Copyright by Xue Li 2020 the Dissertation Committee for Xue Li Certiﬁes That This Is the Approved Version of the Following Dissertation

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Copyright by Xue Li 2020 The Dissertation Committee for Xue Li certiﬁes that this is the approved version of the following dissertation:

Three Essays in Microeconomics

Committee:

Venkataraman Bhaskar, Co-Supervisor

Caroline D. Thomas, Co-Supervisor

Thomas E. Wiseman

Kishore Gawande Three Essays in Microeconomics

Xue Li

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulﬁllment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN May 2020 Dedicated to my parents and friends Acknowledgments

I am deeply grateful to my advisor, Dr. V. Bhaskar and Dr. Caroline Thomas, for their advice and guidance during my graduate study. They have supported me not only in economic research but also in career development. I would also like to express my appreciation to Dr. Thomas Wiseman and Dr. Kishore Gawande for all their valuable comments and suggestions on this dissertation. My gratitude also goes to all other members of UT Austin Theory fac- ulties: Dr. Svetlana Boyarchenko, Dr. Mark Feldman, Dr. John Hatﬁeld, Dr. Vasiliki Skreta, Dr. Dale Stahl, Dr. Maxwell B. Stinchcombe, and Dr. Rodrigo Velez. I wish to give special thanks to my classmates and friends Sangwoo Choi, Jessica Fears, Vasudha Jain, Shaofei Jiang, Anthony Maxam, Yiman Sun, and Mark Whitmeyer for the valuable feedback they provided. I would like to thank my supportive friends Nir Eilam, Ziwei He, Erica Xuewei Jiang, Cao Jin, Yaran Jin, Lisa Yao Liu, Andres Mendez Ruiz, Yong- hang Wang, Shenshen Yang, Chan Yu, and Jiangang Zeng for their help in the last ﬁve years. Finally, I am indebted to my parents for their love, understanding, and support throughout my life.

v Three Essays in Microeconomics

Xue Li, Ph.D. The University of Texas at Austin, 2020

Co-Supervisors: Venkataraman Bhaskar Caroline D. Thomas

This dissertation consists of two chapters in investigating firms’ pricing strategies in different information settings, and one more empirical chapter exploring the effect of low-skilled immigrants on college-educated women. In the first chapter, I investigate the strategic pricing of firms and the innovation adoption processes of consumers in a forward-looking social learning environment. The equilibrium dynamics will depend on both the market structure and the arrival rate of exogenous signals. I show that in perfect good news, with a low arrival rate of exogenous signals, the monopoly firm will encourage learning to end up with a ”free samples” period. In contrast, with a high arrival rate of exogenous signals, the monopoly firm does not sell until there is a good signal. In duopoly under perfect good news, for a low arrival rate of exogenous signals, the market structure will make both firms use ”free samples” strategy in the early stage to compete for the arrival of a

vi breakthrough. On the other hand, for a high arrival rate of exogenous signals, the firms will wait for the good news and only sell after the realization of the state. The concave adoption curves generated in perfect good news are consistent with those in marketing literature. In the second chapter, I investigate the prices and consumer switching when switching costs are consumers’ private information, but firms can observe consumers’ purchase histories. There are two types of equilibrium: poaching equilibrium and non-poaching equilibrium. In a poaching equilibrium, firms will first offer different prices to separate two types of consumers by purchase history and then exploit the two types separately. In a non-poaching equilibrium, firms will keep all their current consumers and charge them low- switching-cost consumers’ prices until the end of the game. Compared with myopic consumers, forward-looking high-switching-cost consumers will consider future high prices when they choose to stay and thus require better prices. Firms will require a more substantial proportion of high-switching-cost consumers to sustain a poaching equilibrium, which improves the efficiency of equilibrium. When consumers’ switching costs can randomly change from period to period, firms will be less willing to pursue poaching equilibrium. In the third chapter, I investigate how the inflow of low-skilled immigrants affects the household decisions: the divorce decision, the fertility decision, and who marries whom. I firstly specify an equilibrium model of matching, fertility decision, and divorce decision over the life cycle. Compared the equilibrium outcomes with and without low-skilled immigrants, I find that with

vii low-skilled immigrants providing household production: (1) marriage market is more positive assortative; (2) couples both with high-education are more likely to have children; (3) couples both with high-education are less likely to get divorced. In the empirical part, I construct an instrumental variable by exploiting the variation in the immigrants’ country of origin. I find that the inflow of low-skilled immigrants does not have a significant effect on women’s marriage rate, fertility rate, and working decision in general. Still, it increases college-educated women’s marriage rate, fertility rate, and labor participation and, at the same time, decreases college-educated women’s divorce rate.

viii Table of Contents

Acknowledgmentsv

Abstract vi

List of Tables xii

List of Figures xiii

Chapter 1. Strategic Pricing and Forward-Looking Social Learn- ing1 1.1 Introduction...... 1 1.2 Model...... 8 1.2.1 Players and Payoﬀs...... 8 1.2.2 Learning...... 9 1.2.3 Equilibrium...... 11 1.3 Equilibrium Analysis...... 14 1.3.1 Perfect Good News Scenario...... 14 1.4 Extensions...... 24 1.4.1 Duopoly in Perfect Good News Scenario...... 24 1.4.1.1 Duopoly Pricing Model...... 25 1.4.1.2 Duopoly Equilibrium Analysis...... 28 1.5 Conclusion...... 34

Chapter 2. Strategic Pricing and Switching in Subscription Mar- kets 36 2.1 Introduction...... 36 2.2 Model...... 41 2.2.1 Firms...... 41 2.2.2 Consumers...... 42

ix 2.2.3 Equilibrium...... 43 2.3 Equilibrium Analysis...... 45 2.3.1 Myopic Consumers...... 45 2.3.2 Forward-looking Consumers...... 49 2.3.3 Equilibrium Eﬃciency...... 52 2.3.4 Changing Switching Costs...... 55 2.4 Conclusion...... 58

Chapter 3. The Impact of Low-skilled Immigration onCollege- educated Women 60 3.1 Introduction...... 60 3.2 Model...... 64 3.2.1 Divorce Decision Stage...... 66 3.2.2 Fertility Stage...... 68 3.2.3 Matching Stage...... 69 3.3 Model Predictions...... 71 3.3.1 Without Low-income Immigrants...... 72 3.3.2 With Low-income Immigrants...... 75 3.4 Reduced-form Evidence...... 77 3.4.1 Constructing Instruments...... 77 3.4.2 First Stage...... 78 3.4.3 General Eﬀects on Women...... 79 3.4.4 Eﬀects on College-Educated Women...... 82 3.5 Conclusion...... 85

Appendices 87

Appendix A. Appendix for Chapter 1 88 A.1 Proof of Lemma 1...... 88 A.2 Proof of Lemma 2...... 91 A.3 Proof of Lemma 3...... 92

x Appendix B. Appendix for Chapter 2 97 B.1 Proof of Proposition 1...... 97 B.2 Proof of Proposition 2...... 99 B.3 Proof of Proposition 6...... 101

Appendix C. Appendix for Chapter 3 104 C.1 Calculation of the Model...... 104 C.1.1 T=2...... 104 C.1.2 T=1...... 107 C.1.3 T=0...... 109 C.2 Equilibrium Without Low-skilled Immigrants...... 110 C.3 Equilibrium With Low-skilled Immigrants...... 113

Bibliography 120

xi List of Tables

2.1 Equilibrium Path with Myopic Consumers...... 47 2.2 Equilibrium Path with Forward-looking Consumers.... 51

3.1 The Eﬀect of Predicted Actual Low-skilled Immigration Flows on Women...... 80 3.2 The Eﬀect of Predicted Actual Low-skilled Immigrants on College-educated Women...... 83

C.1 List of Cities Included in the Sample...... 116 C.2 Predicted Low-skilled Immigration In Selected Cities... 117

xii List of Figures

1.1 Belief, Price and Adoption Curve under PGN With Large ε . 17 1.2 Belief and Price under PGN Without Signal...... 18 1.3 Adoption Curves under PGN With Small ε ...... 22 1.4 Beliefs and Price in Duopoly under PGN With Small ε .... 32

2.1 Comparison of Myopic and Forward-looking Consumers.... 53

3.1 The life cycle of individuals...... 65

C.1 Divorce Rate of Couples Without Low-skilled Immigrants... 112 C.2 Fertility Rate of Couples Without Low-skilled Immigrants.. 112 C.3 Expected Utility of Men Without Low-skilled Immigrants... 112 C.4 Expected Utility of Women Without Low-skilled Immigrants. 113 C.5 Divorce Rate of Couples With Low-skilled Immigrants.... 115 C.6 Fertility Rate of Couples With Low-skilled Immigrants.... 115 C.7 Expected Utility of Men With Low-skilled Immigrants.... 115 C.8 Expected Utility of Women With Low-skilled Immigrants... 116 C.9 Predicted Low-skilled Immigrants...... 118 C.10 The Relationship of Predicted and Actual Numbers of Low- skilled Immigrants...... 119

xiii Chapter 1

Strategic Pricing and Forward-Looking Social Learning

1.1 Introduction

When an innovation is released to the market, it is often the case that the firm is not clear about the quality of the product. The term quality here is general and may include the characteristics of the product and how these characteristics fit the consumers’ preferences. The firm needs to update its information about the quality through consumers’ experimentation. With social media websites such as Yelp, Amazon, and Facebook, both firms and consumers can observe public signals about the quality generated by adopters. The firm needs to decide the amount of consumers’ experimentation by designing an optimal pricing strategy. And with this strategy, forward-looking consumers will choose either to adopt now or to wait for more information. This paper builds a model investigating the strategic pricing of firms and the innovation adoption processes of consumers in a forward-looking social learning environment. The major contribution is the detailed analysis of the firms’ information incentive for different arrival rates of exogenous signals and in different market structures. The first implication of the analysis is that the monopoly firm has different incentives in conducting market learning for dif-

1 ferent arrival rates of exogenous signals. The firm will have a strong incentive to accelerate the learning process when the arrival rate of exogenous signals is low. The second implication is that both the information incentive and the market structure will affect firms’ pricing strategy: the duopoly firms will need to consider the competition pressure as well as the information incentive in pricing. The central ingredient in this model is the firm’s information incentive. The firm will decide how much learning to induce at every point. The equilibrium learning process needs to balance the following trade-off: if too many consumers adopt immediately, the firm will have a higher probability of getting the news, but the population pool is shrinking and the posterior without news at next period may be significantly affected; if too few consumers adopt now, the news will arrive more slowly, but the remaining population pool is still large, and the no-news posterior is not affected much. This trade-off will depend on the arrival rate of exogenous signals and will cause differences in adoption processes.

Summary of Model and Results: Section 2 presents the model of a monopoly ﬁrm and forward-looking social learning consumers. The ﬁrm starts the game by introducing an innovation of uncertain quality to the market. The consumers here are identical with the same preference, discount rate, and prior of belief about the product. The consumers receive stochastic opportunities to adopt the product and stochastic chances to generate public signals. Given the

2 adoption opportunity, each consumer chooses between adopting immediately and waiting. Both the firm and consumers are learning through public signals, which are from both exogenous sources and endogenous consumers’ reviewing sources. The news environment here is based on the ”exponential bandit” framework. And I discuss the strategic pricing and innovation adoption in perfect good news for high and low arrival rates of exogenous signals. Some products such as beauty treatments and movies can be used as natural examples of learning via good news: a recognized award to a movie will make all the consumers confident about the quality. Meanwhile, some other innovations, such as cellphones and new energy automobiles, are learned by the market via bad news. The explosion of cell phones will cause all the consumers to stop adopting. Section 3 analyzes the Markov perfect equilibrium monopoly pricing and adoption behavior in perfect good news models. This section highlights the effect of the different arrival rates of exogenous signals on strategic pricing and adoption processes. Under perfect good news, for high arrival rates of exogenous signals, the monopoly firm will start with waiting for the good news and no selling. The firm will only begin selling after the realization of the signals. Thus, there is a waiting period for the firm and consumers. In contrast, for low arrival rates of exogenous signals, the monopoly firm will encourage consumers’ experiments from the beginning and stop selling after the price is below the marginal cost. The firm would like more consumers to learn to get a breakthrough as early as

3 possible. Even when the price is below the marginal cost, the firm is willing to bear temporary losses in the hope of good news. The information incentive is so strong that the firm will make a ”free samples” period in the end. The ”free samples” period appears at the end of the sale because of the assumption of limited adoption opportunities every time. The information incentive makes the monopoly firm behave like a social planner and achieve social efficiency in perfect good news case.

Section 4 explores the duopoly competition in the good news environment and illustrates how the market structure affects pricing and adoption. When two firms are introducing ex-ante identical products, for high arrival rates of the exogenous signals, the firms will wait for the good news without selling. The only firm that gets the signal will immediately start selling, and then the other starts selling only after it also receives the signal. For low arrival rates of signals, the firms will start with a ”free samples” period, in which they set the same prices below the marginal cost. Then the lucky firm which gets a breakthrough will dominate the market and make a positive profit until both firms get good news. They then go back to Bertrand competition with a price set at the marginal cost level. The competition pressure from the market structure leads to the ”free samples” period in the beginning. Both the competition effect and the information incentive ensure the social efficiency of the MPE in this duopoly scenario.

4 Related Literature: My approach is based on the model of forward-looking social learning innovation adoption in Mira Frick and Yuhta Ishii(2016), where I add the monopoly firm or duopoly firms into the model. Tuomas Laiho and Julia Salmi(2017) also attempt to investigate the monopoly pricing in the forward-looking social learning environment but with a different setup. My duopoly competition part shares some elements with Dirk Bergemann and Juuso Välimäki(1996) about the assumption of firms. The forward-looking social learning environment here generates some interesting conclusions different from theirs. Frick and Ishii(2016)’s study focuses on the innovation adoption behavior of forward-looking social learners, motivated by the rise of social media. They derive two kinds of adoptions curves that are common in the marketing literature: the S-shaped and concave curves. They attribute the different shapes of adoption curves to the differences in informational environments. In perfect good news models, the consumers will adopt at an early stage and stop at a cut-off belief about the product. Thus, there are concave adoption curves in perfect good news case. In the perfect bad news model, the consumers will choose to partially adopt the innovation in the early stage and fully adopt after the belief has reached a cut-off point. Consumers’ behaviors here will lead to S-shaped adoption curves and a saturation effect. In their paper, the difference in the two kinds of adoption curves is mainly from the forward-looking social learning behavior of consumers. In my model, I will add the supply side, such as a monopoly firm and duopoly firms. My finding is that the shapes of

5 adoption curves will be the same as in their paper, but the adoption process will be controlled by the supply side instead of the demand side. Laiho and Salmi(2017) attempt to incorporate the monopoly firm into a forward-looking social learning model. In their setting, a constant rate of consumers arrives in the market every time. The consumers stay in the pool and can decide whenever to buy as they want. The static monopoly pricing strategy is proved as MPE in this setting as in Joel Sobel(1991). There is no exogenous signal in their model. Laiho and Salmi focus on the question of whether the monopoly will sell to the low valuation type to speed up learning or the monopoly only sells to high valuation type to extract the entire surplus. However, I will use the model setup of a fixed pool of consumers and the Poisson process of opportunities to adopt and to signal, in addition to exogenous signals, as in Frick and Ishii’s paper. The question I would like to address is the adoption process of forward-looking social learning consumers under monopoly pricing and duopoly pricing. Bergemann and Välimäki(1996) consider the duopoly competition with market learning and quality uncertainty. They find that the price competition between the sellers will allocate the costs and benefits of learning intertemporally between the buyer and sellers. All the MPEs in their model are efficient. One of them, the cautious equilibrium, has a simple pricing rule. However, their model has only one buyer making purchasing decisions in all discrete periods and the products’ quality random drawing from parameter- ized distributions. I will apply their assumptions about the supply side to

6 the forward-looking social learning environment. I will then investigate the pricing behavior of the firms, the adoption behavior of consumers, and social efficiency. Many papers have studied the dynamic pricing in many myopic learning environments. For monopoly pricing, Kenneth L Judd and Michael H Riordan(1994) investigate the monopoly firm using price as a signal in the consumer learning process. Edward E Schlee(1996) studies the value of public information about quality in a one-period monopoly model. His later paper (Edward E Schlee(2001)) examines the monopoly firm and consumers learning in a two-period learning environment and derives sufficient conditions for introductory pricing. Alessandro Bonatti(2011) explores the dynamic menu pricing of monopoly firms when it releases a new product of uncertain quality to myopic consumers. For duopoly price competition, Dirk Bergemann, Juuso Valimaki et al.(1996) consider a new entrant into the duopoly market. The firm then needs to induce the myopic consumers to experiment and free-ride on the market information externalities. Leonardo Felli and Christopher Harris (1996) solve a problem similar to Bergemann and Välimäki(1996)’s but in the context of wage determination, where human capital acts as the information in the learning models there. Bergemann and Valimaki (1997, 2000 and 2002) all investigate duopoly pricing with myopic consumers in Patrick Bolton and Christopher Harris(1999)’s Brownian bandits learning environment.

7 1.2 Model 1.2.1 Players and Payoﬀs

There are two sets of players in the game: the monopoly firm and the forward-looking social learning consumers. The monopoly firm releases a product of uncertain quality and attempts to maximize the total expected discounted profit. Time t ∈ [0, ∞) is continuous. At t = 0, the firm has a product of unlimited supply but is unsure about the quality θ ∈ {G = 1,B = −1}. Suppose the quality of the product is independent of the cost. The fixed cost is normalized to 0 here. The firm can acquire public information from experimentation done by the consumers and agencies. Based on the information, the firm’s strategy is to post a price pt ∈ R at every period. The firm faces a marginal cost of c when selling every product and discounts future revenue with a discount rate r > 0. The firm is maximizing the expected total discounted revenues across all periods: Z ∞ −rt max e Nt(pt − c)dt (1.1) 0 where Nt is the amount of consumers adopting the product at time t. The consumers are ex-ante identical and maximizing utility by choosing from adopting and waiting. There is a continuum population of consumers ¯ N0 ∈ R. They are identical with common prior in belief α0 ∈ (0, 1) that θ = G and common discount factor r > 0. At every period t, consumers get stochastic chances to adopt the in-

8 novation. The arrival rate of the exogenous Poisson adoption opportunity is ρ > 0, which is independent among the population. Upon receiving the opportunity, the consumer must make a decision between dt = 1 (adopt) and dt = 0 (wait). By waiting, the consumer receives zero flow payoff but expects to learn more public information from the market. By adopting the product now, the consumer will receive the flow payoff Et[θ] − pt immediately and drops out of the market. Each adopting consumer has λ probability of generating a public signal about the quality of the product.

1.2.2 Learning

The monopoly ﬁrm and consumers observe public signals that come from both exogenous and endogenous sources. They learn through perfect Poisson process such as perfect good news and perfect bad news.

The exogenous signals arrive at a ﬁxed rate εθ. These are the information from the independent agencies or government watchdogs and are not aﬀected by the consumers’ adopting decisions.

