ,

You can check that the demarcation locus of V˙ t = 0 below Π∗ , which

becomes greater than (δ + λ)cm , when η > η ∗ . The other way is to check

the sign of v ∗0 : when v ∗0 is negative, Ql cannot be stable, as the dynamic system of the simple reputation model in Figures 4 and 5 implies. QED.

Chapter 3

Group Reputation and the Endogenous Group Formation Joint work with Glenn C. Loury.

3.1

Introduction

In this chapter, we develop an identity switching model that can explain social activities such as passing and selective out-migration among a stereotyped group, loosening the assumption of group identity immutability made in Kim and Loury (2008), which is summarized in Chapter 2. The more talented members of the group, who gain more by separating themselves from the masses, have a greater incentive to pass for the group with the higher reputation (incurring some cost of switching). They also differentiate themselves by adopting the cultural traits of a better-off subgroup in order to send signals of their higher productivity to employers. Also, we show the dynamic process by which elite subgroups emerge out of disadvantaged populations by adopting unique cultural instruments, as discussed in Fang (2001). The most talented members of the stereotyped population have an incentive to develop distinguished cultural indices for differentiation, which are not affordable to other members of the group. As the most talented adopt these indices, an elite “cultural” subgroup grows autonomously, whose members are preferentially treated by employers. 126

127

This work is closely related to statistical discrimination literature. If a worker’s true productivity is not perfectly observable, employers have an incentive to use the collective reputation of the job applicants in the screening process. The individuals who belong to a group with a better collective reputation have a greater incentive to invest in skills because the return for skill investment tends to be greater for them, (and vice versa). With their greater (smaller) skill investment rate, the group maintains a better (worse) collective reputation. Therefore, there are multiple self-confirming equilibria of group reputation (Arrow, 1973; Coate and Loury, 1993). In Chapter 2, Kim and Loury (2008), we discuss this externality of group reputation and the stability of multiple equilibria in a dynamic setting. We identify the balanced dynamic paths to the high and low stable reputation equilibria. When the initial reputation of a group is outside the optimistic (pessimistic) path to the high (low) stable reputation equilibrium, the group’s reputation deteriorates (improves) over time and ends up in the lower (higher) stable equilibrium. We explain the concept of a reputation trap: if a group’s reputation is trapped, the group cannot escape the low skill investment activities without any external interventions such as preferential employers’ treatment and/or affirmative action, and offspring of the group consistently suffer from the developed negative stereotype of their ancestors. In our previous work, there are no implications for multiple social group societies, (except for the policy implication for quota ratio or training subsidy transfer.) An inborn group identity is immutable and each group member is affected only by the collective reputation of his own group. However, when we loosen the immutability assumption, we can explain the relationship between group reputation externality and identity switching between social groups, and the development of an elite group out of a stereotyped population. The first type of identity switching is “passing.” Consider a group in the reputation trap. The talented young members in the group may consider passing for the group with the better collective reputation when the return for passing (such as better treatment in the labor market) outweighs the cost of passing, such as the disconnection from their own ties. A representative historical case is the story of Korean descendants in Japan, who constitute around one percent of the Japanese population.

128

Most of them are the descendants of forced laborers in mines and factories who were brought back by Japan from the peninsula during the period of Japanese imperialism. Their living conditions in Japan were much worse than for Japanese natives, even after the end of World War II. In order to escape negative stereotypes and prejudices, many Korean descendants have passed for native Japanese, changing both surnames and given names at the age when they seek formal employment and marriage. Every year about 10,000 Koreans, out of around 600,000 Korean descendants holding Korean nationality, choose to be naturalized as “official” Japanese, giving up their names and original nationality. Many of the naturalized Koreans conceal their Korean ethnicity, pretending that they have no knowledge about Korean culture and language in order to prevent discrimination in the labor or marriage market (Fukuoka et al., 1998). Other than the case of Korean descendants in Japan, who share a similar appearance with the Japanese, passing is harder for blacks in the United States who were brought to the country as slaves hundreds of years ago, due to their immutable physical marker. However, a meaningful number of the black population consistently passes for White or other races according to the NLS79 National Longitudinal Survey conducted by the Department of Labor of the US. The survey shows that 1.87 percent of those who had originally answered “Black” in 1979 (when they were 14 to 22 years old), switched to answering the interviewer’s race question with either “White,” “I don’t know,” or “other,” before 1998 (Sweet, 2004). The second type of identity switching is “partial passing” or differentiation from others. The term “partial passing” was used first in Loury (2002) to describe the social identity manipulation used by racially marked people to inhibit being stereotyped. When “total passing” for a member of the advantaged group with high reputation is not available due to immutability, the most talented of the stereotyped group are more likely to seek styles of self-presentation that aim to communicate “I’m not one of THEM; I’m one of YOU!” because they are the ones who gain most by separating themselves from the masses (Loury, 2002). That is, they “pass for” the slightly better-off subgroup that maintains a higher reputation than the stereotyped population by adopting the cultural traits

129

of the better-off subgroup. Methods that are known to be used for partial passing among the black population in the US are: affectations of speech, dressing up rather than wearing casual clothes, spending more on conspicuous consumption and so on. For example, blacks earning higher incomes who live in an area where the community income is relatively lower spend more on visible goods to signal their income level and social status, while blacks who live with affluent peers have less need to signal high status (Charles et al., 2007). Also, there is evidence that the more educated (or talented) blacks tend to speak Standard American English rather than African American English (Grogger, 2008). This selective out-migration to the better-off subgroup may undermine solidarity in the disadvantaged population and cause conflicts among them, such as the accusation of “Acting White” against the ones who practice the partial passing methods (Fryer and Torelli, 2006). The collective reputation of the group with the selective out-migration of the most talented may become worse over time. It would be harder for the stereotyped group to move out of the reputation trap even when an external intervention is made. However, there might be a social gain through this practice. Among many subgroups with the unique cultural traits of the stereotyped population, at least some subgroups would be able to recover their reputation when the talented young members gather around the cultural subgroups. The usage of the observable cultural traits in the screening process can cure to some extent the social inefficiency of the reputation trap, which is caused by imperfect information about the true characteristics of workers. Also, using the dynamic model developed in this work, we can explain the emergence of an elite social group out of a stereotyped population. The most talented members of a stereotyped population have an incentive to create a small group with observable distinguished cultural traits so that they can differentiate themselves from the rest in the labor market. The usage of a cultural instrument that is intrinsically irrelevant for productivity to form an elite group is well discussed in Fang (2001) as an explanation for the complexity of elite etiquettes in European (or Confucian) societies and the respect for “Oxford Accent.” Skilled and unskilled workers have different incentives

130

to join a group with unique cultural traits that are “expensive” to obtain. Thus, the small group is preferentially treated by employers due to the higher fraction of the skilled workers, even though the cultural traits of the group are not relevant for productivity. Understanding this mechanism, the talented members of the stereotyped population may develop indices for differentiation, which are not affordable to other members of the group. The indices may include the migration of the most talented to affluent residential areas, spending on luxury goods and designer clothing, showing interest in fine arts, and sending children to a private boarding school. Even when there is no a priori difference in cultural traits among the stereotyped population, we may see an autonomously growing elite subgroup with differentiated cultural traits whose members are preferentially treated by employers and considered as distinguished from their peers. The dynamic model of endogenous group formation in this paper starts with the following basic structure. First, the model is developed based on a dynamic group reputation model in Kim and Loury (2008), using the same notations in Chapter 2. We have two identity groups, group A and group B. The groups are identified by cultural traits (and also by physical marker.) Cultural traits may include speaking standard vs. speaking slang, non-smoker vs. smoker, straight sexually vs. gay, fashionable vs. unfashionable, learning etiquette vs. ignoring etiquette, and living in the suburbs vs. living in the inner city. We assume that a worker’s preference for those traits is irrelevant to his investment cost for skills: the preference distribution is not correlated with the investment cost distribution among a population. Also, we assume that cultural traits, which are observable by employers, are not associated with productivity, as assumed in Fang (2001). Apart from the immutable group identity, which we have assumed in Kim and Loury (2008), this “cultural” group identity is not determined by nature. Newborn individuals can choose which group they belong to at an early stage of their life. Newborns who “switch” from an inborn identity type to another must incur some cost of switching, which varies across individuals. This chapter is organized into the following sections: Section 3.2 describes the basic framework of the model; Section 3.3 examines the dynamic system with no switches and that with switches

131

between two groups, after identifying potential switchers among the population; Section 3.4 provides an analysis of endogenous group formation including passing, partial passing and the emergence of elite subgroup; and Section 3.5 contains the conclusion.

3.2

Framework

In this section, we explain employers’ decision making process under the imperfect information about the workers’ true productivity, together with workers’ decision making process for the skill acquisition and the group identity.

3.2.1

Employers’ Decision

Employers are unable to observe whether a worker is qualified for a task, which is a more demanding and rewarding assignment than other tasks. Employers observe each worker’s group identity and a ¯ The distribution of θ depends on whether or not a worker is qualified. The noisy signal θ ∈ [0, θ]. signal might be the result of a test, an interview, or some form of on-the-job training. The signal is ¯ with uniformly distributed for an unqualified worker in [0, θu ], and for a qualified worker in [θq , θ], θq < θu . In this case, employers will set the hiring standard as either θq or θu . If the signal is below θq , the worker must be unqualified, and, if the signal is above θu , the worker must be qualified. If the signal is between θq and θu , the signal is unable to tell the true characteristic of the worker. Let us denote the probability that, if a worker does invest in skills, his test outcome proves that his is qualified by Pq (=

¯ u θ−θ ¯ q) θ−θ

and the probability that, if a worker does not invest in skills, his test

outcome proves that he is unqualified by Pu (=

θq θu ).

Assumption 3 (Imperfect Information). A qualified worker’s signal is less informative, compared to an unqualified worker’s signal. This is, the payoff uncertainty is greater for qualified workers ¯ compared to for unqualified workers: Pq < Pu , and equivalently, θq + θu > θ.

132

The assumption implies that it is relatively harder to confirm qualification for skilled workers, while it is relatively easier to confirm disqualification for unskilled workers. Employers should make a decision to give the benefit of doubt (BOD) if the signal is unclear. If they give BOD to a group, the hiring standard for the group is θu , but, if not, the hiring standard for the group is θq . Employers’ decision to give BOD is determined by the sign of expected payoff, xq · P rob[qualified|θ] − xu · P rob[unqualified|θ], for θq < θ < θu . Using Bayes’ rule, the posterior probability that the worker with group identity i and an unclear signal (θq < θ < θu ) is qualified is

Πi (1−Pq ) Πi (1−Pq )+(1−Πi )(1−Pu ) .

Thus,

we can find the threshold level Π∗ , above which employers give BOD and below which they do not give BOD, where Π∗ ≡

1−Pu ρ(1−Pq )+1−Pu

with ρ =

xq xu .

Lemma 16. Let us denote ξti as the indicator of employers’ giving BOD to the identity group i at time t: ξti

=

    0,    1,

3.2.2

∀ Πit ∈ [0, Π∗ ) (3.1) ∀

Πit

∗

∈ [Π , 1].

Workers’ Decision

There are two types of identity groups, A and B. Each individual is born a type A or a type B. Let us denote the population size of the type-A born individuals by La and that of the type-B born individuals by Lb . Both La and Lb are constant over time and the total population is La + Lb . Every unit period, λ faction of the total population randomly die and the same fraction are newly born. Thus, λLa (λLb ) is the size of type-A (type-B) newborns in a unit period. A newborn can change his inborn identity with incurring some cost k at an early stage of his life. At the same time, he can choose whether to be qualified or not. In order to be qualified, he must incur some cost c. The c and k are nonnegative and distributed with CDF G(c) and H(k) among the newborns, and the two distributions are independent of each other, which means the switching cost is not relevant to the skill investment cost. Each newborn will choose both identity and qualification at an early stage of his life. Let us

133 denote the lifetime benefits of each choice by Wei , where i ∈ {a, b} and e ∈ {q, u}. Let us denote the return to skill investment (Wqi -Wui ) given the chosen identity i by Ri , and the return to identity switch from i to −i (We−i -Wei ) given the chosen qualification e by Yei : Ri ≡ Wqi − Wui and Yei ≡ We−i − Wei . Note that R−i − Ri ≡ Yqi − Yui . Let us denote vti as the “normalized” lifetime BOD expected to be given to a group i member from time t to infinity: vti

Z = (δ + λ)

∞

ξτi · e−(δ+λ)(τ −t) dτ.

