European Economic Review 75 (2015) 131–155

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European Economic Review journal homepage: www.elsevier.com/locate/eer

How urbanization affect employment and social interactions Yasuhiro Sato a, Yves Zenou b,c,n a

Osaka University, Japan Stockholm University, IFN, Sweden c CEPR, UK b

a r t i c l e i n f o

abstract

Article history: Received 20 February 2014 Accepted 20 January 2015 Available online 2 February 2015

We develop a model where the unemployed workers in the city can find a job either directly or through weak or strong ties. We show that, in denser areas, individuals choose to interact with more people and meet more random encounters (weak ties) than in sparsely populated areas. We also demonstrate that, for a low urbanization level, there is a unique steady-state equilibrium where workers do not interact with weak ties, while, for a high level of urbanization, there is a unique steady-state equilibrium with full social interactions. We show that these equilibria are usually not socially efficient when the urban population has an intermediate size because there are too few social interactions compared to the social optimum. Finally, even when social interactions are optimal, we show that there is over-urbanization in equilibrium. & 2015 Elsevier B.V. All rights reserved.

JEL classification: J61 R14 R23 Keywords: Weak ties Strong ties Urban Economics Labor market

1. Introduction It is well-established that in denser and more populated areas (such as big cities), individuals have more random contacts (weak ties) and thus are more likely to have bigger networks than in less dense areas. Sociologists argue that relationships in large cities are less personal. People in large cities, in comparison with people in small towns or rural areas, experience general deficits in the quality of interpersonal relations (strong ties).1 However, people in small towns or rural areas base their social networks on the limited number of people who live nearby whereas people in large cities have a great deal of choice in constructing their social networks and can seek out others with similar values, interests, and life-styles.2 As a result, urbanites are less likely than rural dwellers to base their personal networks on traditional sources (such as family) and are more likely to include voluntary sources, such as friends, coworkers and club members. The aim of this paper is to propose a simple model that captures and explains these facts and analyze the consequences in the labor market. To be more precise, we develop a model where each agent meet strong and weak ties who can help them find a job.3 We define a weak tie when the social interaction between two persons is transitory (like for example random encounters). On the contrary, a person has a strong-tie when the relationship is repeated over time, for example members of the same n

Corresponding author. Tel.: +468162880. E-mail addresses: [email protected] (Y. Sato), [email protected] (Y. Zenou). 1 This is the perspective of the so-called social disorganization theory and the social capital literature (see e.g. Wirth, 1938; Coleman, 1988; Putnam, 1993, 2001). 2 This is the so-called subculture theory (see e.g. Fischer, 1982). 3 The fact that workers use their friends and relatives (social networks) to find a job is well-documented. See, e.g. Ioannides and Loury (2004). http://dx.doi.org/10.1016/j.euroecorev.2015.01.011 0014-2921/& 2015 Elsevier B.V. All rights reserved.

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family or very close friends.4 Each individual can be in three different states: either she is unemployed and her strong tie is also unemployed or she is unemployed and her strong tie is employed (or the reverse) or both are employed. Workers move between the city and the rural area. In the city, the unemployed workers can find a job either directly or through weak and strong ties and the unemployed workers have to decide how much time (effort) to spend with weak ties. This depends on own effort, on agglomeration economics (since in denser areas, workers tend to meet more people) and on the competition in the labor market (the more employed workers live in the city, the easiest is to meet one of them; the same is true for the unemployed workers). In Proposition 1, we show that, in denser areas (i.e. more populated areas like cities), individuals choose to interact with more people and meet more random encounters (weak ties) than in sparsely populated areas. Although these relationships may not be personal nor strong, yet those weak ties are the ones to matter most for providing social support, in particular in finding jobs. This result is in line with the empirical evidence from the sociological literature (see e.g. Wirth, 1938; Coleman, 1988, and Putnam, 1993, 2001).5 Indeed, Fischer (1982) found that urban dwellers had more dispersed networks containing a higher proportion of non-kin relations than did rural dwellers. This concurs with Wellman's (1979) research in a number of Toronto neighborhoods demonstrating that personal networks are geographically dispersed with large variations in the number of contacts living in the neighborhood. In a review of different studies in the US, Korte (1980) concluded that urbanism positively affects only those relationships which are peripheral; central relationships including ties between families and friends remained unchanged. Palisi and Canning (1986) found that urbanism was positively associated with the frequency of interaction among friends. Although individuals may have fewer strong relationships in cities than in villages, they have more random encounters (weak ties) as shown in Proposition 1, which are more important for support. As Granovetter's (1973) seminal work on the strength of the weak ties argues, weak ties are superior to strong ties for providing support in getting a job. He criticized the assumption that strong ties in close networks were strong in resource terms. Using the example of searching for a job, Granovetter found that neighborhood based close networks were limited in getting information about possible jobs (see also Lin and Dumin, 1986). In a close networks everyone knows each other, information is shared and so potential sources of information are quickly shaken down, the networks quickly becomes redundant in terms of access to new information. In contrast Granovetter stresses the strength of weak ties involving a secondary ring of acquaintances who have contacts with networks outside ego's network and therefore offer new sources of information on job opportunities. The network arrangements in play here involve only partially overlapping networks composed mainly of single-stranded ties. Amato (1993) examines the differences between urban and rural dwellers, as well as between large cities' and smaller towns' inhabitants, in the breadth of assistance received and provided by friends and family. He finds that urbanites receive more help from friends than do rural dwellers, give more help to friends, expect more help from friends, and expect less help from relatives. In other words, he finds little support for the social disorganization theory that argues that urban dwellers receive and provide less support from friends and relatives compared to rural people. In a study on the relationships between health outcomes and social networks in several east London neighborhoods, Cattell (2001) concluded that the most robust networks in terms of health outcomes are those Solidarity Networks that combine positive aspects of dense and loose networks. They consist of a wide range of membership groups, made up of similar and dissimilar people involving strong local contacts of family or local friends and neighbors on the one hand, plus participation in formal and informal organizations on the other. As Cattell concludes “the more varied the network, the greater the range of resources accessible, and the greater the potential benefits to health.” (Cattell, 2001, p. 1513). In economics, there are few papers testing this type of relationship. Wahba and Zenou (2005) is an exception and they find that, in Egypt, in denser cities, people are more likely to find a job through weak ties than in less dense cities. In Proposition 2, we demonstrate that, for low population levels, there is a unique steady-state equilibrium where workers do not interact with weak ties (the so-called No-Interaction Equilibrium), while, for high level of population, there is a unique steady-state equilibrium with full social interactions (the so-called Full-Interaction Equilibrium). We also show in Proposition 3 that the latter equilibrium exhibits lower unemployment rate and higher urbanization than the former equilibrium. This is an important result since it indicates that the size of the total population affects social interactions (with weak ties), which, in turn, have an impact on the unemployment rate and degree of urbanization in the region. In other words, we show that regions with a small population size experience a higher unemployment rate and a lower urbanization rate and workers socially interact less than in regions with a larger population size. There is thus a virtuous circle of agglomeration where the bigger is the population, the higher are the benefits from interacting, which leads to more jobs and more migration to the city. Even though there is no direct evidence on the relation between social interactions and agglomeration, there is clear evidence on the positive relationship between density (i.e. urbanization) and productivity (Ciccone and Hall, 1996; Rosenthal and Strange, 2004; Combes et al., 2010) and on the negative relationship between

4 This is not the precise definition of weak ties first used by Granovetter. In Granovetter's (1973), weak ties are expressed in terms of lack of overlap in personal networks between any two agents; i.e. weak ties refer to a network of acquaintances who are less likely to be socially involved with one another. Formally, two agents A and B have a weak tie if there is little or no overlap between their respective personal networks. Vice versa, the tie is strong if most of A's contacts also appear in B's network. See also Granovetter (1974, 1983). 5 This, in fact, goes back to Tonnies (1957) and Simmel (1995) with the idea of rural gemeinschaft (or community) and urban gesellchaft (or association).

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agglomeration and unemployment rate (Essletzbichler, 2004, 2007; Faberman, 2005; Blien et al., 2006; Mitra and Sato, 2007; Hilber and Voicu, 2009; Micucci and di Giacinto, 2009).6 Conditions that ensure the existence and uniqueness of equilibrium are provided in Proposition 4. In Propositions 5 and 6, we derive some comparative statics results showing how the different parameters of the model (such as, for example, the rate at which the unemployed workers hear directly of a job or the urban wage) affect the urban population size and the urban unemployment rate in each type of equilibrium. Finally, in the last part of the paper, we derive the social optimum in terms of social interactions (Proposition 7) and urbanization level (Proposition 8) and show that the choice of the planner usually does not coincide with that of the market: When the urban population has an intermediate size, there are too few social interactions compared to the social optimum. Even when social interactions are optimal, we show that there is overurbanization in equilibrium. The rest of the paper unfolds as follows. In the next section, we describe the relation to the literature. Section 3 describes the basic environment of the model while, in Section 4, we determine the steady-state equilibrium. In Section 5, we derive the comparative statics properties of our equilibrium and discuss the efficiency results in Section 6. Finally, Section 7 concludes the paper. All the proofs of the propositions can be found in Appendix A.2. 2. Related literature Our paper contributes to the literature on “social interactions and cities”, which is a small but growing field. There are, in fact, few papers that explicitly model both aspects. Urban economics and economics of agglomeration: There is an important literature in urban economics looking at how interactions between agents create agglomeration and city centers.7 However, very few models have put forward the role of social interactions in the agglomeration process. Beckmann (1976) was among the first to propose an urban model with global social interactions. This model describes the urban structure of a single city and shows that, in equilibrium, agents are distributed according to a unimodal spatial distribution. More recently, Mossay and Picard (2011, 2013) propose interesting models in which each agent visits other agents so as to benefit from face-to-face communication (social interactions) and each trip involves a cost which is proportional to distance. The models provide an interesting discussion of spatial issues in terms of use of residential space and formation of neighborhoods and show under which condition different types of city structure emerge. Furthermore, Ghiglino and Nocco (2012) extend the standard economic geography model a la Krugman to incorporate conspicuous consumption. In their model, agents are sensitive to comparisons within their own type group as well as with agents that are outside their own type group. They show that agglomeration patterns depend on the network structure where agents are embedded in. All these models are different from ours since there is no labor market and weak and strong ties are not explicitly modeled. Peer effects, social networks and urbanization: There is a growing interest in theoretical models of peer effects and social networks (see e.g. Akerlof, 1997; Glaeser et al., 1996; Ballester et al., 2006, Calvó-Armengol et al., 2009). However, there are very few papers that explicitly consider the interaction between the social and the geographical space.8 Brueckner et al. (2002), Helsey and Strange (2007), Brueckner and Largey (2008) and Helsley and Zenou (2014) are exceptions but, in all these models either the labor market is not included or weak and strong ties are not modeled. Zenou (2013) is the only paper that has both aspects but the focus is totally different since the paper mainly explains the differences between blacks and whites in terms of labor market outcomes.9 Schelling (1971) is clearly a seminal reference when discussing social preferences and location. Shelling's model shows that even a mild preference for interacting with people from the same community can lead to large differences in terms of location decision. Indeed, his results suggest that total segregation persists even if most of the population is tolerant about heterogeneous neighborhood composition.10 Our model is very different from models a la Schelling since we focus on weak and strong ties and their impact on labor-market outcomes. To the best of our knowledge, our paper is the first one to provide a model that shows how urbanization affects social interactions. We show that workers interact more with their weak ties in more urbanized areas. Thus, the paper provides a first stab at a very important question in both social networks and urban economics. 3. Basic environment The total size of the population is N. People can live either in the city, with population size being Nc, or in the rural area, with population size being Nr, where N r þ Nc ¼ N. 6

For a very complete overview of this literature, see Combes and Gobillon (2015). See Fujita and Thisse (2013) for a literature review. 8 Recent empirical researches have shown that the link between these two spaces is quite strong, especially within community groups (see e.g. Bayer et al., 2008; Hellerstein et al., 2011; Ioannides and Topa, 2010; Patacchini and Zenou, 2012; Topa, 2001). See (Ioannides, 2012, Chapter 5) and Topa and Zenou (2015) who review the literature on social interactions and urban economics. 9 See also Calvó-Armengol et al. (2007), Calvó-Armengol and Jackson (2004) and Zenou (2015) for models of weak and strong ties in the labor market but where the urban space is not modeled. 10 This framework has been modified and extended in different directions, exploring, in particular, the stability and robustness of this extreme outcome (see, for example, Zhang, 2004 or Grauwin et al., 2012). 7

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3.1. Dyads We assume that individuals belong to mutually exclusive two-person groups, referred to as dyads. We say that two individuals belonging to the same dyad hold a strong tie to each other. We assume that dyad members do not change over time. A strong tie is created once and for all and can never be broken. We can think, for example, of a married couple (or members from the same family) so that they tend to stay together for a long time. Each individual can be in either of the two different states: employed or unemployed. Dyads, which consist of paired individuals, can thus be in three different states,11 which are the following: (i) both members are employed –we denote the number of such dyads by d2; (ii) one member is employed and the other is unemployed (d1); (iii) both members are unemployed (d0).