The endogenous signals are from the learning of consumers λθNt. Let

Nt denote the population that adopt the innovation at time t. The ﬂow of people doing experimentation at time t will generate signals at rate λ. Thus, the total endogenous social learning is the term λθNt. The more people adopt at time t, the more probable that a signal will arrive. Therefore, public signals are generated according to Poisson process with arrival rate (εθ + λθNt)dt. As in Frick and Ishii’s paper, here I only consider learning via perfect

9 Poisson process, including perfect good news and perfect bad news. In perfect good news model, I have εB = λB = 0 and εG = ε > 0, λG = λ > 0. This means that when there is a piece of news, it must be conclusive that the product is of good quality. In contrast, in perfect bad news model, with

εG = λG = 0 and εB = ε > 0, λB = λ > 0, I only have conclusive news that the product is of bad quality. Whether the learning process is through perfect good news or perfect bad news depends on the nature of the innovation. Products such as movies, music, books, and herbal remedies are through perfect good news. Winning an award can be seen as a piece of good news that encourages consumers to adopt. On the other hand, cellphones and microwave ovens are through perfect bad news. If one cellphone explodes, people will be extremely skeptical about the quality of the cellphone. 1

When there is a piece of good news, the posterior αt will jump to 1 forever. In contrast, with a piece of bad news, the posterior will jump to 0 forever. Without the arrival of signals, the evolution of belief is through Bayesian updating. The no-news posterior is speciﬁed below as: R t − (ε+λNs)ds α0e 0 αt = R t (1.2) − (ε+λNs)ds α0e 0 + (1 − α0)

α˙τ = (ε + λNτ )ατ (1 − ατ ) in perfect good news model. And

α0 αt = R t (1.3) − (ε+λNs)ds α0 + (1 − α0)e 0

1http://www.samsung.com/us/note7recall/

10 α˙τ = −(ε + λNτ )ατ (1 − ατ ) in perfect bad news model. As can be seen from the equation, the belief is drifting down in the good news case without news, and it is drifting up in the bad news case in the absence of signals.

1.2.3 Equilibrium

In this section, I will deﬁne the Markov Perfect Equilibrium and derive the equilibrium conditions for the monopoly ﬁrm and consumers.

∞ The monopoly ﬁrm chooses the pricing path {pt}t=0 to maximize the total discounted expected proﬁt. The pricing strategy at time t should be ¯ Markov in states (αt, Nt):

pt : [0, 1] × R+ → R+,

¯ where Nt is the population of consumers remaining in the pool of potential ¯ ¯ R t adopters, expressed as Nt := N0 − 0 Nsds. ∞ The pricing path {pt}t=0 will be the equilibrium strategy if it can maximize the total discounted expected proﬁt:

Z ∞ ∞ −rt {pt}t=0 ∈ argmax e Nt(pt − c)dt. (1.4) 0

Obviously, the firm will take different strategies in the two news cases after the news has arrived. In the bad news case, the firm will exit because no one will buy the product with bad news, which means αt = 0. In the

11 good news case, the ﬁrm will be able to act as a monopoly producer without product quality uncertainty. The ﬁrm will charge the static monopoly price, which is 1 when αt = 1, to extract the entire consumer surplus. These two are trivial cases. What I will focus on is the situation when there has not any news arrived.

For the consumers, I consider the aggregate ﬂow of adopters Nt at time t as the consumers’ strategy.

For each individual consumer, she chooses between dt = 1 (adopt) and dt = 0 (wait) at every time period t. The instant utility of adopting now is

2αt − 1 − pt, while the instant utility of waiting is 0.

N I deﬁne the value of waiting to be Wt (σ). Suppose σ is the strategy that the consumer will follow in the future and τ σ is the random time that the consumer will adopt the innovation in the absence of news. In the good news case, the consumer will only choose to adopt if there is

σ a piece of good news before τ . Let τs be the random time that the consumer gets the opportunity to adopt after the news has arrived at time s. Then

N Wt (σ) is:

σ R τ σ N − t (ε+λNs)ds −r(τ −t) Wt (σ) = E[(αte + (1 − αt))e (2ατ σ − 1 − pτ σ )]

After the news has arrived, the consumer will earn no surplus, because the ﬁrm will take all the surplus. In the bad news model, consumers will only adopt if there has not been any news before τ σ:

σ σ R τ N −r(τ −t) N N − t (ε+λNs)ds Wt (σ) = E[e (αt − (1 − αt )e )(2ατ σ − 1 − pτ σ )]

12 N N The value of waiting should be Wt := supσ∈Σt Wt (σ) for all t. For the whole population, if the value of waiting is greater than or equal to the value of adopting instantly, there will be some consumers choose to wait; if the value of waiting is less than or equal to the value of adopting instantly, there will be some consumers choose to adopt now.

∞ I restrict our attention to Markov strategy of {Nt}t=0. Let the state ¯ space be (αt, Nt, pt): the current belief, the remaining population, and the price. The consumers’ strategy is then:

Nt : [0, 1] × R+ × R+ → R+.

Therefore, the equilibrium conditions are:

 N ¯ Wt ≥ 2αt − 1 − pt ⇒ Nt ≤ ρNt, (1.5) N Wt ≤ 2αt − 1 − pt ⇒ Nt > 0.

∞ ∞ Deﬁnition 1: A Markov Perfect Equilibrium (MPE) is a pair ({pt}t=0, {Nt}t=0) such that:

∞ •{ pt}t=0 maximizes the monopoly ﬁrm’s total discounted expected proﬁt ¯ ∞ given the state (αt, Nt) and the consumers’ strategy {Nt}t=0;

∞ ¯ •{ Nt}t=0 maximizes consumers’ expected payoﬀ given (αt, Nt, pt) and

∞ {pt}t=0.

13 To be more explicit, combine (4) and (5), the Markov perfect equilibrium conditions are:  N ¯ ¯ W ≥ 2αt − 1 − pt ⇒ Nt(αt, Nt, pt) ≤ ρNt,  t   N ¯ Wt ≤ 2αt − 1 − pt ⇒ Nt(αt, Nt, pt) > 0, (1.6)  Z ∞  ¯ ∞ −rt  {pt(αt, Nt)}t=0 ∈ argmax e Nt(pt − c)dt. 0 The monopoly firm will not have a commitment problem here because the equilibrium strategy is time-invariant. The firm will always maximize the expected profit from t on, which is consistent with maximizing all the profit at t = 0. However, in Laiho & Salmi (2017), the firm will have a time inconsistency problem.

1.3 Equilibrium Analysis

I will only discuss the perfect good news scenario for now. The perfect bad news scenario is worth discussing, but I have not generated enough results for discussion.

1.3.1 Perfect Good News Scenario

In this section, I will study the Markov perfect equilibria in perfect good news scenario. There exist different equilibria for high and low arrival rates of exogenous signals. I will first find out the equilibrium strategy of players and make some inferences. For low arrival rates of exogenous signals, the equilibrium strategy has an optimal stopping time points. I will then investigate the factors that affect the stopping time of the monopoly firm and

14 how they work. Moreover, I will investigate whether the MPEs are socially efficient. In perfect good news model, the firm will charge the static monopoly prices pt = 1 to extract the entire consumer surplus, after the conclusive good news has arrived. However, before the arrival of the news, the firm may need to encourage consumers to experiment. I only focus on the case that there has not been a breakthrough yet.

Proposition 1: (MPEs under Perfect Good News) For given parameters r, ¯ ρ, N0, and α0: (1) With small ε and large λ, there exists an MPE speciﬁed as below:

∗ ∗ • The monopoly ﬁrm sets pt = 2αt − 1 for t ∈ [0, t ) and pt∗ for t ∈ [t , ∞) when there is no breakthrough. After a breakthrough, the ﬁrm will

charge pt = 1 forever;

• For the consumers’ aggregate strategy Nt,  ¯ ρNt, if pt ≤ 2αt − 1; Nt = (1.7)  0, if pt > 2αt − 1.

(2) With large ε and small λ, there exists an MPE speciﬁed as below:

• The monopoly ﬁrm will set pt = 1;

• The consumer will use the same strategy as above.

15 ∞ Proof: (1) Given the monopoly ﬁrm’s pricing path {pt}t=0, the consumers’

N waiting value will be zero Wt = 0 at any t. The utility of immediate adop-

∗ tion will also be zero for t ∈ [0, t ) and after a breakthrough, 2αt − 1 − pt = 0. Thus, the consumers are indifferent between adopting now or waiting when given an opportunity for t ∈ [0, t∗) and after a breakthrough. Therefore, all consumers choose to adopt as long as pt ≤ 2αt − 1 is a best response for the monopoly firm’s strategy. (2) Given the consumers’ strategy, the firm will only need to make a decision between pt = 2αt − 1 to attract all feasible adopters to adopt or pt > 2αt − 1 to stop selling at t.

By Lemma 1 in appendix, for small ε and large λ, the ﬁrm will set pt = 2αt −1 in the beginning. As the belief drifts down in the absence of signals, the price

∗ pt = 2αt −1 will approach the marginal cost c. At some time t ( pt at or below the marginal cost), the firm will stop selling by setting pt > 2αt − 1.After the arrival of good news, the belief will jump to 1 forever, αt = 1. The firm will use the monopoly power to extract the whole surplus by pricing at 1, pt = 1. In contrast, for large ε and small λ, the firm will be waiting for the signal in the beginning and start selling after the arrival of good news. The firm will be able to sell at pt = 1 as a monopoly.

The proposition states that in equilibrium, the monopoly firm will con- trol the learning process instead of the consumers. The firm will set the price pt = 2αt − 1 to make consumers indifferent about the adoption time. When

16 the arrival rate of the exogenous signals, ε, is high, the firm will wait for the signal to start selling. When the arrival rate of the exogenous signals, ε, is low, the firm will encourage learning in the beginning because more learning will bring a higher possibility of a breakthrough. For large ε, the starting time of selling is the arrival time of the good news, and the selling will not stop. In Figure 1, the firm will not sell before

g g t . When the good news arrives at t , the ﬁrm starts selling at pt = 1 and all consumers choose to adopt from that time on.

Figure 1.1. Belief, Price and Adoption Curve under PGN With Large ε

The more interesting case is when ε is small, the exact stopping time that the ﬁrm decides to cease selling to consumers. I will investigate the stopping time in the following paragraphs.

From Figure 2, the time paths of both the belief and the price under perfect good news scenario with small ε can be seen. Without signals, the belief αt will drift down with time. The price pt = 2αt − 1 will ﬁrstly go down

17 Figure 1.2. Belief and Price under PGN Without Signal

with αt and then stay there from some time on. In the ﬁgure, the stopping time t∗ will not be the same as the time that pt = 2αt − 1 hits the marginal cost c, because the monopoly ﬁrm has the incentive to have more people experimenting in the hope of the good news.

∗ Denote the time that pt hits c as t0. For t ∈ [t0, t ], the ﬁrm is bearing some temporary losses for the arrival of a breakthrough. However, after time t∗, the ﬁrm will stay at pt∗ because the cost of inducing the experiment is too high. The consumers will make adoptions only with a signal after t∗, because

∗ pt > 2αt − 1 for t > t . The exogenous signals are the only possible source of breakthroughs. For the monopoly ﬁrm at time t = 0 to decide at which time t∗ to stop

18 selling, the total revenue from the innovation is: Z t∗ −rt −(1−ρ)t ¯ R = R1 + R2 = e (2αt − 1 − c) ρe N0dt 0 Z ∞ Z ∞ −(s−t∗)ε −rt −(1−ρ)(t−s) ¯ + αt∗ e e (1 − c)ρe Nt∗ dt ds t∗ s (1.8)

∗ ∗ R t −rt Before t , the ﬁrm has earned R1 = 0 e (2αt − 1 − c) Ntdt, where −(1−ρ)t ¯ ∗ Nt = ρe N0. After t , the expected total proﬁt is represented by R2. The arrival of breakthrough will only be possible through exogenous signals. Suppose the breakthrough happens at time s, the expected total rev-

R ∞ −rt −(1−ρ)(t−s) ¯ ∗ enue after s will be s e (1 − c) Nt dt, where Nt = ρe Nt . −(s−t∗)ε The probability of a breakthrough at time s is αt∗ e . Thus, R2 =

R ∞ −(s−t∗)ε R ∞ −rt −(1−ρ)(t−s) ∗ ¯ ∗ t∗ αt e s e (1 − c)ρe Nt dt ds. The optimal t∗ will maximize the total revenue: Z ∞ ∗ −rt∗ −(1−ρ)t∗ ¯ −rt −(1−ρ)(t−t∗) −(1−ρ)t∗ FOC [t ]: e (2αt∗ − 1 − c)ρe N0 − αt∗ e (1 − c)ρe e dt t∗ Z ∞ −(1−ρ)t∗ ¯ −(s−t∗)ε + [−(ε + λρe N0)αt∗ (1 − αt∗ ) + ε + (ρ − 1)]e t∗ Z ∞ −rt −(1−ρ)(t−s) −(1−ρ)t∗ ¯ e (1 − c)ρe e N0 dt ds = 0. s (1.9) By rearranging equation (9), the analytical form solution of t∗ is: (ρ−1−r)t∗ e {(ρ − 1 − r)ρ(2αt∗ − 1 − c) + (1 − c)ραt∗ (1 − c)ρ (1.10) + [ε + ρ − 1 − (ε + λN ∗ )α ∗ (1 − α ∗ )]} = 0. ρ − 1 − r − ε t t t

Denote the time point that the price hits the marginal cost c at t0. The condition t0 should satisfy is that 2α0 − 1 = c. I solve this and get: λN¯ ρ λN¯ ρ (1 + c)(1 − α ) 0 (ρ−1)t0 0 0 εt0 + e = − ln . (1.11) ρ − 1 ρ − 1 (1 − c)α0

19 ∗ Supposedly, I should have t > t0 and I call this period ”free samples period.” Intuitively, the monopoly firm is expecting the arrival of good signals from the market. However, the breakthrough has not arrived when the price is approaching the marginal cost. The firm will have the incentive to sell to consumers even when the price is below the marginal cost to encourage market experiments. For some products, such as beauty products, the firm will distribute free samples at the beginning to attract consumers to review the product. However, in the model here, the adoption opportunity is exogenous for consumers. The firm cannot sell to as many consumers as it would like to in the beginning. The monopoly firm then employs free samples strategy near the end of the selling period to encourage experimentation. The below- marginal-cost sale here works the same way as free samples. When deciding to stop at t∗, the firm will be indifferent between stopping at t∗ or t∗ + dt if dt is very small. This is because the value function is continuous, composed of the sum of integrals of polynomials. The value of

∗ R ∞ −(s−t∗)ε R ∞ −rt −(1−ρ)(t−s) ∗ ¯ ∗ stopping at t is:R2 = t∗ αt e s e (1 − c)ρe Nt dt ds. If the firm is to stop at t∗ + dt, the expected profit at t∗ is: Z ∞ 0 −rt −(1−ρ)(t−s) ¯ R2 =(2αt∗ − 1 − c)Nt∗ + αt∗ (ε + λNt∗ ) dt e (1 − c)ρe Nt∗ dt t∗+dt Z ∞ Z ∞ −(s−t∗−dt)ε −rt −(1−ρ)(t−s) ¯ + [1 − αt∗ (ε + λNt∗ ) dt] αt∗+dte e (1 − c)ρe Nt∗ dt ds. t∗+dt s The main trade-off faced by the monopoly firm lies in the first line of

0 0 R2, since the second line integral part of R2 should not be very diﬀerent from

R2. pt∗ = 2αt∗ − 1 is less than the marginal cost c, thus the ﬁrm is bearing

20 losses now. The ﬁrm is doing so in exchange for a hope of the immediate

∗ R ∞ −rt ∗ ∗ arrival of the good news at t + dt, that is αt (ε + λNt ) dt t∗+dt e (1 − −(1−ρ)(t−s) ¯ c)ρe Nt∗ dt. Therefore, for comparative statics:

• Discount rate r: Higher discount rate will lead to greater t∗. Since the firm will care more about future good news profit and less about current losses, the firm will be willing to experiment more. This will cause t∗ to be greater.

• Adoption opportunity arrival rate ρ: Greater ρ will lead to smaller t∗. Greater ρ means more people will adopt at time t∗, which will cause larger current losses. The ﬁrm will then have the incentive to stop selling earlier.

• Exogenous signal arrival rate ε: Greater ε will lead to greater t∗. Greater ε means more chances of getting good news at time t∗ + dt, thus the ﬁrm will be more willing to bear current losses. This will lead to greater t∗.

• Endogenous signal arrival rate λ: Greater λ will lead to greater t∗ by the same logic as above.

¯ ¯ • Initial population N0: Initial population N0 will not inﬂuence the stopping time t∗, since it will be canceled out in the equation.

21 • Marginal cost c: Higher marginal cost c will make t∗ smaller because the current losses of the ﬁrm are greater.

• Initial belief α0: The eﬀect of an initial belief is still not clear in the equation since it inﬂuences both the current losses and the chances of the arrival of good news at t∗ + dt.

Figure 1.3. Adoption Curves under PGN With Small ε

The adoption curves will be the same as in Frick and Ishii (2016) paper. However, the stopping time t∗ here is endogenously set by the monopoly ﬁrm.

Proposition 2: The MPE here is socially eﬃcient.

R ∞ −rt Proof: The monopoly ﬁrm’s value function at time t = 0 is: Vf = 0 e (pt− R ∞ −rt c) Nt dt. And the consumers’ value function at time t = 0 is: Vb = 0 e (2αt−

1 − pt) Nt dt. Thus, the social value at the beginning is: Z ∞ −rt W = Vf + Vb = e (2αt − 1 − c) Nt dt 0

22 The social value does not depend on the price path of the product. It depends

∞ ∞ only on {αt}t=0 and {Nt}t=0. After the arrival of good news, it is obvious that all people should adopt at the first opportunity. Before the arrival of good news, for small ε and large λ, consumers need to experiment as much as possible. Thus, the consumers will choose to adopt at the first opportunity. t∗ is the socially efficient stopping time since the social planner is facing the same trade-off as the monopoly firm. This is because the monopoly firm is pricing at pt = 2αt − 1, which makes the social value the same as the monopoly’s value function at t = 0. For large ε and small λ, even though the consumers would like to adopt as soon as possible, the firm will not sell before the arrival of good news. The firm’s strategy is socially optimal because before the signal, the firm knows it needs to price exactly at pt = 2αt −1 to sell. By maximizing the total expected revenue, the firm is maximizing social welfare.

This proposition states that the monopoly ﬁrm internalizes all the learning costs and beneﬁts, and it acts as a social planner in the adoption process.

∞ The price path {pt}t=0 will induce the eﬃcient learning and adoption processes separately for large ε and small ε.

23 1.4 Extensions 1.4.1 Duopoly in Perfect Good News Scenario

In this section, I will consider duopoly firms with forward-looking consumers in perfect good news scenario. The duopoly competition under learning and uncertainty has been investigated in depth by Bergemann and Valimaki (1996b). In that paper, they discuss a dynamic equilibrium model with products of uncertain quality from two sellers. The sellers need to gain information about the quality from experimentation, which can only be conducted by consumers. They find that the price competition will allocate the costs and benefits of learning between buyers and sellers intertemporally. All MPEs in their model are efficient, and they identify a cautious equilibrium, which has a straightforward pricing rule. The firms will sometimes set prices below the marginal cost to sustain experimentation. However, their paper employs the discrete-time bandit model with the random quality realization from pa- rameterized distributions. There is only a single consumer making purchasing decisions every period in their model. Here, inspired by their idea, I will investigate the situation with a large fixed population of identical consumers and duopoly firms in a continuous-time bandit model. I will first set up the duopoly pricing with a forward-looking innovation adoption model and then discuss the equilibria in the model.