(3.2)

t

Note that vti = 1 when ξτi = 1, ∀τ ∈ [t, ∞]. (Let vt−i denote the normalized lifetime BOD expected to be given to the members of the other group.) By virtue of normalization, the evolution rule of vti is simplified with v˙ ti = (δ + λ)[vti − ξti ].

(3.3)

Using the notation of vti , the lifetime benefits of each choice (i, e), Wei , is expressed as   R∞   Wqi = t {wξτi + wPq (1 − ξτi )} · e−(δ+λ)(τ −t) dτ =   R  Wui = ∞ w(1 − Pu )ξτi · e−(δ+λ)(τ −t) dτ = t

w(1−Pu ) δ+λ

wPq δ+λ

·

+

w(1−Pq ) δ+λ

· vti (3.4)

vti .

i Thus, Rti and Ye,t evaluated by the time t newborn are

Rti

=

i Ye,t

=

wPq w(Pu − Pq ) i + · vt δ+λ δ+λ w(1 − Pe ) · (vt−i − vti ). δ+λ

(3.5) (3.6)

Consider a type-i born individual with the cost set (c, k). The net payoff for each choice (i∗ , e∗ )

134 ∗

denoted by Nei∗ , given {i, c, k}, is

∗

∗

Net Payoff for Choice (i , e )i,c,k

     Nui          Nqi

= Wui = Wqi − c (3.7)

    Nu−i          N −i

=

Wu−i

−k

= Wq−i − c − k

q

∗

Comparing the net payoff ( Nei∗ ) for each choice (i∗ , e∗ ), we can determine the best response, (i∗ , e∗ )i,c,k , for type i newborns with the cost levels of c and k. Lemma 17. When v −i > v i , the identity and skill decision for a type-i newborn with the cost set (c, k) is

∗

∗

(i , e )i,c,k =

     (i, u)          (i, q)     (−i, u)          (−i, q)

if c > Ri , k > Yui and k + c > R−i + Yui if c < Ri and k > Yqi (3.8) if c > R

−i

and k <

Yui

if c < R−i , k < Yqi and k + c < R−i + Yui ,

and, when v −i ≤ v i , no type-i newborn switches his inborn type: (i∗ , e∗ )i,c,k = (i, u), ∀c ∈ (0, Ri ), and (i∗ , e∗ )i,c,k = (i, q), ∀c ∈ (Ri , ∞). Proof. When v −i > v i , we know that R−i > Ri and Yqi > Yui , as described in Panel A of Figure ∗

3.1. The result is confirmed when comparing Nei∗ for each range of (c, k). For example, Nqi > max{Nui , Nq−i , Nu−i } if c < Ri and k > Yqi . When v −i ≤ v i , we know that Yqi ≤ Yui ≤ 0. Thus, no type-i newborn has a willingness to pay k to switch his inborn type. His choice of qualification depends on Ri .  The lemma is described in Panel A of Figure 3.1 for the case of v −i > v i . The lemma directly proves the following proposition.

135

Proposition 20. Under Assumption 3, the more talented the newborn, the more likely that he will switch from his inborn identity type to the other identity type. The more talented, the more likely that he will invest in skills. The less talented, the more likely that he will not invest in skills. The return to identity switch for a qualified worker is greater than that for an unqualified worker under Assumption 3: Yqi > Yei given v −i > v i . (This is because that the payoff uncertainty is greater for qualified workers than unqualified workers: 1 − Pq > 1 − Pu . The switch to the group with the better collective reputation can reduce the uncertainty.) Thus, the more talented, the more likely that he will switch to the other type whose members will receive the better treatment by employers.

3.3

Dynamic Systems

For the purpose of the dynamic analysis, we will simplify both G(c) and H(k). Each cohort of either group is composed of Πl fraction of low investment cost newborns, Πh − Πl fraction of medium investment cost newborns and 1 − Πh fraction of high investment cost newborns. Denoting those cost levels by cl , cm and ch , they satisfy the following condition: Assumption 4. cl <

wPq δ+λ

< cm <

wPu δ+λ

< ch .

With this assumption, we ensure that cl is small enough that the Πl faction of low cost newborns always invest in skills, and ch is big enough that the 1−Πh fraction of high cost newborns never invest in skills, regardless of Ri ,R−i and Yei . Also, each cohort of either group is composed of η fraction of high switching cost newborns and 1 − η fraction of low switching cost newborns. Denoting those cost levels by kh and kl , they satisfy the following condition: Assumption 5.

w δ+λ

− cm < kl <

w(1−Pq ) δ+λ

< kh .

With this assumption, we ensure that kh is big enough that the η fraction of high switching cost never switch their inborn identity types, regardless of their investment cost c. Also, kl is big enough

136

that the newborns with an investment cost of either cm or ch never switch their inborn identity types. However, any newborn with the low investment cost cl and the low switching cost kl will switch his identity type and join the other group as long as the return for the identity switch (Yqi ) is greater than the switching cost kl . The population distribution that satisfies the two assumptions is depicted in Panel B of Figure 3.1. Lemma 18. Under Assumptions 4 and 5, the newborns with investment cost cl always invest in skills and the newborns with investment cost ch never invest in skills. The newborns with switching cost kh never switch their inborn types. The newborns with investment cost either cm or ch never switch their inborn identity types. Proof. See the proof in the appendix.  The above lemma implies: Proposition 21 (Potential Switcher). Under Assumptions 4 and 5, newborns with the cost set (cl , kl ) are the only potential switchers from their inborn identity types to the other type. Type i born potential switchers switch if and only if Yqi is greater than kl . Proof. Lemma 18 implies that newborns with the cost set (cl , kl ) are the only potential switchers. Also, they will invest in skills whether or not they switch to the other type according to the lemma. Since Yqi is the extra benefits of switching for the newborns who will invest in skills (Wq−i − Wqi ), they switch if Yqi > kl . Otherwise, the switching cost is greater than (or equal to) the benefits of the switching for the potential switchers. Thus, they do not switch. 

3.3.1

Dynamics with Identity Switches Restricted

Before moving to the identity switch dynamics, let us analyze the simplest situation in which no newborn switches his inborn group identity. We can do this by simply imposing a condition that identity switch is prohibited by an authority, or the fraction of newborns with the highest switching cost kh is one (η = 1). Each variable in this section is expressed with the superscript “n”, symbolizing

137 the condition of identity switch restriction. By equations (3.3) and (3.5), we can describe how Rtn evolves over time:

R˙ tn

=

w(Pu − Pq ) n v˙ t δ+λ

w(Pu − Pq )(vtn − ξtn )   w(Pu − Pq ) n wPq − ξt . = (δ + λ) Rtn − δ+λ δ+λ

=

(3.9)

Let φnt denote the fraction of time t born workers who invest and become qualified:

φnt

=

     0,           Πl ,

∀Rtn ∈ [0, cl ) ∀Rtn ∈ [cl , cm ) (3.10)

    Πh , ∀Rtn ∈ [cm , ch )          1, ∀Rtn ∈ [ch , 1]. Since λ fraction of the total population is replaced with newborns in a unit period, Πnt evolves in short time interval ∆t in the following way.

Πnt+∆t

 ≈ λ∆t ·

φnt + φnt+∆t 2



+ (1 − λ∆t) · Πnt .

(3.11)

By the rearrangement of the equation, we have

 n  Πnt+∆t − Πnt φt + φnt+∆t ∆Πnt n ≡ ≈λ − Πt . ∆t ∆t 2

Taking ∆t → 0, we can express how Πnt evolves over time:

˙ nt = λ[φnt − Πnt ]. Π

(3.12)

138

Therefore, the dynamic system is summarized with

R˙ tn

  wPq w(Pu − Pq ) n n = (δ + λ) Rt − − ξt δ+λ δ+λ

˙ nt Π

= λ[φnt − Πnt ],

(3.13)

in which ξtn is a function of Πnt and φnt is a function of Rtn , according to equations (3.1) and (3.10).   wP Panel A of Figure 3.2 describes the dynamic paths toward the two stable equilibria, Ql δ+λq , Πl   n n u and Qh wP δ+λ , Πh . Knowing that Rt is a linear function of vt in equation (3.5), we have

φnt =

     Πl ,

∀vtn ∈ [0, v ∗ )

with v ∗ ≡

   Πh , ∀vtn ∈ [v ∗ , 1],

(δ + λ)cm − wPq . w(Pu − Pq )

(3.14)

The usage of vtn , instead of Rtn , can further simplify the dynamic system. Proposition 22. The dynamic system with a flow variable Πnt and a jumping variable vtn is

v˙ tn

=

(δ + λ)[vtn − ξtn ]

˙ nt Π

= λ[φnt − Πnt ],

with demarcation loci of

v˙t n = 0 Locus : vtn = ξtn ˙ n = 0 Locus : Π t

Πnt = φnt .

Panel B of Figure 3.2 describes the dynamic paths to two stable equilibria, Qnl (0, Πl ) and Qnh (1, Πh ). Let us denote π o as the level of reputation at v n = v ∗ with which the group at the state (v ∗ , π o ) can directly reach the upper equilibria Qnh along the optimistic path. Also, denote π p

139 as the level of reputation at v n = v ∗ with which the group at the state (v ∗ , π p ) can directly reach the lower equilibria Qnl along the pessimistic path. Using the differential equations in Proposition 22, we can find

λ

πo

=

Πh + (Π∗ − Πh )v ∗− δ+λ ,

πp

=

Πl + (Π∗ − Πl )(1 − v ∗ )− δ+λ .

(3.15) λ

(3.16)

In this chapter, we assume that δ is big enough that two economically stable states are “separate” from each other. (Refer to Lemma 3 in Kim and Loury (2008) for the definition of separation.) With the separated two equilibria, a group in the lower equilibrium Qnl is in a reputation trap, which means the group cannot escape the status of low skill investment activities, owing to the negative influence of the group’s bad reputation. A group in the upper equilibrium Qnh enjoys the secured BODs given by employers and maintains the high skill investment activities, owing to the positive influence of the group’s good reputation. If the two equilibria are separated, the size of overlap Ln is simply the difference between π p and π o :

λ

λ

Ln = Πl − Πh + (Π∗ − Πl )(1 − v ∗ )− δ+λ + (Πh − Π∗ )v ∗− δ+λ .

(3.17)

Inside the overlap, the expectation about the future determines the final state, either Qnl or Qnh . Outside the overlap, the initial reputation is critical: if it is below the overlap, the final state should be the lower equilibrium Qnl , and, if it is above the overlap, the final state should be the upper one Qnh . (Kim and Loury, 2008)

3.3.2

Dynamics with Identity Switches from Type j to Type i

Imagine a situation that some fraction of type-j newborns switch to type i consistently since some fixed point of time X. Until the incidence, both group sizes have been constant as Li and Lj . Under

140 the given assumptions, the exact (1 − η)Πl fraction of type-j newborns, whose cost set is (cl , kl ), will switch their inborn identity types to type i, according to Lemma 18 and Proposition 21. Thus, the population sizes of group i and group j eventually arrive Li + (1 − η)Πl Lj and Lj − (1 − η)Πl Lj for each. In the following sections, we will address the dynamic system for group i which benefits from the inflows of skilled workers from type-j newborns, and the dynamic system for group j which loses some of the most talented newborns to group i. Let us denote the size ratio of group j and group i ˜ i (≡ by L

3.3.2.1

Lj Li ).

Dynamic System of Group i with Inflows from Group j

Let us denote the size of the type-i skilled workers at time t by Zti , and the total size of the type-i i is Li , and Mti increases consistently over time with the workers at time t by Mti . Note that MX

inflows from the type-j newborns since time X. Thus, Mti changes in short time interval ∆t:

i Mt+∆t = (1 − λ∆t)Mti + Li λ∆t + Lj λ∆t · Π0l .

