Discussion Papers In Economics
And Business

How Urbanization Affect Employment and Social Interactions

Yasuhiro Sato and Yves Zenou

Discussion Paper 13-32

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHow Urbanization Affect Employment and Social Interactions

Yasuhiro Sato and Yves Zenou

Discussion Paper 13-32

December 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHow Urbanization Affect Employment and Social Interactions
Yasuhiro Sato∗

Yves Zenou†

December 16, 2013

Abstract
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 choose to interact withmore 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 uniquesteady-state equilibrium with full social interactions. We show that these equilibria are usually notsocially 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, weshow that there is over-urbanization in equilibrium.

Keywords: Weak ties, strong ties, social interactions, urban economics, labor market.
JEL Classification: J61, R14, R23.

∗†

Osaka University, Japan. Email: ysato@econ.osaka-u.ac.jp.
Stockholm University, Research Institute of Industrial Economics (IFN), Sweden,yves.zenou@ne.su.se.

1

and CEPR. E-mail: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 thelimited number of people who live nearby whereas people in large cities have a great deal of choice inconstructing 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 traditionalsources (such as family) and are more likely to include voluntary sources, such as friends, coworkers andclub 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 that can help
them find a job.3 We define a weak tie when social interactions between two persons is transitory (likefor example random encounters). On the contrary, a person has a strong-tie when the relationship isrepeated over time, for example members of the same family or very close friends.4 Each individual canbe in three different states: either she is unemployed and her strong tie is also unemployed (dyad d0 ) or
she is unemployed and her strong tie is employed (or the reverse) (dyad d1 ) or both are employed (dyadd2 ). Workers move between the city and the rural area. In the city, unemployed workers can find a job
either directly or through weak and strong ties and the unemployed worker belonging to a d1 dyad has todecide how much time (effort) to spend with weak ties. As a result, an unemployed worker in a d1 dyad
can find a job either directly or through their strong ties (who are employed) or through their weak ties.
The latter effect 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 the1

This is the perspective of the so-called social disorganization theory and the social capital literature (see e.g. Wirth, 1938,

Coleman, 1988, and Putman, 1993, 2001).
2
This is the so-called subculture theory (see e.g. Fisher 1976, 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).
4This is not the precise definition of weak ties first used by Granovetter. In Granovetter’s (1973), weak ties are expressedin 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 orno overlap between their respective personal networks. Vice versa, the tie is strong if most of A’s contacts also appear in B’s
network.

2

city, the easiest is to meet one of them; the reverse is true for the unemployed workers).
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 those relationships may not be personal nor strong, yet those weak ties are the ones to matter mostfor providing social support, in particular in finding jobs. We also demonstrate that, for low urbanizationlevels, there is a unique steady-state equilibrium where workers do not interact with weak ties, while,for high level of urbanization, there is a unique steady-state equilibrium with full social interactions.
We finally show that these equilibria are usually not socially efficient when the urban population has anintermediate size because there are too few social interactions compared to the social optimum. Even
when social interactions are optimal, we show that there is over-urbanization 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 theefficiency results in Section 6. Finally, Section 7 concludes. All proofs of propositions can be found in
the Appendix.

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.5 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 interestingmodels 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 andNocco (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
5

See Fujita and Thisse (2013) for a literature review.

3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 isno 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.6 Brueckner et al. (2002), Helsley 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 paperthat 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.7 Schelling (1971) is clearly a seminalreference 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 oflocation decision. Indeed, his results suggest that total segregation persists even if most of the population
is tolerant about heterogeneous neighborhood composition. Our model is very different from models ala 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, which population size is Nc , orin the rural area, with population size of Nr , with Nr + Nc = N .
6

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 also Ioannides (2012, Chap. 5) who reviews the literature on social interactions and urban economics.
7
See also Calvó-Armengol et al. (2007), Calvó-Armengol and Jackson (2004) and Zenou (2014) for models of weak andstrong ties in the labor market but where the urban space is not modeled.

4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 two different states: employed or unemployed. Dyads, which
consist of paired individuals, can thus be in three different states,8 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 (t) = 2d2 (t) + d1 (t)⎩ U (t) = 2d (t) + d (t)
c

0

(1)

1

Since total urban population is Nc , we have

Ec (t) + Uc (t) = Nc

(2)

or, alternatively,
d2 (t) + d1 (t) + d0 (t) =

Nc
2

(3)

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, stayfor ever in the city c. We can easily show that this is optimal in equilibrium.
8

The inner ordering of dyad members does not matter.