24 1.4.1.1 Duopoly Pricing Model

The basic setup of the monopoly pricing model is not changed. The ¯ timing is still the same, as well as the fixed population N0 of consumers. The difference is that now there are two firms, namely firm 1 and firm 2. Each firm releases a product of uncertain quality at t = 0. Both firms need to get public information about the quality from the market, and they can get information about the competitor’s public signals. They are attempting to maximize their own discounted total expected profit by setting an optimal price path:

Z ∞ −rt i i max e Nt (pt − c)dt, (1.12) 0

i where i = 1, 2 and Nt is the number of consumers adopting the innovation from firm i at time t. The consumers are having the same exogenous Poisson adoption opportunity arrival rate ρ as before. Now each consumer has to make a decision among dt = 0 (wait), dt = 1 (adopt from firm 1) and dt = 2 (adopt from firm 2). The immediate value of waiting is still 0. The flow payoff of adopting now

i i i is Et[θ ] − pt, where θ is the quality of the innovation by ﬁrm i. The learning process is still the same for each product with public signals from both exogenous source at rate εθ and endogenous source at rate

i λθNt . For each innovation, the public signals is Poisson process with arrival

i rate (εθ +λθNt )dt. Here, in the perfect good news scenario, I have εB = λB = 0 and εG = ε > 0, λG = λ > 0. The consumer population is Bayesian updating

25 the belief of quality for each product:

R t i i − (ε+λNs)ds i α0e 0 αt = R t i (1.13) i − (ε+λNs)ds i α0e 0 + (1 − α0) ˙i i i i ατ = (ε + λNτ )ατ (1 − ατ )

i where α0 is the initial belief of quality for each product. For simplicity, I set

1 2 α0 = α0, which means that consumers have the same initial beliefs about both products. After the arrival of good news for ﬁrm i, its belief will jump to 1

i forever, αt = 1. From the evolution of beliefs, I can see that the competing firms are relying on the consumers to conduct experimentation for them. At every pe- ¯ riod, there are only ρNt consumers acquired the opportunity to adopt. If more of them choose to experiment for firm 1, there will be fewer consumers left for firm 2. Thus, firms are competing for consumers not only for immediate revenue but also for the chance of a breakthrough. The competition pressure is what will be different from the monopoly model. For equilibrium concept, I still use the Markov Perfect Equilibrium and characterize the firms’ and consumers’ strategies as below. Firm i’s strategy should be Markov in the remaining population, the be-

i −i −i ¯ liefs of both innovations, and the price set by the competitor (αt, αt , pt , Nt):

pt : [0, 1] × [0, 1] × R+ × R+ → R+,

i ∞ The pricing path {pt}t=0 should maximize ﬁrm i’s total discounted expected proﬁt: Z ∞ i i −i −i ¯ ∞ −rt i i {pt(αt, αt , pt , Nt)}t=0 ∈ argmax e Nt (pt − c) dt. (1.14) 0

26 i For the consumers, I still use the aggregate Nt to specify their strategy.

N The value of waiting Wt is deﬁned in the same way, which represents the supremum of value for all the strategies that can be applied in waiting.

i ∞ The adoption path {Nt }t=0 should be Markov in the beliefs about the quality of both products, the price of the two products, and the remaining

1 2 1 2 ¯ population (αt , αt , pt , pt , Nt):

i Nt : [0, 1] × [0, 1] × R+ × R+ × R+ → R+.

Therefore, the equilibrium conditions are:

 N i i 1 2 ¯  Wt ≥ maxi∈{1,2} (2αt − 1 − pt) ⇒ Nt + Nt ≤ ρNt,

N i i i i j Wt ≤ maxi∈{1,2} (2αt − 1 − pt)& j ∈ argmaxi∈{1,2} (2αt − 1 − pt) ⇒ Nt > 0. (1.15) The conditions state that when the value of waiting is greater than or equal to adopting either of the products, some consumers will choose to wait now. When adopting either of the product can oﬀer more value than waiting, some consumers will choose to adopt the innovation that provides higher util-

j j −j −j ity, that is 2αt − 1 − pt ≥ 2αt − 1 − pt . The deﬁnition of MPE is similar as before:

1 2 ∞ 1 2 ∞ Deﬁnition 2: A Markov Perfect Equilibrium (MPE) is ({pt , pt }t=0, {Nt ,Nt }t=0) such that:

i ∞ •{ pt}t=0 maximizes the ﬁrm i’s total discounted expected proﬁt given the i −i −i ¯ 1 2 ∞ state (αt, αt , pt , Nt) and the consumers’ strategy {Nt ,Nt }t=0;

27 1 2 ∞ 1 2 1 2 ¯ •{ Nt ,Nt }t=0 maximizes consumers’ expected payoﬀ given (αt , αt , pt , pt , Nt)

1 2 ∞ and {pt , pt }t=0.

1.4.1.2 Duopoly Equilibrium Analysis

First, I will discuss the MPEs in the duopoly scenario, and then derive the eﬃciency of the MPEs.

Proposition 3: (Duopoly MPEs under Perfect Good News) For given pa- ¯ rameters r, ρ, N0, and α0: (1) With small ε and large λ, there exists an MPE speciﬁed as below:

1 2 • The duopoly ﬁrms will set the same price pt = pt before either of them

i get a breakthrough. Then firm i with good news will set price pt = −i 2 − 2αt + c when it is the only firm with good news, the other firm will

1 2 not sell in this period. Both ﬁrms will set pt = pt = c after both get good news;

i • For the consumers’ aggregate strategy Nt :  ¯ i i −i −i ρNt, if 2αt − 1 − pt > 2αt − 1 − pt ;  ρN¯ N i = t , if 2αi − 1 − pi = 2α−i − 1 − p−i; (1.16) t 2 t t t t   i i −i −i 0, if 2αt − 1 − pt < 2αt − 1 − pt .

(2) With large ε and small λ, there exists an MPE speciﬁed as below:

• The duopoly ﬁrms will not sell before either of them get a breakthrough.

i −i Then ﬁrm i with good news will set price pt = 2 − 2αt + c when it is

28 the only ﬁrm with good news, the other ﬁrm will not sell in this period.

1 2 Firms will set pt = pt = c after both get good news;

• The consumers will use the same strategy as above.

Proof: Use backward induction: (1) Both ﬁrms have good news: this is the same as Bertrand competition without uncertainty. Firms will set the price

1 2 at marginal cost level pt = pt = c. The consumers will adopt at the first opportunity. (2) Firm i has good news and firm j does not have good news: firm i has

i αt = 1 and tries to attract all the consumers. The consumer will be indiﬀerent

i j j j between adopting either i or j if 1 − pt = 2αt − 1 − pt . The lowest pt that firm j is willing to set is the marginal cost c, because even with good news, firm j will not expect to make a profit in Bertrand competition. Firm j will not be willing to bear losses now in the hope of good news. Thus, firm i just needs to

i j set pt = 2 − 2αt + c − δ, where δ is very small, to get all the consumers. Notice

i j that pt = 2 − 2αt + c − δ > c. Therefore, firm i is selling to all consumers and making a positive profit, while firm j is not selling. Consumers will all adopt from firm i at the first opportunity by lemma 2. (3) Neither firm gets good news: the firms are expecting positive profit when only itself gets the good news. Detailed proof in lemma 3. 1o) For small ε and large λ, the firms are competing for consumers to do experimentation for them. In the Bertrand competition case, the firms will end up

1 2 with equally sharing the consumers with equal pricing at every time pt = pt .

29 The ﬁrms will need to price at the level such that the consumers will choose

i i N to adopt immediately, which requires that 2αt − 1 − pt > Wt . The pricing

i ∞ path {pt}t=0in the first stage should be a fixed point that maximizes the firm’s

i i N i profit such that 2αt − 1 − pt > Wt and E(πt) ≥ 0. Since the firms are sharing the population equally, the beliefs of both firms will evolve in the same way in the absence of good news. This will continue until at least one of them gets the good news. 2o) For large ε and small λ, both firms will wait for the signals to start selling. The logic is similar to that in the monopoly case. When the arrival rate of the exogenous signals is large, the endogenous learning process will lower the posterior and thus the expected profit after the good news. The firms will wait for the good news only through exogenous signals.

This proposition states that for large ε, the firms will wait for the good news and then start selling only after the signals. When a firm is the only one with good news, the firm will be able to make a profit. In contrast, for small ε, the competing firms will price at the same level and share equally the consumers in the beginning. Then, if one of them gets the good news, the lucky one will be able to charge a price above the marginal cost and make a positive profit during this period. Consumers will only adopt a good-quality product when given a chance. The price will go up as the belief about the quality of the competing product is drifting down. At last, when both firms get good news, the firms will both charge the marginal cost and share equally

30 the consumer population just like in Bertrand competition. The next proposition concerns about ﬁrms’ proﬁt in the process:

Proposition 4: (Expected Profit in Duopoly MPE under Perfect Good News) For small ε, the expected profit from the product will be zero for both firms at time t = 0; For large ε, the firms will have positive expected profit at time t = 0.

Proof: For small ε, the firms will get zero profit when both of them get good news. However, one of them is receiving a positive profit when it is the only with good news. With this expectation, both firms will charge below the marginal cost level in the beginning to attract consumers. They will set

1 2 pt = pt < c when neither of them gets good news. They will both get zero expected profit at time t = 0. Otherwise, one of them will have the incentive to lower the current price by a small δ to attract all consumers to do experiments only for it. For large ε, the firms will get zero profit when both of them get good news. And the only firm with the good news will receive a positive profit before both of them get signals. Because neither of them sells to the consumers before signals, the firms are expecting positive profit when they are the only lucky one with signals. Therefore, in this case, the firms are expecting positive profit at time zero.

31 For large ε, the firms will start with waiting for the signals and no selling. The beliefs of both products will be drifting down at the same pace. Then the lucky firm gets the signal and starts selling until the other firm gets the signal. The process is intuitive. I will investigate more about the other case. For small ε, I find that both firms have zero expected profit in the beginning. If one of them ends up with a positive profit, it is gaining by luck.

1 2 The other ﬁrm will bear losses. The prices will be pt = pt < c in the beginning. Then, the evolution of beliefs and prices in the duopoly case should be:

Figure 1.4. Beliefs and Price in Duopoly under PGN With Small ε

From Figure 4, at first, both firms charge the same price below the marginal cost and share equally the consumer population. The beliefs of both firms are drifting down in the same way during this period. Then, one of the firms gets good news. It charges a price above the marginal cost and attracts all the consumers. The other firm’s belief is still drifting down, but only by

32 exogenous source of signals until suddenly it gets a piece of good news. After both ﬁrms have good news, they will behave as in Bertrand competition, setting the price at the marginal cost level. The adoption curve will be concave without an interval of no learning. The following proposition considers the social eﬃciency of the MPE:

Proposition 5: The MPEs in duopoly are socially eﬃcient.

i R ∞ −rt i i Proof: Firm i’s profit: Vf = 0 e (pt − c) Nt dt. And the consumers’ value R ∞ −rt 1 1 1 2 2 2 function at time t = 0 is: Vb = 0 e [(2αt −1−pt ) Nt +(2αt −1−pt ) Nt ] dt. Thus, the social value at the beginning is: Z ∞ 1 2 −rt 1 1 2 2 W = Vf + Vf + Vb = e [(2αt − 1 − c) Nt + (2αt − 1 − c) Nt ] dt. 0 The social value function does not depend on the pricing path of the firms. I can see that, after both firms get the signal, all the consumers should adopt at the first opportunity from either firm. When only one firm has good news, consumers should adopt at the first opportunity from the proven good-quality firm. When there is no news for either firm, the consumers will adopt from the firm with a higher belief. For large ε, the firms have considered future profit and decide not to sell in the beginning. For small ε, the firms start with the same initial belief, and consumers will equally choose from the two products. When all people adopt from firm 1 in dt, if there is no good news, then the belief of firm 2 is higher. All people will choose to adopt from firm 2 in the next dt. When dt is very small, the above should be equivalent to half of the

33 people adopting each product all the time.

Therefore, the MPEs I get in duopoly maximize the social value function.

This proposition states that firms will act as a social planner in duopoly competition. For small ε, with the competition, the firms will have the incentive to encourage learning and bear the cost of experimentation. In a monopoly, the firm is only motivated by information incentives. By contrast, in a duopoly, the firms face both competition pressure and information incentive.

1.5 Conclusion

This paper adds the supply side to the forward-looking social learning environment and investigates the strategic pricing as well as the adoption processes. The adoption curves derived here are similar to those without the supply side: concave for products in perfect good news model. However, the driving force of the curve is different here. The firm is now controlling the learning process instead of the consumers. By strategic pricing, the firm makes consumers indifferent about the adopting time and achieves the optimal learning process for the firm. In perfect good news, when the arrival rate of exogenous good news ε is low, the monopoly firm will encourage early learning in the hope of a breakthrough. This process leads to a concave adoption curve. For high arrival rates of exogenous good news ε, the firm will not sell in the beginning and waits for the good news. There will be

34 a flat period before the concave part of the adoption curve. For duopoly firms in perfect good news, the firms are competing for consumers to get a higher probability of a breakthrough. The adoption curves are also concave here as in the monopoly counterpart. For low arrival rates of exogenous good news ε in both monopoly and duopoly under perfect good news model, there are periods of pricing below the marginal cost, or ”free samples” periods. However, the main incentives of ”free samples” are different in the two cases. The monopoly is pricing below the marginal cost to encourage more experiments only for the arrival of a breakthrough. Nevertheless, for the competing firms in a duopoly, they need to compete in prices to seize consumers from the competitor. Therefore, the duopoly firms are facing the pressure of competition as well as the incentive for good news when using ”free samples” strategy. The same mechanism ensures the social efficiency of MPEs in both monopoly and duopoly under perfect good news. The incentive for the timing of a breakthrough makes the monopoly firm behave like a social planner. In contrast, both the competition pressure and information incentive motivate the duopoly firms to achieve social efficiency in perfect good news.

35 Chapter 2

Strategic Pricing and Switching in Subscription Markets

2.1 Introduction

In this article, I explore firms’ competition and consumers’ switching in subscription markets, for example, internet access and cell phone plans. Specifically, I investigate how the consumers behave when they face some introductory offers from the competitors of their current suppliers. I find that high-switching-cost consumers will want to mimic their low-switching- cost peers when they consider future utility. The firms will need to offer better prices to keep the high-switching-cost consumers and thus less inclined to separate low-switching-cost consumers from high-switching-cost ones. This will improve the efficiency of equilibrium. The business model in subscription markets is that users pay a subscription fee for the access to certain products or services for a period. Some traditional subscription markets include cable television, cell phone plans, internet access, and magazines. The recent development of e-commerce has witnessed the success of many online subscription markets, from streaming- media service, including Netflix, to delivery services such as Amazon Prime, and to meal kit service like Blue Apron. Some insurance markets, e.g., car in-

36 surance and health insurance, though often have complicated pricing schemes, can be viewed as users paying subscription fees for coverages during a period. In subscription markets, firms usually directly deliver products to consumers, which gives firms some information about who their consumers are. They can at least distinguish between new users and returning users, and track how long a consumer has been theirs. This information allows firms to discriminate consumers based on their purchase histories, which is a form of behavior-based price discrimination. Consumers can switch suppliers after their term period. However, there is empirical evidence supporting that consumers face switching costs when they change suppliers. For example, Victor Stango(2002) quantifies the switching costs in the credit card market, where researchers observed customer lock-in. Consumers can have different switching costs due to differences in the time and the effort they would like to spend in searching and switching suppliers (Paul S Calem and Loretta J Mester(1995)). Most of the time, switching costs are private information observed only by consumers themselves. In markets with high switching costs, firms generally prefer to charge a higher price to existing consumers and offer a lower introductory price to new consumers to gain future profit from these consumers. However, introductory offers are not equally attractive to every consumer. If firms can observe the switching costs of all the consumers, firms will find that introductory offers often attract consumers with low switching costs, who will then easily be lured away by other firms. Therefore, firms would want to infer the con-

37 sumers’ switching costs based on their purchase histories. Then firms can practice behavior-based price discrimination by charging consumers different prices based on the firms’ beliefs about the consumers’ switching costs. This paper builds a model investigating the prices and consumer switching when switching costs are private information, but firms can observe consumers’ purchase histories. The central ingredient in this model is the firms’ incentives to separate the two types of consumers. In the last period of the game, a firm can benefit from keeping a large proportion of high-switching-cost consumers. However, the firm needs to offer high-switching-cost consumers low enough prices to keep them stay and, at the same time, forgo the low- switching-cost consumers to its competitor in previous periods. Only when there are enough high-switching-cost consumers in the firm’s pool, will the firm set a price to separate the two types of consumers.

Summary of Model and Results: Section 2 presents the model of two firms and a continuum of consumers in a subscription market. At the beginning of each period, a firm knows a consumer’s purchase history, forms a belief about the consumer’s switching cost, and posts a price. A consumer’s history contains information about past prices offered by both firms and the consumer choice of suppliers in each period. Consumers are only different in switching costs, which is a consumer’s private information. A consumer has unit demand for the product in each period. She can also observe her history (past prices and suppliers) and choose

38 a supplier each period. In equilibrium, firms and consumers should maximize their expected total payoffs and firms should update their beliefs about the consumers’ switching costs whenever possible. Section 3 analyzes the firms’ price paths and consumers’ supplier choices in Perfect Bayesian Equilibrium in three cases: (1) consumers are myopic; (2) consumers are forward-looking; (3) consumers’ types are randomly changing. When there are enough high-switching-cost consumers, firms will first offer different prices to separate two types of consumers by purchase history and then exploit the two types differently. Otherwise, firms will keep all their current consumers and charge them low-switching-cost consumers’ prices until the end of the game. When consumers are forward-looking, firms will require a more substantial proportion of high-switching-cost consumers to be willing to separate two types of consumers, which improves the efficiency of equilibria. When consumers’ switching costs can randomly change, firms are less willing to separate high-switching-cost and low-switching-cost consumers, but the effect on the efficiency is unclear.

Related Literature: My model is related to several strands of the literature. One is the classic literature on behavior-based price discrimination. Drew Fudenberg and Jean Tirole(2000), J Miguel Villas-Boas(1999), and Greg Shaﬀer and Z John Zhang(2000) study poaching in the market for a horizontally diﬀerentiated product where there are no explicit costs of changing suppliers. Firms are poaching customers from their competitors by

39 offering them low introductory offers when they can recognize their customers. Fudenberg and Tirole(2000) allow firms to provide a menu of long-term contracts and short-term contracts. Villas-Boas(1999) extends the model into an infinite-horizon, overlapping-generations model and investigates Markov- perfect equilibrium prices and market shares. Shaffer and Zhang(2000) extend the analysis into asymmetric markets. However, their studies are for consumers with different preferences in differentiated-product markets. Another strand is the literature on switching costs in homogeneous- product markets. Yongmin Chen(1997) and Curtis R Taylor(2003) find out that because of switching costs, firms will attract new customers with low introductory prices below cost while simultaneously exploiting their monopoly power over current subscribers in subscription markets. However, consumers draw their switching costs randomly at the beginning of each period in both models, which make consumers behave the same when they are myopic and forward-looking. Yet another is the literature on consumer recognition. Yuxin Chen and Z John Zhang(2009), Yongmin Chen and Jason Pearcy(2010), and Rosa- Branca Esteves(2010) develop various theories of dynamic pricing in which firms may offer separate prices to different consumers based on their past purchases. Esteves(2010) finds that forward-looking firms may avoid consumer recognition to make more profit. Chen and Zhang(2009) point out that competing firms may benefit from consumer recognition more than a monopoly, even when consumers can switch strategically. They are all inves-

40 tigating diﬀerentiated-product markets by including consumers’ brand preferences for suppliers, as in Fudenberg and Tirole(2000).