3.2. Aggregate state By denoting the employment level and the unemployment level in the city at time t by Ec(t) and Uc(t), we have ( Ec ðtÞ ¼ 2d2 ðtÞ þd1 ðtÞ U c ðtÞ ¼ 2d0 ðtÞ þ d1 ðtÞ Since the total city population is Nc, we have Ec ðtÞ þ U c ðtÞ ¼ N c or, alternatively, d2 ðt Þ þ d1 ðt Þ þ d0 ðt Þ ¼

Nc 2

3.3. Rural versus urban areas We assume that only people belonging to a d0 dyad can freely choose where to live between the urban and the rural area. The other people, belonging either to a d1 or d2 dyad, who are mostly employed, stay for ever in the city c. This is because workers in a d0 dyad are all unemployed and thus are ready to move to improve their utility. On the contrary, workers from the other dyads, who are mostly employed, will not find it optimal to move to the rural area. In Appendix A.1, we show that this is optimal in equilibrium. 3.4. Information transmission in the city The labor market in the city is not perfectly competitive because, for example, of search frictions. Let us now describe the information transmission about jobs in the city. Each job offer is taken to arrive to both employed and unemployed workers at rate λ. If an employed worker hears about a job, she automatically direct it to her strong tie. This is a convenient modelling assumption, which stresses the importance of on-the-job information (Ioannides and Loury, 2004). All jobs and all workers are identical (unskilled labor) so that all employed workers obtain the same wage wc. Therefore, the employed workers, who hear about a job, pass this information on to their current matched partner since they cannot use this information for themselves. They can also transmit this job information to a weak tie if they meet one. As stated above, we assume that only members of a d0 dyad can migrate. Since they are newcomers in the city relatively to other dyad members, an unemployed worker in a d0 dyad is assumed to have no social connections, i.e. no contact with weak ties. This is because it takes time to meet weak ties and people who are stuck in a d0 dyad tend to interact mostly with their strong ties and isolate themselves from the urban community (see Zenou (2013), who demonstrates such a result in an explicit urban framework). As a result, the only way they can find a job is by hearing directly about it at rate λ. On the contrary, an unemployed worker in a d1 dyad can meet weak ties and obtain job information from them. We denote by γ A ½0; γ  the effort of interacting with weak ties for an unemployed worker in a d1 dyad. There is an effort cost, which is given by βγ (β 4 0). We also assume that there are agglomeration effects so that the higher is the city population Nc, the easier is to meet weak ties (see, e.g. Desmond, 2012, and the Introduction for evidence). The agglomeration effect is captured by ϕðN c ðtÞÞ, with ϕ0 ð:Þ 4 0, ϕ″ r 0 and 0 r ϕ r ϕðÞ r ϕ r 1. The concavity of ϕ means that the smaller the marginal agglomeration effect gets the larger the city population is because of the increasing duplication or congestion of weak 11

The inner ordering of dyad members does not matter.

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ties. We make this assumption to eliminate the possibility that the sufficient condition for unique equilibrium, which will be given below, never holds true. As a result, at time t, an unemployed worker in a d1 dyad will hear about a job from a weak tie at rate γϕðNc ðtÞÞ2λd2 ðtÞ=d1 ðtÞ. Indeed, the rate at which an unemployed worker obtains a job from a weak tie is increasing in her own effort γ, in the city population Nc(t) and depends on 2λd2 ðtÞ=d1 ðtÞ, the fraction of employed workers who are aware about a job. In other words, 2λd2 ðtÞ=d1 ðtÞ captures something similar to the matching function in the search literature (Pissarides, 2000) where the number of employed workers in the d2 dyad is 2d2 ðtÞ while the number of unemployed workers who are in competition for job information is d1 ðtÞ.12 Hence, an unemployed worker in a d1 dyad can find a job either directly at rate λ or through her strong tie at rate λ or through her weak tie at rate γϕðNc ðtÞÞ2λd2 ðtÞ=d1 ðtÞ. Letting g(t) be the rate at which an unemployed worker in a d1 dyad finds a job (or equivalently the rate at which a d1 dyad becomes a d2 dyad), we have g ðt Þ  2λ þγϕðNc ðtÞÞ2λ

d2 ðtÞ d1 ðtÞ

ð1Þ

Finally, we assume that each dyad “dies” at rate δ. If we think of the married couple interpretation of a dyad, this means that the couple in the dyad retires and leave the economy. In that case, they are replaced by a new d0 dyad.13 For example, if a d1 dyad “dies”, then a new d0 dyad will be created. The rational for this is that a new dyad is composed of young workers who have not worked yet. This is an overlapping generation model. As a result, in the city, at each period of time t, δðd2 ðtÞ þd1 ðtÞ þd0 ðtÞÞ die and δðd2 ðtÞ þ d1 ðtÞ þ d0 ðtÞÞ ¼ δN c ðtÞ=2 are born as a d0 dyad. Observe that λ, the rate at which a person hears from a job, is individual specific while δ, the rate at which a dyad dies, is dyad specific. This information transmission protocol defines a continuous time Markov process. The state variable is the relative size of each type of dyad. Transitions depend on labor market turnover and the nature of social interactions as captured by γ. Because of the continuous time Markov process, the probability of a two-state change is zero (small order) during a small interval of time t and t þdt. This means, in particular, that both members of a dyad cannot change their status at the same time. For example, two unemployed workers cannot find a job at the same time, i.e. during t and t þdt, the probability assigned to a transition from a d0 -dyad to a d2 -dyad is zero. Similarly, two employed workers (d2 -dyad) cannot both become unemployed, i.e. switch to a d0 -dyad during t and t þ dt. This applies to all other dyads mentioned above. 3.5. Rural labor market The rural labor market is perfectly competitive and has no friction. In other words, everybody can obtain a job in the rural area and it is assumed that the rural wage is flexible enough to guarantee that there is full-employment.14 The wage in the rural area wr is thus determined by the marginal productivity of workers, i.e. 0

wr ¼ f ðN  Nc Þ

ð2Þ 0

″

0

As usual, we assume that f ð:Þ 4 0 and f ð:Þ r0 and that the Inada conditions hold (i.e., f ð0Þ ¼ 0, limNr -0 f ðN r Þ ¼ 1, and 0 limNr -1 f ðNr Þ ¼ 0). These assumptions reflect the implicit assumption that the land endowment is limited and the agricultural sector exhibits decreasing returns with respect to labor input. In rural areas, people live in family within a dyad. As in the urban area, a dyad that dies (at rate δ) is automatically replaced by a new dyad (but as a d2 dyad since there is full employment). As a result, in rural areas, all workers are in a d2 dyad forever. When they migrate to the city, they switch from a d2 dyad from the rural area to a d0 dyad in the city. 3.6. Flows of dyads between states It is readily checked that the net flow of dyads in the city from each state between a small interval of time t and t þ dt is given by 8  > > d2 ðtÞ ¼ gðtÞd1 ðtÞ  δd2 ðtÞ > > <  ð3Þ d1 ðtÞ ¼ 2λd0 ðtÞ  ðδ þgðtÞÞd1 ðtÞ > >  > >  δN c ðtÞ 1 : d0 ðt Þ ¼ 2  2λd0 ðt Þ  δd0 ðt Þ þ 2N c ðt Þ

12 Observe that, in this expression, we take into account the effort level of other job searchers. Let γa denote the average effort level in the city. We assume that (i) each worker's job arrival rate increases with the relative effort level, γ=γ a and (ii) a higher γa results in more matches with weak ties in aggregate. The job arrival rate from weak ties then becomes

γ d2 ðtÞ d2 ðtÞ ¼ γϕðN c ðtÞÞ2λ ϕðN c ðtÞÞ γ a 2λ γa d1 ðtÞ d1 ðtÞ 13 The assumption that dyad re-birth occurs in the same location is made for expositional simplicity. Alternatively, we can assume that a new dyad can choose the place to enter. As long as we focus on the steady-state, the equilibrium allocation is unaltered under this alternative assumption. 14 This is a standard assumption in the migration literature (see e.g. Zenou, 2011) and it does make sense, especially in developing countries, where jobs in rural areas are mostly from the agricultural sector and easy to obtain. Also many rural firms are family related and thus coordination failures and thus search frictions should not be too large (see e.g. Yamada, 1996; Marcouiller et al., 1997; Maloney, 1999).

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d

d

g (t )

d

2 freely mobile

Rural employment Fig. 1. Flows in the labor market in steady state.

where g(t) is defined by (1) and    Nc ðtÞ ¼ α W u0c ðtÞ  W r

ð4Þ

W u0c ðtÞ

where is the lifetime expected utility of an unemployed worker in a d0 dyad living in the city, Wr is lifetime expected utility of a rural worker and α4 0 is a parameter. As stated above, all rural workers who migrate to the city start as unemployed in a d0 dyad. Because people in the same dyad are married (or family members), the migration decision is always made together (within the dyad) and not individually. In other words, it is not possible for two persons from the same dyad to live in different areas. Let us explain (3). The positive change of d0 dyads over time is due to the fact that, at each period of time t,  δðd2 ðtÞ þ d1 ðtÞ þ d0 ðtÞÞ die and δðd2 ðtÞ þd1 ðtÞ þ d0 ðtÞÞ ¼ δNc ðtÞ=2 are born as a d0 dyad and that 12Nc ðt Þ workers migrate to the city in a d0 dyad. The negative change of d0 dyads over time is due to the fact that one of the workers from a d0 dyad finds a job directly (2λ) and that the dyad dies (δ). For d1 dyads, they increase their size because one of the workers from a d0 dyad got a job directly (2λ) but they lose people because either the dyad died (δ) or because the unemployed worker from the d1 dyad obtained a job (g(t)). Finally, for d2 dyads, they gain people from d1 dyads (g) but lose people since the dyad can die at rate δ. Remember that the number of employed workers in the city at time t, Ec(t), is given by Ec ðtÞ ¼ 2d2 ðtÞ þ d1 ðtÞ whereas the number of unemployed workers, Uc(t), is given by U c ðtÞ ¼ 2d0 ðtÞ þd1 ðtÞ. As a result, N c ðtÞ ¼ Ec ðtÞ þUðtÞ ¼ 2ðd0 ðtÞ þ d1 ðtÞ þ d2 ðtÞÞ. 







In steady state, d2 ðtÞ ¼ 0, d1 ðtÞ ¼ 0, d0 ðtÞ ¼ 0 and Nc ðtÞ ¼ 0, and the flows in the labor market can be described in Fig. 1 By solving (3) in steady state, we obtain15 n

d0 ¼ n

δNnc 2ðδþ 2λÞ n



d1 ¼ d2 n

d2 ¼

   δ γϕ N nc 2λ

2λ2 N nc ðδ þ2λÞ½δþ 2λð1  γϕðNnc ÞÞ

ð5Þ ð6Þ

ð7Þ

n

For d1 to be strictly positive, we assume that 2λϕγ oδ

ð8Þ n

Moreover, for d2 to be positive, we assume that δþ 2λð1  γ ϕÞ 4 0, which can be rearranged as 2λϕγ oδ þ 2λ

ð9Þ

Combining (8) and (9) leads to 2λϕγ oδ

ð10Þ

3.7. Steady-state asset value equations Let us write the steady-state lifetime expected utilities of all workers (i.e. the Bellman equations). For a rural worker, we have rW r ¼ wr  δW r

ð11Þ

where Wr is lifetime expected utility of a rural worker, wr is the rural wage defined by (2), and r is the discount rate. The second term of the right-hand side of (11) represents the capital loss from exiting the economy. 15

A variable with a star indicates that it is a steady-state equilibrium variable.