(3.18)

Taking ∆t → 0, we have the evolution rule of Mti :

M˙ ti = λ[Li + Lj Π0l − Mti ].

(3.19)

i Then, since MX is Li , Mti can be expressed explicitly:

Mti = Li + Lj Π0l · [1 − e−λ(t−X) ].

(3.20)

The Zti changes in short time interval ∆t, denoting (1 − η)Πl by Π0l :

i Zt+∆t = (1 − λ∆t)Zti + Li λ∆t ·

φit + φit+∆t + Lj λ∆t · Π0l . 2

(3.21)

141 Taking ∆t → 0, we have the evolution rule of Zti :

Z˙ ti = λ[Li φit + Lj Π0l − Zti ].

(3.22)

i As far as φit is constant over time (φit = φ¯i ), Zti can be expressed explicitly, knowing ZX = ΠiX · Li :

Zti = Li φ¯i + Lj Π0l + [Li ΠiX − Li φ¯i − Lj Π0l ]e−λ(t−X) .

(3.23)

Therefore, using equations (3.20) and (3.23), we can express the reputation of group i at time t:

Πit



Zi = ti Mt

 =

Li φ¯i + Lj Π0l + [Li ΠiX − Li φ¯i − Lj Π0l ]e−λ(t−X) . Li + Lj Π0l · [1 − e−λ(t−X) ]

(3.24)

Since we already know that v˙ ti = (δ + λ)[vti − ξti ], as far as ξti is constant (ξti = ξ¯i ),

i vti = ξ¯i + (vX − ξ¯i )e(δ+λ)(t−X) .

(3.25)

After the rearrangement, we have the following useful outcome:

e

−λ(t−X)

λ

 i − ξ¯i δ+λ vX . = vti − ξ¯i 

(3.26)

From equations (3.24) and (3.26), we can achieve the following useful lemma: Lemma 19. Suppose the Π0l fraction of type-j newborns consistently switch to type i since t = X. i Given constant ξ¯i and φ¯i , we can express the relationship between the initial state (vX , ΠiX ) and the

state at time t (vti , Πit ):



λ

 i vX − ξ¯i δ+λ Li φ¯i + Lj Π0l − (Li + Lj Π0l )Πit = . i vt − ξ¯i −Li ΠiX + Li φ¯i + Lj Π0l − Lj Π0l Πit

(3.27)

142

Also, we can evaluate the following, using equations (3.18) and (3.21),

Πi − Πit ∆Πit ≡ t+∆t ∆t ∆t

=

  i Zt+∆t 1 i − Πt . · i ∆t Mt+∆t

Taking ∆t → 0, we have the evolution rule of Πit : i 0 0 i ˙ it = λ[(Li φt + Lj Πl ) − (Li + Lj Πl )Πt ] . Π Mti

(3.28)

Proposition 23. Suppose that the Π0l fraction of type-j newborns switch to type i consistently since time X. Then, the dynamic system with a flow variable Πit and a jumping variable vti is

v˙ ti

=

(δ + λ)[vti − ξti ]

˙i Π t

=

λ[(Li φit + Lj Π0l ) − (Li + Lj Π0l )Πit ] , Mti

with demarcation loci of

v˙t i = 0 Locus : vti = ξti ˙ it = 0 Locus : Π

Πit =

Li φit + Lj Π0l . Li + Lj Π0l

Corollary 8. In the dynamics of group i which is growing with the inflows of the most talented type-j newborns, the reputation of group i improves faster (or deteriorates slower) compared to that ˙ it > Π ˙ nt , ∀φit ∈ {Πl , Πh }, ∀t ∈ (X, ∞), except when Πi = 1. (Note of the no-switches dynamics: Π X ˙i =Π ˙ n .) that when ΠiX = 1, Π X X Proof. See the proof in the appendix. 

The dynamics generates two stable equilibria: Q0l (0, L0i ) and Q0h (1, Hi0 ), where L0i = and Hi0 =

Li Πh +Lj Π0l Li +Lj Π0l

Li Πl +Lj Π0l Li +Lj Π0l

. Both of them are positioned higher than stable equilibria in no-switches

143 dynamics, Qnl (0, Πl ) and Qnh (1, Πh ). Let us denote π o0i as the time-X reputation level ΠiX at v n = v ∗ with which group i at the time-X state (v ∗ , π o0i ) can directly reach the upper equilibria Q0h along the optimistic path. Also, denote π p0i as the level of reputation at v n = v ∗ with which group i at the time-X state (v ∗ , π p0i ) can directly reach the lower equilibrium Q0l along the pessimistic path. Using i Lemma 19, we can compute both of them. For the first, apply ξ¯i = 0, φ¯i = Πh , (vX , ΠiX ) = (v ∗ , π o0i )

and (vti , Πit ) = (1, Π∗ ):

λ ˜ i Π0 (1 − Π∗ ) − [L ˜ i Π0 (1 − Π∗ ) + (Πh − Π∗ )] · v ∗− δ+λ . π o0i = Πh + L l l

(3.29)

i , ΠiX ) = (v ∗ , π p0 ) and (vti , Πit ) = (0, Π∗ ): For the second, apply ξ¯i = 1, φ¯i = Πl , (vX

λ ˜ i Π0 (1 − Π∗ ) − [L ˜ i Π0 (1 − Π∗ ) + (Πl − Π∗ )] · (1 − v ∗ )− δ+λ π p0i = Πl + L . l l

(3.30)

Comparing π o0i and π p0i with π o and π p in equations (3.15) and (3.16), we have the following result. Corollary 9. Both π o0i and π p0i in group i dynamics with the inflows of the most talented type-j newborns are smaller than π o and π p in the dynamics with no identity switches: π o0i < π o and π p0i < π p . The optimistic path from (v ∗ , π o0i ) and the pessimistic path from (v ∗ , π p0i ) are described in Figure 3.3.

3.3.2.2

Dynamic System of Group j with Outflows to Group i

According to Proposition 21, outflows to group i should be among the most talented type-j newborns with the lower switching cost kl . Note that when the potential switchers start to switch at time X, the reputation level of group j should be lower than π o , which is the lower boundary of the optimistic path in the no-switches dynamics: ΠjX < π o .

144 Lemma 20. When type-j potential switchers start to switch at time X, ΠjX < π o and, consequently, j vX = 0.

Proof. See the proof in the appendix. 

j As the Πl (1 − η) fraction of type-j newborns switches to type i since time X, MX = Lj and Mtj

decreases over time. Then, Mtj changes in the short time interval ∆t:

j = (1 − λ∆t)Mtj + Lj λ∆t[1 − Π0l ]. Mt+∆t

(3.31)

M˙ tj = λ[Lj (1 − Π0l ) − Mtj ].

(3.32)

Taking ∆t → 0, we have

j Then, since MX is Lj , Mtj can be expressed explicitly:

Mtj = Lj (1 − Π0l ) + Lj Π0l e−λ(t−X) .

(3.33)

Also, the size of skilled workers among group j changes over time:

j Zt+∆t = (1 − λ∆t)Ztj + Lj λ∆t · Πl η.

(3.34)

Taking ∆t → 0, we have the evolution rule of Ztj :

Z˙ tj = λ[Lj Πl η − Ztj ].

(3.35)

j Since ZX = ΠjX · Lj , the Zit can be expressed explicitly:

Ztj = Lj Πl η + Lj (ΠjX − Πl η)e−λ(t−X) .

(3.36)

145

Thus, we can evaluate the following, using equations (3.34) and (3.31), Πj − Πjt ∆Πjt ≡ t+∆t ∆t ∆t

=

" j # Zt+∆t 1 j − Πt . · j ∆t Mt+∆t

Taking ∆t → 0, we have the evolution rule of Πjt : j 0 ˙ jt = λLj [Πl η − (1 − Πl )Πt ] . Π Mtj

(3.37)

Therefore, using the above lemma, we can reach the following results: Proposition 24. Suppose that the Πl (1 − η) fraction of type-j newborns switch to type i consistently since time X. Then, the dynamic system with a flow variable Πjt and a jumping variable vtj is

v˙ tj

=

(δ + λ)[vtj − ξtj ]

˙ jt Π

=

λLj [Πl η − (1 − Π0l )Πjt ] Mtj

,

in which vtj = ξtj = 0, ∀t ∈ (X, ∞), and Πjt approaches monotonically L00j (≡

Πl η 1−Πl (1−η) ),

which is

smaller than Πl . Proof. Since π o > Πl and Πl > L00j (≡

Πl η 1−Πl (1−η) ),

π o > L00j . For any ΠjX < π o , Πjt approaches

L00j . Under the no-switches dynamics, the reputation recovery path is not available for any initial reputation level Πn0 ∈ (0, π o ). Therefore, the reputation recovery path should not be available to group j which is losing their most talented newborns to the other group, which implies vtj = ξtj = 0, ∀t ∈ (X, ∞), as vtn = ξtn = 0, ∀t, for any Πn0 ∈ (0, π o ).  The dynamics of group j is displayed in Figure 3.4. Note that whenever the most talented type-j newborns switch to type i, group j is positioned on the pessimistic path with ξtj = 0, vtj = 0 and Rtj =

wPq δ+λ ,

∀t ∈ (X, ∞). The state of group j losing the most talented to group i converges to

Q00 (0, L00j ), where L00j =

Πl η 1−Πl (1−η) ,

which is smaller than Πl .

146

Corollary 10. In the dynamics of group j which is losing some of the most talented newborns to group i, the reputation of group j deteriorates faster (or improves slower) compared to that of the ˙ jt < Π ˙ nt , ∀t ∈ (X, ∞). no-switches dynamics: Π Proof. See the proof in the appendix. 

3.4

Endogenous Group Formation

In order to analyze the endogenous process of group formation, we impose the following reasonable assumptions about the behaviors of group members: 1) Group members can make a consensus for the group state that will be realized in the far future, within a reasonably short period. They can agree quickly with the path to be taken, when multiple equilibrium paths (optimistic and pessimistic) are available. 2) Whenever multiple equilibria are possible for the future group state, group members tend to choose the equilibrium with the higher group reputation. Whenever multiple paths are available, the group tends to choose the (optimistic) path that leads to the higher group reputation. 3) Once group members agree with a future group state, they behave in a way to arrive there as early as possible. Once group members choose the path to take, they determine the level of a jumping variable in a way that the group state reaches the equilibrium as fast as possible. 4) When two groups hold expectations about the future that conflict with each other, they can reach “social consensus” toward the future within a reasonably short period. For example, when it is impossible that both groups take the optimistic path, one group gives up the option to take the optimistic path within a reasonably short period. We assume that the overlap in the no-switches dynamics is placed within the two stable equilibria: Qnl and Qnh . Under the constraints that newborns cannot switch, any group in the lower equilibrium Qnl is in the reputation trap, and cannot escape the low skill investment activities. j By Lemma 20, we know that vX = 0 when the Π0l fraction of type-j newborns start to join group

147 j i i. Since they switch their inborn types only when Yq,t > kl , we can find the threshold level of vX :

j Yq,X

=

w(1 − Pq ) i w(1 − Pq ) i j (vX − vX )= vX > kl . δ+λ δ+λ

i i Therefore, the threshold level of vX , denoted by vˆX , is

(δ+λ)kl w(1−Pq ) ,

j given vX = 0. We impose the

following condition that is not critical in the structure of the given dynamic model, but useful to achieve the main results more effectively: i Condition 1. The level of kl ensures vˆX > v ∗ : kl >

1−Pq Pu −Pq



cm −

wPq δ+λ



.

It is notable that there always exists a positive range of kl that satisfies the condition and Assumption 5 for any cm satisfying Assumption 4: the range of kl is



 max

for given cm ∈



wPq wPu δ+λ , δ+λ

w 1 − Pq − cm , δ+λ P u − Pq

   w(1 − Pq ) wPq , , cm − δ+λ δ+λ



. Readers may try and confirm that the following results in this chapter   1−P wP i can be replicated for the case of vˆX ≤ v ∗ : kl ≤ Pu −Pqq cm − δ+λq .