2.2 Model

There are three periods in the game: time t = 0, 1, 2 is discrete. There are two sets of players: two ﬁrms (A and B) and a continuum of consumers with total mass normalized to 1.

2.2.1 Firms

Two firms are competing in a subscription market, where they have the same technology and produce homogeneous products at zero marginal cost. Firms are risk-neutral expected profit maximizers with a discount factor δ. A firm can infer each consumer’s purchase history by retrieving records of whether the consumer purchased from this firm in each period. It then practices behavior-based price discrimination based on beliefs of consumers’ switching costs. At the beginning of each period, a firm can infer a consumer’s purchase history, form a belief about the consumer’s switching cost, and post a price. At the beginning of t = 0, each firm proposes a price without observing any history. At the beginning of t = 1, 2, each firm can observe past prices offered to each consumer and the consumer’s past actions. A history here is defined

A B as ht = (p0, ··· , pt−1, a0, ··· , at−1), where pt = {pt , pt } is a vector containing prices proposed by both ﬁrms at time t. Firms simultaneously post the prices

41 oﬀered to each consumer and the prices become a part of history.

i 2 Each firm chooses the pricing path {pt}t=0 to maximize its total discounted expected profit. As a firm can charge different prices to consumers

i with different purchase histories, pt can be vector. A firm’s pricing strategy for i + + t t + a consumer with a specific history should be pt :(R × R ) × {A, B} → R , for i ∈ {A, B}. A firm’s current profit is the sum of all revenues from its current customers. A firm will maximize its total payoff, which is the sum of discounted

i proﬁt in all the periods. Let Nt (ht) be the number of consumers buying from supplier i with history ht. Firm i’s objective function is: 2 X t X i i max δ pt(ht) ∗ Nt (ht) (2.1) t=0 ht

2.2.2 Consumers

There are two types of consumers: high-switching-cost consumers with switching cost λH and low-switching-cost consumers with switching cost λL.

Consumer types are randomly drawn with P r(λ = λH ) = α, α ∈ (0, 1). Con- sumers are identical in all other aspects. A consumer has unit demand at each period, which means she values one unit of the product at v and an additional unit at zero. We can assume v to be a large enough number that makes consumers always purchase in each period. Consumers enter the game with switching costs as private information. Entering each period, a consumer observes her history and prices posted by both ﬁrms and then chooses a supplier. At the beginning of t = 0, a consumer

42 will choose by observing prices offer by two firms. At the beginning of t = 1, 2, each consumer can observe past prices offered p0, ··· , pt−1, her own past ac-

A B tions a0, ··· , at−1, and current prices oﬀered by two ﬁrms pt , pt . Therefore, a

A B consumer can observe history ht and current prices oﬀered to her pt , pt before choosing a supplier.

+ + t t + + A consumer’s strategy: at : λ × (R × R ) × {A, B} × R × R → {A, B} specifies the consumer’s choice of a supplier after considering all ob- servable information and her private information. If the consumer’s current supplier is firm i, her current payoff will be

i −i v − pt if her supplier was also ﬁrm i in last period, or v − pt − λ if not, for i ∈

at {A, B}. Myopic consumers only care about current utility v−pt −λ∗1at6=at−1 . Forward-looking consumers have discount factor δ and will maximize the sum of discounted utility for all the periods:

2 a0 X t at max v − p0 + δ [v − pt − λ ∗ 1at6=at−1 ] (2.2) t=1

2.2.3 Equilibrium

In this section, I will deﬁne the Perfect Bayesian Equilibrium and derive the equilibrium conditions for ﬁrms and consumers.

i 2 A ﬁrm’s pricing path {pt}t=0 will be an equilibrium strategy if it can maximize the total discounted expected proﬁt:

3 i 2 X t X i i {pt}t=0 ∈ argmax δ pt(ht) ∗ Nt (ht) (2.3) t=0 ht

43 For a consumer, her equilibrium strategy will maximize her current

at utility v − pt − λ ∗ 1at6=at−1 if she is myopic, and will maximize the sum of discounted utility in all the periods:

3 3 a0 X t at {at}t=0 ∈ argmax v − p0 + δ [v − pt − λ ∗ 1at6=at−1 ] (2.4) t=1 if she is forward-looking. Then, I deﬁne a Perfect Bayesian Equilibrium as follows:

Deﬁnition 1: A Perfect Bayesian Equilibrium is an assessment (at(ht), pt(ht), µt(λH |ht)) such that:

• at(ht) maximizes each consumer’s expected continuation payoﬀ given the

history ht and ﬁrms’ prices pt(ht), for all ht;

• pt(ht) maximizes each ﬁrm’s expected continuation payoﬀ given belief

µ(λH |ht), for all ht;

• whenever possible, beliefs are deﬁned by Bayes’ rule: if α ∗ P r(ht|λH ) +

(1 − α) ∗ P r(ht|λL) > 0, then

α ∗ P r(ht|λH ) µ(λH |ht) = α ∗ P r(ht|λH ) + (1 − α) ∗ P r(ht|λL)

for all t = 0, 1, 2, 3.

A history ht = (p0, ··· , pt−1, a0, ··· , at−1) contains all the past prices posted by two ﬁrms and all past actions taken by the consumer.

44 2.3 Equilibrium Analysis

Here I will analyze how firms set prices and how consumers choose suppliers when consumers’ switching costs are private information, but firms can observe purchase histories. I proposed that consumers will behave differently when they consider future payoffs. Therefore, I will investigate and compare consumers’ equilibrium behaviors when they are myopic and when they are forward-looking. I will also investigate the equilibrium when consumers’ types can change from one period to another.

2.3.1 Myopic Consumers

In this section, I will discuss the equilibrium when firms are forward- looking, but consumers are myopic. Consumers will only consider payoffs in the current period when making decisions about suppliers. I will first characterize the off-path beliefs in divine equilibrium and then characterize the equilibrium.

Lemma 1: Each ﬁrm believes that a consumer chooses to switches a supplier oﬀ-path should be of type λL in divine equilibrium.

Proof: For a consumer with history ht, ∀ht: if her current supplier is ﬁrm i with

i j i price pt at this period. The other ﬁrm j will oﬀer a price pt ≤ pt −λL to attract

λL-type consumers to switch. If ﬁrm j would like to attract λH -type consumers

j i i i to switch, it need to offer pt ≤ pt − λH . Since (−∞, pt − λH ] ⊂ (−∞, pt − λL), the firm’s belief about the consumer switching off-path should be type λL by

45 D1-Criterion.

This lemma states that consumers who choose to switch oﬀ-the-equilibrium are supposed to have λL switching cost. The reason is that λL-switching-cost consumers are more likely to be attracted by the other supplier due to these consumers’ low cost in switching. Then I will deﬁne two types of equilibrium that will appear here.

Deﬁnition 2: A Poaching Equilibrium: Firms poach only λL-type consumers from the competitor. Two types of consumers are then completely separated by purchase histories. Firms will then perform Bertrand competition for each group of consumers.

In a poaching equilibrium, both low-switching-cost and high-switching- cost consumers start with the same history ht and the same supplier firm i. However, the other supplier attracts low-switching-cost consumers away but not high-switching-cost consumers. The two types of consumers then have different purchase histories and can be distinguished by the firms. The firms will know the consumers’ true switching costs and perform Bertrand competition for each type.

Deﬁnition 3: A non-Poaching Equilibrium: No poaching happens. All consumers will stay with the same supplier for all the periods. Consumer types will not be recognized by ﬁrms.

In a non-poaching equilibrium, if both low-switching-cost and high-

46 switching-cost consumers have the same history ht and the same supplier firm i, they will stay with the same supplier to the end. Two types of consumers then have exactly the same purchase histories and can not be distinguished by the firms. The firms will offer prices based on their beliefs about the prior of the two types.

Proposition 1: (PBE with Myopic Consumers) When ﬁrms are forward- looking but consumers are myopic:

• If α ∈ [ λL , 1): there is a poaching equilibrium; (1+δ)λH −δλL

• If α ∈ (0, λL ]: there is a non-poaching equilibrium. (1+δ)λH −δλL

Table 2.1 Equilibrium Path with Myopic Consumers

prior t = 0 t = 1 t = 2 unswitched: existing: λL −δα((1 + λH ; α ∈ [ (1+δ)λ −δλ , 1) price λH − δλL; H L δ)λH −δλL) switched: (poaching new: −δλL λL; new: 0 equilibrium) choose ran- λ consumers stay stay H domly choose ran- λ consumers switch stay L domly existing: λL existing: α ∈ (0, (1+δ)λ −δλ ] price −δλL λL − δλL; H L λL; new: 0 (non-poaching new: −δλL choose ran- equilibrium) λ consumers stay stay H domly choose ran- λ consumers stay stay L domly

47 When there is a high proportion of high-switching-cost consumers (α ≥

λL ), firms will prefer to focus on only high-switching-cost consumers (1+δ)λH −δλL and give up low-switching-cost consumers, which will lead to a poaching equilibrium. Both firms will first offer the same introductory offer −δα((1+δ)λH −

δλL) because they expect to earn δα((1 + δ)λH − δλL) from each consumer. Consumers will be indiﬀerent between the two suppliers and choose randomly. Each supplier will end up with half the population with the same prior α.

Then at t = 1, each firm will charge its own consumers λH − δλL and the competitor’s consumers −δλL. Here two types of consumers will make different choices. High-switching-cost consumers will stay with the current supplier, but low-switching-cost consumers will switch. Then both firms perform Bertrand competition for each type of consumer and make no profit. When the proportion of high-switching-cost consumers is low (α <

λL ), firms will choose to keep all the current consumers instead of just (1+δ)λH −δλL exploiting high-switching-cost consumers, which will lead to a non-poaching equilibrium. Both firms will first offer the same introductory offer −δλL because they expect to earn δλL from each consumer. Consumers will be indifferent between the two suppliers and choose randomly. Each supplier will end up with half the population with the same prior α. Then at t = 1, each

ﬁrm will charge its own consumers (1 − δ)λL and the competitor’s consumers

−δλL. No consumer will choose to switch, and no information on the consumers’ switching costs is revealed. Both ﬁrms perform Bertrand competition until the end of the game without knowing consumers’ switching costs. Each

48 firm will earn zero profit. Therefore, when there are many high-switching-cost consumers, firms will first offer different prices to separate two types of consumers by purchase history and then exploit the two types differently. However, when there are not many high-switching-cost consumers, firms will be better off keeping all their current consumers and charge them low-switching-cost consumers’ prices. Due to Bertrand competition, firms will make introductory offers so attractive that they will make no profit in the game.

2.3.2 Forward-looking Consumers

In this section, I discuss the equilibrium when both firms and consumers are forward-looking. Consumers will consider not only current payoffs but also future payoffs when making decisions. I will again first characterize the off- path beliefs in divine equilibrium and then characterize the equilibrium.

Lemma 2: Each ﬁrm believes that a consumer chooses to switches a supplier oﬀ-path should be of type λL in divine equilibrium.

Proof: For a consumer with history ht, ∀ht: if her current supplier is ﬁrm

i j i with price pt at this period. Suppose the other firm j offers a price ptH j to attract λH -type consumers to switch. This means that v − ptH − λH + i continuation value(switching) ≥ v − pt + continuation value(staying). j j Then firm j can to offer ptL = ptH +(λH −λL) to attract λL-type consumers to switch now. Because λL-type consumers can always choose to mimic λH -type

49 consumers, λL-type consumers can get at least the continuation value that λH -

j type consumers get. Therefore, v−ptL −λL +continuation value(switching) ≥ i v − pt + continuation value(staying).

i i Since (−∞, pt − λH ] ⊂ (−∞, pt − λL), the ﬁrm’s belief about the consumer switching oﬀ-path should be type λL by D1-Criterion.

This lemma states that consumers who choose to switch oﬀ-path are supposed to have λL switching costs. That means λL-switching-cost consumers are more likely to be attracted away by the other supplier. The reason is that λL-switching-cost consumers incur lower switching cost in the current period and can get all what λH -switching-cost consumers’ continuation utility by mimicking them. Thus, the introductory prices that can make λH -type consumers switch will always be able to attract λL-type consumers away. The reserve is not true. Therefore, λL-type consumers are more likely to send oﬀ- the-equilibrium signals by switching.

Proposition 2: (PBE with Forward-looking Consumers) When both ﬁrms and consumers are forward-looking:

• If α ∈ [ λL , 1): there is a poaching equilibrium; λH

• If α ∈ (0, λL ]: there is a non-poaching equilibrium. λH

50 Table 2.2 Equilibrium Path with Forward-looking Consumers

prior t = 0 t = 1 t = 2 switched: existing: λL λH ; α ∈ [ , 1) price −δαλH (1 − δ)λH ; λH unswitched: (poaching new: −δλ L λ ; new: 0 equilibrium) L choose ran- λ consumers stay stay H domly choose ran- λ consumers switch stay L domly existing: λL existing: α ∈ (0, λ ] price −δλL (1 − δ)λL; H λL; new: 0 (non-poaching new: −δλL equilibrium) choose ran- λ consumers stay stay H domly choose ran- λ consumers stay stay L domly

When there is a high proportion of high-switching-cost consumers (α ≥

λL ), ﬁrms will prefer to focus on only high-switching-cost consumers and give λH up low-switching-cost consumers, which will lead to a poaching equilibrium.

Both firms will first offer the same introductory offer −δαλH because they expect to earn δαλH from each consumer. Consumers will be indifferent between the two suppliers and choose randomly. Each supplier will end up with half the population with the same prior α. Then at t = 1, each firm will charge its own consumers (1 − δ)λH and the competitor’s consumers −δλL. Here two types of consumers will make different choices. High-switching-cost consumers will stay with the current supplier, but low-switching-cost consumers will switch. Then both firms perform Bertrand competition for each type of consumer and make no profit.

51 When the proportion of high-switching-cost consumers is low (α < λL ), λH ﬁrms will choose to keep all the current consumers instead of just exploiting high-switching-cost consumers, which will lead to a non-poaching equilibrium.

Both firms will first offer the same introductory offer −δλL because they expect to earn δλL from each consumer. Consumers will be indifferent between the two suppliers and choose randomly. Each supplier will end up with half the population with the same prior α. Then at t = 1, each firm will charge its own consumers (1−δ)λL and the competitor’s consumers −δλL. No consumer will choose to switch, and no information on the consumers’ switching costs is revealed. Both firms perform Bertrand competition until the end of the game without knowing consumers’ true switching costs. Each firm will earn zero profit. Therefore, when there are many high-switching-cost consumers, firms will first offer different prices to separate two types of consumers by purchase history and then exploit the two types differently. However, when there are not many high-switching-cost consumers, firms will be better off keeping all their current consumers and charge them low-switching-cost consumers’ prices. Due to Bertrand competition, firms will make introductory offers so attractive that they will make no profit in the game.

2.3.3 Equilibrium Eﬃciency

From the analysis above, if we compare the two cases when consumers are myopic and are forward-looking, we find that the cutoffs are different

52 ( λL < λL ). When consumers are forward-looking, more pooling of (1+δ)λH −δλL λH high-switching-cost consumers and low-switching-cost consumers will happen. High-switching-cost consumers realize that they will be exploited in the last period in a poaching equilibrium and require better prices to stay with the current supplier. The ﬁrms are thus less willing to pursue a poaching equilibrium. Therefore, less switching will happen with forward-looking consumers.

Myopic Consumers

λ Both λL and λH L Separate λL and λH (1+δ)λH −δλL

λL α Both λL and λH λH Separate λL and λH

Forward-looking Consumers

Figure 2.1. Comparison of Myopic and Forward-looking Consumers

The result is stated below:

Proposition 3: Firms will be less willing to pursue a poaching equilibrium when consumers are forward-looking than when they are myopic. Forward- looking consumers, on average, stay longer than when they are myopic.

From the two cases, firms will make no profit due to Bertrand competition. Firms will make no profit from poaching the competitor’s consumers because they have to make the discounted future profit as introductory offers to attract consumers away from the competitor. Firms also make no profit from their current consumers. Because firms’ consumers are the poaching

53 target of the competitor, they have to offer a price making consumers indifferent between switching and not switching. The possible future profit from consumers has to be offered at t = 0 to attract consumers due to Bertrand competition. The result is stated below:

Proposition 4: (No Profit in Bertrand Competition) For both myopic and forward-looking consumers, firms will make no profit.

As Taylor (2003) has mentioned, switching suppliers is never efficient in a subscription market with switching costs. Here suppliers provide homogeneous goods to customers, and consumers pay directly to suppliers. Costs associated with switching are dead-weight loss here. Inefficient switching oc- curs because switching costs are the private information of consumers, and firms would like to use switching as a signal to distinguish consumers’ types. We see that more pooling of high-switching-cost consumers and low-switching- cost consumers will happen when consumers are forward-looking than when consumers are myopic. If myopic consumers become forward-looking, inefficient switching will be avoided when α ∈ [ λL , λL ]. Therefore, the (1+δ)λH −δλL λH equilibrium there will be more efficient. The result is stated below:

Proposition 5: (Eﬃciency Comparison) The equilibrium will be more eﬃ- cient when myopic consumers become forward-looking by avoiding some switching.

54 2.3.4 Changing Switching Costs

As Taylor (2003) assumed, a consumer’s switching cost can vary from period to period indicating the consumer’s busy periods and slack periods. In this section, I will explore the equilibrium where consumers’ switching costs can randomly change from one period to another and investigate the implication on the time a consumer stays with current suppliers.

Assume two types of consumers diﬀer only in switching costs: λH high- switching cost consumers and λL low-switching cost consumers. Their switching costs can randomly change from period to period. At the end of each period, 1 − β1 of high-switching-cost consumers will become low-switching cost consumers, and 1 − β2 of low-switching-cost consumers will become high- switching cost consumers in the next period. The change is random, and the probability is identical for all consumers with the same current switching costs.

I assume β1 and β2 are close to 1 so that the majority of users keep their switching costs in the next period. I also assume that α∗β1 +(1−α)∗(1−β2) = α so the proportion of high-switching-cost consumers are constant in each period.

λH λL λH β1 1 − β1 λL 1 − β2 β2

I assume firms and consumers are all forward-looking and have the same discount factor δ as in Section 3.2. We can also know that firms will believe the consumers who choose to switch off-path are of type λL with the same logic

55 in lemma 2. Then I can state the equilibrium here:

Proposition 6: (PBE with Consumers Changing Types) When both ﬁrms and consumers are forward-looking and consumers’ switching costs are randomly changing as stated above:

• If α ∈ [ λL+β2λH −λH , 1): there is a poaching equilibrium; λH (β1+β2−1)

• If α ∈ (0, λL+β2λH −λH ): there is a non-poaching equilibrium. λH (β1+β2−1)

The two types of equilibrium are the same as in previous analysis: when there are many high-switching-cost consumers (α ∈ [ λL+β2λH −λH , 1)), firms will first λH (β1+β2−1) offer different prices to separate two types of consumers by purchase history and then exploit the two types differently. However, when there are not many high-switching-cost consumers (α ∈ (0, λL+β2λH −λH )), firms will be better off λH (β1+β2−1) keeping all their current consumers and charging them low-switching-cost consumers’ prices. Due to Bertrand competition, firms will make introductory offers so attractive that they will make no profit in the game. The differences are: (1) firms take future type changes into consideration when setting prices at t = 1, which mean firms are less willing to offer prices to separate the two types because they are aware of the possible type changes of high-switching cost consumers in the last period; (2) Low-switching- costs consumers at t = 1 who change to λH type will benefit from pooling with other low-switching-cost consumers as they only needs to pay λL at t = 2; (3)

However, for high-switching-costs consumers at t = 1 who change to λL type

56 later, the reverse is true. The consumer will be charged at λH because she is pooling with other high-switching-costs consumers and will need to switch to the other supplier charging a small amount below λH − λL. The effect on efficiency is ambiguous. Though firms are less willing to offer prices to separate the two types, high-switching-cost consumers at t = 1 who change to λL type at t = 2 will switch in the last period because they suffer from pooling with other high-switching-cost consumers in the previous period. The effect on the time that consumers stay with suppliers is clear: low-switching cost consumers are more likely to stay with only one supplier than when their types are deterministic, and high-switching-cost consumers are staying with their suppliers for a shorter period of time. The reason is that firms are less willing to separate the two types at t = 1, which gives low- switching cost consumers fewer chances to switch. However, high-switching- cost consumers, who never switch when their types are deterministic, will choose to switch if their types change at t = 2. Therefore, though the same two types of equilibrium still exist here, firms are less willing to separate the two types, and low-switching-cost (high- switching-cost) consumers may benefit (suffer) from pooling with their peers at t = 1. The effect of changing types on efficiency is not clear.