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e

Let us now write Wc, the lifetime expected utility of an urban worker who is employed in the city. It is given by rW ec ¼ wc  δW ec where wc is the (exogenous) urban wage. Observe that being employed does not depend on which dyad you belong to. This is because, in this model, people are employed for ever until they exit the economy (at rate δ) either because they retire or they die.16 To avoid the assumption that workers are employed forever, one can interpret δ in a different way. We can assume that when δ hits a dyad, then the two workers belonging to the dyad (a married couple, for example) lose their job and migrate to a new city (which is not modeled here) and are replaced by new workers coming to the city. In this interpretation, workers are not anymore employed for life but keep a job until it is hit by a negative shock. This means that workers within the same dyad always work in the same firm or the same sector. Let us now write the expected utility of the unemployed workers. Since their chance of escaping unemployment depends on which dyad they belong to, we need to define two different expected utilities. Denote by W u0c , the lifetime expected utility of an urban unemployed worker in a d0 dyad. Then, we have rW u0c ¼ b þ λðW ec  W u0c Þ þ λðW u1c W u0c Þ  δW u0c Indeed, when someone is unemployed in a d0 dyad, she obtains an unemployment benefit of 0 ob o wc . Then, she can hear directly about a job at rate λ and become employed and thus obtains a surplus of W ec  W u0c 40 or her partner in the dyad, who is also unemployed, finds a job at rate λ, and thus switches to a d1 dyad so that the surplus is now W u1c  W u0c 4 0. Finally, the dyad can die at rate δ and the negative surplus is 0 W u0c o 0. Finally, the lifetime expected utility of an urban unemployed worker in a d1 dyad is equal to rW u1c ¼ b  βγ þg n ðW ec W u1c Þ  δW u1c where gn is the steady-state rate at which the unemployed worker from a d1 dyad finds a job and is equal to   dn g n ¼ 2λ þ γϕ Nnc 2λ 2n d1

ð12Þ

Indeed, an unemployed worker in a d1 dyad earns an unemployment benefit of b and pays a cost of βγ for interacting with weak ties in the city where β 40 is the marginal cost (remember that γ is the worker's effort). This worker can leave unemployment at rate gn and obtain a surplus of W ec  W u1c 40 or the dyad can die at rate δ. We assume that γ rb=β, which ensures that b βγ Z 0.17 Since b=β is the natural upper bound of γ, we set γ ¼ b=β without loss of generality.18 As a result, condition (10) can be written as b 2λϕ oδ β

ð13Þ

Solving for the Bellman equations, we easily obtain W u0c ¼

W u1c ¼

b þ λðW ec þ W u1c Þ r þδ þ 2λ g n wc r þδ r þ δþ g n

b  βγ þ

ð14Þ

ð15Þ

W ec ¼

wc r þδ

ð16Þ

Wr ¼

wr r þδ

ð17Þ

where Nc, gn, and wr are endogenous. 16 This is clearly a simplified assumption, which helps us solve the model. In Appendix A.3, we relax this assumption and introduce a job-destruction rate σ 40. The Bellman equations become more complicated and the resolution of the model more cumbersome but we show that the main results of the model remain qualitatively unchanged. 17 γ can be interpreted as time spent on encountering weak ties. Then, γ represents time endowment, b is the utility in money term when spending all time endowment with strong ties, and the effort cost βγ is interpreted as opportunity costs of interactions with weak ties. 18 Even if γ a b=β, our main results are unaltered. However, the derived equations become more lengthy. For the expositional simplicity, we specify γ ¼ b=β.

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4. Steady-state equilibrium 4.1. Definition    Definition 1. A steady-state equilibrium Nnc ; γ n is such that the equilibrium migration condition is satisfied N c ðtÞ ¼ 0, i.e.

W r ¼ W u0c

ð18Þ n

which determines Nnc , and the unemployed workers in a d1 dyad optimally choose γ that maximizes W u1c , taking as given N nc , n n n d0 , d1 , and d2 . 4.2. Optimal choice of social interactions n

Let us determine the optimal γ chosen by the unemployed worker in a d1 dyad. When deciding on γ, workers take the city population, Nc, as given. Hence, we here characterize the workers' decision on γ for a given Nc, which is determined endogenously later on. Define   δβðr þ δ þ2λÞ ð19Þ N nc  ϕ  1 4λ2 ðwc bÞ þ2λγ βðr þ δþ 2λÞ n

By examining the optimal choice of γ by the unemployed worker in a d1 dyad, we have the following result: Proposition 1. Assume (13): n

(i) If the city population is sufficiently small, i.e. Nc oN nc , then the unemployed workers in a d1 dyad choose not to interact with weak ties, i.e. γ n ¼ 0. (ii) If the city population is sufficiently large, i.e. N c ZN nc , then the unemployed workers choose to fully interact with weak ties, i.e. γn ¼ γ .

This is an interesting result that links city population to social interactions. The intuition of this result is as follows. When choosing their optimal social-interaction effort γ, workers trade off the long-run benefits of increasing γ, which is finding a job more quickly, and the short-run costs, which is simply the effort cost βγ. Moreover, the benefits from increasing γ is captured by gn, the rate at which the workers leave unemployment, which increases with γ but depends on the size of the city population Nc (agglomeration effect). When the population size is too small, the benefits are lower than the costs and workers are better off not interacting with weak ties and only relying on direct methods and their strong ties. When urbanization increases above a certain population size, the benefits outweigh the costs and it becomes optimal for workers to fully interact with weak ties.19 This result implies that the ratio between the number of jobs obtained through weak ties and that obtained through direct job information flows or strong ties, Rweak, is higher as Nc becomes larger. Indeed, Rweak is given by d2 γϕðN c Þ2λ d1 d1 ; Rweak ¼ ð2d0 þ d1 Þλ where γϕðNc Þ2λd2 =d1 is the rate at which unemployed workers in a d1 dyad find a job and ð2d0 þ d1 Þλ ¼ U c λ. Rweak is zero when workers choose not to interact at all with weak ties. As Nc becomes larger and workers choose to interact with weak ties, this ratio becomes positive. Moreover, we can readily checked that ∂Rweak =∂Nc 40 whenever γ 40. This means that a larger Nc (i.e. more urbanization) results in a larger Rweak. Because Nc represents the population density in our framework, it implies that Rweak is higher in a denser city, which is consistent with the empirical finding provided by Wahba and Zenou (2005). Observe that N nc depends on the different parameters of the model. Since ϕð:Þ is an increasing function, it can be seen that n N c is decreasing with wc and λ and increasing with b, β, δ. Take, for example, λ, δ, and wc  b. The higher is λ (or the lower is δ or the higher is wc b), the lower is N nc and the more likely the unemployed workers will choose γ ¼ γ . A higher λ results in a larger fraction of employed workers who are aware about a job, implying higher returns on job finding rate, gn, from increasing γ while the cost is unaffected. A lower δ implies that workers are less likely to exit the economy and an employed worker can expect a longer employment period. A higher wc b implies a larger difference in flow income between employed and unemployed workers. Thus, a lower δ or a higher wc  b results in larger gains once employed while it does not affect effort costs. 19

This bang–bang property is a result from the assumptions made on the effort cost and the job arrival rate from weak ties, which are both linear in γ.

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4.3. Characterization of equilibrium Here, we characterize the equilibrium. Depending on the effort level chosen by workers, we distinguish equilibrium into two as follows:   n NIn Definition 2. A No-Interaction Equilibrium NNI is when all workers in the city choose γ NIn ¼ 0 while a Full  c ;γ n FI n is when all workers in the city choose γ FIn ¼ γ . Interaction Equilibrium NFI c ;γ n

n

n

In the following, we provide, for each equilibrium, the equilibrium values of the numbers of dyads, d0 , d1 and d2 , the equilibrium city population size Nnc , the level of urban unemployment U nc , the job arrival rate of an unemployed worker in a d1 dyad, gn, and all the equilibrium value functions for all workers. This confirms our previous result showing how urbanization (capture by Nc) affects the social behavior of workers. Let us first characterize the No-Interaction Equilibrium for which all workers provide zero effort, i.e. γ n ¼ 0. Using (5)–(7) and the fact that γ ¼ 0, we easily obtain: n

d0 ¼

δN nc ; 2ðδþ 2λÞ

n

d1 ¼

λδNnc ðδþ 2λÞ

2

;

n

d2 ¼

2λ2 N nc

ð20Þ

ðδ þ2λÞ2 n

n

Since the urban unemployment level is U nc ¼ 2d0 þ d1 , we obtain Un ¼

δðδþ 3λÞNnc

ð21Þ

ðδþ 2λÞ2

From (12), we obtain g n ¼ 2λ

ð22Þ

From (14)–(17), we have  ðr þ δÞðr þ δ þ3λÞb þλðr þ δ þ4λÞwc W u0cn γ ¼ 0 ¼ ðr þδÞðr þ δ þ2λÞ2

ð23Þ

 ðr þ δÞb þ 2λwc W u1cn γ ¼ 0 ¼ ðr þ δÞðr þ δ þ2λÞ

ð24Þ

wc r þδ  0 f N  Nnc Wr ¼ r þδ W ecn ¼

ð25Þ ð26Þ

 Finally, the migration equilibrium condition (18) is given by f ðN N c Þ=ðr þδÞ ¼ W u0cn γ ¼ 0 . By using (23), this equation can be written as 0

 ðr þ δÞðr þ δþ 3λÞb þ λðr þ δþ 4λÞwc 0 f N  Nnc ¼ ðr þ δþ 2λÞ2

ð27Þ

which implicitly determines Nnc . Let us now characterize the Full-Interaction Equilibrium for which all workers provide maximal effort, i.e. γ n ¼ γ . Proceeding exactly as above, we easily obtain the equilibrium values as follows:     δN nc 2λ2 N nc δ n n  γ ϕ N nc ; d1 ¼ d0 ¼ ð28Þ n 2ðδþ 2λÞ ðδþ 2λÞ½δ þ2λð1  γ ϕðN c ÞÞ 2λ d2 ¼

2λ2 N nc ðδþ 2λÞ½δ þ 2λð1 γ ϕðN nc ÞÞ

ð29Þ

Un ¼

δðδþ 3λÞ  2λγ ϕðN nc Þðδ þλÞ n N ðδþ 2λÞ½δ þ2λð1  γ ϕðN nc ÞÞ c

ð30Þ

gn ¼

2δλ δ 2λγϕðN nc Þ

ð31Þ

n

W u0c ¼





βδ bðr þ δÞðr þ δþ 2λÞ þ λwc ðr þ δþ 4λÞ  2bλϕðN nc Þðr þ δÞ bðr þ δÞ þ λwc

ðr þδÞðr þδ þ 2λÞ δβðr þ δþ 2λÞ  2λbϕðNnc Þðr þ δÞ

ð32Þ

W u1c ¼

2βδλwc

; ðr þ δÞ βδðr þ δþ 2λÞ  2bλϕðN nc Þðr þ δÞ

 0 f N  Nnc r þδ

ð33Þ

W ec ¼

wc ; r þδ

Wr ¼

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 βδ bðr þδÞðr þ δ þ2λÞ þλwc ðr þ δþ 4λÞ 2bλϕðN nc Þðr þ δÞ bðr þδÞ þλwc 0

f N  Nnc ¼ n ðr þ δþ 2λÞ δβðr þ δ þ2λÞ 2λbϕðN c Þðr þ δÞ

ð34Þ

Next, we provide conditions under which each equilibrium is possible. The city population, N nc , is determined by the decision of the new entrants to migrate to the city. This requires that (18) holds true, i.e., the asset value of a rural employed worker to be equal to that of an urban unemployed worker in the d0 dyad. Eq. (14) implies that the asset value of an e u u u  unemployed worker in the d0 dyad, W 1c   0c , increases with W  . Because Wc does not depend on γ, we know that W 0c γ ¼ γ is u  u  u  larger than W 0c γ ¼ 0 if and only if W 1c γ ¼ γ is larger than W 1c γ ¼ 0 , i.e., the city population is sufficiently large so that N c 4N nc . Hence, from (14), the equilibrium migration condition (18) becomes  0 bþ λðW ec þW u1c γ ¼ 0 Þ  f ðN N c Þ u  ¼ W 0c γ ¼ 0 ¼ if and only if Nc o N nc ; r þδ r þ δþ 2λ  0 bþ λðW ec þW u1c γ ¼ γ Þ  f ðN N c Þ ¼ W u0c γ ¼ γ ¼ ð35Þ if and only if Nc Z N nc : r þδ r þ δþ 2λ n From (27), we know that the city population in the No-Interaction Equilibrium, N NI c , is given by !  1 ðr þ δÞðr þδ þ 3λÞb þ λðr þδ þ 4λÞwc n ¼ N f NNI c ðr þ δþ 2λÞ2 n n increases with the total population, N. Moreover, changes in N are perfectly absorbed by changes in NNI implying that N NI c c .  u  This property comes from the fact that N does not affect W 0c γ ¼ 0 : even after changes in N, the equilibrium migration 0 0 condition f ðN N c Þ=ðr þ δÞð ¼ f ðN r Þ=ðr þ δÞÞ ¼ W u0c γ ¼ 0 results in the same rural population as before, implying   that any n u  u  changes in N must be offset by changes in N NI c . Because of the above stated fact that W 0c γ ¼ γ is larger than W 0c γ ¼ 0 if and only if Nc 4N nc , we readily know that N nc ⋛N nc is equivalent to N⋛N, where N is defined as !  1 ðr þ δÞðr þ δþ 3λÞbþ λðr þ δþ 4λÞwc N  N nc þ f : ðr þ δþ 2λÞ2