3.4.1

Group i Equilibrium Path with Skill Inflows from Group j

Since kl is less than

w(1−Pq ) δ+λ

i by Assumption 5, we know that v ∗ < vˆX < 1 under Condition

ˆ i , above which the 1. Then, we find the corresponding threshold level of group i reputation, Π X initial reputation of group i can lead the low cost talented type-j newborns to switch to type i i ˆi ˆ i is computed applying ξ¯i = 0, φ¯i = Πh , (v i , Πi ) = (ˆ immediately. Using Lemma 19, Π vX , ΠX ) X X X

and (vti , Πit ) = (1, Π∗ ):

λ ˆ i = Πh + L ˜ i Π0 (1 − Π∗ ) − [L ˜ i Π0 (1 − Π∗ ) + (Πh − Π∗ )] · vˆi − δ+λ . Π X l l X

(3.38)

148

However, if group i members can expect the inflows of the talented type-j newborns in the future, they may increase their skill investment rate much earlier even before the incidence of the skill inflows, as described in Panel A of Figure 3.5. Before the incidence of the skill inflows, the evolution rules for Πit and vti should follow the rules in the no-switches dynamics summarized in Proposition i ˆi vX , ΠX ) to the differential 22. Thus, applying ξtn = 0, φnt = Πh , (v0i , Πi0 ) = (v ∗ , π o00i ) and (vti , Πit ) = (ˆ

equations in the proposition, we can find the group i reputation level π o00i at v i = v ∗ with which the i ˆi group state (vti , Πit ) reaches (ˆ vX , ΠX ) that initiates the switching of the talented type-j newborns:

λ i ˆ i ) · [ˆ π o00i = Πh − (Πh − Π vX /v ∗ ] δ+λ . X

(3.39)

The following lemma summarizes the relative size of πio , π o0i , and π o00i . i < 1, π o0i < π o00i < π o . Lemma 21. Since v ∗ < vˆX

Proof. Compare π o , π o0i and π o00i , using equations (3.15), (3.29) and (3.39) and applying equation (3.38):

λ

λ

π o00i − π o0i

=

λ i δ+λ ˜ i Π0l (1 − Π∗ ) · (ˆ L vX − v ∗ δ+λ )v ∗ − δ+λ > 0

π o − π o00i

=

λ i δ+λ ˜ i Π0l (1 − Π∗ ) · (1 − vˆX L )v ∗ − δ+λ > 0.

λ

(3.40)

 i ˆi If π o00i is below Πl , the optimistic path that can reach (ˆ vX , ΠX ) is extended further up to the

Πi = 0 horizontal line and, thus, the group i even with zero reputation can take the optimistic path to Q0h (1, Hi0 ), as described in Panel B of Figure 3.5. Therefore, we can find the effective threshold ˜ i , above which the optimistic path to the higher equilibrium Q0 (1, H 0 ) is of group i reputation Π i h available to group i members. j ˜ i is as follows, Proposition 25. Given vX = 0, the effective threshold of group i reputation Π

above which group i can move out of the reputation trap and reach the high reputation equilibrium

149 Q0h (1, Hi0 ): ˜i = Π

  λ  i  ˆ i ) · [ˆ Πh − (Πh − Π vX /v ∗ ] δ+λ (= π o00i ) X

if π o00i ≥ Πl , (3.41)

   0

if

π o00i

< Πl .

The following corollary shows the important role of relative size between two groups in the ˜ i ): determination of the effective reputation threshold for group i (Π ˜ i in (0, ∞), above which Π ˜ i = 0 and below which ˜ ∗ of L Corollary 11. There exists the threshold L i ˜ i = π o00 : Π i λ

λ

0 ∗ i δ+λ )v ∗ − δ+λ ˜ ∗i = Πl (1 − Π ) · (1 − vˆX L . λ Πh − Πl + (Π∗ − Πh )v ∗− δ+λ

(3.42)

˜ ∗ that satisfies π o00 = Πl . L ˜ ∗ is positive Proof. Using equations (3.38) and (3.39), we can get L i i i λ

because the denominator is positive: Πh − Πl + (Π∗ − Πh )v ∗− δ+λ = π o − Πl > 0.

3.4.2

Search For Final State Given Initial State (Πb0 , Πa0 )

Now let us get return to the original question, the dynamics of groups A and B. Using the findings in the previous sections, we can search for the final state for each initial state (Πb0 , Πa0 ) under the imposed assumptions and a condition. First, check the state evolution under the constraints that type-B newborns can switch to the other type but the switches of type-A newborns to type B are not permitted. In this case, only type-B potential switchers with the cost set (cl , kl ) may consider switching (Proposition 21). By Lemma 20, there is no switching of type-B newborns when Πb0 ≥ π o . When Πb0 < π o , vtb = 0, ∀t, whether or not the type-B potential switchers switch, according to the dynamics summarized in b Propositions 22 and 24. According to Proposition 25, since vX = 0 with Πb0 < π o given, group A

˜ a can reach the high reputation equilibrium Q0 . Below Π ˜ a , the with its initial reputation above Π h group’s reputation ends up with Πl . Therefore, we can summarize the basin of attraction for each

150

potential attractor in the following way:

Basins of Attraction I

     {(Πb0 , Πa0 )|π o ≤ Πb0 ≤ 1, π o ≤ Πa0 ≤ 1}          {(Πb0 , Πa0 )|π o ≤ Πb0 ≤ 1, 0 ≤ Πa0 < π o }    ˜ a ≤ Πa ≤ 1}  {(Πb0 , Πa0 )|0 ≤ Πb0 < π o , Π  0         ˜ a} {(Πb0 , Πa0 )|0 ≤ Πb0 < π o , 0 ≤ Πa0 < Π

for attractor (Πh , Πh ), for attractor (Πh , Πl ), (3.43) for attractor

(L00b , Ha0 ),

for attractor (Πl , Πl ).

˜ a = π o00 (that is, These basins of attractions are displayed in Panel A of Figure 3.6 for the case of Π a π o00a > Πl ). Any initial position in the basin of attraction for the attractor (L00b , Ha0 ) that is colored in yellow in the panel will lead the type-B potential switchers to start to join group A at time X in order to obtain the benefits of superior collective reputation of group A in the labor market. Note that, given 0 ≤ Πb0 < π o , the type-B potential switchers immediately start to switch when the initial ˜ a , the ˆ a , though Πa is greater than Π ˆ a ; If Πa is smaller than Π reputation Πa0 is greater than Π 0 0 X X type-B potential switchers do not switch right away, but wait until the group A’s reputation Πat ˆa . improves up to Π X Second, check the state evolution under the constraints that type-A newborns can switch to the other type but the switching of type-B newborns to type A are not permitted. In this case, only type-A potential switchers with the cost set (cl , kl ) may consider switching (Proposition 21). With the same logic above, we can summarize the basin of attraction for each potential attractor:

Basins of Attraction II

     {(Πb0 , Πa0 )|π o ≤ Πa0 ≤ 1, π o ≤ Πb0 ≤ 1}          {(Πb0 , Πa0 )|π o ≤ Πa0 ≤ 1, 0 ≤ Πb0 < π o }    ˜ b ≤ Πb ≤ 1}  {(Πb0 , Πa0 )|0 ≤ Πa0 < π o , Π  0         ˜ b} {(Πb0 , Πa0 )|0 ≤ Πa0 < π o , 0 ≤ Πb0 < Π

for attractor (Πh , Πh ), for attractor (Πl , Πh ), (3.44) for attractor (Hb0 , L00a ), for attractor (Πl , Πl ).

˜ b = π o00 (that is, These basins of attractions are displayed in Panel B of Figure 3.6 for the case of Π b

151 π o00b > Πl ). Any initial position in the basin of attraction for the attractor (Hb0 , L00a ) that is colored in orange in the panel will lead the type-A potential switchers to start joining group B at time X. Note that, given 0 ≤ Πa0 < π o , the type-A potential switchers immediately start to switch when Πb0 ˆ b . Otherwise, they do not switch until group B’s reputation Πbt improves further is greater than Π X ˆ b . From Basins of Attraction I and II summarized above, we have the following lemma. up to Π X Lemma 22 (Basin of Attraction with Switching). For any type i ∈ {A, B}, the following is true: under the constraints that only type i newborns can switch and the type -i newborns are restricted i o ˜ −i not to switch, any initial position in {(Πi0 , Π−i ≤ Π−i 0 )|0 ≤ Π0 < π , Π 0 ≤ 1} will lead the low-cost

type i potential switchers to start to join group −i at some point of time X. An initial position in other areas never initiate the switching of the type i potential switchers in the future. By lifting both constraints above, we can obtain the full dynamic picture: the overlap of Panel ˜ a = π o00a and A and Panel B of Figure 3.6 generates Panel A of Figure 3.7 (for the case that Π ˜ b = π o00 ). First, when both Πa and Πb are greater than π o , no potential switcher considers the Π 0 0 b switching (Lemma 20), and the state (Πat , Πbt ) approaches the high reputation symmetric equilibrium ˜ a and Π ˜ b for each, no potential switcher considers (Πh , Πh ). Also when Πa0 and Πb0 are smaller than Π switching (Proposition 25), and the state (Πat , Πbt ) approaches the low reputation symmetric equilibrium (Πl , Πl ). For the other initial positions, (Πb0 , Πa0 ) belongs to either the basin of attraction for ˜ a or the basin of attraction for (H 0 , L00a ), (L00b , Ha0 ), which is the area satisfying Πb0 < π o and Πa0 ≥ Π b ˜ b , or to both areas. For the former case, type B which is the area satisfying Πa0 < π o and Πb0 ≥ Π potential switchers consistently join group A from the time X and the state (Πbt , Πat ) approaches (L00b , Ha0 ). For the latter case, type A potential switchers consistently join group B from the time X and the state (Πbt , Πat ) approaches (Hb0 , L00a ). The basin of attraction for (L00b , Ha0 ) and that for ˜ b ≤ Πb < π o and Π ˜ a ≤ Πa < π o }, in which both type-A (Hb0 , L00a ) are overlapped in the area: X ≡ {Π 0 0 potential switchers and type-B potential switchers have an incentive to switch to the other type as long as the potential switchers of the other type do not switch. Therefore, in this case, the social

152

consensus about the future among the whole population may determine the future state: if people believe that group A will grow as an elite group with a higher reputation, only type-B potential switchers may switch to the other type and, consequently, the future state approaches (L00b , Ha0 ). If people believe that group B will grow as an elite group with a higher reputation, only type-A potential switchers may switch to the other type and, consequently, the future state approaches (Hb0 , L00a ). Lemma 23 (Overlap of Basins of Attraction). In an overlapped area of the basin of attraction for (L00b , Ha0 ) and that for (Hb0 , L00a ), the expectation about the future among the whole population determines the final state. In the areas other than the overlap, the initial state (Πb0 , Πa0 ) decisively determines the final state. Example 1 (Expectation - Point A in Figure 3.7). In a point A, in which the initial statuses of two groups are identical in terms of the group size and the group reputation, the difference between two groups’ reputations grows over time as the potential switchers of one group consistently migrate to the other group. The expectation (social consensus) about the future determines which group grows as an elite group and which one keeps on losing its reputation. Using Lemma 22 and Lemma 23, we can summarize the future state for each initial state (Πb0 , Πa0 ) in the following way: Theorem 5. For given initial state (Πb0 , Πa0 ), the final state limt→∞ (Πbt , Πat ) is      (Πh , Πh )           (Πl , Πl )      lim (Πbt , Πat ) = (L00 , H 0 ) a t→∞ b         (Hb0 , L00a )          (L00b , Ha0 ) or (Hb0 , L00a )

if (Πb0 , Πa0 ) ∈ {π o ≤ Πb0 ≤ 1, π o ≤ Πa0 ≤ 1} ˜ a , 0 ≤ Πb < Π ˜ b} if (Πb0 , Πa0 ) ∈ {0 ≤ Πa0 < Π 0 ˜ a ≤ Πa ≤ 1} − X if (Πb0 , Πa0 ) ∈ {0 ≤ Πb0 < π o , Π 0 ˜ b ≤ Πb ≤ 1} − X if (Πb0 , Πa0 ) ∈ {0 ≤ Πa0 < π o , Π 0 if (Πb0 , Πa0 ) ∈ X,

(3.45)

153 ˜ b ≤ Πb < π o and Π ˜ a ≤ Πa < π o }, L00a = L00 = in which X ≡ {Π 0 0 b Hb0 =

Πl η 1−Πl (1−η) ,

Ha0 =

La Πh +Lb Π0l La +Lb Π0l

and

Lb Πh +La Π0l Lb +La Π0l .