57 2.4 Conclusion

This paper builds a model investigating the prices and consumer switching when switching costs are consumers’ private information, but firms can observe consumers’ purchase histories. Firms will infer consumers’ switching costs from purchase histories and practice behavior-based price discrimination. When consumers are myopic, there are two types of equilibrium: poaching equilibrium and non-poaching equilibrium. In a poaching equilibrium, firms will first offer different prices to separate two types of consumers by purchase history and then exploit the two types differently. In a non-poaching equilibrium, firms will keep all their current consumers and charge them low- switching-cost consumers’ prices until the end of the game. Firms will pursue a poaching equilibrium only if there are a large portion of high-switching-cost consumers to make a profit from. When consumers are forward-looking, the same two types of equilibrium will still exist. However, high-switching-cost consumers will consider future high prices when they choose to stay and thus require better prices. Firms will require a more substantial proportion of high-switching-cost consumers to sustain a poaching equilibrium. The result is that low-switching cost consumers are more likely to stay with the same supplier until the end of the game. Consumers being forward-looking also reduces inefficient switching, which improves the efficiency of equilibrium. When consumers’ switching costs can randomly change from period to period, firms will account for future type changes and less willing to sepa-

58 rate high-switching-cost and low-switching-cost consumers. This makes low- switching-cost consumers even more likely to stay with the same supplier throughout the game. However, the eﬃciency improvement may be countered by high-switching-cost consumers who change to low-switching-cost type in the last period and decide to switch.

59 Chapter 3

The Impact of Low-skilled Immigration onCollege-educated Women

3.1 Introduction

This paper investigates how the inflow of low-skilled immigrants affects the household decisions: the divorce decision, the fertility decision, and who marries whom. The influx of low-skilled immigrants leads to lower prices of services that are close substitutes for household production and increases the supply of market work among highly skilled women. Previous work has shown the significant effect of low-skilled immigrants on gender pay equality (Cortes and Pan (2017)). However, the inflow of immigrants also affects the marriage of high-income women and, thus, all people. This paper provides a model as well as empirical investigation of the marriage market equilibrium effects of the influx of low-skilled immigrants. By uncovering the underlying mechanisms that link the changes in the number of low-skilled immigrants to marriage patterns and welfare, this paper offers a framework for thinking about the designing of policies aiming to improve social welfare in an economy with low-skilled immigrants. Low-skilled immigrants were supposed to take those low-income jobs, which was criticized by people as competing jobs against local workers. However, my results reveal

60 that the inflow brings more than just the labor force. I show that more high- income families will choose to have children with more low-skilled immigrants working as nannies. And the high fertility rate will stabilize the high-income marriage. Moreover, the matching patterns will become more positive assortative. This paper, hence, extends the discussion about the welfare effects of low-skilled immigrants and shows that they are changing more than people have imagined. To understand the link between substitutes for household production and the equilibrium in the marriage market, I specify an equilibrium model of household formation, fertility decision, and divorce over the life cycle. In the model, individuals first enter a matching market to decide whether to get married and the education level of their partner. After they form families, single and married individuals enter a household life cycle. Married couples decide whether to have children. Only when couples have children, can they consume public goods in marriage. However, they have to look after the children by themselves or by hiring nannies. A match quality shock happens at the end of the fertility decision stage. Couples make divorce decisions in the next stage. Children will be assigned to single mothers. Fathers will pay child support transfer and enjoy partial utility from public goods. Marriage decisions depend on the expected utility from fertility and divorce. The model predicts that the inflow of low-skilled immigrants will benefit college-educated women by increasing their opportunities to marry college-educated men, increasing their fertility rate, and decreasing their divorce rate.

61 Then I conduct an empirical analysis to test the effect of the inflow of low-skilled immigrants on college-educated women. The analysis shows that the inflow of low-skilled immigrants does not have a significant impact on women’s marriage rate, fertility rate, and working decision in general. Still, it increases college-educated women’s marriage rate, fertility rate, and labor participation and, at the same time, decrease college-educated women’s divorce rate. To exploit the exogenous part of the immigration data, I construct an instrumental variable by predicting immigration using the historical distribution of immigrants from the same country of origin (Patricia Cortes(2008)). Then I identify the effect of low-skilled immigrants on college-educated women by exploiting the changes in marriage, fertility, divorce, and labor force participation over time in cities with different immigration inflows. The empirical results support my prediction in the model.

Related Literature: This paper contributes to various strands in the literature. First, by focusing on the eﬀect of low-skilled immigrants on the family decisions, I extend the literature that studies how the low-skilled immigrants impact the labor market (Cortes(2008), Patricia Cortes and Jos´e Tessada(2011), and Patricia Cortes and Jessica Pan(2019)). They investigated whether providing highly skilled women with low-cost substitutes for household production reduces the gender earning gap by increasing their labor supply. I extend the discussion by embedding a life cycle model of household behavior into an equilibrium model of household formation, which allows me

62 to quantify the change in welfare in the marriage market. In this sense, this paper is close to a recent contribution by Ana Reynoso(2018) that analyzes how the adoption of unilateral divorce affects the gains from marriage and who marries whom. Unlike her paper, I simplify the model into three periods and analyze the impact of low-skilled immigrants on family decisions: fertility and divorce. Second, by embedding a model of household behavior into an equilibrium framework, I also extend the literature that empirically quantifies marital welfare. Eugene Choo and Aloysius Siow(2006) empirically estimate the marital gains with a model and observed matching patterns data. A multi-market environment extension by Pierre-AndréChiappori, Bernard Salaniéand Yoram Weiss(2015) investigate the marital college premium with US matching patterns data.Venkataraman Bhaskar(2019) analyzes the age gap at the marriage in response to persistent imbalances in cohort size with census data in Asia and Africa. Importantly, these papers rely exclusively on matching patterns data. In my framework, the marital gains are derived not only from the observed marriage patterns but also from observed fertility rate and divorce rate. I am not the first to extend the literature by combining an equilibrium model of marriage with the collective model of household behavior. Pierre- AndréChiappori, Monica Costa Dias and Costas Meghir(2018) develop an equilibrium life cycle model of education, marriage, and labor supply and consumption in a transferable utility context. I add the divorce stage to make the life cycle complete and more realistic.

63 The rest of the paper is organized as follows. Section 2 will present an equilibrium model of matching, fertility decision, and divorce decision over the life cycle. Section 3 will state the model predictions in equilibrium to show the eﬀect of low-skilled immigrants. Section 4 will show some reduced-form evidence from US Census and American Community Survey data. Section 5 concludes.

3.2 Model

To study the general eﬀect of low-skilled immigration and to understand the mechanism behind the consequences, I specify an equilibrium model of matching, fertility decision, and divorce decision over the life cycle. Then, I will compare the equilibrium results when there are no low-skilled immigrants, and when there are low-skilled immigrants. The economy is populated by college-educated people with high income

YH and not college-educated people with low income YL. Anyone can choose to take care of the children at home and receive part-time income Y 0. Assume

0 0 YH > YL > Y and YH − YL > Y . Women are either high-income type wH or low-income type wL; similarly, men are either high-income type mH or low- income type mL. Agents live for three periods: matching, fertility, and divorce. Figure 1 illustrates the life cycle of individuals.

64 match quality ε

t = 0 t = 1 t = 2

Fertility decision Divorce decision Matching period period period

Figure 3.1. The life cycle of individuals

In the matching stage at t = 0, the individuals meet in the marriage market and choose the education level of the other sex. Women can choose from {mH , mL, ∅} and men can choose from {wH , wL, ∅}. The life of singles or couples develops in period 1 and 2. In the fertility decision stage at t = 1, couples decide whether to have children k ∈ {0, 1}. If they decide to have children, they will have to have someone taking care of the children and receiving part-time wage Y 0. But the couple will enjoy utility from public goods Q multiplied by their private consumption. Otherwise, the couple consumes their income without children. At the end of t = 1, the match quality ε of each couple was realized. In the divorce decision stage at t = 2, couples decide whether to divorce d ∈ {0, 1}. Based on the value of ε, the couple will decide whether they want to stay in the marriage. If they decide to divorce, the wife will get the children (as in most cases) and receive a child support transfer τ. The wife would then enjoy the public goods from children but manage to take care of the children

65 on her own. The father will only be able to enjoy discounted utility from public goods and needs to pay child support. If the couple chooses to stay in the marriage, they enjoy and support the children together. If the couple does not have children, they consume their income.

In the matching stage, every woman chooses from {mH , mL, ∅} and every man chooses from {wH , wL, ∅}. Thus, there are eight possible matches in the model: {mH &wH , mH &wL, mH &∅, mL&wH , mL&wL, mL&∅, ∅&wH , ∅&wL} denoted as: {HH, HL, mH , LH, LL, mL, wH , wL}.

Deﬁnition: A Stable Matching is pair (m, w) such that for m ∈ {mH , mL} and w ∈ {wH , wL}:

• there is no pair (m, w0) such at m strictly prefers w0 to w and w0 strictly

0 prefers m to her current partner, for w ∈ {wH , wL, ∅} (no blocking pairs);

• there is no pair (m0, w) such at w strictly prefers m0 to m and m0 strictly

0 prefers w to his current partner, for m ∈ {mH , mL, ∅} (no blocking pairs).

We begin to solve the model using backward induction starting from the last stage:

3.2.1 Divorce Decision Stage

In this stage, after the realization of match quality ε ∼ U[−M,M], formed couples decide whether to divorce by comparing the utility of staying

66 in or leaving the marriage. Depending on their decision in the last stage (fertility stage), there are two situations:

(1) The couple do not have children: if they divorce, they will each consume their own income. However, if they stay in marriage, they will share the income and match quality ε. They solve a Nash bargaining problem in their marriage:

argmaxcm (cm + ε − Ym) ∗ (cw + ε − Yw) c is the consumption of each partner, and Y indicates his/her income. (2) The couple has children: if they divorce, the wife will get the children and take care of them with a child support transfer from the husband. The single mother will solve the problem:

maxcw,Q cw ∗ Q

s.t. cw + Q = Yw2 + τ cw is the private consumption of the single mother, and Q is the public goods.

0 The single mother maximizes utility under budget constraints. Yw2 = Y for low-income women because they would rather stay at home and look after the

0 children themselves. However, Yw2 = YH − YL > Y for high-income women, because they would prefer working and hiring someone to take care of the children if they can ﬁnd nannies. The father enjoys discounted utility from children and needs to pay the

67 single mother child support transfer τ, so he solves the following problem:

maxcm,τ δ ∗ cm ∗ Q

s.t. cm + τ = Ym2 cm is the private consumption of the divorced man. If they choose to stay in the marriage, they enjoy the additional match quality and solve the Nash equilibrium:

t=2,k=1 t=2,k=1 argmaxcm (cm ∗Q+ε−EUm (divorce))∗(cW ∗Q+ε−EUw (divorce))

By comparing the utility they get with, and without marriage, couples make divorce decisions.

3.2.2 Fertility Stage

In this stage, every formed couple will decide whether to have children k ∈ {0, 1} by comparing the utility with children and without children. If the couple chooses not to have children now, they will never have children. They do not share public goods and only consume their income. If couples choose to have children, they solve the following problem:

t=2,k=1 maxcm,cw,Q cm ∗ Q + EUm

0 s.t. cm + cw + Q = Y + Y

t=2,k=1 ¯ 00 cw ∗ Q + EUw ≥ Uw

They will allocate their income between public goods and private consumption under the budget constraint. The couples can hire someone to take

68 care of the baby if there are nannies available; otherwise, they need to stay at home themselves to look after the children. The choice of staying at home this period will depreciate their human capital and lead to a discounted factor K ∈ (0, 1) on wage if they choose to work in the next period. The couples make fertility decisions taking into consideration the expected utility of having

t=2,k=1 children in the divorce decision stage, EUm . For couples choosing to have children, they allocate private consumption by solving the Nash bargaining problem:

t=2,k=1 t=2,k=0 t=2,k=1 t=2,k=0 argmaxcm (cm ∗ Q + EUm − EUm ) ∗ (cW ∗ Q + EUw − EUw )

By comparing the utility they get with, and without children, the couples make fertility decisions.

3.2.3 Matching Stage

In this stage, every man and woman is making a matching decision. The men can choose from {wH , wL, ∅} and the women can choose from {mH , mL, ∅}. They make matching decisions by comparing the utility of matching with each type of the other sex or staying single. If the agent chooses to stay single (by choosing ∅), the agent will consume her/his income for the next two periods. She/he will not derive utility from children or marriage and needs not make any fertility or divorce decisions. For a man with a college education, he can choose to marry women with

69 or without a college education. If a mH &wL (HL) match is formed, the low- income woman will stay at home looking after children if they decide to have one. If they get divorced, the woman will take the children and stay at home

0 taking care of them, receiving only Y . However, if a mH &wH (HH) match is formed and in a situation without nannies, with half of the probability, the man will stay at home looking after the children. If they choose to divorce at t = 2, the man will only receive KYH because of human capital depreciation. The woman taking the children will have to stay at home taking care of the children because no nannies are available. With half of the probability, the woman will remain at home, taking care of the children at t = 1. And the

0 man will make YH with the woman only receiving Y if divorced. However, the situation changes if there are nannies available. The high-income couple can hire a nanny and pay her/him YL for taking care of the children. Since

0 YH − YL > Y , the couple will both choose to work and hire a nanny at t = 1. The single mother will also hire a nanny at t = 2 if they get divorced. For a man without a college education, he can choose to marry women with or without a college education. If a mL&wH (LH) match is formed, the low-income man will stay at home looking after children if they decide to have one. If they get divorced, the woman will take the children and stay at home taking care of them receiving only Y 0 if no nanny exists; she will hire a nanny and pay YL if nannies are available. The man will go back to the labor market, earning KYL. However, if a mL&wL (LL) match is formed, with half of the probability, the man will stay at home looking after the children. The

70 man will earn KYL if he goes back to work after divorce. With half of the probability, the woman will look after the children, which will enable the man to earn YL after divorce. The presence of low-income immigrants (nannies) will not affect the LL match because they can not afford to hire nannies. For women with a college education and without a college education, their choices are the same, i.e., men with and without a college education. They can form matches with different types of men. The patterns will be the same as described above.

In summary, in a matching market, each type of men and women, mH , mL, wH , and wL are selecting the type of partners that give them the most utility. We can derive the actual matching patterns by considering each type of matches:

{mH &wH , mH &wL, mH &∅, mL&wH , mL&wL, mL&∅, ∅&wH , ∅&wL}.

Stable matching satisﬁes the no-blocking-pair condition. If college-educated men and college-educated women prefer each other but are not matched together, they can break their current match and form new pairs to improve utility.

3.3 Model Predictions

Here I predict the equilibrium outcomes when there are no low-skilled immigrants, and when there are low-skilled immigrants. The main diﬀerence

71 is that high-income individuals can choose to work and hire nannies to take care of their babies with the presence of low-skilled immigrants. The available nanny option may change people’s decisions in divorce, fertility, and matching.

3.3.1 Without Low-income Immigrants

I summarize the results of the possible matches in the divorce decision stage in the following proposition:

Proposition 1: Divorce rateLH > Divorce rateLL > Divorce rateHH >

Divorce rateHL.

The divorce rate here is deﬁned as the probability that the realization of the match quality will not be able to support a marriage. By staying in the marriage, couples will enjoy the realized match quality ε. However, if the realization of ε is too low, the couple will choose to divorce. Since ε ∼ U[−M,M], the probability can be calculated as in Appendix C. The result shows that the LH and LL match have the highest divorce rate, then HH and HL follows in the divorce rate. The HL couples have the lowest divorce rate. The phenomenon can be explained by the Nash bargaining that happened within families. For HH couples, half of the time, they will act like HL couples when women stayed at home at t = 1. For the other half of the probability, the husband stayed at home at t = 1. The couple will have lower utility after divorce compared to HL match because the ex-husband will only earn KYH . Thus, their divorce rate will be between HL and LH couples.

72 For LL couples, their divorce rate is nearly 0.5, which largely depends on the realization of match quality. If they stay in the marriage, the husband

0 will make YL, and the wife will make Y . They will make the same if they divorce. The marriage does not provide a signiﬁcant improvement in utility. Therefore, LL couples are more aﬀected by the realization of match quality in divorce decisions. Divorce decisions can be related to fertility decisions. Couples consider the expected utility in the divorce stage when making fertility decisions. I summarize the results of matches in the fertility decision stage in the following proposition:

Proposition 2: F ertility rateHL > F ertility rateHH > F ertility rateLH >

F ertility rateLL.

The fertility decision is deterministic for each type of couples since everything else is deterministic or can be calculated in expectation. Here I am using the diﬀerence in utility of having children or not having children

EU t=1,k=1 − EU t=1,k=0 as a proxy for fertility rate. The diﬀerence is the same for men and women. Intuitively, if having children gives more utility than not having children, couples are more likely to have children in the presence of all other kinds of shocks. Proposition 2 states that the LL couples will be the least likely to have children. The reason may be that LL couples expect a very high expected divorce rate at t = 2. They may prefer not to have children now to avoid the Y 0 after divorce.

73 For the other three types of matches, HL and LH have clear house specialization if they have children, which is that the low-income person stays at home looking after the children. LH couples will get lower utility than HL couples because the divorced man only earns KYL in LH. Also, LH couples expect a higher divorce rate in the next stage. Therefore, the LH couples have a lower fertility rate than HL couples. HH couples sacriﬁce the most for having children because one of them must forgo the high income at t = 1 in the world without nannies. This explains why HH couples have a lower fertility rate than HL couples. With all the predictions in the fertility and divorce stages, we can derive the matching patterns at t = 0: Proposition 3: High-income men and low-income women are stable matches.

In ﬁgure 3 and ﬁgure 4 in Appendix C, we see that high-income men would like to match with low-income women, while low-income men would like to match with high-income women; High-income women would like to match with high-income men, while low-income women also would like to match with high-income men. High-income men and low-income women are stable matches due to household specialization. High-income men prefer low-income women because women would like to stay at home and have children. However, high-income women face higher opportunity cost when having children, and are thus less likely to have children. Both high-income and low-income women prefer matching with high-

74 income men. Because a single mother will get very little if her ex-husband was low-income. In summary, high-income men and low-income women form stable matches when there are no low-skilled immigrants. Household specialization dominates in the matching market.