Therefore, the equilibrium migration condition can be rewritten as 0

 f ðN N c Þ ¼ W u0c γ ¼ 0 r þδ 0

 f ðN N c Þ ¼ W u0c γ ¼ γ r þδ

if and only if N o N; if and only if N Z N:

Summarizing the above arguments, we obtain the following proposition. Proposition 2. Assume (13):   n NIn (i) If the total population is sufficiently small, i.e., N oN , the city is characterized by the No-Interaction Equilibrium NNI c ;γ such that all urban workers provide effort γ NIn ¼ 0.   n FI n such (ii) If the total population is sufficiently large, N Z N, the city is characterized by the Full-Interaction Equilibrium N FI c ;γ that all urban workers exert effort γ FIn ¼ γ .

When the level of urbanization is low, the contact rate with weak ties, ϕðNc Þ, is also low so that it is not worth devoting much effort to interact with weak ties. As the city population grows, the agglomeration effect regarding the contact rate gets n larger and it becomes rewarding to social interact with weak ties. The city population size, NNI c , increases with the total n population size (rural plus urban), N, in the No-Interaction Equilibrium. Moreover, N NI has the property that c n limN-1 NNI ¼ 1. Hence, there exists a threshold value of N under which we observe the No-Interaction Equilibrium and c beyond which we observe the Full-Interaction Equilibrium. Thus, if we consider a continuous growth process of total population size, there will be a regime change from the No-Interaction Equilibrium to the Full-Interaction Equilibrium in the city.

4.4. Comparing equilibria Let us discuss the effect of γ, the social interactions with weak ties, on the unemployment rate and the city population size by comparing these values in the No-Interaction Equilibrium and the Full-Interaction Equilibrium. We are aware that these two equilibria occur for different population size N (see Proposition 2) but we want to understand how N affects the unemployment rate and city population size.

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141

Let u denote the unemployment rate in the city, i.e., u ¼ U=Nc . Then, from (21) and (30), we have U NIn ¼

U FIn ¼

δðδþ 3λÞNNI c ðδþ 2λÞ2

thus uNIn ¼

δðδ þ 3λÞ

ð36Þ

ðδ þ 2λÞ2

n δðδþ 3λÞ  2λγ ϕðN FI c Þðδ þλÞ n ðδþ 2λÞ½δ þ2λð1  γ ϕðNFI c ÞÞ

n NFI thus uFIn ¼ c

n δðδ þ3λÞ 2λγ ϕðN FI c Þðδ þ λÞ n ðδ þ2λÞ½δ þ 2λð1 γ ϕðN FI c ÞÞ

ð37Þ

As a result, we observe that uNIn 4uFIn 3

δðδþ 3λÞ ðδþ 2λÞ

2

4

n δðδ þ3λÞ 2λγ ϕðN FI c Þðδ þ λÞ n ðδþ 2λÞ½δ þ 2λð1 γ ϕðN FI c ÞÞ

which is equivalent to n 2λϕðNFI c Þγ oδ þ 2λ: n n FI n This is always true by (9). Moreover, (35) implies that NNI c oN c r N c , that is, the city population size is always larger in the Full-Interaction Equilibrium than in the No-Interaction Equilibrium. The following proposition summarizes these findings.

Proposition 3. Given the conditions guarantying each equilibrium (Proposition 2), the unemployment rate is lower and the city population size is larger in the Full-Interaction Equilibrium than in the No-Interaction Equilibrium. These are comparisons between equilibria under different total population size, N. We can also compare these equilibrium values with the values under some hypothetical scenarios. Indeed, suppose that N Z N and we have the FullInteraction Equilibrium. We can then ask the following question: if workers had chosen not to interact at all, i.e. γ FIn ¼ 0, what would have been the levels of the unemployment rate and the city population size? Such comparisons would allow us to grasp the potential effects of social interactions. In that case, the unemployment rates would be given by uNIn ¼ uFIn ðγ ¼ 0Þ ¼

δðδþ 3λÞ ðδþ 2λÞ2

4uFIn ðγ ¼ γ Þ

which implies that the unemployment rate in the Full-Interaction Equilibrium is lower than the one observed if workers had  0 chosen not to interact. As described in (35), the city population size is determined by f ðN  Nc Þ ¼ ðr þδÞW u0c γ ¼ γ and we   n FI n FI n know that W u0c γ ¼ γ Z W u0c γ ¼ 0 . From the assumption that f ″ð:Þ r0, we can see that NFI c ðγ ¼ γ Þ 4N c ðγ ¼ 0Þ, i.e. N c is larger than the city population size observed if workers had chosen not to interact (Nc that is determined by  0 f ðN N c Þ ¼ ðr þ δÞW u0c γ ¼ 0 ). Similar arguments show that when N o N and the economy is in the No-Interaction Equilibrium, the levels of unemployment rate and city population size are higher than those observed if workers had chosen to fully interact. These thought experiments indicate that the social interactions with weak ties, once chosen by workers, result in a lower unemployment rate and higher urbanization. 4.5. Existence and uniqueness of equilibrium Since, in Proposition 2, we showed only the condition under which each equilibrium is possible, we need to examine under which condition there exists an equilibrium and if it is unique. The determination of the city population Nc is described in Fig. 2(a) and (b). In Fig. 2(a) and (b), the horizontal axis represents the city population Nc. The rural population Nr is represented by the difference between N and Nc. The vertical axis shows the asset values Wr and W u0c . Wr is represented by a upward sloping curve (with respect to the city population Nc) because dW r =dN c ¼  dW r =dN r ¼  f ″=ðr þ δÞ 40. From (45), (46) and (48), we    know that W u0c γ ¼ 0 is independent of Nc whereas W u0c γ ¼ γ is increasing in Nc. Observe first that Wr and W u0c γ ¼ 0 has a  0 unique interaction if W u  4 f ðNÞ=ðr þδÞ because we assumed the Inada conditions for f ðÞ. In Fig. 2(a), when the total 0c γ ¼ 0

n n population N increases, the city population, N NI c , also increases. When the city population reaches N c , the regime switches

from the No-Interaction Equilibrium (γ NIn ¼ 0) to the Full-Interaction Equilibrium (γ FIn ¼ γ ) and the equilibrium population    distribution is determined by W r ¼ W u0c γ ¼ γ . Observe second that W u0c γ ¼ γ ZW u0c γ ¼ 0 for Nc ZN nc and the opposite hold true for Nc oN nc as long as Fig. 2(b) is relevant (i.e., N Z N ). The Inada conditions for f ðÞ combined with the fact that    0 if W u0c γ ¼ 0 4 f ðNÞ=ðr þ δÞ. W u0c γ ¼ γ ;ϕ ¼ ϕðNÞ has a finite value, ensure that Wr and W u0c γ ¼ γ has at least one intersection  0 u  NI n To show that there is a unique equilibrium with γ ¼ 0, we need to show that W 0c γ ¼ 0 4f ðNÞ=ðr þδÞ. This is equivalent to 0

ðr þ δÞðr þ δþ 3λÞb þλðr þ δþ 4λÞwc 4 ðr þ δ þ2λÞ2 f ðNÞ

ð38Þ

Condition (38) is satisfied e.g., if income levels of urban workers (b or wc or both b and wc) are sufficiently large. To show  that there is a unique equilibrium with γ FIn ¼ γ , we further need to show that the slope of Wr is steeper than that of W u0c γ ¼ γ

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with an increase in N

W0uc

W0uc

γ =γ ,φ =φ ( N ) with an increase in

W0uc

γ =γ

γ =0

0

N cFI *

N Nc N rFI *

Fig. 2. Equilibrium population distribution when there are (a) no social interactions (γ NIn ¼ 0) and (b) are full social interactions (γ FIn ¼ γ ).

    n at all intersections of these two curves, i.e., W 0r ðNc ÞNc ¼ NFIn 4∂W u0c γ ¼ γ =∂Nc  for all Nc ¼ N FI c . This is equivalent to FIn c  Nc¼ Nc n   2   4λ3 wc ð2λ þ δÞðr þ δÞγ ϕ0 NFI c ″ n FI n FI n 4 f N  N ð39Þ  ðr þ δþ 2λÞ 1 þγ ϕ N FI 0 1 2 32 for all Nc ¼ Nc c c FI n δ 2λγ ϕðN Þ 4ðr þ δÞ@  c  A þ 4λ2 5 n 1 þ γ ϕ NFI c  Note that Wr is increasing in Nc because of decreasing marginal productivity in the rural area whereas W u0c γ ¼ γ is increasing  in Nc because of the agglomeration economies in encountering weak ties. The condition that W 0r ðNc ÞNc ¼ NFIn 4 c    requires the latter effect is smaller than the former effect. Condition (39) is satisfied e.g., if we ∂W u0c γ ¼ γ =∂Nc  FIn Nc ¼ N c

specify ϕðN c Þ as ϕNc =ðN c þ 1Þ.20 Therefore, we have the following proposition: Proposition 4. Assume (13):   n NIn , (i) If N is small enough, i.e. N o N, and Condition (38) holds, then there exists a unique No-Interaction Equilibrium N NI c ;γ which is given by (20)–(27) (see Fig. 2(a)). (ii) If N is large enough, i.e. N Z N, and Conditions (38) and (39) hold, then there exists a unique Full-Interaction Equilibrium   n FI n , which is given by (28)–(34) (see Fig. 2(b)). NFI c ;γ

5. Comparative statics In this section, we provide the results of basic comparative statics with respect to the city population size, Nnc , and the unemployment rate, unc ¼ U n =N. 5.1. No-Interaction Equilibrium   n NI n Let us start with the No-Interaction Equilibrium N NI , where Nnc is determined by (27) and is described in Fig. 2(a). c ;γ In order to focus on the unique No-Interaction Equilibrium, we assume that condition (38) holds. We provide here 20

 When ϕðN c Þ ¼ ϕN c =ðN c þ 1Þ, we can show that W u0c γ ¼ γ is an increasing and concave function of Nc.

Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

143

explanations based on the figure and relegate the formal proofs to the Appendix. As shown in Fig. 2(a), an increase in the urban wage, wc, in the unemployment  benefit, b, or in the direct job arrival rate, λ, raises the asset value of an urban unemployed worker in a d0 dyad, W u0c γ ¼ 0 , whereas it does not affect the asset value of a rural employed worker, Wr. Hence, by increasing these variables, cities will attract more people and the city population size, N nc , will become larger. Furthermore, an increase in the exit rate, δ, or the discount rate, r, lowers the asset values, but the reduction is larger for an urban unemployed worker. As a result, Nnc decreases. An increase  in the total population size, N, reduces Wr, leading to a larger N nc . Note first that an increase in N does not affect W u0c γ ¼ 0 . If increased population is absorbed by the rural population, Nr, it decreases the rural marginal productivity and hence Wr, which violates the equilibrium migration  condition, W r ¼ W u0c γ ¼ 0 . In order for this condition to hold true, an increase in N must be absorbed by Nnc . The effects on unemployment rate can be readily examined. Because the number of unemployed workers is given by (21), the effect of an increase of wc, b, or r on the unemployment rate, unc ¼ U n =N, is proportional to that of Nnc . An increase in N affects both the numerator, Un, and the denominator, N, of unc ¼ U n =N. Still, we can see from the Appendix that ∂N nc =∂N ¼ 1, implying that ∂ðN nc =NÞ=∂N ¼ ðN  Nnc Þ=N2 4 0. Hence, we know that an increase in N raises unc . However, an increase in δ or λ has additional impacts on unc , making its total effect on unemployment rate ambiguous. The following proposition summarizes our findings:   n NI n , an increase in wc, b, N, or λ, or Proposition 5. Suppose that Condition (38) holds. In the No-Interaction Equilibrium N NI c ;γ a decrease in δ or r increases the city population size, Nnc . Moreover, an increase in wc, b, or N, or a decrease in r increases the unemployment rate, unc . An increase in δ or λ has an ambiguous effect on unc . 5.2. Full-Interaction Equilibrium We next examine the Full-Interaction Equilibrium, where the city population size is determined by (34) and is described in Fig. 2(b). Here, we assume that conditions (38) and (39) hold so that the model has the unique Full-Interaction   n FI n Equilibrium N FI . As shown in Fig. 2(b), a higher value of wc shifts the asset value of an urban unemployed worker in a c ;γ  u  d0 dyad, W 0c γ ¼ γ , upwards whereas it does not change the asset value of a rural employed worker, Wr. Thus, it increases Nnc .  Furthermore, an increase in N lowers Wr whereas it does not change W u  , resulting in a larger Nn . Unfortunately, the c

0c γ ¼ γ

effects of a change in other parameters are ambiguous. The effects of unemployment rate, unc , are somewhat more complicated in this case because now the city population size (and hence density) has agglomeration effects on the contact rate with weak ties described by ϕðNc Þ. From (30), we can see that, on the one hand, a larger N nc has an direct effect of raising unc . On the other hand, it indirectly lowers unc by raising ϕðN c Þ. Such an indirect effect is not sufficient to dominate the direct effect in the case of an increase in wc, but it is so in the case of an increase in N.   n FI n Proposition 6. Suppose that conditions (38) and (39) hold. In the Full-Interaction Equilibrium N FI , an increase in wc or N c ;γ increases the city population size, Nnc . Moreover, an increase in wc increases the unemployment rate, unc , whereas an increase in N decreases it. An increase in b, δ, λ, or r has ambiguous effects on Nnc and unc . The following table summarizes all our comparative statics results: 6. Efficiency In this section, we explore the efficiency properties of each equilibrium. Our questions are as follows: (i) are the decisions in terms of social interactions, γ n , efficient? and (ii) is the degree of urbanization, Nnc , efficient? We follow the search and matching literature (Pissarides, 2000) by defining the social welfare, SW, as the sum of utilities of all workers, i.e. Z 1

SW ¼ e  rt f ðN N c ðtÞÞ þ Ec ðtÞwc þ U c ðtÞb βd1 ðtÞγðtÞ dt ð40Þ 0

The dynamics of the number of dyads are given by (3). Note that Ec ðtÞ ¼ 2d2 ðtÞ þd1 ðtÞ

ð41Þ

U c ðtÞ ¼ 2d0 ðtÞ þd1 ðtÞ

ð42Þ

Nc ðtÞ ¼ 2ðd0 ðtÞ þ d1 ðtÞ þ d2 ðtÞÞ

ð43Þ

and

The planner chooses γðtÞ and Nc(t) that maximize (40) under the flow constraint (3). We obtain the present-value Hamiltonian: H t ¼ e  rt ðf ðN N c ðtÞÞ þ Ec ðtÞwc þ U c ðtÞb  βd1 ðtÞγðtÞÞ þ μ1 ðtÞ½2λd0 ðtÞ ðδþ gðtÞÞd1 ðtÞ þ μ2 ðtÞðgðtÞd1 ðtÞ  δd2 ðtÞÞ 



(or co-state variables) corresponding to d1 ðtÞ and d2 ðtÞ. We do not need where μ1 ðtÞ and μ2 ðtÞ are the Lagrangian multipliers  to write the constraint corresponding to d0 ðtÞ because we use (43). The control variables of the social planner are γðtÞ and Nc(t) and the state variables are d1 ðtÞ and d2 ðtÞ. We have the following result:

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Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

Fig. 3. Equilibrium versus optimal social interactions.

Table 1 Comparative statics for both equilibria. b

N

δ

λ

r

No-Interaction Equilibrium (γ n ¼ 0) þ N nc þ unc

þ þ

þ þ

 ?

þ ?

 

Full-Interaction Equilibrium (γ n ¼ γ ) þ N nc þ unc

? ?

þ –

? ?

? ?

? ?

wc

Proposition 7. When the city population is sufficiently small (Nc r N oc ), there is no social interactions in equilibrium and it is optimal from a social welfare viewpoint, i.e. γ n ¼ γ o ¼ 0. When the city population is sufficiently large (N c 4N nc ), there is full social interactions in equilibrium and it is optimal from a social welfare viewpoint, i.e. γ n ¼ γ o ¼ γ . When the city population takes intermediate values ðN oc oN c rN nc Þ, there is no social interaction in equilibrium (γ n ¼ 0) while full social interactions are optimal ðγ o ¼ γ Þ. b o as Define ϕ bo  ϕ

βδðδþ 2λÞ : 2λðbδþ 2wc λÞ

ð44Þ o

o

b and γ ¼ 0 is optimal if ϕ o ϕ b . Define next N o as As shown in the proof of Proposition 7, γ ¼ γ is optimal if ϕ 4 ϕ c  o b : N oc  ϕ  1 ϕ Then, that γ ¼ γ is optimal if Nc 4 N oc and γ ¼ 0 is optimal if Nc o N oc . Because (44) and (47) result in  nwe oobtain  b b sgn ϕ  ϕ ¼ sgnðwc bÞ, we know that N nc 4N oc , implying Proposition 7. Fig. 3 describes when the equilibrium decision on social interactions is optimal and when it is not. From the comparative statics results (Table 1), we know that the city population size becomes larger when the total population size increases. Thus, the results of Proposition 7 imply that, as the total population grows, the economy experiences steady urbanization. During this process, the social-interaction decision is efficient only at an early and a late stage of urbanization but not at an intermediary stage of urbanization. The source of this inefficiency is due to (positive) externalities related to the decision on interacting with weak ties. Indeed, social interactions with weak ties make it more likely for an unemployed worker belonging to a d1 dyad to find a job and thus to switch to a d2 dyad. This implies that a higher level of social interactions increases the number of employed workers in d2 dyads, which, in turn, raises the possibility of job information transmission through weak ties (i.e. increases gn) for d1 dyads. However, an unemployed worker in a d1 dyad ignores this positive externality on other unemployed workers when making her social-interaction decision γ, resulting in inefficiency of this decision. Next, we move to the efficiency analysis of city population size, Nnc . In order to focus on the properties of urbanization, we mainly restrict our attention to the cases where the equilibrium decision on social interactions is efficient (i.e., γ n ¼ γ o , or equivalently, Nnc r N oc or Nnc 4 N nc ). In such cases, the equilibrium condition for N nc , (18), is equal to 8 u < W 0c γ ¼ 0  Wr ¼ : Wu  0c γ ¼ γ

for γ ¼ 0 for γ ¼ γ

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145

Fig. 4. Equilibrium versus optimal population distribution when there are (a) no social interactions (γ NIn ¼ γ o ¼ 0) and (b) full social interactions (γ FIn ¼ γ o ¼ γ ).

whereas the optimal condition is given by 8  < Θot γ ¼ 0  Wr ¼ : Θot γ ¼ γ

for γ ¼ 0 for γ ¼ γ

where Θot jγ ¼ 0 and Θot jγ ¼ γ are defined in (60). As shown in the proof of Propositions 7 and 8 in the Appendix, we have that W u0c jγ ¼ 0 4 Θot jγ ¼ 0 . Moreover, if the discount rate, r, is sufficiently small and the exit rate, δ, is not extremely high (δo 2λ), we also have W u0c jγ ¼ γ 4Θot jγ ¼ γ .21 These results imply that the curves of W u0c jγ ¼ 0 and W u0c jγ ¼ γ are above the curves of Θot jγ ¼ 0 and Θot jγ ¼ γ , as can be seen in Fig. 4(a) and (b), respectively. We have the following proposition. o

Proposition 8. When Nnc r N oc , the equilibrium urbanization level N nc is higher than the optimal level one Nc . When Nnc 4 N nc , Nnc o is higher than Nc if r is sufficiently small and δo 2λ. Such over-urbanization partly arises from the search frictions in the urban labor market. Indeed, when deciding whether to migrate or not, each migrant compares the discounted sum of expected income. Given the existence of urban unemployment, such decision leads to over-urbanization (as in, e.g., Harris and Todaro, 1970). In addition, in the FullInteraction Equilibrium, there exist agglomeration (positive) externalities and congestion (negative) externalities from finding jobs via weak ties. Indeed, when a worker decides to migrate to the city, it increases the size of the city population and thus the possibility of contacting a weak tie, which is represented by the term ϕðNc Þ in g(t). This creates positive agglomeration externalities. At the same time, the decision to migrate increases the number of workers in d0 dyads, which, in turn, increases the number of workers in d1 dyads. This, in turn, decreases the possibility of job information transmission from a weak tie, which is represented by the term d2 =d1 in g(t). This results in the congestion negative externality. Here, the effect of the negative externality dominates the effect of the positive one when workers do not discount the future and the unemployment rate is not extremely high, and this is why we observe over-urbanization even in the Full-Interaction Equilibrium as compared to the social optimum. Finally, we can briefly comment on the case for which γ n a γ o (i.e., N oc oN c r N nc ). In this case, although we cannot determine analytically the efficiency properties of N nc , simple numerical examples indicate that W u0c jγ ¼ γ 4 Θot jγ ¼ 0 .22 Therefore, it would be safe to conclude that this economy experiences too much urbanization in general. In summary, there are two market failures. The first one stems form social interactions (at intermediate levels of city population) so that there are too few social interactions in equilibrium. The second comes from urbanization. When social 21

If δ Z 2λ, the unemployment rate exceeds at least 50%. We believe that such a case should be treated as an exception. We tried two specifications of ϕðN c Þ (i.e., ϕðN c Þ ¼ N c =ð1 þ N c Þ and ϕðN c Þ ¼ 1 exp½  N c ). We also tried various sets of parameter values. For any combination, we obtained that W u0c jγ ¼ γ 4Θot jγ ¼ 0 . 22

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interactions are optimal, which corresponds to either small or large cities, there is too much agglomeration or urbanization in equilibrium. 7. Conclusion We develop a model where unemployed workers in the city can find a job either directly or through weak or strong ties. We show that, in denser areas, individuals interact with more people and have more random encounters (weak ties) than in sparsely populated areas. We also demonstrate that, for low urbanization levels, there is a unique steady-state No-Interaction Equilibrium where workers do not interact with weak ties, while, for high level of urbanization, there is a unique steady-state Full-Interaction Equilibrium with full social interactions. Thus, if we consider a continuous growth process of total population size, there will be a regime change from the No-Interaction Equilibrium to the Full-Interaction Equilibrium in the city. We show that these equilibria are usually not socially efficient when the city population has an intermediate size because there are too few social interactions compared to the social optimum. When the equilibrium level of social interactions is optimal, the equilibrium urbanization level is always higher than the optimal level one, leading to over-urbanization. There are many empirical studies that try to measure agglomeration economies in different cities (see, for example, Glaeser, 2010). However, few studies have put forward the role of social interactions and social networks in agglomeration and urbanization. Usually, following Marshall (1890) and Jacobs (1969), authors have emphasized the role that cities can play in speeding the flow of ideas. The interactions of smart and skilled people in urban areas enhance the development of person-specific human capital and increase the rate at which new ideas are formed. We believe that the role of social interactions and the fact that people tend to extend their social networks by meeting more weak ties in bigger cities that help them find a job are crucial in explaining agglomeration. We also believe that this can lead to over-urbanization, which would imply that cities are oversized. These issues certainly need more thorough empirical investigations.