First, note that both (Πl , Πh ) and (Πh , Πl ) are not stable. The newborn potential switchers in a disadvantaged group may switch their inborn types (thus incurring the cost of switching) and try to join the advantaged group to take the benefits of their superior reputation, which is often called a “passing” behavior. Corollary 12 (Instability of (Πl , Πh ) and (Πh , Πl )). Both (Πl , Πh ) and (Πh , Πl ), which are stable in a no-switches dynamics (Proposition 22), are not stable in a dynamic system with switches allowed. Example 2 (“Passing” - Points B or B 0 in Figure 3.7). Given the initial points B(Πl , Πh ) or B 0 (Πh , Πl ) in Figure 3.7, the talented newborns with the lower switching cost among a stereotyped disadvantaged population consistently pass for the advantaged group. Consequently, the reputation of the disadvantaged group becomes even worse and that of the advantaged group becomes even better. The above theorem implies the followings: 1) when both groups’ reputations are good enough (Πb0 > π o , Πa0 > π o ), the two groups’ reputations tend to converge to the high reputation level Πh ; 2) ˜ b , Πa < Π ˜ a ), they tend to converge to the low reputation level Πl ; when both are very bad (Πb0 < Π 0 and 3) otherwise, the little better-off group’s reputation tends to improve over time and approach a reputation even higher than Πh , while the little worse-off group’s reputation tends to deteriorate over time to a reputation level even worse than Πl , as the potential switchers of the worse-off group consistently differentiate themselves from their own group and join the little better-off group that is expected to grow as an elite group. Let us define this behavior as “partial passing” (Loury 2002). Those who commit partial passing might be blamed by their peer members for the differentiation from them, which is often convicted as “Action White.” Corollary 13 (Divergence among Disadvantaged Population). Unless the reputations of two dis˜ b , Πa < Π ˜ a ), the reputations of the two advantaged groups (Πb0 < π o , Πa0 < π o ) are very bad (Πb0 < Π 0

154

groups tend to diverge over time, as the potential switchers of the little worse-off group consistently migrate to the little better-off group. Example 3 (“Partial Passing” - Points C or C 0 in Figure 3.7). Consider two subgroups with distinguished cultural traits of the stereotyped population. Assume a small difference in their initial reputations. The talented newborns of the worse-off subgroup have an incentive to differentiate themselves from the other members of the group and join the better-off subgroup, by collectively adopting the cultural traits of the better-off subgroup. Owing to the “partial passing” activities of the talented young members of the worse-off subgroup, the reputation of the slightly better-off subgroup may improve significantly over time, as the percentage of the qualified workers among them grows continuously. This partial passing or differentiation activities can help the talented young members in the stereotyped population to be less influenced by the negative stereotype in the labor market. Now, let us consider two disadvantaged groups with the different group sizes. When group sizes are different, the low cost talented newborns of the bigger group have greater incentive to switch to the smaller group than the other way around, because the talented newborns’ switching of the bigger group can make a significant improvement in the reputation of the smaller group, but the newborns’ switching of the smaller group to the bigger group would not make enough difference. Thus, with the others being equal, the smaller group is more likely than the bigger group to become an “elite” group. Panel B of Figure 3.7 displays this tendency: π o00b > π o00a when La < Lb . Corollary 14 (Selective Out-Migration from Bigger Group to Smaller Group). Among the two disadvantaged groups A and B (Πb0 < π o , Πa0 < π o ), the smaller group is more likely to grow as an “elite” group and the bigger one is more likely to remain as a disadvantaged group in the reputation trap. Proof. See the proof in the appendix. 

Example 4 (Separating from Masses - Point D in Figure 3.7). Consider two subgroups A and B

155

in the stereotyped population, with the different group sizes (La < Lb ), which means the cultural traits of B-type are more popular a priori than those of A-type. The talented newborns of the majority subgroup B, who suffer from the group’s negative stereotype in the market, may seek a way to differentiate themselves from the masses. One way to do this is to collectively join the less popular cultural group A while incurring the costs of adopting the new cultural traits. Even when the initial reputation of the minority cultural group A is worse than that of the majority group B as noted in Point D in Figure 3.7 (Πa0 < Πb0 ), the selective out-migration of the most talented of the larger group B to the smaller group A can improve the smaller group A’s reputation fast, and thus the talented newborns of group B can escape the reputation handicap in the labor market. The feasibility of identity-switching is represented by the parameter η: the greater η is, the more newborns never consider switching due to the very high switching cost. Also, the feasibility is represented by the parameter kl : the greater kl is, the less affordable the switching is for the newborns who may consider switching. Corollary 15 (Switching Feasibility). The less feasible the switching is, the less likely the divergence of group reputations arises. Proof. See the proof in the appendix. 

If the social identity manipulation such as partial passing is not available, any subgroup in the stereotyped population may not recover its reputation, moving out of the reputation trap. The usage of the identity manipulation can help some disadvantaged subgroups to build up their reputations with the inflows of the most talented from other subgroups, and with their greater skill investment activities. In this sense, the identity manipulation or the usage of the cultural instrument in the labor market (Fang 2001) can improve the social efficiency. However, with the selective out-migration, other disadvantaged subgroups losing the most talented may suffer further from having the worse collective reputation, which may undermine solidarity in the disadvantaged population and cause

156

conflicts between the subgroups (Loury 2002). Corollary 16 (Social Efficiency). The behavior of the social identity manipulation such as partial passing may improve the social efficiency, and the usage of the observable cultural traits in the screening process may cure to some extent the social inefficiency caused by the imperfect information in the labor market. Proof. Suppose both Πa0 and Πa0 are below π o . Without the usage of the cultural traits in the screening process in the labor market, the total size of skilled workers would be (La + Lb )Πl . With the usage of the cultural traits, the final state may approach either (L00b , Ha0 ) or (Hb0 , L00a ), which means the total size of skilled workers may approach either Lb Πl + La Πh or La Πl + Lb Πh . Both are greater than (La + Lb )Πl . 

3.4.3

Autonomous Emergence of an Elite Group among the Stereotyped

In this section, we show how a small elite subgroup with the unique cultural traits can emerge autonomously among a negatively stereotyped population. First, the emergence is feasible whenever there exists a sufficiently small subgroup with unique cultural traits that are expensive enough to obtain for the other members in the stereotyped population except the most talented (Panel A in Figure 3.8). Second, the most talented members of a stereotyped population have an incentive to create a small group with the observable distinguished cultural traits so that they can differentiate themselves from the rest in the labor market (Panel B of Figure 3.8).

3.4.3.1

Small Cultural Group’s Growing as an Elite Group

We know that a symmetric initial position (Πl , Πl ) is stable if the identity switch between the groups is not allowed in the given model (Proposition 22). Once the switch is feasible, the symmetric initial position (Πl , Πl ) is not stable any more if the size disparity between the groups is sufficiently big, which is implied in Corollary 11:

157 ˜a > L ˜ ∗a or L ˜b > L ˜∗. Lemma 24. The (Πb0 , Πa0 ) is not stable at (Πl , Πl ), when either L b ˜ i is zero when L ˜i > L ˜ ∗ . According to Theorem 5, the basin of Proof. According to Corollary 11, Π i ˜ a , 0 ≤ Πb < Π ˜ b }. Thus, with L ˜i > L ˜ ∗ , (Πl , Πl ) cannot belong attraction for (Πl , Πl ) is {0 ≤ Πa0 < Π 0 i to the basin of attraction for (Πl , Πl ).  ˜ i (≡ Lj /Li ) is greater than Suppose (Πi0 , Πj0 ) = (Πl , Πl ) at time zero. If Li is small enough that L ˜∗, Π ˜ i is equal to zero (Corollary 11) and, consequently, v i > v ∗ , which means the medium investL 0 i ment cost newborns with type i immediately invest in skills expecting the better group reputation and the more preferential treatment in the future. Therefore, the group i’s reputation improves immediately beyond the low reputation level Πl . The unequal reputations between two groups emerge autonomously as the talented type-j newborns join group i (Corollary 14), and the final state approaches the asymmetric stable state (Πi∞ , Πj∞ ) = (Hi0 , L00j ), as displayed in Panel A of Figure 3.8. Note that the instability of (Πl , Πl ) has arisen because the most talented of the disadvantaged population have an incentive to join a small subgroup i expecting the fast reputation improvement of the group in the future with their joining the subgroup and the increased skill investment activities among the type i newborns. Thus, the above lemma implies the following proposition. Proposition 26 (A Small Group’s Growing as Elite Group). Imagine a negatively stereotyped population at the low reputation level Πl , which is a stable equilibrium in the no-switches dynamics. As far as there exists a sufficiently small subgroup with unique cultural traits for which the switching cost is large enough that it satisfies Assumption 5, the most talented young members of the stereotyped population have an incentive to differentiate themselves from the masses joining the small cultural subgroup. The small subgroup emerges as an elite group out of the stereotyped population.

3.4.3.2

Endogenous Creation of Elite Group

So far, we have assumed that groups with different cultural traits are exogenously given. Imagine a negatively stereotyped group B. If we allow that the most talented members of the stereotyped group can find proper cultural indices for differentiation and create a distinguished cultural group A by

158 ˜a ≈ ∞ > L ˜ ∗a , adopting specific indices, the condition in Lemma ?? is immediately satisfied because L as far as the chosen indices are rare in nature (La ≈ 0). Therefore, an elite group consisting of the most talented can emerge immediately and the size of the group will grow over time, as described in Panel B of Figure 3.8: the size of the created cultural group is close to zero in the beginning, but increases up to Π0l Lb . From the beginning, the reputation of the created group is one, which means that most members of the group are skilled workers. The reputation of the stereotyped population will become even worse over time as they lose the most talented newborns of the group to the distinguished cultural group A. A real life example would be the migration of talented members of a stereotyped population to specific residential areas that are not affordable to less talented peer members. Spending money on luxury goods and designer clothing that are not affordable to other members of the stereotyped population would be another example of differentiation by the most talented. They might also commit to fine arts or send their children to a private boarding school to signal their higher social status to outsiders. Corollary 17 (Emergence of Elite Group). If the talented young members of a stereotyped group can find proper cultural indices for differentiation, which are not affordable to other members of the group, they will form an elite subgroup based on the indices, incurring a cost to obtain them. Through this, they can immediately escape statistical discrimination practices and will be preferentially treated in the labor market.

3.5

Conclusion

The externality of group reputation is important to explain the discriminatory practices by employers and the different skill investment activities across social groups. In the previous chapter, Kim and Loury (2008), we suggest the concept of a reputation trap by developing a dynamic model of group reputation. Once a group’s collective reputation enters the range below a threshold, the group cannot escape the lower skill investment activities and the negative stereotype. A group with the

159

reputation above the threshold can build up its reputation through young members’ optimism about the future and their collective action for skill achievement. This work discusses the identity-switching behaviors of the most talented young members of a stereotyped population in a reputation trap, who have greater incentives to separate themselves from the stereotyped masses. We have presented three different identity-switching activities for the differentiation: passing, partial passing and elite culture development. Passing for an advantaged group would be the most efficient way for differentiation if the identity switching cost is not large. The most talented who succeed in passing can take advantage of the superior collective reputation of the group immediately. We often observe the passing activities among the stereotyped population who “fortunately” share a similar appearance with the advantaged population (eg. Korean descendants in Japan). When passing is not available, the talented members of the stereotyped population may consider the “partial” passing. They “pass” for the better-off subgroup with the unique cultural traits in order to send signals of their higher productivity to employers. The partial passing is a common activity among physically marked stereotyped people (eg. Blacks in the United States). Finally, the most talented individuals may develop distinguished cultural indices that are not affordable to the less talented members of the stereotyped population. The talented young members adopting the indices may form an elite cultural subgroup, whose members are distinguished from the rest in the population and who will be preferentially treated by employers. Note that, in the given dynamic identity-switching model, we have simplified the composition of a population in the following way. Group members are classified into three categories: the most talented, the medium talented and the least talented. Each talent group is classified again into individuals with higher switching cost and those with lower switching cost. Making some assumptions, we argue that the most talented members with the lower switching cost are identified as the only potential switchers who may consider the identity switching. The identification of the potential switchers was an important starting point for the analysis of endogenous group formation. We were able to find the exact dynamic paths tracing the decision-process of the potential switchers.