3.3.2 With Low-income Immigrants

The presence of low-income immigrants changes the results in the divorce stage and fertility stage. I summarize the results with nannies in the divorce decision stage in the following proposition:

Proposition 4: Divorce rateLL > Divorce rateLH > Divorce rateHL >

Divorce rateHH .

Proposition 4 states the results in the divorce stage with the inflow of low-income immigrants. We see that LL couples remain unaffected by the inflow of people. They get divorced if the realized match quality is negative. LH and HL couples behave similarly in divorce decisions. The intuition is that by allowing high-income women in LH to work after a divorce, LH and HL couples will have similar income before and after divorce. They, therefore, face similarly trade-offs in making divorce decisions. HH couples are now the least likely to get divorced. Their total family income does not change after divorce, but their families are very stable to match quality shock. This may be explained by the high fertility rate and, thus, high utility from public goods (children).

75 To better understand the change in equilibrium in the fertility decision stage, I summarize the results in the following proposition:

Proposition 5: F ertility rateHH > F ertility rateHL > F ertility rateLH >

F ertility rateLL.

Proposition 5 shows that the inﬂow of nannies does not change the situation of LL couples. They are still the least likely to have children. However, HH couples are now most likely to have children because the college-educated wives can choose to work and hire nannies if they have children. The opportunity cost of children decreases for HH couples. LH and HL couples behave similarly in fertility decisions because both types of matches have household specialization at t = 1 and expect similar divorce rate at t = 2. Combining Proposition 4 and 5, it is clear that children as public goods can increase utility and stabilize the marriage for high-income people. With the help of nannies, more HH couples can have children to enjoy utility from public goods, which also prevents them from being divorced. Therefore, children as public goods improve family utility and stability. Proposition 6: High-income men and high-income women are stable matches.

In Figure 7 and Figure 8 in Appendix C, we see that both high-income and low-income men would like to match with how-income women; Both high- income and low-income women would like to match with high-income men. High-income men and high-income women are stable matches in the

76 market now. Because of the existence of low-skilled immigrants, high-income women can enjoy both work and children. Income dominates in the matching market.

In summary, the eﬀects of low-skilled immigrants are: (1) marriage market is more positive assortative; (2) couples both with high-education are more likely to have children; (3) couples both with high-education are less likely to get divorced. These changes signiﬁcantly improve the situation of high-educated women.

3.4 Reduced-form Evidence

From the model, we see that the inflow of low-skill immigration will affect women’s marriage, fertility, divorce, and labor force participation decisions. To identify these effects, we exploit cross-city variation induced by low-skilled immigration. I restrict my sample to 36 large cities in the US. The data are from the US Censuses, and the American Community Survey (ACS) accessed through IPUMS USA. I choose the sample years to be 1970, 1980, 1990, 2000, and 2010. A list of the cities I choose is attached in Table 1 in Appendix C.

3.4.1 Constructing Instruments

Low-skilled immigrants are deﬁned as those without a high school degree and who are not US citizens in the Census/ACS. To account for the potential endogeneity in the location choice of low-skilled immigration, I con-

77 struct a measure of predicting low-skilled immigration. The measure isolates a plausibly exogenous component in the cross-city distribution of low-skilled immigrants by exploring the tendency of immigrants to settle in a city with an existing enclave of immigrants from a speciﬁc country. I choose 1970 as a benchmark to calculate the predicted inﬂow of immigration. There are 66 countries counted as immigration origin. The formula is:

X Immigrantspc,1970 P redicted Immigrants = ∗ T otal Immigrants c,t Immigrants pt p p,1970 (3.1) p denotes the country of origin and c denotes the city. To make diﬀerent cities comparable, I divide our instrument by the size of the city’s labor force. Table 2 and Figure 9 in Appendix C show the predicted immigrants.

3.4.2 First Stage

Here I compare the predicted number of low-skilled immigrants and the actual number of low-skilled immigrants. We see that the Pearson R is 0.72, and the P value is 0 in Figure 10 in Appendix C. This means that the predicted number and the actual number of low-skilled immigrants are highly positively correlated. The regression result is as follows:

78 Actual No. Low-skilled Immigrants =135777.8*** +0.817*** Predicted No. Low-skilled Immigrants (345.01) (701.04) N=342400 adj. R-sq=0.518 t statistics in parentheses =”* p¡0.05 ** p¡0.01 *** p¡0.001”

We see that the predicted number of low-skilled immigrants can explain more than half of the variation in the actual number of low-skilled immigrants. The two variables are highly positively correlated. Therefore, I will use the predicted number of low-skilled immigrants as an instrument for the actual number of low-skilled immigrants. I predict the actual number low-skilled immigrants Actual Immigrationˆ with the instrument as in the ﬁrst stage of 2SLS and use in all the following regressions.

3.4.3 General Eﬀects on Women

ˆ Yict = α + δ ∗ ln( Actual Immigrationct ) + γXict + θc + µt + ct (3.2)

Yict are: P rob(Get married), P rob(Give birth to children), P rob(Get divorced), P rob(W orking) and log(W orking hours).

Xict: education dummies, race dummies, age, age square.

θc and µt are city and time ﬁxed eﬀects. Standard errors are clustered at the city level to allow for the possibility of serial correlation within cities across years. The results are below:

79 Table 3.1 The Eﬀect of Predicted Actual Low-skilled Immigration Flows on Women

(1) (2) (3) (4) (5) (6) (7) Prob(Married) Prob(Having Children) Prob(Giving Birth to Children) Prob(Divorced) Prob(Working) ln(Work Hours) ln(Wage) (1980 and 1990 Samples) (2010 Sample)

Actual Immigrationˆ 0.124 -0.593 -0.885 1.257* -0.366 0.125 -0.533 [0.183] [1.136] [1.109] [0.560] [0.301] [0.0934] [0.396]

High School Graduates 0.589*** 0.194*** 0.0335 -0.120 1.053*** 0.916*** -0.652*** [0.0353] [0.0370] [0.100] [0.0976] [0.0291] [0.0362] [0.0483]

Some College 0.513*** -0.202** -0.205* -0.342** 1.443*** 1.255*** -0.0983** [0.0457] [0.0634] [0.0963] [0.120] [0.0251] [0.0108] [0.0352]

College Graduates 0.672*** -0.751*** -0.258 -0.688*** 1.636*** 1.407*** 0.672*** [0.0904] [0.127] [0.134] [0.0985] [0.0823] [0.0507] [0.0786]

College Graduates Plus 0.566*** -1.105*** 0.145 -0.915*** 1.949*** 1.583*** 1.750*** [0.0542] [0.0893] [0.109] [0.232] [0.0561] [0.0315] [0.0567]

Age 0.272*** 0.320*** 0.671*** 0.147*** 0.253*** 0.108*** -0.355*** [0.00761] [0.0128] [0.0392] [0.0168] [0.00209] [0.00101] [0.00252]

Age-square -0.00266*** -0.00281*** -0.0111*** -0.00146*** -0.00314*** -0.00136*** 0.00343*** [0.0000721] [0.000107] [0.000632] [0.000194] [0.0000185] [0.0000163] [0.0000319]

White -0.0460 -0.303*** -0.0903 -0.513*** 0.0848*** -0.00218 0.231*** [0.0328] [0.0434] [0.0950] [0.0657] [0.0222] [0.0162] [0.0303]

Black -1.149*** 0.0686 0.0147 0.507*** -0.0551 -0.0162 0.329*** [0.0461] [0.0591] [0.100] [0.117] [0.0487] [0.0251] [0.0396]

City FE X X X X X X X

Year FE X X X X X X X

Constant X X X X X X X

N 159675 76370 45236 159675 159675 159675 159543

t statistics in parentheses =”* p¡0.05 E**ˆ p¡0.01 E***ˆ p¡0.001”

The first column shows the effect of predicted low-skilled immigrants on women’s marriage rates. Generally, more low-skilled immigrants occur with more women being married, but the coefficient is not significant at 5% level.

80 Education and age contribute significantly to women’s marriage rate as the result suggests. Column 2 and 3 exhibit the effect of predicted low-skilled immigrants on women’s fertility decisions. In 1980 and 1990 US Census samples, the variable that relates to fertility is children ever born. The reduced-form result shows that more low-skilled immigrants slightly and insignificantly decrease the fertility rate of women. College education and being white are the two main factors that contribute to women’s low fertility rate. In the 2010 ACS sample, there is a variable that reflects children born within the last year. I study the effect of low-skilled immigrants on this variable and get a similar result as that from children ever born variable. Column 4 presents the effect of predicted low-skilled immigrants on the couple’s divorce decisions. The effect of predicted low-skilled immigrants on divorce rate is positive and statistically significant at the 5% level, showing that more low-skilled immigrants occur with more divorce. White women and women with higher education are more likely to have a stable marriage as shown in the estimation. Column 5 to 7 exhibit the effect of predicted low-skilled immigrants on women’s labor decisions. Low-skilled immigrants have not changed the labor decision of women in general. The estimation results show that women with higher education are more likely to work and receive higher incomes. In summary, the inflow of low-skilled immigrants does not have a significant effect on women’s marriage rate, fertility rate, and working decision.

81 More low-skilled immigrants are related to a higher divorce rate. The factors that contribute to women’s marriage, fertility, and working decisions are education, age, and race. However, this is for the whole local women population. As the model predicts, the inflow of low-skilled immigrants will influence high- educated women’s marriage, fertility, and working decisions. I will investigate these effects in the following section.

3.4.4 Eﬀects on College-Educated Women

ˆ Yict = α + δ ∗ ln( Actual Immigrationct ) + λ ∗ Collegeict + γXict

ˆ + β ∗ ln( Actual Immigrationct ) ∗ Collegeict + θc + µt + ct (3.3)

Yict are: P rob(Get married), P rob(Give birth to children), P rob(Get divorced), P rob(W orking) and log(W orking hours).

Xict: race dummies, age, age square.

θc and µt are city and time ﬁxed eﬀects. Standard errors are clustered at the city level to allow for the possibility of serial correlation within cities across years. The results are below:

82 Table 3.2 The Eﬀect of Predicted Actual Low-skilled Immigrants on College-educated Women

(1) (2) (3) (4) (5) (6) (7) Prob(Married) Prob(Having Children) Prob(Giving Birth to Children) Prob(Divorced) Prob(Working) ln(Work Hours) ln(Wage) (1980 and 1990 Samples) (2010 Sample)

Actual Immigrationˆ 0.312 -0.691 -1.488 1.042 -0.431 0.0767 -0.663 [0.223] [1.181] [1.084] [0.553] [0.285] [0.0918] [0.402]

College Graduates and Plus 0.216*** -0.936*** -0.0962 -0.649*** 0.739*** 0.635*** 1.347*** [0.0392] [0.0800] [0.108] [0.0594] [0.105] [0.0818] [0.0820]

Actual Immigrationˆ *College Graduates and Plus 0.737*** 0.793* 1.275* -0.934*** 0.512* 0.211 0.823*** [0.169] [0.357] [0.572] [0.180] [0.229] [0.130] [0.205]

Age 0.286*** 0.320*** 0.663*** 0.138*** 0.307*** 0.152*** -0.371*** [0.00918] [0.0138] [0.0336] [0.0134] [0.00233] [0.00290] [0.00237]

Age-square -0.00281*** -0.00280*** -0.0109*** -0.00137*** -0.00374*** -0.00180*** 0.00358*** [0.0000874] [0.000119] [0.000553] [0.000154] [0.0000320] [0.0000304] [0.0000344]

White 0.0219 -0.310*** -0.0877 -0.549*** 0.218*** 0.0937*** 0.205*** [0.0372] [0.0487] [0.0928] [0.0850] [0.0340] [0.0264] [0.0263]

Black -1.108*** 0.0559 0.0111 0.481*** 0.0300 0.0406 0.324*** [0.0525] [0.0625] [0.101] [0.129] [0.0446] [0.0217] [0.0376]

City FE X X X X X X X

Year FE X X X X X X X

Constant X X X X X X X

N 159675 76370 45236 159675 159675 159675 159543

t statistics in parentheses =”* p¡0.05 ** p¡0.01 *** p¡0.001”

The results for the reduced form specification, as detailed in equation (2) are reported in Table 2. All the regressions include year and city fixed effects. I cluster standard errors simultaneously at the city levels. If my hypothesis is correct, I expect the coefficient estimates on the interaction term (Actual Immigrationˆ ∗ College Graduates and P lus) to be positive for

83 marriage, fertility and working decisions – that is, an increase in predicted actual low-skilled immigrant flows should substitute high-educated women’s housework, thereby increasing the marriage rate, fertility rate and labor participation of college-educated women. The first column shows the effect of predicted low-skilled immigrants on college-educated women’s marriage rate. The estimate of the effect of predicted low-skilled immigrants on the college-educated women’s marriage decision is positive and statistically significant at the 0.1% level. College education, older age and being black all contribute to higher marriage rate of women. Column 2 and 3 exhibit the effect of predicted low-skilled immigrants on college-educated women’s fertility decisions. In 1980 and 1990 US Cen- sus samples, the variable that relates to fertility is children ever born. The reduced-form result shows that more low-skilled immigrants positively and significantly increase the fertility rate of college-educated women. The results are the same in the 2010 ACS sample. College education and being black is related to a lower fertility rate. And older women tend to have more children. Column 4 presents the effect of predicted low-skilled immigrants on college-educated women’s divorce decisions. The effect of predicted low-skilled immigrants on divorce rate is negative and statistically significant at the 0.1% level, showing that more low-skilled immigrants occur with fewer college- educated women divorced. White women and women with higher education are more likely to have a stable marriage as shown in the estimation. Column 5 to 7 exhibit the effect of predicted low-skilled immigrants on

84 college-educated women’s working decisions. Low-skilled immigrants have a positive and statistically significant effect on college-educated women’s labor participation and wage. The estimation results show that education, race, and age all contribute to more working and higher income. In summary, the inflow of low-skilled immigrants substitutes high- educated women’s housework, thereby increases the marriage rate, fertility rate, and labor participation of college-educated women. College-educated women are also less likely to be divorced with the inflow. The other factors that contribute to women’s marriage, fertility, and working decisions are education, age, and race.

3.5 Conclusion

This paper investigates how the inflow of low-skilled immigrants affects the household decisions: the divorce decision, the fertility decision, and who marries whom. I provide a model as well as an empirical investigation of the marriage market equilibrium effects of the influx of low-skilled immigrants. I firstly specify an equilibrium model of matching, fertility decision, and divorce decision over the life cycle. The model predicts that the effects of low-skilled immigrants are: (1) marriage market is more positive assortative; (2) couples both with high-education are more likely to have children; (3) couples both with high-education are less likely to get divorced. These changes significantly improve the situation of high-educated women. In the empirical part, I use data from US Censuses in 1970,1980, 1990,

85 2000, and the American Community Survey (ACS) in 2010 to analyze the effect of low-skilled immigrants. I construct an instrumental variable by exploiting the variation in the country of origin in the immigrants in 1970 and predict the actual low-skilled immigration in 1980, 1990, 2000, and 2010. The regression results show that inflow of low-skilled immigrants does not have a significant effect on women’s marriage rate, fertility rate and working decision in general, but it increases college-educated women’s marriage rate, fertility rate, and labor participation and at the same time decreases college-educated women’s divorce rate. Therefore, my empirical results support my model predictions. The inflow of low-skilled immigrants is changing more than people have imagined and has some implications for related policies.

86 Appendices

87 Appendix A

Appendix for Chapter 1

A.1 Proof of Lemma 1

Lemma 1: Under perfect good news, for small ε and large λ, the monopoly ﬁrm selling to consumers as early as possible is an SPE strategy before t∗. For large ε and small λ, the ﬁrm will only start selling after the good news.

Proof: (1) After the arrival of a breakthrough, the firm will set price pt = 1 and make 1 − c profit from each consumer.The value of good news at t = s will be: Z ∞ G −rt (1−ρ)(t−s) ¯ Vs = e (1 − c) ρe Ns dt. s Here, the firm will have the incentive to make the consumer adopt at the first given opportunity, because it is discounting the profit with r.

(2) Without a breakthrough, the ﬁrm will decide at every t whether to sell to the consumers. Consider an one shot deviation, that is at time t, the ﬁrm decide not to sell to consumers by setting pt > 2αt − 1. Then the continuation

88 value is: G ¯ 0 ¯ 0 αtεdt Vt+dt(Nt, αt+dt) + (1 − αtεdt) Vt+dt(Nt, αt+dt),

0 where αt+dt = αt − εαt(1 − αt). −rt ¯ Without deviation, the ﬁrm is earning e (pt − c)ρNt now with continuation value:

¯ G ¯ ¯ Vt(Nt, αt) = αt(λNt+ε)dt Vt+dt(Nt+dt, αt+dt)+[1−αt(λNt+ε)dt] Vt+dt(Nt+dt, αt+dt),

¯ ¯ where αt+dt = αt − (λNt + ε)αt(1 − αt) and Nt+dt = Nt − Nt. The value function of the ﬁrm is then:

¯ G ¯ 0 ¯ 0 Vt(Nt, αt) = max { αtεdt Vt+dt(Nt, αt+dt) + (1 − αtεdt) Vt+dt(Nt, αt+dt), −rt ¯ G ¯ ¯ e (pt − c)ρNt + αt(λNt + ε)dt Vt+dt(Nt+dt, αt+dt) + [1 − αt(λNt + ε)dt] Vt+dt(Nt+dt, αt+dt)} The diﬀerence in continuation value of the two options is:

−rt ¯ G ¯ ¯ ∆Vt =e (pt − c)ρNt + αt(λNt + ε)dt Vt+dt(Nt+dt, αt+dt) + [1 − αt(λNt + ε)dt] Vt+dt(Nt+dt, αt+dt) G ¯ 0 ¯ 0 − αtεdt Vt+dt(Nt, αt+dt) + (1 − αtεdt) Vt+dt(Nt, αt+dt)

When ∆Vt ≥ 0, the ﬁrm will choose to sell immediately; while for ∆Vt ≤ 0, the ﬁrm will choose to wait for the news.

o 1 ) For small ε and large λ, ∆Vt ≥ 0. Intuitively, when the rate of the exogenous signals is very small compared with the rate of endogenous signals, the ﬁrm will have incentive to let the market learn the product to acquire the good news as soon as possible. When ε = 0, the only source of the signals is from the consumers. The ﬁrm will sell immediately to accelerate the market

89 learning process.

o 2 ) For large ε and small λ, ∆Vt ≤ 0. Intuitively, when the arrival rate of exogenous signals is so high that market learning will have no importance in deciding when the good news will come, the endogenous learning process will only the posterior. In the limiting case, when λ = 0, the only trade-off faced by the firm will be to get no-news profit or good-news profit afterwards. For high discount factor r → 1, the firm will choose to wait for the good news and start selling at pt = 1.

¯ To solve for the value function Vt(Nt, αt), we assume the public signal as Poisson process q.  0 with probability (1 − λNt + ε); dq =  1 with probability (λNt + ε).

Then, the change of state variable αt is:

dαt = (λNt + ε)αt(1 − αt)dt + (1 − αt)dq

Then:

G dVt = (λNt + ε)dt[Vt+dt − Vt] + [1 − (λNt + ε)dt][Vt+dt − Vt]

We get a partial diﬀerential equation here:

¯ −rt ¯ G dVt ρVt(αt, Nt) = e (pt − c)ρNt + (λNt + ε)[Vt+dt − Vt] + (λNt + ε)αt(1 − αt) dαt

90 V G is given. V (α , N¯ ) has two state variables and dVt is the partial deriva- t+dt t t t dαt tive. I am not able to solve the partial diﬀerential equation for now. Thus, I do not get the explicit value function.