Acknowledgments We thank the editor, Joerg Oechssler, an associate editor, two anonymous referees, Keisuke Kawata, Akihiko Matsui, Hiroshi Ohashi, Pascal Mossay, Yasuyuki Sawada, Takatoshi Tabuchi, Kenichi Ueda, Noriyuki Yanagawa, and the participants of various seminars and conferences for very useful comments and discussions. This study was conducted as a part of the Project “Spatial Economic Analysis on Regional Growth” undertaken at the Research Institute of Economy, Trade and Industry (RIETI). We acknowledge the financial support from the Japan Society for the Promotion of Science through a Grant-in-Aid for Young Scientists (B). Yves Zenou gratefully acknowledges the financial support from the French National Research Agency grant ANR-2011-BSH1-014-01. Appendix A A.1. Optimal migration decision in equilibrium As we show in Proposition 1, we have the No-Interaction Equilibrium with γ ¼ 0 and the Full-Interaction Equilibrium  with γ ¼ γ . Then, by comparing the asset values given by (14)–(16), we can see that W ec  W u0c γ ¼ 0 4 0 and    W u1c γ ¼ 0 W u0c γ ¼ 0 40. Because we have W r ¼ W u0c γ ¼ 0 in the No-Interaction Equilibrium, workers in d1 and d2 dyads do not have incentive to move to the rural area. In the Full-Interaction Equilibrium, note that from (14) and (15),   W u1c  W u0c ¼ ½ðr þδ þ λÞW u1c  b λW ec =ðr þδ þ 2λÞ and W u1c γ ¼ γ 4W u1c γ ¼ 0 hold true. Hence, we obtain that      W e W u  40 and W u  Wu  4Wu  Wu  4 0, implying that workers in d1 and d2 dyads again do c

0c γ ¼ γ

1c γ ¼ γ

0c γ ¼ γ

1c γ ¼ 0

0c γ ¼ 0

not have incentive to move to the rural area. A.2. Proofs n

Proof of Proposition 1. Let us determine the optimal γ. Unemployed workers in a d1 dyad choose γ that maximize W u1c , n n taking as given Nnc , d1 and d2 . Using (12) and (15), we obtain !    n dn2 wc  n dn2 wc n n r þ δþ g 2λϕ Nc n  β þ 2λϕ N ð Þ  b βγ þ g c n r þδ d1 r þδ d1 ∂W u1c ¼ 2 n ∂γ ðr þ δ þg Þ  dn  dn  dn wx wc n 2λϕ Nnc 2n βðr þ δþ g Þ þ ðr þ δ þg n Þ2λϕ N nc 2n  ðb  βγ Þ2λϕ N nc 2n  g n r þδ r þδ d1 d1 d1 ¼ ðr þδ þ g n Þ2

Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

¼

 dn  β r þ δþ 2λ þγ2λϕ N nc 2n d1

!

n

þ

147

n

 d  d wc ðr þ δÞ2λϕ N nc 2n  ðb  βγ Þ2λϕ N nc 2n r þδ d1 d1

ðr þ δþ g n Þ2  d  dn  βðr þδ þ 2λÞ þ wc 2λϕ Nnc 2n b2λϕ N nc 2n d1 d1 n

¼

ðr þδ þ g n Þ2

n   d 2λϕ N nc ðwc bÞ β 1n ðr þ δþ 2λÞ d d2 ¼ 2n d1 ðr þ δþ g n Þ2 n

This implies that !   n   ∂W u1c d ¼ sgn 2λϕ N nc ðwc  bÞ  β 1n ðr þδ þ2λÞ sgn ∂γ d2 Using (6) and (7), we see that n   d1 δ  γϕ Nnc n ¼ d2 2λ n

n

Plugging d1 =d2 into the above equation, we can see that           δ ∂W u1c ðr þ δ þ2λÞ : ¼ sgn 2λϕ Nnc ðwc bÞ þβ γϕ Nnc  sgn 2λ ∂γ This leads to ∂W u1c δ 2λðwc  bÞ  ⋛0 3 γ⋛ ∂γ 2λϕðNnc Þ βðr þ δþ 2λÞ n

This condition only depends on N nc . We therefore know that an unemployed worker in the d1 dyad chooses either γ ¼ 0 (no interaction at all with weak ties) or γ ¼ γ (maximum interaction with weak ties), i.e. we have only corner solutions. Let us characterize all the solutions of this maximization problem. (i) If δ 2λðwc  bÞ ; o 2λϕðNnc Þ βðr þ δþ 2λÞ which is equivalent to   δβðr þ δ þ2λÞ ϕ N nc 4 4λ2 ðwc  bÞ then γ n ¼ γ . Since ϕ0 ð:Þ 40, this means that if Nnc 4 N c , then γ n ¼ γ , where   δβðr þδ þ 2λÞ Nc  ϕ1 4λ2 ðwc  bÞ (ii) If δ 2λðwc  bÞ 4γ;  2λϕðNnc Þ βðr þ δþ 2λÞ which is equivalent to   ϕ N nc o

δβðr þ δþ 2λÞ 4λ2 ðwc  bÞ þ 2λγ βðr þδ þ 2λÞ

then γ n ¼ 0. Since ϕ0 ð:Þ 4 0, this means that if N nc oN nc , then γ n ¼ 0, where   δβðr þ δþ 2λÞ N nc  ϕ  1 4λ2 ðwc  bÞ þ 2λγ βðr þ δþ 2λÞ (iii) If 0o

δ 2λðwc  bÞ oγ  2λϕðNnc Þ βðr þ δþ 2λÞ

then both cases γ n ¼ 0 and γ n ¼ γ can arise. Fig. A1 illustrates these different cases.

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Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

Fig. A1. Optimal choice of social interactions.

Let us now deal with case (iii) only since the other cases are straightforward. Then, using the fact that γ ¼ b=β, W u1c can be written as 

W u1c γ ¼ 0

2λwc r þδ ¼ r þ δ þ2λ

 W u1c γ ¼ γ ¼

bþ

ð45Þ

2λδwc ðr þ δÞ½βδðr þ δþ 2λÞ  2bλðr þ δÞϕðNnc Þ

ð46Þ

Comparing the asset values under no social interaction (γ ¼ 0) to that under full social interaction (γ ¼ γ ), we can see that  W u1c γ ¼ 0 ¼ W u1c γ ¼ γ if and only if   b ϕ Nnc ¼ ϕ

βδðr þ δ þ2λÞ 2λ½bðr þ δÞ þ 2wc λ

Furthermore, we have that     ∂ Wu  W u  1c γ ¼ γ

∂ϕðNc Þ

1c γ ¼ 0

¼

   ∂ W u1c γ ¼ γ ∂ϕðNc Þ

ð47Þ

¼

4bβwc δλ2

2 40 βδðr þ δþ 2λÞ  2bλϕðNc Þðr þ δÞ

ð48Þ

n b and chooses γ n ¼ 0 if and As a result, an unemployed worker in the d1 dyad always chooses γ n ¼ γ if and only if ϕðN c Þ 4 ϕ b c  ϕ  1 ðϕÞ, b From the assumption that ϕ0 ðNc Þ 4 0, we have a threshold regarding the city population: N b i.e. only if ϕðNc Þ o ϕ.   βδðr þ δþ 2λÞ bc  ϕ1 N 2λ½bðr þδÞ þ2wc λ

b c ¼ N n , given by (19). Observe also that Since γ ¼ b=β, then it is easily verified that N c b c ¼ N n oN c N c n

which implies that an unemployed worker in the d1 dyad always chooses γ ¼ γ in case (iii). Proposition 1 summarizes these results. □ Proof of Proposition 5. In the No-Interaction Equilibrium, the city population size, Nnc , is determined by (34). By totally differentiating (34), we obtain, after straightforward but tedious calculations, the following results of basic comparative statics:

∂Nnc bðr þ δÞ2 ðr þδ þ 4λÞ þ 2wc λ r 2 þ δ2 þ 6δλ þ4λ2 þ 2rðδþ 3λÞ ¼ o0 ″ ∂δ f ðN  Nnc Þðr þδÞðr þ δ þ2λÞ3 ∂Nnc ðwc  bÞðr þ δÞðr þ δþ 6λÞ ¼ ″ 40 ∂λ f ðN  Nnc Þðr þ δþ 2λÞ3 ∂Nnc ðr þδÞðr þδ þ3λÞ ¼ ″ 40 ∂b f ðN N nc Þðr þ δþ 2λÞ2 ∂Nnc λðr þ δþ 4λÞ ¼ ″ 40 ∂wc f ðN  N nc Þðr þ δþ 2λÞ2 ∂Nnc ¼ 140 ∂N

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149



∂N nc bðr þ δÞ2 ðr þ δ þ4λÞ þ2wc λ r 2 þδ2 þ 6δλþ 4λ2 þ 2rðδ þ 3λÞ ¼ o0 ″ ∂r f ðN N nc Þðr þ δÞðr þ δþ 2λÞ3 The equilibrium unemployment rate is defined by un ¼

Un N

where Un is given by (21). Hence, we readily obtain ∂N nc n ∂un Nc λðδþ 6λÞ þ δðδ þ 2λÞðδ þ3λÞ ∂δ ¼ ∂δ N ðδ þ 2λÞ3   ∂Nnc δ  Nnc ðδ þ 6λÞ þ δ2 þ 5δλ þ 6λ2 n ∂u ∂λ ¼ ∂λ Nðδþ 2λÞ3 ∂un δðδþ 3λÞ ∂Nnc ¼ 40 ∂b Nðδþ 2λÞ2 ∂b ∂un δðδ þ 3λÞ ∂Nnc ¼ 40 ∂wc N ðδ þ 2λÞ2 ∂wc   ∂Nnc n   δ ð δ þ 3λ Þ  N þ N c δðδ þ 3λÞ N N nc ∂un ∂N ¼ ¼ 40 ∂N N2 ðδ þ2λÞ2 N 2 ðδþ 2λÞ2 ∂un δðδþ 3λÞ ∂Nnc ¼ o0 ∂r Nðδþ 2λÞ2 ∂r This completes the proof.