160

However, some researchers may generalize our findings using continuous distribution functions of skill investment costs and switching costs, without introducing the potential switchers. As we have discussed the endogenous group formation under the existence of group reputation externality, we may develop a similar work under the existence of social network externality. Kim (2008), which is summarized in Chapter 1, developed a dynamic model of group inequality through the channel of social network externalities. He suggested the concept of a network trap. Once the quality of a group’s social network enters some range below a threshold, the group cannot escape the low skill investment activities due to the negative influence of the network effects. The most talented of a disadvantaged social group in the network trap may consider switching to other social groups with a better network quality. There are several practices of this kind that we may observe in the real world, such as people moving to a residential area with more members of advantaged social groups, sending their children to a private school in which descendants of the advantaged group are prevalent, and attending social clubs where they can associate with members of advantaged social groups. If switching to advantaged social groups is not possible, the most talented may build up an elite subgroup with entering barriers, separating themselves from the low quality social network. Because the most talented individuals of a disadvantaged social group have a greater incentive to separate themselves from their peers than the most talented of an advantaged social group, we might observe a more divisive culture among the disadvantaged group than among the advantaged group. Some researchers may develop a dynamic model of endogenous group formation through this channel of social network externality.

161

3.6

Appendix: Proofs

Proof of Lemma 18. The results are driven using Lemma 17. The first argument is obvious because wPq δ+λ

wPu δ+λ

is max{Ri } and

is min{Ri }, which means cl < min{Ri , R−i } and ch > max{Ri , R−i }. The second argument is

obvious as well because

w(1−Pq ) δ+λ

is max{Yqi }, which means kh > max{Yqi }. By the above argument,

we need to check the following two newborn sorts for the third argument: (cm , kl ) and (ch , kl ). [Newborns with Cost Set (cm , kl )] First, let us prove that they never switch their inborn types when they have chosen to be unqualified, (i∗ , e∗ )|i,cm ,kl 6= (−i, u). Suppose they can switch. Then, by Lemma 17, kl < Yui ≤ cm <

wPu δ+λ ,

w(1−Pu ) δ+λ .

which implies kl >

However, by Assumptions 4 and 5, cm + kl >

w(1−Pu ) δ+λ .

w δ+λ

and

wPq δ+λ

<

Thus, they contradict each other. Second, let us prove that

they never switch their inborn types when they have chosen to be qualified: (i∗ , e∗ )|i,cm ,kl 6= (−i, q). Suppose they can switch. Then, by the lemma and equations (3.5) and (3.6), cm and kl should satisfy cm + kl < R−i + Yui =

wPq δ+λ

+

w(1−Pq ) −i δ+λ v

−

w(1−Pu ) i δ+λ v ,

which implies cm + kl <

w δ+λ .

This

contradicts Assumption 5. [Newborns with Cost Set (ch , kl )] First, let us prove that they never switch their inborn types when they have chosen to be unqualified: (i∗ , e∗ )|i,ch ,kl 6= (−i, u). Suppose they can switch. Then, by Lemma 17, kl < Yui ≤

w(1−Pu ) δ+λ .

As shown already, kl >

w(1−Pu ) δ+λ

by Assumptions 4 and 5. So,

there is a contradiction. Second, let us prove that they never switch their inborn types when they have chosen to be qualified: (i∗ , e∗ )|i,ch ,kl 6= (−i, q). Suppose they can switch. Then, by the lemma, ch < R−i ≤

wPu δ+λ ,

which contradicts Assumption 4. QED.

Proof of Corollary 8.

162 Replacing Mti with Li + ρLj Π0l in equation (3.28), where ρ ∈ [0, 1], we have

˙i Π t

=

λ[(Li φit + Lj Π0l ) − (Li + Lj Π0l )Πit ] Li + ρLj Π0l

=

λ[Li (φit − Πit ) + Lj Π0l (1 − Πit )] Li + ρLj Π0l

=

λ[Li (φit − Πit ) + Lj Π0l (1 − Πit ) + ρLj Π0l (φit − Πit ) − ρLj Π0l (φit − Πit )] Li + ρLj Π0l

= λ(φit − Πit ) +

λ[Lj Π0l (1 − Πit − ρ(φit − Πit ))] Li + ρLj Π0l

0 i i ˙ nt + λLj Πl [(1 − ρ)(1 − Πt ) + ρ(1 − φt )] = Π Li + ρLj Π0l

˙ it > Π ˙ nt except when Πit = 1 and ρ = 0. Since ρ = 0 when time is X, Since φit is either Πl or Πh , Π ˙ nt except when Πi = 1. QED. ˙ it > Π Π X

Proof of Lemma 20. j Suppose that ΠjX ≥ π o . Then, vX ≥ v ∗ because the optimistic path to Qnh is available to group j.

Then, we have the following:

j Yq,X

= ≤ =

j Thus, Yq,X +

w(1−Pu ) δ+λ

w(1 − Pq ) j i · (vX − vX ) δ+λ w(1 − Pq ) · (1 − v ∗ ) δ+λ w w(1 − Pu ) ∗ − cm − ·v δ+λ δ+λ

· v∗ ≤

w δ+λ

− cm . Since

w δ+λ



(δ + λ)cm − wPq ∵v ≡ w(Pu − Pq ) ∗

 .

j − cm < kl by Assumption 4, Yq,X < kl , which

j contradicts to Proposition 21 that Yq,X < kl when type-j potential switchers switch. QED.

Proof of Corollary 10.

163 Replacing Mtj with Lj − ρLj Π0l in equation (3.37), where ρ ∈ [0, 1], we have

˙ jt Π

=

λLj [Πl η − (1 − Π0l )Πjt ] Lj − ρLj Π0l

=

λLj [Πl − Πjt + Π0l (−1 + Πjt )] Lj − ρLj Π0l

=

λLj [Πl − Πjt + Π0l (−1 + Πjt ) − ρΠ0l (Πl − Πjt ) + ρΠ0l (Πl − Πjt )] Lj (1 − ρΠ0l )

= λ(Πl − Πjt ) +

λLj Π0l [−1 + Πjt + ρ(Πl − Πjt )] Lj (1 − ρΠ0l )

j j 0 ˙ n + λΠl [−1 + Πt + ρ(Πl − Πt )] = Π t 1 − ρΠ0l

˙ jt < Π ˙ nt . Since Πj < π o (Lemma 20) and Πjt monotonically approaches Therefore, as far as Πjt < 1, Π X L00j , Πjt < 1, ∀t ∈ (X, ∞). QED.

Proof of Corollary 14. First, you may check the difference between π o00a and π o00b , using equations (3.38) and (3.39):

π o00a

−

π o00b



 λ λ Lb La i i − δ+λ = − ) · [ˆ vX /v ∗ ] δ+λ . · Πl (1 − Π∗ )(1 − vˆX La Lb

(3.46)

Thus, when La < Lb , π o00a < π o00b . Also, you may check the partial derivative of π o00i with respect to Li :

∂π o00i ∂Li

=

ˆ i ∂L ˜i λ ∂Π i X · · [ˆ vX /v ∗ ] δ+λ ˜ ∂L ∂ Li i λ

i − δ+λ = −Π0l (1 − Π∗ )(1 − vˆX )·

λ Lj i · [ˆ vX /v ∗ ] δ+λ > 0 L2i

Thus, the bigger the size of the group Li , the greater π o00i , that is, the smaller the basin of attraction for the attractor with Hi0 . QED.

Proof of Corollary 15.

164 The divergence occurs either in the basin of attraction for (L00b , Ha0 ) or in that for (Hb0 , L00a ). We can show that the basins tend to shrink with the greater η, or with the greater kl , using equations (3.38) and (3.39):

∂π o00i ∂η

∂π o00i ∂kl

=

ˆi λ ∂Π 0 · [ˆ v0i /v ∗ ] δ+λ ∂η

=

˜ i Πl (1 − Π∗ )(1 − vˆi − δ+λ ) · [ˆ −L v0i /v ∗ ] δ+λ > 0 0

=

λ v0i δ+λ ˜ i Π0 (1 − Π∗ ) · v ∗ − δ+λ · ∂ˆ L l ∂kl

=

λ v0i δ+λ ˜ i Π0l (1 − Π∗ ) · v ∗ − δ+λ · ∂ˆ L > 0. ∂kl

λ

λ

λ

λ

QED.

  (δ + λ)kl i ∵ vˆ0 = w(1 − Pq )

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Figure g 1.1 Steadyy States in the Homogeneous g Economyy

. Panel A Finding the st=0 Locus ã

Iso-π curves: ã=A(s, π)

s=1-G(ã)

π Increasing

πd

πu st=0

su

sd

st=1

. . Panel B Steady States with the st=0 and πt=0 loci πt . st =0 (su,πu)

πu

πd

Eh Em

El

st=0

. πt =0

(sd,πd)

st=1

172

Figure g 1.2 Equilibrium q Paths in the Homogenous g Economy y

173

πt . st =0 0

Optimistic Equilibrium Path

Eh

. πt =0

Em El Pessimistic Equilibrium Path

st=0

sl

eo

ep

sh

st=1

174

Figure g 1.3 Comparison p of Three Distinct Cases

Panel A Lifetime Network Externality Only Panel B Education Period Externality Only

πt

πt

. st =0

. st =0

. πt =0 Optimistic Path

(su,πu)

Eh δ

Em

El

Eh

(α+ρ)

El

sl

(sd,πd)

Pessimistic Path

sh

Panel C Both Education Period and Lifetime Network Externality

sh

sl πt

. st =0

(su,πu) Optimistic Path

El

. Eh πt =0

Em (sd,πd) Pessimistic Path

sl

eo

ep

sh

175

Figure g 1.4 Economicallyy Stable States with Total Segregation g g

π2

s2 sh

Qlh

Qmh

Qhh

s2=1 sh

ep

sm Qlm

Qmm

Qhm

Qll

Qlm

Qhl sh s1

sl sl

sm

π1

s1=0

sl

eo

ep

sh

s1=1

eo

sl s2=0

Figure g 1.5 Manifold Ranges g and Overlaps p with Total Segregation g g

Panel A Equilibrium path of group 1

176

Panel B Equilibrium path of group 2

s2

s2

sh

sh ep eo

sl

sl sl

eo

sh

ep

s1

sl

Panel C Manifold Ranges to Qhh and Qll s2

sh

s1

Panel D Manifold Ranges to Qhl and Qlh s2

sh

sh Mhh

ep

Mlh

ep eo

eo Mll

Mhl

sl

sl sl

eo

sh

ep

s1

sl

eo

ep

sh

s1

Panell E M P Manifold if ld R Ranges and d Folded F ld d O Overlaps l 2 s sh Four-folded overlap

Mhh

Mlh ep

Two-folded overlap

eo

Deterministic Range

Mhl

Mll

sl sl

eo

ep

sh

s1

Figure g 1.6 Partial Steadyy States

(s ( i*,,σi*)

Given

177

sj

Panel A Introduction of Ã(σi*) function

ã

si*=1-G(ã)

iso-π curves

( l, ãl | πl) (s

sa

Ã(σi*) π Increasing (sm, ãm | πm)

sb

σt=0 st=0

(sh, ãh | πh)

σb

σa

σt=1 st=1

Panel B Identification of (si*,σi*) given sj

si

Panel C Smoothing condition for G(Ã(σi*))

G(Ã(σ Ã i*)) si=1-G(Ã(σi)) locus

(sh, sh)

Point of Inflection Concave

(sm, sm)

sb Convex

(sl, sl) sa

σa i i i σ =k s +(1-ki)sj

σb

σi

σi*

178

Figure g 1.7 Global Steadyy States with η and β Given .