A.2 Proof of Lemma 2

Lemma 2: In duopoly competition under perfect good news, consumers will adopt from the ﬁrm with good news at the ﬁrst given opportunity.

Proof: Suppose ﬁrm i has the good news and ﬁrm j does not. Firm i will

i j set pt = 2 − 2αt + c. The consumer with an opportunity to adopt must choose among adopting from i, adopting from j and waiting. The consumer will get more utility from ﬁrm i than from ﬁrm j. Then the consumer is comparing

i N 1 − pt and Wt .

N Wt is deﬁned as the supremum of value from all the strategies the can be applied in waiting. If the strategy σ is to adopt at some time before ﬁrm j

i gets good news, the utility will lower than 1−pt since the belief of product j is

i j drifting down and pt = 2 − 2αt + c is drifting up accordingly. If the strategy σ is to adopt at the ﬁrst chance after ﬁrm j gets good news, the expected utility

j αt ρε(1−c) will be (r+ρ)(ε+(r+ρ) . With small ε, the value will nearly be zero. i N Therefore, I get 1−pt > Wt . The consumer will be better-oﬀ adopting from

ﬁrm i.

91 A.3 Proof of Lemma 3

Lemma 3: In duopoly competition under perfect good news, when neither of the firms gets signals, for low arrival rates of exogenous signals ε, both firms will choose to sell immediately; for high arrival rates of exogenous signals ε, both firms will wait for the signals to start selling.

Proof: In duopoly competition, the firms can make decisions to form four kinds situations: 1 firm i sells, firm j does not sell; 2 both firms sell; 3 firm i does not sell, firm j sells; 4 neither of the firms sells. After making each decision, there are also four kinds of consequences: (1) firm i gets a good signal, firm j does not; (2) both firms get good signals; (3) firm i does not get a good signal, firm j gets a good signal; (4) neither of the firms gets good signals.If firm i gets the signal alone, it will be able to make profit: Z ∞ Z m G −i −(m−s)ε −rt i (1−ρ)(t−s) ¯ Vs = αs e e (pt − c) ρe Ns dt dm s s when we suppose the good signals of the other firm comes at time m. If both firms get the signal, they will enter Bertrand competition with zero expected profit. If only firm j gets the signal, firm i will get zero expected profit because with or without the signal, firm i will not make positive profit from now on. If neither of the firm gets a signal, the game continues to time t + dt. Therefore, for firm i, the continuation value of strategy 1 firm i sells, firm j does not sell is: 1 −rt i ¯ i −i G ¯ 00 V = e (pt − c)ρNt + αt(ε + λNt)dt[1 − αt εdt]Vt+dt(Nt+dt, αt+dt)) i −i ¯ 00 +[1 − αt(ε + λNt)dt][1 − αt εdt]Vt+dt(Nt+dt, αt+dt)).

92 For ﬁrm i, the continuation value of strategy 2 both ﬁrms sell is:

e−rt(pi − c)ρN¯ λN λN V 2 = t t + αi(ε + t )dt[1 − α−i(ε + t )dt]V G (N¯ , α0 )) 2 t 2 t 2 t+dt t+dt t+dt λN λN +[1 − αi(ε + t )dt][1 − α−i(ε + t )dt]V (N¯ , α0 )). t 2 t 2 t+dt t+dt t+dt For firm i, the continuation value of strategy 3 firm i does not sell, firm j sells is:

3 i −i G ¯ V = αtεdt[1 − αt (ε + λNt)dt]Vt+dt(Nt+dt, αt+dt)) i −i ¯ +[1 − αtεdt][1 − αt (ε + λNt)dt]Vt+dt(Nt+dt, αt+dt)). For ﬁrm i, the continuation value of strategy 4 neither of the ﬁrms sells is:

4 i −i G ¯ V = αtεdt[1 − αt εdt]Vt+dt(Nt, αt+dt)) i −i ¯ +[1 − αtεdt][1 − αt εdt]Vt+dt(Nt, αt+dt)).

0 λNt 00 Where αt+dt = αt − εαt(1 − αt), αt+dt = αt − ( 2 + ε)αt(1 − αt), αt+dt = ¯ ¯ αt − (λNt + ε)αt(1 − αt) and Nt+dt = Nt − Nt. The value function of the ﬁrm should be:

¯ 1 2 3 4 Vt(Nt, αt) = max{V ,V ,V ,V }

1o) For small ε and large λ, V 1 > V 2 > V 4 > V 3 . That means immediate selling is better than waiting. The logic is the same as that in the monopoly case. Intuitively, when the rate of the exogenous signals is very small compared with the rate of endogenous signals, the ﬁrm will have incentive to let the market learn the product to acquire the good news as soon as possible. When ε = 0, the only source of the signals is from the consumers. The ﬁrm will sell immediately to accelerate the market learning process.

93 However, the two firms are symmetric in all ways. Thus, both firms will 2 2 choose to sell immediately for small enough ε. Therefore, Vt = V . V is the continuation value when both firms use the strategy to sell immediately. To solve for the value function for small ε, we assume the public signal as Poisson process q.

 λNt 0 with probability (1 − + ε); dq = 2 λN  1 with probability ( t + ε). 2 i Then, the change of state variable αt is: λN dαi = ( t + ε)αi(1 − αi)dt + (1 − αi)dq t 2 t t t

Then:

λN λN dV = ( t + ε)dt[V G − V ] + [1 − ( t + ε)dt][V − V ] t 2 t+dt t 2 t+dt t

We get a partial diﬀerential equation here:

−rt i ¯ i ¯ e (pt − c)ρNt λNt G dVt λNt i i ρVt(αt, Nt) = + ( + ε)[Vt+dt − Vt] + i ( + ε)αt(1 − αt) 2 2 dαt 2

G i ¯ dVt Vt+dt is given. Vt(αt, Nt) has two state variables and i is the partial deriva- dαt tive. I am not able to solve the partial diﬀerential equation for now. Thus, I do not get the explicit value function for both ﬁrms immediately selling.

2o) For large ε and small λ, V 3 > V 4 > V 2 > V 1 . That means waiting is better than immediate selling. The logic is the same as that in the monopoly case. Intuitively, when the arrival rate of exogenous signals is

94 so high that market learning will have no importance in deciding when the good news will come, the endogenous learning process will only the posterior. When λ = 0, the only trade-off faced by the firm will be to get no-news profit or good-news profit afterward. For high discount factor r → 1, the firm will choose to wait for the good news and start selling at pt = 1. However, the two firms are symmetric in all ways. Thus, both firms will wait 4 4 for the signals to start selling for large enough ε. Therefore, Vt = V . V is the continuation value when both firms use the strategy to wait for the signal. To solve for the value function for a large ε, we assume the public signal as Poisson process q.  0 with probability (1 − ε); dq =  1 with probability ε.

i Then, the change of state variable αt is:

i i i i dαt = εαt(1 − αt)dt + (1 − αt)dq

Then:

G dVt = εdt[Vt+dt − Vt] + [1 − εdt][Vt+dt − Vt]

We get a partial diﬀerential equation here:

i ¯ G dVt i i ρVt(αt, Nt) = ε[Vt+dt − Vt] + i εαt(1 − αt) dαt

95 In summary, in duopoly competition under perfect good news, when neither of the firms gets signals, for small arrival rate of exogenous signals ε, both firms will choose to sell immediately; for large arrival rate of exogenous signals ε, both firms will wait for the signals to start selling.

96 Appendix B

Appendix for Chapter 2

B.1 Proof of Proposition 1

Proposition 1: (PBE with Myopic Consumers) When ﬁrms are forward- looking but consumers are myopic:

• If α ≥ λL : there is a poaching equilibrium; (1+δ)λH −δλL

• If α < λL : there is a non-poaching equilibrium. (1+δ)λH −δλL

Proof: I will prove by backward induction starting at the last period t = 2. At t = 2, if the consumer is ﬁrm i’s current consumer and the belief about her being λH -type is µ, then the PBE will be:

λL • If µ ∈ [ , 1]: ﬁrm i will charge the consumer λH , and the competitor λH

will charge λH − λL. If the consumer is λH -type, she will stay will the

current supplier and get v − λH . If the consumer is λL-type, she will

switch to the competitor and get still v − λH . Firm i expects to earn

µ ∗ λH from the consumer and ﬁrm j expects to earn (1 − µ) ∗ (λH − λL);

λL • If µ ∈ [0, ]: ﬁrm i will charge the consumer λL, and the competitor λH will charge 0. The consumer will stay will the current supplier and get

v − λL. Firm i will earn λL from the consumer and ﬁrm j will earn 0.

97 At t = 1, if the consumer is ﬁrm i’s current consumer and the belief about her being λH -type is µ, then the PBE will be:

• If µ = 1: ﬁrm i will charge the consumer (1 − δ)λH , and the competitor

will charge −δλH . The consumer will stay will the current supplier and

get (1 + δ)v − λH for t = 1. Firm i will earn λH from the consumer and ﬁrm j will earn 0.

λL • If µ ∈ [ , 1): ﬁrm i will charge the consumer λH −δλL, and the (1+δ)λH −δλL

competitor will charge −δλL. If the consumer is λH -type, she will stay

will the current supplier and expect to get (1+δ)v−(1+δ)λH +δλL from

t = 1 on. If the consumer is λL-type, she will switch to the competitor

and expect to get (1 + δ)v − λL from t = 1 on. Firm i expects to earn

µ ∗ ((1 + δ)λH − δλL) from the consumer and ﬁrm j will earn 0;

λL • If µ ∈ [0, ]: ﬁrm i will charge the consumer λL − δλL, and the (1+δ)λH −δλL

competitor will charge −δλL. The consumer will stay will the current

supplier and get (1 + δ)v − λL for t = 1. Firm i will earn λL from the consumer and ﬁrm j will earn 0.

At t = 0, the two firms are symmetric and both have no consumer. The consumer will choose one supplier from the two. If both firms offer the same price, the consumers will randomly purchase from one. The firms weakly prefer selling to consumers than shutting down. The PBE here will be:

98 • If α ∈ [ λL , 1): both ﬁrms will charge the consumer −δα((1 + (1+δ)λH −δλL

δ)λH − δλL). Consumers will randomly purchase from a supplier;

λL • If α ∈ (0, ]: both ﬁrms will charge the consumer −δλL. Con- (1+δ)λH −δλL sumers will randomly purchase from a supplier.

B.2 Proof of Proposition 2

Proposition 2: (PBE with Forward-looking Consumers) When ﬁrms are forward-looking but consumers are myopic:

• If α ≥ λL : there is a poaching equilibrium; λH

• If α < λL : there is a non-poaching equilibrium. λH

λL • If µ ∈ [ , 1]: ﬁrm i will charge the consumer λH , and the competitor λH

will charge λH − λL. If the consumer is λH -type, she will stay will the

current supplier and get v − λH . If the consumer is λL-type, she will

switch to the competitor and get still v − λH . Firm i expects to earn

µ ∗ λH from the consumer and ﬁrm j expects to earn (1 − µ) ∗ (λH − λL);

λL • If µ ∈ [0, ]: ﬁrm i will charge the consumer λL, and the competitor λH will charge 0. The consumer will stay will the current supplier and get

99 v − λL. Firm i will earn λL from the consumer and ﬁrm j will earn 0.

At t = 1, if the consumer is ﬁrm i’s current consumer and the belief about her being λH -type is µ, then the PBE will be:

• If µ = 1: ﬁrm i will charge the consumer (1 − δ)λH , and the competitor

will charge −δλH . The consumer will stay will the current supplier and

expect to get (1 + δ)v − λH for t = 1. Firm i will earn λH from the consumer and ﬁrm j will earn 0.

λL • If µ ∈ [ , 1): ﬁrm i will charge the consumer (1 − δ)λH , and the com- λH

petitor will charge −δλL. If the consumer is λH -type, she will stay will

the current supplier and expect to get (1 + δ)v − λH for t = 1. If the

consumer is λL-type, she will switch to the competitor and expect to get

(1 + δ)v − λL for t = 1. Firm i expects to earn µ ∗ λH from the consumer and ﬁrm j will earn 0;

λL • If µ ∈ [0, ]: ﬁrm i will charge the consumer λL −δλL, and the competi- λH

tor will charge −δλL. The consumer will stay will the current supplier

and expect to get (1 + δ)v − λL for t = 1. Firm i will earn λL from the consumer and ﬁrm j will earn 0.

100 λL • If α ∈ [ , 1): both ﬁrms will charge the consumer −δαλH . Consumers λH will randomly purchase from a supplier;

λL • If α ∈ (0, ]: both ﬁrms will charge the consumer −δλL. Consumers λH will randomly purchase from a supplier.

B.3 Proof of Proposition 6

Proposition 6: (PBE with Consumers Changing Types) When both ﬁrms and consumers are forward-looking and consumers’ switching costs are randomly changing as stated above:

• If α ∈ [ λL+β2λH −λH , 1): there is a poaching equilibrium; λH (β1+β2−1)

• If α ∈ (0, λL+β2λH −λH ): there is a non-poaching equilibrium. λH (β1+β2−1)

λL • If µ ∈ [ , 1]: ﬁrm i will charge the consumer λH , and the competitor λH

will charge λH − λL. If the consumer is λH -type, she will stay will the

current supplier and get v − λH . If the consumer is λL-type, she will

switch to the competitor and get still v − λH . Firm i expects to earn

µ ∗ λH from the consumer and ﬁrm j expects to earn (1 − µ) ∗ (λH − λL);

101 λL • If µ ∈ [0, ]: ﬁrm i will charge the consumer λL, and the competitor λH will charge 0. The consumer will stay will the current supplier and get

v − λL. Firm i will earn λL from the consumer and ﬁrm j will earn 0.

At t = 1, if the consumer is ﬁrm i’s current consumer and the belief about her being λH -type is µ, then the PBE will be:

• If µ = 1: ﬁrm i will charge the consumer (1 − δ)λH , and the competitor

will charge −δλH . The consumer will stay will the current supplier and

expect to get (1 + δ)v − λH for t = 1. Firm i will earn λH from the consumer and ﬁrm j will earn 0.

λL+β2λH −λH • If µ ∈ [ , 1): ﬁrm i will charge the consumer (1 − δ)λH , and λH (β1+β2−1)

the competitor will charge −δλL. If the consumer is λH -type, she will

stay will the current supplier and expect to get (1 + δ)v − λH for t = 1.

If the consumer is λL-type, she will switch to the competitor and expect

to get (1 + δ)v − λL for t = 1. Firm i expects to earn (1 − δ + β1δ)µ ∗ λH from the consumer and ﬁrm j will earn 0;

λL+β2λH −λH • If µ ∈ [0, ]: ﬁrm i will charge the consumer λL −δλL, and the λH (β1+β2−1)

competitor will charge −δλL. The consumer will stay will the current

supplier and expect to get (1 + δ)v − λL for t = 1. Firm i will earn λL from the consumer and ﬁrm j will earn 0.

At t = 0, the two firms are symmetric and both have no consumer. The consumer will choose one supplier from the two. If both firms offer the same

102 price, the consumers will randomly purchase from one. The ﬁrms weakly prefer selling to consumers than shutting down. The PBE here will be:

• If α ∈ [ λL+β2λH −λH , 1): both ﬁrms will charge the consumer −αδ(1 − δ + λH (β1+β2−1)

β1δ)λH . Consumers will randomly purchase from a supplier;

λL+β2λH −λH • If α ∈ (0, ]: both ﬁrms will charge the consumer −δλL. Con- λH (β1+β2−1) sumers will randomly purchase from a supplier.

103 Appendix C

Appendix for Chapter 3

C.1 Calculation of the Model

I will calculate the model by backward induction and derive some nu- merical results here.

I denote the income of men at t = 0 as Ym, the income of women at t = 0 as

Yw, the income of men at t = 1 as Ym1 if they have children, the income of women at t = 1 as Yw1 if they have children, the income of men at t = 2 as

Ym2 if they divorce and the income of women at t = 2 as Yw2 if they divorce.

C.1.1 T=2

In this period, the decision for the couple is {Divorse, stay}. Couples without children:

If they choose to divorce, both of them will only get their income, Ym and Yw. If they choose to stay in the marriage, they will share the income and match quality ε. The Nash Bargaining problem for the couple is:

argmaxcm (cm + ε − Ym) ∗ (cW + ε − Yw)

The solution is that the couple just consume their own income in the marriage

∗ ∗ t=2,k=0 with cm = Ym and cw = Yw. Their utility from marriage are: EUm (stay) =

104 t=2,k=0 Ym + ε and EUw (stay) = Yw + ε. t=2,k=0 t=2,k=0 The couple will choose divorce if and only if: EUm (stay) ≤ EUm (divorce) t=2,k=0 t=2,k=0 and EUw (stay) ≤ EUw (divorce). The conditions are equivalent to 1 ε ≤ 0, which is of probability 2 . The couple’s expected utility when entering T = 2 are:

1 1 U t=2,k=0 = P r(ε ≤ 0)∗Y +(1−P r(ε ≤ 0))∗(Y +ε) = ∗Y + ∗(Y + [ε|ε > 0]) E m m m 2 m 2 m E 1 1 = Y + ∗ [ε|ε > 0] = Y + M m 2 E m 4 1 U t=2,k=0 = Y + M E w w 4 Couples with children: If the couple with children choose to divorce, the woman will get the children and child support transfer τ by law. The single mother solves the problem:

maxcw,Q cw ∗ Q + τ

s.t. cw + Q = Yw2 + τ

Yw2+τ We get cw = Q = 2 . The father will get less utility from the children and solve the problem of:

maxcm,τ δ ∗ cm ∗ Q − τ

s.t. cm + τ = Ym2

∗ Ym2−Yw2 Ym2+Yw2 This will give us τ = 2 and cm = 2 . 2 t=2,k=1 δ(Ym2+Yw2) (Ym2−Yw2) t=2,k=1 Therefore, we get EUm (divorce) = 8 − 2 and EUw (divorce) =

105 2 (Ym2+Yw2) (Ym2−Yw2) 16 + 2 . If the couple choose to stay in the marriage, they will solve the problem:

maxcm,cw,Q cm ∗ Q + ε

s.t. cm + cw + Q = Ym1 + Yw1

¯ 0 cw ∗ Q + ε ≥ Uw

∗ Ym1+Yw1 Ym1+Yw1 we get Q = 2 and cm + cw = 2 . The household solves the Nash bargaining problem of:

t=2,k=1 t=2,k=1 argmaxcm (cm ∗Q+ε−EUm (divorce))∗(cW ∗Q+ε−EUw (divorce))

4Y 2 −16Y −Y 2 +2δY 2 +8Y Y +4Y 2 +16Y −2Y Y +4δY Y −Y 2 +2δY 2 This gives c∗ = m1 m2 m2 m2 m1 w1 w1 w2 m2 w2 m2 w2 w2 w2 m 16(Ym1+Yw1) 4Y 2 −16Y −Y 2 +2δY 2 +8Y Y +4Y 2 +16Y −2Y Y +4δY Y −Y 2 +2δY 2 and c∗ = Ym1+Yw1 −( m1 m2 m2 m2 m1 w1 w1 w2 m2 w2 m2 w2 w2 w2 ). w 2 16(Ym1+Yw1) This gives the couple’s utility when staying in the marriage:

4Y 2 − 16Y − Y 2 + 2δY 2 + 8Y Y + 4Y 2 + 16Y − 2Y Y U t=2,k=1(stay) = m1 m2 m2 m2 w1 m1 w1 w2 m2 w2 E m 32 4δY Y − Y 2 + 2δY 2 + m2 w2 w2 w2 + ε 32

4Y 2 + 16Y + Y 2 − 2δY 2 + 8Y Y + 4Y 2 − 16Y U t=2,k=1(stay) = m1 m2 m2 m2 w1 m1 w1 w2 E w 32 2Y Y − 4δY Y + Y 2 − 2δY 2 + m2 w2 m2 w2 w2 w2 + ε 32

t=2,k=1 The couple will choose to divorce if and only if: EUm (stay) ≤ t=2,k=1 t=2,k=1 t=2,k=1 EUm (divorce) and EUw (stay) ≤ EUw (divorce). The conditions

106 are equivalent to

−Y + Y δ(Y + Y )2 −4Y 2 + 16Y − 4Y 2 − 16Y ε ≤ m2 w2 + m2 w2 + m1 m2 w1 w2 2 8 32 Y 2 − 2δY 2 − 8Y Y + 2Y Y − 4δY Y + Y 2 − 2δY 2 + m2 m2 w1 m1 m2 w2 m2 w2 w2 w2 = cutoff 32

2 1 −Ym2+Yw2 δ(Ym2+Yw2) , which is of probability 2 + 4M + 16M 2 2 2 2 2 2 −4Ym1+16Ym2+Ym2−2δYm2−8Yw1Ym1−4Yw1−16Yw2+2Ym2Yw2−4δYm2Yw2+Yw2−2δYw2 + 64M = pd. The couple’s expected utility when entering T = 2 are:

Z M Z cutoff t=2,k=1 t=2,k=1 t=2,k=1 EUm = EUm (divorce) dε + EUm (stay) dε cutoff −M

Z M Z cutoff t=2,k=1 t=2,k=1 t=2,k=1 EUw = EUw (divorce) dε + EUw (stay) dε cutoff −M

C.1.2 T=1

In this time period, the decision for the couple is {k = 1 (have children), k = 0 (do not have children)}. Couples without children: If the couple do not have children at T = 1, they will never have children.