□

Proof of Proposition 6. Here, we assume that (39) holds, i.e.,   ∂W u0c γ ¼ γ   W 0r ðNc ÞNc ¼ Nn  40  c ∂N c  n

ð49Þ

N c ¼ Nc

to ensure that the model has the unique Full-Interaction Equilibrium. In the Full-Interaction Equilibrium, the city population size is determined by (34). By totally differentiating (34), we can obtain !  1(  ∂W u0c jγ ¼ γ  ∂N nc bðr þδÞ2 ¼ W 0r ðNc ÞjNc ¼ Nnc   ∂δ ∂Nc Nc ¼ Nnc ðr þ δÞ2 ðr þδ þ 2λÞ2 h i 2wc λβ2 δ2 ðr þ δþ 2λÞ ðr þδÞ2 þ6ðr þδÞλ þ 4λ2 

2 ðr þδÞ2 ðr þδ þ 2λÞ2 βδðr þδ þ2λÞ  2bðr þ δÞλϕðN nc Þ h i 4wc λ2 bβðr þδÞ 2δðr þδÞ2  ðr  8δÞðr þδÞλ  2ðr  3δÞλ2 ϕðNnc Þ þ

2 ðr þ δÞ2 ðr þδ þ 2λÞ2 βδðr þ δ þ2λÞ 2bðr þδÞλϕðN nc Þ )  2 2 8wc λ3 b ðr þ δÞ2 ðr þ δ þλÞϕ Nnc 

2 ðr þδÞ2 ðr þδ þ 2λÞ2 βδðr þδ þ2λÞ  2bðr þ δÞλϕðN nc Þ 0 11   ∂W u0c γ ¼ γ   ∂N nc @ 0  A ¼ W r ðN c Þ Nc ¼ Nn   c ∂λ ∂Nc  Nc ¼ N nc (

β2 δ2 ðr þ δþ 2λÞ wc ðr þ δþ 6λÞ  2bðr þ δþ 2λÞ 

2 ðr þ δþ 2λÞ2 βδðr þδ þ 2λÞ  2bðr þ δÞλϕðN nc Þ

4bβδðr þ δÞλ 2bðr þ δ þ2λÞ wc ðr þδ þ3λÞ ϕðN nc Þ þ  2 ðr þ δþ 2λÞ2 βδðr þ δþ 2λÞ  2bðr þ δÞλϕðNnc Þ )  2 2 4b ð2b  wc Þðr þδÞ2 λ2 ϕ Nnc   2 ðr þ δ þ2λÞ2 βδðr þ δþ 2λÞ  2bðr þ δÞλϕðNnc Þ

150

Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

0   ∂W u0c γ ¼ γ   ∂Nnc @ 0  ¼ W r ðNc Þ Nc ¼ Nn   c ∂b ∂Nc 

11 A

N c ¼ N nc

(

β2 δ2 ðr þ δþ 2λÞ2 þ4βδλ wc λ2 bðr þ δÞðr þ δþ 2λÞ ϕðNnc Þ

2 ðr þ δþ 2λÞ βδðr þδ þ 2λÞ  2bðr þ δÞλϕðN nc Þ )  2 2 4b ðr þδÞ2 λ2 ϕ Nnc þ

2 ðr þδ þ 2λÞ βδðr þ δþ 2λÞ  2bðr þ δÞλϕðNnc Þ



0   ∂W u0c γ ¼ γ   ∂Nnc @ 0  ¼ W r ðNc Þ Nc ¼ Nn   c ∂wc ∂Nc 

11 A

Nc ¼ N nc



λ βδðr þ δ þ4λÞ 2bðr þδÞλϕðN nc Þ

40  ðr þ δÞðr þ δ þ2λÞ βδðr þδ þ 2λÞ  2bðr þ δÞλϕðNnc Þ 0   ″ ∂W u0c γ ¼ γ   ∂Nnc f ðN N nc Þ@ 0 W r ðN c ÞNc ¼ Nn  ¼  c r þδ ∂N ∂Nc 

11

0   u ″ ″ f ðN  Nnc Þ@ f ðN N nc Þ ∂W 0c γ ¼ γ    ¼  r þδ r þδ ∂N c 

11

ð50Þ

A

Nc ¼ N nc

A

41

ð51Þ

N c ¼ Nnc

0   ∂W u0c γ ¼ γ   ∂Nnc @ 0 ¼ W r ðNc ÞNc ¼ Nn   c ∂r ∂Nc 

11 A

N c ¼ N nc

9 8 4wc β2 δ2 λ3 ðr þδ þ2λÞ 2wc βδλ2 ð3ðr þ δÞ þ 4λÞ > > > >  > > < ðr þ δÞ bðr þ δÞ þ w λ þ w λðr þ δþ 2λÞ 2 βδðr þ δ þ2λÞ 2bðr þδÞλϕ = ðβδðr þ δþ 2λÞ 2bðr þδÞλϕÞ c c   þ 2 2 > > ðr þ δÞ2 ðr þ δ þ2λÞ ðr þδÞ2 ðr þ δþ 2λÞ > > > > ; :

where the inequality in (50) comes from assumptions (13) and (49) which states that δ4 2γ ϕλ, and the inequality in (51) ″ comes from (49) and the assumption that f ðÞ r 0. n In the Full-Interaction Equilibrium, U is given by (30). Hence, we can see that



∂Nnc n n 3 n 0 βδðδ þ 2λÞðδ þ3λÞ 4bλ β ð δþ 2λ Þ βλN ð δþ 6λ Þ þ N ϕ ðN Þ c c c ∂un ∂δ ¼

2 n 2 ∂δ Nðδþ 2λÞ βðδ þ2λÞ 2bλϕðN c Þ   n   ∂N n 4bβλϕ Nc λN nc ðδþ 3λÞ þ c ðδ þ2λÞ δ2 þ 3δλþ λ2 ∂δ 

2 Nðδþ 2λÞ2 βðδ þ2λÞ 2bλϕðN nc Þ   2 ∂Nn 2 4b λ2 ϕ N nc λN nc þ c ðδ þ λÞðδ þ 2λÞ ∂δ þ

2 Nðδþ 2λÞ2 βðδ þ2λÞ 2bλϕðN nc Þ





∂N nc ðδ þ2λÞ βðδþ 2λÞ  2bλϕðN nc Þ βδðδþ 3λÞ  2bλðδþ λÞϕðNnc Þ ∂u ∂λ ¼

2 ∂λ Nðδ þ2λÞ2 βðδ þ 2λÞ  2bλϕðNnc Þ  ∂Nn   Nnc β 4bλ3 c ϕ0 N nc þ βδðδþ 6λÞ ∂λ 

2 Nðδþ 2λÞ βðδ þ 2λÞ  2bλϕðNnc Þ n

Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

þ

151



4bδλNnc ϕðN nc Þ βðδþ 3λÞ  bλϕðN nc Þ

2 N ðδ þ 2λÞ2 βðδþ 2λÞ  2bλϕðNnc Þ

 n 2 ∂N nc n 2 2 2 n 3 ∂un  4Nc βλ ϕ þ ∂b β δðδþ 2λÞðδ þ 3λÞ þ 4b λ ðδ þλÞϕ Nc ¼

2 ∂b Nðδ þ2λÞ βðδþ 2λÞ  2bλϕðN nc Þ 2 n 0 n  2

  4bβλ λ Nc ϕ ðNc Þ þ δ þ3δλ þ λ2 ϕðNnc Þ 

2 Nðδþ 2λÞ βðδþ 2λÞ  2bλϕðNnc Þ (   n ∂un ∂N nc β2 δðδ2 þ 5δλ þ4λ2 Þ  4bβλ δ2 þ 3δλ þ λ2 ϕðNc Þ ¼

2 n ∂wc ∂wc Nðδ þ 2λÞ βðδ þ 2λÞ  2bλϕðNc Þ h i9  2 2 2 2λ 2b ðδ þλÞϕ Nnc þ β2 δ 2bβλNnc ϕ0 ðNnc Þ = 40 þ

2 ; Nðδ þ 2λÞ βðδ þ 2λÞ  2bλϕðNnc Þ 

 ∂un 1 ∂N n δβðδ þ3λÞ ðδþ λÞ2bλϕðN nc Þ ¼ 2 N nc N c ∂N ∂N βðδ þ2λÞ 2bλϕðN nc Þ N ðδ þ 2λÞ ) n 0 n n 3 4bβλ NN c ϕ ðN c Þ ∂Nc o0 þ 2 βðδ þ2λÞ 2bλϕðN n Þ ∂N

ð52Þ

ð53Þ

c

 2 ∂N nc n 2 2 β δðδþ 2λÞðδ þ 3λÞ þ 4b λ2 ðδ þλÞϕ Nnc ∂un ∂r ¼

2 ∂r Nðδþ 2λÞ βðδ þ 2λÞ  2bλϕðNnc Þ    4bβλ λ2 N nc ϕ0 ðN nc Þ þ ϕðNnc Þ δ2 þ3δλ þ λ2 

2 Nðδ þ2λÞ βðδ þ2λÞ 2bλϕðN nc Þ where the inequality in (52) comes from assumption of concavity of ϕðÞ and assumption (13) which states that δ4 2γ ϕλ, and the inequality in (53) comes from the facts that ∂Nnc =∂N 4 1 and N 4 Nnc , and assumption (13). □ Proof of Propositions 7 and 8. Using (41) and (42), the present-value Hamiltonian can be written as

H t ¼ e  rt f ðN  Nc ðtÞÞ þ ð2d2 ðtÞ þ d1 ðtÞÞwc þ ðNc ðtÞ 2d2 ðtÞ  d1 ðtÞÞb βd1 ðtÞγðtÞ

      N c ðtÞ γd2 ðtÞ  d1 ðt Þ  d2 ðt Þ  δ þ 2λ 1 þ ϕðNc ðt ÞÞ d1 ðt Þ þμ1 ðt Þ 2λ 2 d1 ðtÞ    γd2 ðtÞ ϕðNc ðt ÞÞ d1 ðt Þ  δd2 ðt Þ þμ2 ðt Þ 2λ 1 þ d1 ðtÞ

ð54Þ

Moreover, we need to consider the inequality constraints γ Z 0 and γ Z γ. The Lagrangian for the maximization problem becomes Lt ¼ H t þξ0 γ þ ξ1 ðγ γÞ: The first-order conditions for the maximization are Nc ðt Þ: γ ðt Þ: d1 : d2 :

∂Lt ; ∂N c ðtÞ ∂Lt ; 0¼ ∂γðtÞ ∂Lt  ; μ1 ðt Þ ¼  ∂d1 ðtÞ ∂Lt  ; μ2 ðt Þ ¼  ∂d2 ðtÞ 0¼

This is equivalent to 0¼

 0    ∂H t ¼ e  rt  f ðN  Nc ðtÞÞ þ b þμ1 ðt Þλ þ μ2 ðt Þ  μ1 ðt Þ 2λγd2 ðt Þϕ0 ðN c ðt ÞÞ ∂N c ðtÞ

0¼



∂H t þ ξ  ξ1 ¼ e  rt βd1 ðt Þ þ 2λϕðNc ðtÞÞd2 ðt Þ μ2 ðtÞ  μ1 ðtÞ þ ξ0  ξ1 ∂γðtÞ 0



μ1 ðtÞ ¼  e  rt ðwc  b βγÞ þ μ1 ðtÞðδ þ4λÞ 2μ2 ðtÞλ

ð55Þ

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μ2 ðtÞ ¼  2e  rt ðwc  bÞ þ 2μ1 ðtÞλð1 þγϕðNc ðtÞÞÞ þ μ2 ðtÞðδ  2γλϕðNc ðtÞÞÞ We evaluate the optimal allocation at a steady state. Then, the last two differential equations yield  e  rt 2λðwc  bþ βγÞ wc b  βγ þ μ1 ðt Þ ¼ r þ δ þ2λ  2γλϕðNc Þ r þ δ þ2λ μ2 ðt Þ ¼ 2e  rt

rwc þ βγλð1 þ γϕðN c ÞÞ þ wc ½δ þλð3 γϕðNc ÞÞ  b½r þ δþ λð3  γϕðN c ÞÞ ðr þ δ þ2λÞðr þδ þ2λ  2γλϕðN c ÞÞ

ð56Þ

ð57Þ

ð58Þ

Eq. (56) can be written as ∂H t ¼ ξ1  ξ0 ; ∂γðtÞ which implies that the optimal γ is a corner solution. If ξ0 40 and ξ1 ¼ 0, then γ o ¼ 0, whereas if ξ0 ¼ 0 and ξ1 4 0, then γ o ¼ γ . Substituting (57) and (58) into (56), we obtain       ∂H t δ 2λϕðN c Þðwc  b βγ Þ þ ¼ sgn β γϕðNc Þ  : sgn 2λ r þδ þ 2λ  2γλϕðN c Þ ∂γðtÞ Because we have assumed (13), i.e. δ4 2γ ϕλ, then ∂H t =∂γðtÞ 40 can be possible only when wc b  βγ ¼ wc 2b 40. Moreover, plugging the steady state conditions (5)–(7), into the Hamiltonian (54), we obtain H t jγ ¼ γ  H t jγ ¼ 0 ¼

bNc λ½ βδðδþ 2λÞ þ 2λðbδþ 2wc λÞϕðN c Þ ðδþ 2λÞ2 ½βðδ þ2λÞ 2bλϕðN c Þ

ð59Þ

Define bo  ϕ

βδðδþ 2λÞ 2λðbδþ 2wc λÞ

From (59) and using (13), we know that if bo H t jγ ¼ γ ⋛H t jγ ¼ 0 3 ϕ⋛ϕ Let N oc denote the city population that satisfies b ϕðN oc Þ ¼ ϕ

o

which means that   βδðδ þ2λÞ N oc ¼ ϕ  1 2λðbδþ 2wc λÞ b n was the equilibrium threshold defined by (47). Simple comparison yields Remember that ϕ bo ¼ bn  ϕ ϕ

rβδðwc bÞ

4 0; ðbδ þ2wc λÞ bðr þδÞ þ2wc λ

implying that N nc 4 N oc . Next, we examine whether the urbanization level is optimal when the equilibrium level of social interaction is efficient. Substituting μ1 ðtÞ and μ2 ðtÞ in (57) and (58) and using (55), we obtain that ( o Θt jγ ¼ 0 for γ ¼ 0 Wr ¼ Θot jγ ¼ γ for γ ¼ γ ; where Θot jγ ¼ 0 and Θot jγ ¼ γ are defined by b λðwc  bÞðr þ δ þ4λÞ  ; r þ δ ðr þ δÞðr þ δþ 2λÞ2  b λ 2βwc λ  wc  2bþ  r þ δ ðr þδÞðr þ δ þ2λÞ cðr þ δþ 2λÞ  2bλϕðNc Þ