.

.

Panel A Find s1*(s2) locus with s1=Π1=0

k2 σ1

s2 (sh, sh)

(sh, sh)

(sm, sm)

(sm, sm)

k1

s2=1 (sl, sl)

s2=sm

(sl, sl) s1

s2=0

s2 locus :D2(s1*))

σ2

s1=0 s1=sm s1=1

s2 s1*(s2)

(sh, sh)

s2*(s1) locus :D1(s2*)

(sm, sm)

(sh, sh)

(sm, sm)

(sl, sl)

(sl, sl) s1

s1 Region 1

Panel C Global steady states

.

Panel B Find s2*(s1) locus with s2=Π2=0

Region 2

s2 (sh, sh)

(sm, sm)

(sl, sl) Region 4

Region 3

s1

Figure g 1.8 Steadyy States for Each Level of η ((Given Small

β1)

- Segregation level η declines in Panel A, B, C, D, E, F order, with η=1 in Panel A and η=0 in Panel F.

Panel A Nine steady states with η=1

Panel B Nine steady states

sh

sh

sm

sm

sl

sl sl

eo sm ep

sh

sl

Panel C Nine steady states sh

sm

sm

sl

sl sm

sh

Panel E Five steady states

sl

sh

sm

sm

sl

sl sm

sh

sm

sh

Panel F Three steady states with η=0

sh

sl

sh

Panel D Seven steady states

sh

sl

sm

sl

sm

sh

179

. . Figure g 1.9 s1t and Π1t Demarcation Surfaces

180

- The sliced segments of the surfaces for each level of s2 are depicted in the second picture of each panel. s2 Panel A Panel B

(sh, sh)

Panel C

(sm, sm)

Panel D Panel E Panel A

s=1-G(ã) Iso-π curves

ã

(sl, sl)

s1 . s1t =0 0

π1

(S2=1) 1)

Panel B

st=1 1

st=0

0

σ1

ã

0

s1=11 σ1

s1=0

π1

(sh, πh)

(S2=Sh)

Panel C (S2=Sm)

1

σ1

ã

0

s1=1 1 σ1

s1=0

π1 (sm, πm)

st=1

st=0

0

Panel E (S2=Sl)

st=1

st=0

0

. π1t =0

1

σ1

ã

0

s1=0

s1=1

1 σ1

π1

(sl, πl) 0 Panel F (S2=0)

st=0

st=1

1

σ1

ã

0

s1=0

s1=1

0 s1=0

s1=1

1 σ1

π1

st=0 0

st=1

σ1 1

1 σ1

Figure g 1.10 Stable States and Location of Stable Manifold

. π2t =0 S2 Qlh(sl’, sh’)

Qmh

Qlm

Qll(sl, sl)

π1

Qhh(sh, sh)

Qmm((sm, sm)

Qml

Qhl(sh’’, sl’’) S1

. π1t =0

S1=0

2 1 σ

S2=1

Qlm

. S1t =0

0

. S2t =0

181

S1=1 1 σ1

π2

S2=0 0

182

Figure g 1.11 Stable Manifolds and Folded Overlaps p

Panel B A Stable Manifold to Qlh

Panel A Manifold Range Mlh S2

Mlh

Qmh

Qlh(sl’, sh’) Qlm

Qhh(sh, sh)

Qmm(sm, sm)

Qlm S2

(sl’, sh’) π2 π1

Qml

Qll(sl, sl)

Qhl(sh’’, sl’’) S1

Panel C Manifold Ranges and N-Folded Overlaps Network Trap of Group 1 S2

Three-folded Three folded overlap

Qmh

Qlh

Qhh

Mhh

Mlh

Four-folded overlap Two-folded overlap

Qlm

Qmm Mll

Qll

Deterministic Range

Qlm Mhl

Qml

Network Trap of Group 2

Qhl S1

S1

183

Figure g 1.12 Sable States and Manifold Ranges g for Each Level of η 1 (Given Small β ) - Segregation level η declines in Panel A, B, C, D, E, F order, with η=1 in Panel A and η=0 in Panel F.

Panel A Four stable states with η=1

Panel B Four stable states

sh

sh

sm

sm

sl

sl sl

eo sm ep

sh

sl

sh

sh

sm

sm

sl

sl sm

sh

Panel E Two stable states

sl

sh

sm

sm

sl

sl sm

sh

sm

sh

Panel F Two stable states with η=0

sh

sl

sh

Panel D Three stable states

Panel C Four stable states

sl

sm

sl

sm

sh

184

Figure 1.13 Integration Effect: Economic State Move as η Declines

Given an initial State: (S1,S2)=(Sl,Sh) Segregation η Deterministic Range for Qlh

η*(β1)

Overlap of Mlh and Mll p of Mlh and Mhh Overlap Overlap of Mlh, Mll and Mhh η**(β1)

η^(β1)

η^(β1)

Deterministic Range for Qhh

Deterministic Range for Qll

Overlap Mll and Mhh

β**

β^

β*

Population Share of Disadvantaged Group 1, β1

185

Figure g 1.14 Affirmative Action Policies

Panel A Asymmetric Training Subsidy Network Trap

Two‐Folded Overlap

sh

sh

sm

sm

sl

sl sl

sm

sh

sl

(Group 1 in Network Trap)

sm

sh

(Qlh Moves into Overlap)

Panel B Quota System Network Trap at Qlh(sl’,sh’)

Two‐Folded Overlap

s = β1sl’+β2sh’ sh

sh

sm

sm

sl

sl sl

sm

sh

(Group 1 in Network Trap)

sl

sm

sh

(Economic State Placed in Overlap)

186

Figure g 1.15 Equalization q Policy y Implementation p Panel As: Economy with Small β1

Panel Bs: Economy with Large β1 Panel B1 High η

Panel A1 High η

sh

Z’’

X

X

Z’

Mhh

Mhh

sm s = β1sl’+β2sh’

Mll

Mll

sl sl

sm

sh s = β1sl’+β2sh’ Panel B2 Medium η

Panel A2 Medium η X’

Z’ ’

Y’ X’

Z’

s = β1sl’’+β2sh’’ Panel B3 Low η

Panel A3 Low η X’’

X’’

187

Figure g 1.16 Macroeconomic Effects of Inequality q y

Panel A Positive Effect of Equality

Panel B Positive Effect of Inequality

Mhh

Mlh

Mll

Mhl Equality promotes economic growth.

Resource Constraint: R = β1sl+β2sh

Panel C Multiple Stable States in Three Group Economy

(h h h) Most Developed Economy  (h,h,h):  l d

Least Developed Economy 

Figure g 1.17 [[Application] pp ] Urbanization and Regional g Groups p

North  Korea

Seoul ★ Metropolitan Population ‐ 20.1% (5.0 mil, 1960) ‐ 35.5% (13.3 mil, 1980) ‐ 48.3% (23.8 mil, 2007) 48.3% (23.8 mil, 2007)

Chungcheong

Honam

Jeju

Kangwon

Youngnam

188

Figure g 1.18 [[Application] pp ] Evolution of Regional g Group p Disparity p y

Panel A Positioned at A in 50s Youngnam

Panel B Positioned at A’ in mid 70s Youngnam

sh

sh

sm

sm

A

sl

189

sl

sl

sm

sh

A’

A sl

Honam

sm

sh

Honam

(Low‐skilled Equal Society)

(Emergence of Initial Disparity)

Panel C Positioned at A’’ in 80s

Panel D On the move to A’’’ since early 90s

Youngnam sh

Youngnam

sm

A’’’

sh

A’’

A’’

sm

A’

sl

sl sl

sm

sh Honam

(Youngnam’s Skill Advance And Network Trap of Honam)

sl

sm

sh Honam

(Honam’s Skill Advance  And Convergence Between Groups)

Figure g 1.19 [[Application] pp ] College g Advancement Rate

190

College Advancement Rate 100

Youngnam’s Skill Advance

Honam’s Skill Advance

90 80 70 60 50 40 30 20

Youngnam

Honam

• Source: Statistical Yearbook of Education (South Korea) • The statistics rule out the vocational high schools. Note that an equalized public school system was established in the mid 1970s. Before then, the Seoul’s college advancement rate was the highest because the major prestigious private schools were located in Seoul.

Appendix pp Figure g 1.1 Size of Overlap p in a Simplified p Economy y

πt

. st=0 Optimistic Equilibrium Path

Eh . πt =0

El Pessimistic Equilibrium Path

st=0

eo

ep

st=1

191

192

Appendix pp Figure g 1.2 Manifold Ranges g with No Lifetime Externalities - Segregation level η declines in Panel A, B, C, D, E, F order, with η=1 in Panel A and η=0 in Panel F.

Panel A Four stable states with η=1

Panel B Four stable states

sh

sh

sm

sm

sl

sl sl

eo sm ep

sh

sl

sh

sh

sm

sm

sl

sl sm

sh

Panel E Two stable states

sl

sh

sm

sm

sl

sl sm

sh

sm

sh

Panel F Two stable states with η=0

sh

sl

sh

Panel D Three stable states

Panel C Four stable states

sl

sm

sl

sm

sh

Appendix pp Figure g 1.3 Regional g Group p Disparity p y in S. Korea

193

1.4 1.2 1 0.8 0.6 0.4 0.2 0

1.6 1.4 1.2 1 08 0.8 0.6 0.4 0.2 0

Honam Rep

Youngnam Rep

• Source: Chosun Daily Leaders’ Database www.dbchosun.com; Eui-Young Yu (2003) • Rep index: Rep index is the ratio between leader's birthplace percentage in 2002 and newborn percentage of the birthplace in 1970. The newborn distribution is a proxy of the regional distribution of young families in 1970. Rep index is calculated excluding Seoul born natives (about 5% of the population).

Appendix pp Figure g 1.4 Regional g Voting g in Presidential Elections

Election Year Parties Candidates Honam Youngnam Nationwide Birthplace of Candidate

1963

1967

Democratic Industrial Democratic BS Yoon

CH Park

Industrial CH Park

1987

Democratic Industrial Democrat1 Democrat2 DJ Kim

CH Park

DJ Kim

YS Kim

Industrial TW Roh

41

59

52

48

64

36

88.4

1.2

9.9

36

64

27

73

25

75

5.0

41.6

48.8

49

51

45

55

46

54

27.1

28

38.6

Chung cheong

Youngnam

Chung cheong

Youngnam

Honam

Youngnam

Honam

Youngnam

Youngnam

Election Year

1992

Parties Candidates Honam Youngnam Nationwide

Democratic Industrial

Birthplace of Candidate

BS Yoon

1971

194

1997 Democratic

2002

2007

Industrial Democratic Industrial Democratic Industrial

DJ Kim

YS Kim

DJ Kim

HC Lee

MH Roh

HC Lee

DY Jung

MB Lee

91

4.2

93.5

3.8

92.5

5.4

79.5

9.0

10

98

12.3

58.4

24.5

70.3

9.1

62

33.8

42

40.3

38.7

48.9

46.6

26.1

48.7

Honam

Youngnam

Honam

North Korea

Youngnam

North Korea

Honam

Youngnam

• Source: National Election Commission (South Korea ) • [Anecdote] The Democratic Party has been based on Honam region and the Industrial Party based on Youngnam region since 1971 election. There was no presidential election between 1971 and 1987 due to President Park’s dictatorship and the second military coup by President Chon in 1980. In 1987 election, the democratic party was split into two. “Democratic 1” gave the candidacy to DJ Kim born in Honam, and “Democratic 2” to YS Kim born in Youngnam. In 1992 election the “Democratic election, Democratic 2 2” party was merged into the Industrial Party and two democratic leaders, DJ Kim and YS Kim, competed for the presidency. In 1997 and 2002, the honam-based Democratic Party won the presidential elections.