∗ They do not share any public goods, and eat their own income, cm = Ym and

∗ cw = Yw. Their expected utility are:

1 U t=1,k=0 = U t=1(k = 0) = Y + U t=2,k=0 = 2Y + M E m E m m E m m 4 1 U t=1,k=0 = U t=1(k = 0) = 2Y + M E w E W w 4

107 Couples with children: If the couple choose to have children, they will solve the problem:

t=2,k=1 maxcm,cw,Q cm ∗ Q + EUm

s.t. cm + cw + Q = Ym1 + Yw1

t=2,k=1 ¯ 00 cw ∗ Q + EUw ≥ Uw

∗ Ym1+Yw1 Ym1+Yw1 We get Q = 2 and cm + cw = 2 . The household solves the Nash bargaining problem of:

t=2,k=1 t=1,k=0 t=2,k=1 t=1,k=0 argmaxcm (cm ∗ Q + EUm − EUm ) ∗ (cw ∗ Q + EUw − EUw )

This gives

2 2 2 ∗ 32Ym + 4Ym1 + 16Ym2 + Ym2 − 2δYm2 − 32Yw + 8Ym1Yw1 cm = 16(Ym1 + Yw1) 4Y 2 − 16Y + 2Y Y − 4δY Y + Y 2 − 2δY 2 + w1 w2 m2 w2 m2 w2 w2 w2 16(Ym1 + Yw1)

∗ Ym1+Yw1 ∗ and cw = 2 − cm. The couple’s expected utility of having children will be:

M ((3 − 2δ)2(Y + Y )4) U t=1,k=1 = c ∗ Q + U t=2,k=1 = + m2 w2 E m m E m 4 4096M 64Y + 8Y 2 + 5Y 2 + 2δY 2 − 64Y + 16Y Y + m m1 m2 m2 w m1 w1 64 8Y 2 + 10Y Y + 4δY Y + 5Y 2 + 2δY 2 + w1 m2 w2 m2 w2 w2 w2 64 M ((3 − 2δ)2(Y + Y )4) U t=1,k=1 = c ∗ Q + U t=2,k=1 = + m2 w2 E w w E w 4 4096M

108 −64Y + 8Y 2 + 5Y 2 + 2δY 2 + 64Y + 16Y Y + m m1 m2 m2 w m1 w1 64 8Y 2 + 10Y Y + 4δY Y + 5Y 2 + 2δY 2 + w1 m2 w2 m2 w2 w2 w2 64

The couple’s expected utility when entering T = 1 are:

t=1 t=1,k=0 t=1,k=1 EUm = max{EUm , EUm }

t=1 t=1,k=0 t=1,k=1 EUw = max{EUw , EUw }.

C.1.3 T=0

In the matching market at T = 0, consider each type of men and women and their possible matches. mH in matching market

For high type of men mH , they can choose to match with high-type women wH , or low-type women wL, or stay single ∅.

t=0 EUm (mH , ∅) = 2YH

I will discuss the utility of men mH match with high-type women wH or low- type women wL in detail in the following section. mL in matching market

For high type of men mL, they can choose to match with high-type women wH , or low-type women wL, or stay single ∅.

t=0 EUm (mL, ∅) = 2YL

109 I will discuss the utility of men mL match with high-type women wH or low- type women wL in detail in the following section.

The situations will be the same for high-type women wH and low-type women wL when making decisions in the matching market.

C.2 Equilibrium Without Low-skilled Immigrants

There are eight possible matches in the model:

{mH &wH , mH &wL, mH &∅, mL&wH , mL&wL, mL&∅, ∅&wH , ∅&wL}

denoted as: {HH, HL, mH , LH, LL, mL, wH , wL}. HL couples

When mH and wL get married, the woman will stay at home looking after children and earn Y 0 if they have children. If the couple chooses to divorce after having children, the single mother will get only Y 0 and child support transfer from the high-income ex-husband. LH couples

When mL and wH get married, the man will stay at home looking after children and earn Y 0 if they have children. If the couple chooses to divorce after having children, the single mother will get only Y 0 and child support transfer from the discounted low-income ex-husband (KYL). HH couples

When mH and wH get married, with 0.5 probability, the woman will stay at

110 home looking after children and earn Y 0 if they have children. If the couple chooses to divorce after having children, the single mother will get only Y 0 and child support transfer from the high-income ex-husband. With 0.5 probability, the man will stay at home looking after children and earn Y 0 if they have children. If the couple chooses to divorce after having children, the single mother will get only Y 0 and child support transfer from the discounted high-income ex-husband (KYH ). LL couples

When mL and wL get married, with 0.5 probability, the woman will stay at home looking after children and earn Y 0 if they have children. If the couple chooses to divorce after having children, the single mother will get only Y 0 and child support transfer from the low-income ex-husband. With 0.5 probability, the man will stay at home looking after children and earn Y 0 if they have children. If the couple chooses to divorce after having children, the single mother will get only Y 0 and child support transfer from the discounted low-income ex-husband (KYL). The solution of the equilibrium depends on the value of the variables and gets

0 very complicated to solve. Thus, I set Y = 0, YL = 10, YH = xYL, M = 100 (match quality ε ∼ U[−M,M] ), K = 0.7 and δ = 0.8 to numerically solve the equilibrium.

In all the graphs below, the x-axis represents the ratio YH . YL The divorce rate of these matches in the divorce decision stage is:

111 Divorce Rate 0.50 Divorcelh Divorcell 0.45

0.40

0.35 Divorcehh

Yh/YlDivorcehl 1.5 2.0 2.5 3.0

Figure C.1. Divorce Rate of Couples Without Low-skilled Immigrants

The fertility rate of these matches in the fertility decision stage is:

Fertility Rate

Fertilityhl 0.25

Fertilityhh 0.20

0.15 Fertilitylh 0.10

0.05

Yh/YlFertilityll 2 4 6 8 10

Figure C.2. Fertility Rate of Couples Without Low-skilled Immigrants

The expected utility of men in these matches in the matching stage is:

Expected Utility

250 ExpectedUtilityhl

200 ExpectedUtilityhh

150 ExpectedUtilitylh 100 ExpectedUtilityho 50 ExpectedUtilityll Yh/YlExpectedUtilitylo 1.5 2.0 2.5 3.0

Figure C.3. Expected Utility of Men Without Low-skilled Immigrants

112 The expected utility of women in these matches in the matching stage is:

Expected Utility ExpectedUtilityhl 200 ExpectedUtilityhh ExpectedUtilitylh 150

100 ExpectedUtilityoh 50 ExpectedUtilityll ExpectedUtilityol Yh/Yl 1.5 2.0 2.5 3.0

Figure C.4. Expected Utility of Women Without Low-skilled Immigrants

C.3 Equilibrium With Low-skilled Immigrants

Again, for the eight possible matches in the model:

{mH &wH , mH &wL, mH &∅, mL&wH , mL&wL, mL&∅, ∅&wH , ∅&wL}

denoted as: {HH, HL, mH , LH, LL, mL, wH , wL}. HL couples

113 support transfer from the discounted low-income ex-husband (KYL). HH couples

When mH and wH get married, they will hire a nanny and pay YL if they have children. If the couple chooses to divorce after having children, the single mother will hire a nanny paying YL, but she will get child support transfer from the high-income ex-husband. LL couples

0 very complicated to solve. Thus, I set Y = 0, YL = 10, YH = xYL, M = 100 (match quality ε ∼ U[−M,M] ), K = 0.7 and δ = 0.8 to numerically solve the equilibrium.

In all the graphs below, the x-axis represents the ratio YH . YL The divorce rate of these matches in the divorce decision stage is:

114 Divorce Rate

0.5 Divorcell

0.4

Divorcelh

0.3 Divorcehl

0.2

Yh/YlDivorcehh 0.1 1.5 2.0 2.5 3.0 Figure C.5. Divorce Rate of Couples With Low-skilled Immigrants

The fertility rate of these matches in the fertility decision stage is:

Fertility Rate

Fertilityhh

0.3

0.2 Fertilityhl Fertilitylh 0.1

Yh/YlFertilityll 1.5 2.0 2.5 3.0

Figure C.6. Fertility Rate of Couples With Low-skilled Immigrants

The expected utility of men in these matches in the matching stage is:

Expected Utility

400 ExpectedUtilityhh

300 ExpectedUtilityhl 200 ExpectedUtilitylh

100 ExpectedUtilityho ExpectedUtilityll Yh/YlExpectedUtilitylo 1.5 2.0 2.5 3.0

Figure C.7. Expected Utility of Men With Low-skilled Immigrants

115 The expected utility of women in these matches in the matching stage is:

Expected Utility

400 ExpectedUtilityhh

300 ExpectedUtilitylh 200 ExpectedUtilityhl

100 ExpectedUtilityoh ExpectedUtilityll Yh/YlExpectedUtilityol 1.5 2.0 2.5 3.0

Figure C.8. Expected Utility of Women With Low-skilled Immigrants

Table C.1 List of Cities Included in the Sample

Baltimore, MD Hartford-Bristol-Middleton- New Britain, CT Rochester, NY Boston, MA/NH Honolulu, HI Sacramento, CA Bridgeport, CT Houston-Brazoria, TX San Antonio, TX Buﬀalo-Niagara Falls, NY Los Angeles-Long Beach, CA San Diego, CA Chicago, IL Miami-Hialeah, FL San Francisco-Oakland-Vallejo, CA Cleveland, OH New Haven-Meriden, CT San Jose, CA Dallas-Fort Worth, TX New York, NY-Northeastern NJ Seattle-Everett, WA Detroit, MI Philadelphia, PA/NJ Ventura-Oxnard-Simi Valley, CA El Paso, TX Providence-Fall River-Pawtucket, MA/RI Washington, DC/MD/VA Fresno, CA Riverside-San Bernardino, CA Worcester, MA

116 Table C.2 Predicted Low-skilled Immigration In Selected Cities

City Name 1980 1990 2000 2010 Baltimore, MD 13118.98402 11855.69169 10624.64755 9449.344235 Boston, MA/NH 45842.92668 51851.7118 33987.01152 35660.24196 Bridgeport, CT 7978.621902 9462.076727 10141.08786 6984.190853 Buﬀalo-Niagara Falls, NY 72085.30769 109529.8852 165903.0363 174523.1092 Chicago, IL 152653.96 176518.6516 204091.1998 179404.9279 Cleveland, OH 17244.71924 20331.72334 17700.64885 15356.60692 Dallas-Fort Worth, TX 8310.354399 16520.31568 23915.04616 24272.61682 Detroit, MI 24140.95236 22988.81975 27736.54618 18997.25715 El Paso, TX 12725.85401 33244.0536 51139.05403 53448.90671 Fresno, CA 17183.19658 20928.27094 33846.67988 35295.91623 Hartford-Bristol-Middleton- New Britain, CT 102751.9899 157788.2044 248440.4475 265298.8545 Honolulu, HI 54396.00739 83136.73727 89018.87747 81098.54912 Houston-Brazoria, TX 13063.85786 28211.58269 33584.94725 32639.37247 Los Angeles-Long Beach, CA 359126.7824 572951.5759 667533.981 681510.4967 Miami-Hialeah, FL 500714.8474 732067.4829 993260.6385 1008131.596 New Haven-Meriden, CT 20128.08478 34233.65655 6779.374438 4746.376109 New York, NY-Northeastern NJ 759313.7721 1025914.525 1143946.281 1133924.898 Philadelphia, PA/NJ 170498.2525 253334.6129 358681.2987 369071.2159 Providence-Fall River-Pawtucket, MA/RI 38686.67776 58165.30329 83487.26465 88470.57173 Riverside-San Bernardino, CA 11541.35414 23745.98063 33570.25115 34802.85311 Rochester, NY 13677.42811 18177.00446 22131.97865 25968.64287 Sacramento, CA 16018.44277 19763.17401 26633.70045 22340.96143 San Antonio, TX 11311.79944 29215.53222 43923.35081 46428.68654 San Diego, CA 15566.27333 30485.72989 40636.76505 39833.15137 San Francisco-Oakland-Vallejo, CA 310235.237 482088.4722 625691.1385 660872.7853 San Jose, CA 26163.17997 36767.1662 37824.98556 37084.0847 Seattle-Everett, WA 54300.36685 80765.49614 39918.47163 16697.25095 Ventura-Oxnard-Simi Valley, CA 9167.621104 22307.59606 33528.77398 34765.74093 Washington, DC/MD/VA 34966.5894 51938.76297 25602.17216 25284.00081 Worcester, MA 153166.5589 123917.6837 108266.2434 116175.0435

117 Figure C.9. Predicted Low-skilled Immigrants

118 Figure C.10. The Relationship of Predicted and Actual Numbers of Low-skilled Immigrants

119 Bibliography

Bergemann, Dirk, and Juuso V¨alim¨aki. 1996. “Learning and strategic pricing.” Econometrica: Journal of the Econometric Society, 1125–1149.

Bergemann, Dirk, and Juuso Välimäki. 1997. “Market diffusion with two-sided learning.” The RAND Journal of Economics, 773–795.

Bergemann, Dirk, and Juuso V¨alim¨aki. 2000. “Experimentation in markets.” The Review of Economic Studies, 67(2): 213–234.

Bergemann, Dirk, and Juuso Välimäki. 2002. “Entry and vertical differ- entiation.” Journal of Economic Theory, 106(1): 91–125.

Bergemann, Dirk, Juuso Valimaki, et al. 1996. “Market experimentation and pricing.” Cowles Foundation for Research in Economics, Yale University.

Bhaskar, Venkataraman. 2019. “The demographic transition and the po- sition of women: A marriage market perspective.” The Economic Journal, 129(624): 2999–3024.

Bolton, Patrick, and Christopher Harris. 1999. “Strategic experimentation.” Econometrica, 67(2): 349–374.

Bonatti, Alessandro. 2011. “Menu pricing and learning.” American Eco- nomic Journal: Microeconomics, 3(3): 124–163.

120 Calem, Paul S, and Loretta J Mester. 1995. “Consumer behavior and the stickiness of credit-card interest rates.” The American Economic Review, 85(5): 1327–1336.

Chen, Yongmin. 1997. “Paying customers to switch.” Journal of Economics & Management Strategy, 6(4): 877–897.

Chen, Yongmin, and Jason Pearcy. 2010. “Dynamic pricing: when to entice brand switching and when to reward consumer loyalty.” The RAND Journal of Economics, 41(4): 674–685.

Chen, Yuxin, and Z John Zhang. 2009. “Dynamic targeted pricing with strategic consumers.” International Journal of Industrial Organization, 27(1): 43–50.

Chiappori, Pierre-Andr´e,Bernard Salani´e,and Yoram Weiss. 2015. “Partner choice and the marital college premium: Analyzing marital patterns over several decades.”

Chiappori, Pierre-Andr´e,Monica Costa Dias, and Costas Meghir. 2018. “The marriage market, labor supply, and education choice.” Journal of Political Economy, 126(S1): S26–S72.

Choo, Eugene, and Aloysius Siow. 2006. “Who marries whom and why.” Journal of political Economy, 114(1): 175–201.

Cortes, Patricia. 2008. “The eﬀect of low-skilled immigration on US prices: evidence from CPI data.” Journal of political Economy, 116(3): 381–422.

121 Cortes, Patricia, and Jessica Pan. 2019. “When time binds: substitutes for household production, returns to working long hours, and the skilled gender wage gap.” Journal of Labor Economics, 37(2): 351–398.

Cortes, Patricia, and Jos´eTessada. 2011. “Low-skilled immigration and the labor supply of highly skilled women.” American Economic Journal: Applied Economics, 3(3): 88–123.

Esteves, Rosa-Branca. 2010. “Pricing with customer recognition.” Interna- tional Journal of Industrial Organization, 28(6): 669–681.

Felli, Leonardo, and Christopher Harris. 1996. “Learning, wage dynamics, and ﬁrm-speciﬁc human capital.” Journal of Political Economy, 104(4): 838–868.

Frick, Mira, and Yuhta Ishii. 2016. “Innovation Adoption by Forward- Looking Social Learners.”

Fudenberg, Drew, and Jean Tirole. 2000. “Customer poaching and brand switching.” RAND Journal of Economics, 634–657.

Judd, Kenneth L, and Michael H Riordan. 1994. “Price and quality in a new product monopoly.” The Review of Economic Studies, 61(4): 773–789.

Laiho, Tuomas, and Julia Salmi. 2017. “Social learning and monopoly pricing with forward looking buyers.”

122 Reynoso, Ana. 2018. “The impact of divorce laws on the equilibrium in the marriage market.” Unpublished manuscript.

Schlee, Edward E. 1996. “The value of information about product quality.” The RAND Journal of Economics, 803–815.

Schlee, Edward E. 2001. “Buyer experimentation and introductory pricing.” Journal of Economic Behavior & Organization, 44(3): 347–362.

Shaﬀer, Greg, and Z John Zhang. 2000. “Pay to switch or pay to stay: preference-based price discrimination in markets with switching costs.” Journal of Economics & Management Strategy, 9(3): 397–424.

Sobel, Joel. 1991. “Durable goods monopoly with entry of new consumers.” Econometrica: Journal of the Econometric Society, 1455–1485.

Stango, Victor. 2002. “Pricing with consumer switching costs: Evidence from the credit card market.” The Journal of Industrial Economics, 50(4): 475–492.

Taylor, Curtis R. 2003. “Supplier surﬁng: Competition and consumer behavior in subscription markets.” RAND Journal of Economics, 223–246.

Villas-Boas, J Miguel. 1999. “Dynamic competition with customer recognition.” The Rand Journal of Economics, 604–631.

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