Θot jγ ¼ 0  Θot jγ ¼ γ

4wc λ3 ϕ0 ðN c ÞNc b=β ðr þ δÞðδ þ 2λÞðδ þ2λ  2bλϕðNc Þ=βÞðr þδ þ2λ  2bλϕðNc Þ=βÞ   0 and Wr is given by (17) W r  f ðN  Nc Þ=ðr þδÞ . In contrast, the equilibrium urbanization level is determined by ( u W 0c jγ ¼ 0 for γ ¼ 0 ; Wr ¼ W u0c jγ ¼ γ for γ ¼ γ þ

ð60Þ

Y. Sato, Y. Zenou / European Economic Review 75 (2015) 131–155

153

where W u0c jγ ¼ 0 and W u0c jγ ¼ γ are given by W u0c jγ ¼ 0 ¼

bðr þδÞðr þδ þ3λÞ þwc λðr þ δþ 4λÞ ðr þδÞðr þδ þ2λÞ2

and 2

W u0c jγ ¼ γ ¼

βwc δλðr þδ þ4λÞ 2b λϕðNc Þðr þ δÞ2 þ bðr þδÞ½βδðr þ δþ 2λÞ  2wc λ2 ϕðNc Þ : ðr þ δÞðr þ δ þ2λÞ½βδðr þ δþ 2λÞ  2bλϕðNc Þðr þ δÞ

Taking the difference, we have W u0c jγ ¼ 0  Θot jγ ¼ 0 ¼

2ðwc  bÞλðr þδ þ 4λÞ ðr þ δÞðr þ δþ 2λÞ2

4 0:

Moreover, from the concavity of ϕðÞ and 0 o ϕ r ϕðÞ, we know that ϕðN c Þ 4ϕ0 ðNc ÞN c . Then, we have W u0c jγ ¼ γ  Θot jγ ¼ γ 4 W u0c jγ ¼ γ  

 

bðr þ δ þ3λÞ λ wc  b þ2λwc = r þ δ þ2λ  2bλϕðNc Þ=β ðr þ δÞðr þδ þ 2λÞ

4wc λ3 ϕðNc Þb=β : ðr þ δÞðδ þ 2λÞðδþ 2λ 2bλϕðNc Þ=βÞðr þ δ þ2λ  2bλϕðNc Þ=βÞ

Hence, we obtain that



  w 2b eβwc λ βðδþ 2λÞ  3bλϕðNc Þ

40 lim W u0c jγ ¼ γ  Θot jγ ¼ γ ¼ þ δ þ 2λ ðδþ 2λÞ βðδþ 2λÞ  2bλϕðNc Þ r-0

where the last inequality comes from the fact that γ ¼ γ can be optimal only when wc 2b 40, and assumptions δo 2λ and (13) which states that δ 42γ ϕλ. From Fig. 4(a) and (b), we obtain the results of the proposition. □ A.3. Introducing the job-destruction rate in the model We introduce the possibility of job destruction rate, which arrives to each employed worker in the city at an exogenous rate σ( 40). The law of motion of the dyads is not anymore given by (3) but by 8  > > > > d2 ðtÞ ¼ gðtÞd1 ðtÞ  ð2σ þ δÞd2 ðtÞ > <  d1 ðtÞ ¼ 2λd0 ðtÞ þ2σd2 ðtÞ  ðσ þ δþ gðtÞÞd1 ðtÞ > > >  δN c ðtÞ 1  > > þσd1 ðt Þ  2λd0 ðt Þ  δd0 ðt Þ þ Nc ðt Þ : d0 ðt Þ ¼ 2 2 







The steady-state values of dyads are then obtained by solving d2 ðtÞ ¼ d1 ðtÞ ¼ d0 ðtÞ ¼ Nc ðtÞ ¼ 0 (combined with Nc =2 ¼ d0 ðtÞ þ d1 ðtÞ þ d2 ðtÞ) and we obtain 2λ2 Nc δ c ÞÞ þ2λσð2  γϕðN c ÞÞ þ δ½3σ þλð4 2γϕðN c ÞÞ λN c ðδ þ2σ  2γλϕðN c ÞÞ d1 ¼ 2 δ þ 2σ 2 þ λ2 ð4  4γϕðNc ÞÞ þ2λσð2  γϕðNc ÞÞ þ δ½3σ þλð4 2γϕðN c ÞÞ   Nc δ2 þ 2σðσ  γλϕðN c ÞÞ þ δ½3σ þ λð2  2γϕðNc ÞÞ d0 ¼  2  2 δ þ 2σ 2 þ λ2 ð4 4γϕðN c ÞÞ þ 2λσð2  γϕðN c ÞÞ þ δ½3σ þ λð4  2γϕðNc ÞÞ d2 ¼

2

þ 2σ 2 þ λ2 ð4  4γϕðN

The dynamics of the city population, Nc, remain unaltered and is still given by (4). The Bellman equations can now be written as rW r ¼ wr δW r rW e1c ¼ wc þσðW u0c  W e1c Þ þ g n ðW e2c  W e1c Þ  δW e1c rW e2c ¼ wc þσðW u1c  W e2c Þ þ σðW e1c  W e2c Þ  δW ec rW u0c ¼ b þ λðW e1c  W u0c Þ þλðW u1c  W u0c Þ  δW u0c rW u1c ¼ b  βγ þg n ðW e2c  W u1c Þ þ σðW u0c  W u1c Þ  δW u1c where the major differences between the equations given in the main text and the ones above lie in the asset values of employed workers in the city, that is rW e1c and rW e2c . When σ ¼ 0, these two values were equal but now they are different. The former rW e1c is the asset value of an employed worker in a d1 dyad and the latter rW e2c is that of an employed worker in a d2 dyad. If an employed worker in a d1 dyad is hit by a negative shock (at rate σ), she loses her job and the d1 dyad becomes a d0 dyad. This leads to a reduction in capital gains by W e1c  W u0c . If, on the contrary, her partner becomes employed (at rate gn),

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then the d1 dyad becomes a d2 dyad, and there is an increase in capital gains by W e2c  W e1c . The interpretation is similar for rW e2c .    A steady-state equilibrium Nnc ; γ n is such that the equilibrium migration condition is satisfied Nc ðtÞ ¼ 0, i.e. W r ¼ W u0c n

which determines Nnc , and the unemployed workers in a d1 dyad optimally choose γ that maximizes W u1c , taking as given N nc , n n n d0 , d1 , and d2 . For the ease of the exposition, we take the limit of r-0 (i.e., no discounting) to see the equilibrium property regarding γ. If we define ΦðγÞ as   ∂W u1c ¼ sgnðΦðγÞÞ sgn ∂γ then after tedious calculations, we obtain that Φ0 ðγÞ 40 This implies that, as in the main text, we have only two possibilities as equilibrium outcomes: either γ ¼ 0 or γ ¼ γ , which is what we obtained when σ ¼ 0 (see Proposition 2). Because of the complexity of the model, in the remaining part of this Section, we resort to numerical analysis for further characterization of equilibrium. For this purpose, we specify the function of the agglomeration effect, ϕðNc Þ, as ϕðNc Þ ¼ ϕ

Nc 1 þ Nc

and the rural production function f ðN r Þ, as f ðNr Þ ¼ μlnð1 þ Nr Þ where α and μ are positive constants. We then set parameters of the model as β ¼ 1=5, δ ¼ 1=10, ϕ ¼ 1=20, λ ¼ 1=5, μ ¼ 4, σ ¼ 1=10, b ¼ 1=3, r¼ 0.05, and wc ¼ 5.23 Under these parameter values, job seekers will choose not to interact, i.e., γ ¼ 0, if and only if Nc o 8:04412 whereas they choose to fully interact, i.e., γ ¼ γ , if and only if Nc Z 8:04412. Furthermore, once we fix the total population, N, the no-migration condition, W u0c ¼ W r , determines the equilibrium city population, N nc . When the n total population is small, N ¼3, the equilibrium city population size is NNI ¼ 2:54, and hence we observe the No-Interaction c n Equilibrium. When the total population is large and N ¼10, the equilibrium city population size is NFI c ¼ 9:54, and we observe the Full-Interaction Equilibrium. In a similar way described in the text, we can derive conditions for the constrained optimum. Under the parameter values set above, it is optimal to fully interact when Nc is larger than 3.89, which is smaller than the threshold value for equilibrium. Hence, we obtain a result similar to Proposition 7. Finally, the optimal size of city population is N NIo c ¼ 2:5 when N ¼3 and NFIo c ¼ 9:49 when N ¼10. Hence, we also observe over-urbanization in this numerical exercise, which is in line with Proposition 8. References Akerlof, G., 1997. Social distance and social decisions. Econometrica 65, 1005–1027. Amato, P.R., 1993. Urban-rural differences in helping friends and family members. Soc. Psychol. Q. 56, 249–262. Ballester, C., Calvó-Armengol, A., Zenou, Y., 2006. Who's who in networks. Wanted: the keyplayer. Econometrica 74, 1403–1417. Bayer, P., Ross, S.L., Topa, G., 2008. Place of work and place of residence: informal hiring networks and labor market outcomes. J. Polit. Econ. 116, 1150–1196. Beckmann, M.J., 1976. Spatial equilibrium and the dispersed city. In: Papageorgiou, Y.Y. (Ed.), Mathematical Land Use Theory, Lexington Books, Lexington, MA, pp. 117–125. Blien, U., Suedekum, J., Wolf, K., 2006. Productivity and the density of economic activity. Labour Econ. 13, 445–458. Brueckner, J.K., Largey, A.G., 2008. Social interaction and urban sprawl. J. Urban Econ. 64, 18–34. Brueckner, J.K., Thisse, J.-F., Zenou, Y., 2002. Local labor markets, job matching and urban location. Int. Econ. Rev. 43, 155–171. Calvó-Armengol, A., Jackson, M.O., 2004. The effects of social networks on employment and inequality. Am. Econ. Rev. 94, 426–454. Calvó-Armengol, A., Verdier, T., Zenou, Y., 2007. Strong and weak ties in employment and crime. J. Public Econ. 91, 203–233. Calvó-Armengol, A., Patacchini, E., Zenou, Y., 2009. Peer effects and social networks in education. Rev. Econ. Stud. 76, 1239–1267. Cattell, V., 2001. Poor people, poor places, and poor health: the mediating role of social networks and social capital. Soc. Sci. Med. 52, 1501–1516. Ciccone, A., Hall, R.E., 1996. Productivity and the density of economic activity. Am. Econ. Rev. 86, 54–70. Coleman, J.S., 1988. Social capital in the creation of human capital. Am. J. Sociol. 94, S95–S120. Combes, P.-P., Gobillon, L., 2015. The empirics of agglomeration economies. In: Duranton, G., Henderson, V., Strange, W. (Eds.), Handbook of Regional and Urban Economics, vol. 5. , Elsevier Publisher, Amsterdam. forthcoming. Combes, P.-P., Duranton, G., Gobillon, L., Roux, S., 2010. Estimating agglomeration effects with history, geology, and worker fixed-effects. In: Glaeser, E.L. (Ed.), Agglomeration Economics, Chicago University Press, Chicago, pp. 15–65. Desmond, M., 2012. Disposable ties and the urban poor. Am. J. Sociol. 117, 1295–1335. Essletzbichler, J., 2004. The geography of job creation and job destruction in the U.S. manufacturing sector 1967-1997. Ann. Assoc. Am. Geogr. 94, 602–619.

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