195

Figure g 2.1 Multiple p Steady y States with Noisy y Signals g

Panel A. Noisy Signals

fu (θ) Unclear  Signal

Unclear  Signal

θˉ

0

fq (θ)

θˉ

0

Panel B. Three Steady States Π

Π 1

1 s*(Π) •

W

~β(s*(Π))

“Prior‐Belief”

~ G(β(s)) •

~ G(β)

• •

0

W•

B θˉ standard (s)

(Multiple steady states in (Π, s) domain)

B • 0

~ Benefits of Inv. (β)

~ (Multiple steady states in (Π, β) domain)

196

Figure g 2.2 Unique q Steady y State with Better Signals g

Panel A. Less Noisy Signals

fu (θ)

fq (θ)

Unclear Signal

Unclear Signal

θˉ

0

θˉ

0

Panel B. Unique Steady State Π 1

B, W s*(Π) •

0

Π 1

~ β(s*(Π))

~ G(β(s))

θˉ standard (s)

(Unique steady state in (Π, s) domain)

0

B, W •

~ G(β)

~ Benefits of Inv. (β)

~ (Unique steady state in (Π, β) domain)

197

Figure g 2.3 Phase Space p in Simple p Model Panel A. Phase Diagram Πt 1

.

Qh

Vt=0 Locus: β 0L βt(Πt)

Πh

• Qm

Π*

.

Πt=0 Locus: φt(Vt)

• Ql

Πl

• ωPq

0

ωPu

(δ+λ)Cm

Vt

Panel B. Equilibrium Paths Πt 1

.

Qh

Vt=0 Locus

Πh

•

.

Πt=0 Locus Π*

• Ql

Πl 0

• ωPq

(δ+λ)Cm

ωPu

Vt

Figure g 2.4 Differential Equations q in Modified Simple p Model

Πt

1 . Πt=λ(Πl‐Πt) . vt=(δ+λ)(vt‐1)

Π*

. Πt=λ(Πh‐Πt) . vt=(δ+λ)(vt‐1) Ⅱ

Ⅰ

Ⅲ

Ⅳ

. Πt=λ(Πl‐Πt) . vt=(δ+λ)vt

0 Vt= ωPq

. Πt=λ(Πh ‐Πt) . vt=(δ+λ)v (δ+λ)vt

v* Vt=(δ+λ)Cm

1 vt Vt= ωPu

198

199

Figure g 2.5 Phase Space p in Modified Simple p Model

Πt 1 φt=Πh

• Qh (1, Πh)

πp Pessimistic Path Overlap

•

Π*

Optimistic Path πo Ql (0, Πl) • Reputation Trap

0 Vt= ωPq

φt=Πl

v* Vt=(δ+λ)Cm

1 vt Vt= ωPu

200

Figure g 2.6 Phase Diagram g in Generalized Model

Panel A. Demarcation Loci Πt

Πt

1

1

. Vt=0 Locus: Vt=β(s*(Πt))

. Πt=0 Locus: Πt=G(Vt/(δ+λ))

Vt

0

Vt

0

Panel B. Phase Diagram Πt 1

. Vt=0 Locus Qh(Vh, Πh) •

Qm(Vm, Πm) Ql(Vl, Πl) 0

. Πt=0 Locus Saddle Point

• Unstable Point

• Saddle Point Vt

201

Figure g 2.7 Equilibrium q Paths in Generalized Model

Panel A. Equilibrium Paths Πt  1

Πt 1 • Q (V , Π ) h h h a

πo

•

•

•

πp c

b Ql(Vl, Πl)

Optimistic Path

0

Vt

d

•

Pessimistic Path •

0

Vt

Panel B. Overlap and Reputation Trap Πt 1 Qh(Vh, Πh) • πp Overlap

• πo

• Q (V , Π ) l l l

0

Vt Reputation Trap

202

Figure g 2.8 Development p of US Racial Disparity p y

Πt 1 •

White

πp Π(to) for White • πo Π(to) for Black 

• 0

Black Vt

Figure g 2.9 Male-Female Disparity p y in Patriarchal Society y

203

Panel A. Investment Rate Difference  1 Φt  (Male) Φt  (Female)

Vt

0

Panel B. Reputation Map for Male and Female Groups Πt 1 Πhmale

•

Male

(a) Male Group

•

Π(to) Πlmale

•

0 Πt

(b) Female Group

Vt

1

Πhfemale

• •

Π(to) Πlfemale 0

•

Female l Vt

204

Figure g 2.10 Strategy gy of Monopolistic p Principals p

Panel A. Adjustment of Reputation Threshold Πt 1

Πt 1

B

• Qh

•

Π*

Ql •

0

• Loss Area (Πloss)

Π* Π*’ Π Ql •

B: Trapped in Ql 1 vt v* (Group B in Reputation Trap)

• Qh

0

*c *b

*a 1 vt v* (Adjustment of Π* to Π*’)

Panel B. Training Subsidy Πt 1

Πt 1

B

• Qh

•

Π*

b

πo Ql •

Ql •

B: Trapped in Ql

0

v*

1 vt

(Group B in Reputation Trap)

c

•

Π*

0

a *

• Qh

*

* Subsidy Area

1 vt v*’ v* (Adjustment of v* to v*’)

205

Figure g 2.10 Strategy gy of Monopolistic p Principals p ((Continued))

Panel C. Improvement of Screening Process p g Πt 1

. Vt=0 Locus

Qh

Πh Π*

•

Πl 0

Ql •

Ql’

ωPq

ωPq’

•

•

. Πt=0 Locus

*b

*a (δ+λ)Cm

ωPu

(Improvement of Screening from Pq to Pq’)

Vt

206

Figure g 2.11 Egalitarian g Policies

Panel A. Colorblind Hiring

ΠtW 1

Group W

ΠtB W

Πw*

Group B 1 B

• Qh

• Qh c *

•

Π Π*

b*

•

Π* Unstable

a Ql • * ΠB* 1 vt v* 0 (Threshold Adjustment of Π* to Π (Threshold Adjustment of Π to ΠB*))

Ql • 1 vt v* 0 (Threshold Adjustment of Π* to Π (Threshold Adjustment of Π to Πw*))

Panel B. Strict Quota System

β’non

β’’non

βinv

Depending  Depending on Belief

Improving

GR*

Patronizing

GR**

Πh

GR

207

Figure g 2.11 Egalitarian g Policies ((Continued))

Panel C. Asymmetric Training Cost Group W

ΠtW 1

ΠtB W

1 B

•Qh

•

Π*

Group B

b

πo Ql • 0

Ql • ( ) v* v*(W) (Cost Increase for Group W)

1 vt

c

•

Π*

0

a

•Qh

*

* Training Subsidy

*

v*(B) ( )

v*

(Cost Decrease for Group B)

1 vt

208

Appendix pp Figure g 2.1 Market Learning g Process

Panel A. Market Learning in General Model

.

Πt

Vt=0 0 Locus: Shift more with higher η Locus: Shift more with higher η

1 β(s*(Πt))

. Πt=0 Locus

•

πp •

πo

• 0

Vt

Panel B. Market Learning in Simple Model . Vt=0 Locus: Shift more with higher η =0 Locus: Shift more with higher η Πt 1 • πp Π*

. Πt=0 Locus

•

πo • 0

wPq

(δ+λ)Cm

wPu

Vt

209

Appendix pp Figure g 2.2 Proof of Lemma 1

Panel A  Phase Path Passing a Πt 1

Ⅱ

Ⅰ • Qh b(v*,Πb) a(va,Π*)

c(vc,Π*)

Π*

a’(va’,Π*) d(v*,Πd) Ql •

Ⅲ

Ⅳ v*

0

1 vt

Panel B  Compare va and va’ va va’(vc) va(vc)

v*

v*

vc

210

Figure g 3.1 Costs Distribution and Newborns’ Decision

Panel A. Identity and Skill Decision of Type i Newborns (given v‐i>vi) k (i,q)

(i,u)

Yiq (‐i,q) Yi

u

(‐i,u)

0

c

R‐ii

Ri

Panel B. Costs Distribution under Assumptions 2 and 3 k kh w(1 − Pq )

δ +λ kl

w(1 − Pu ) δ +λ

Potential Switchers

0

cl

wPq

δ +λ

cm

wPu δ +λ

ch

c

211

Figure g 3.2 Dynamics y with No Switches between Types yp

Panel A. Equilibrium Paths in (Rnt, Πnt) coordinates Πnt 1 Qnh

Πh

•

Π*

• Qnl

Πl

• cl

0

wPq

cm

δ +λ

ch

wPu δ +λ

Rnt

Panel B. Equilibrium Paths in (vnt, Πnt) coordinates Πnt

1 φnt=Πh

• Qnh (1, Πh)

πp

Overlap

•

Π* πo

Qnl (0, Πl) • 0

φnt=Πl

v*

1

vnt

212

Figure g 3.3 Dynamics y of Group p i with Inflows from Group pj

Panel A. Equilibrium Paths in (Rit, Πit) coordinates Πit 1 Q’h

Πh

•

Π* Q’l Πl 0

cl

•

wPq

cm

ch

wPu δ +λ

δ +λ

Rit

Panel B. Equilibrium Paths in (vit, Πit) coordinates Πit

1 φt=Πh

• Q’h(1, H’)

πp’ Π*

πo’• Q’l(0, L’) 0

φt=Π Πl v*

1

vit

213

Figure g 3.4 Dynamics y of Group p j with Outflows to Group pi

Panel A. Equilibrium Paths in (Rjt, Πjt) coordinates Πjt 1 Πh

Π* πo Πl

Ql’’ cl

0

•

wPq

cm

δ +λ

ch

wPu δ +λ

Panel B. Equilibrium Paths in (vjt, Πjt) coordinates Πjt

1 φt=Πh

Π* πo

Ql’’(0,L’’) ’’(0 L’’) • 0

φt=Πl v*

1

vjt

Rjt

Figure g 3.5 Group p i Equilibrium q Path with Inflows from Group pj

Panel A. Equilibrium Path with πo’’ > Πl No Inflows Yet Inflows from Group j

Πt 1

φt=Πh

• Q’h

Π* ^i Π X

•

πo’’i Qnl •

φt=Πl

0

v*

^vi x

1 vt

Panel B. Equilibrium Path with πo’’ < Πl No Inflows Yet Inflows from Group j

Πt 1

φt=Πh

• Q’h

Π* ^i Π X

• φt=Πl •

πo’’i 0

v*

^vi X

1 vt

214

Figure g 3.6 State Evolution From Initial State

((Πb0, 0

Πa0)

Panel A. Under Constraints that Only Type‐B Newborns Switch Πa0 1 (L’’b, H’a) (Πh, Πh)

“Type B to A”  Switching Area πo ^a Π X

πo’’a πo’a (Πl, Πl)

(Πh, Πl) 1 Πb0

πo

Panel B. Under Constraints that Only Type‐A Newborns Switch Πa0 1

(Πh, Πl)

(Πh, Πh)

πo “Type A to B”  Switching Area (H’b, L’’a)

(Πl, Πl) πo’b πo’’b

^b Π X

πo

1 Πb0

215

216

Figure g 3.7 Endogenous g Group p Formation

Panel A. Skill Divergence with Equal Group Sizes (Given La=Lb) Πa0 1

( ’’b, H’’a) (L’’ B

(Πh, Πh)

Overlap πo ~ a(=πo’’ ) Π a

A

C

C’ (Πl, Πl)

B' (H’b, LL’’a) (H

~ b(=πo’’ ) πo Π b

1 Πb0

Panel B. Skill Divergence with Unequal Group Sizes (Given La
D

~ a(=πo’’ ) Π a (Πl, Πl)

~ b(=πo’’ ) πo Π b

(H’b, L’’a) 1 Πb0

Figure g 3.8 Autonomous Emergence g of Elite Group p

Panel A. Small Cultural Group's Growing as Elite Group (La << Lb) Πa0 1 (L’’b, H’a) (Πh, Πh)

πo

πo’’a ~ a=0 Π

(H’b, L’’a)

Unstable  (Πl, Π Πl) πo’’b πo

1 Πb0

Panel B. Endogenous Creation of Elite Group (Given La≈ 0) Πa0 1

(L’’b, 1)  Ma∞≈ Πl’Lb (Πh, Πh)

πo

~ a=0 Π

Unstable  (Πl, Πl)  Ma0≈ 0 ≈0 πo

(Πh, L’’a) 1 Πb0

217