5

3.4 Labor market
The rural labor market is perfectly competitive and has no friction. In other words, everybody can obtaina job in the rural area and it is assumed that the rural wage is flexible enough to guarantee that there is
full-employment.9 The wage in the rural area wr is thus determined by the marginal productivity ofworkers, i.e.
wr = f 0 (N − Nc )(4)

As usual, we assume that f 0 (.) > 0 and f 00 (.) ≤ 0 and the Inada conditions. These assumptions reflect
the implicit assumption that the land endowment is limited and the agricultural sector exhibits decreasing
returns with respect to labor input.

3.5 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 employedworker hears about a job, she automatically direct it to her strong tie. This is a convenient modellingassumption, which stresses the importance of on-the-job information (Ioannides and Loury, 2004). Alljobs and all workers are identical (unskilled labor) so that all employed workers obtain the same wage
wc . Therefore, employed workers, who hear about a job, pass this information on to their current matched
partner since they cannot use this information themselves.
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 nosocial connections, i.e. no contact with weak ties. This is because it takes time to meet weak ties andpeople stuck in d0 dyads tend to interact mostly with their strong ties and isolate themselves from theurban community. As a result, the only way she 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 fromthem. We denote by γ ∈ [0, 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 cγ (c > 0). We also assume that there are agglomeration
effects so that the higher is the urban population Nc , the easier is to meet weak ties (see, e.g. Desmond,
9This is a standard assumption in the migration literature (see e.g. Zenou, 2011) and it does make sense, especially indeveloping countries, where jobs in rural areas are mostly from the agricultural sector and easy to obtain. Also many ruralfirms 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).

6

2012, for evidence). The agglomeration effect is captured by φ(Nc (t)), with φ0 (.) > 0, φ00 ≤ 0 and
0 < φ ≤ φ(·) ≤ φ ≤ 1.
As a result, an unemployed worker in a d1 dyad will hear about a job from a weak tie at rate
γφ (Nc (t)) 2λd2 (t)/d1 (t). Indeed, the rate at which an unemployed worker obtains a job from a weak tie
is increasing in her own effort γ, in the urban population Nc (t) and depends on 2λd2 (t)/d1 (t), the fraction of employed workers who are aware about a job. In other words, 2λd2 (t)/d1 (t) captures somethingsimilar to the matching function in the search literature (Pissarides, 2000) where the number of employedworkers in the d2 dyad is 2d2 (t) while the number of unemployed workers who are in competition forjob information is d1 (t). 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 γφ (Nc (t)) 2λd2 (t)/d1 (t). Lettingg(t) be the rate at which an unemployed worker in a d1 dyad finds a job (or equivalently the rate at whicha d1 dyad becomes a d2 dyad), we haveg(t) ≡ 2λ + γφ (Nc (t)) 2λd2 (t)
d1 (t)(5)

Finally, we assume that each dyad “dies” at rate δ. If we think of the married couple interpretationof 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. 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, at each period of time t, δ [d2 (t) + d1 (t) + d0 (t)] die andδ [d2 (t) + d1 (t) + d0 (t)] = δNc (t)/2 are born as a d0 dyad. Observe that λ, the rate at which a personhears 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 ofsocial interactions as captured by γ. Because of the continuous time Markov process, the probabilityof a two-state change is zero (small order) during a small interval of time t and t + dt. This means, inparticular, 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 + dt, 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 + dt. This applies to all other dyads
mentioned above.

7

3.6 Flows of dyads between states
It is readily checked that the net flow of dyads from each state between t and t + dt is given by:⎧ •⎪d (t) = g(t)d1 (t) − δd2 (t)
⎪
⎪
⎨ 2
•

d1 (t) = 2λd0 (t) − [δ + g(t)] d1 (t)⎪⎪⎪⎩ d• (t) = δNc (t) − 2λd (t) − δd (t) + 1 N •(t)
0

0

2

0

2

(6)

c

where g(t) is defined by (5) and•u
Nc (t) = α [W0c(t) − Ω]

(7)u (t) is the lifetime expected utility of an unemployed worker in a d dyad living in the city,where W0c0

Ω is lifetime expected utility of a rural worker and α > 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.
Let us comment (6). The positive change of d0 dyads over time is due to the fact that, at each periodof time t, δ [d2 (t) + d1 (t) + d0 (t)] die and δ [d2 (t) + d1 (t) + d0 (t)] = δNc (t)/2 are born as a d0 dyad•

and that 12 Nc (t) workers migrate to the city in a d0 dyad. The negative change of d0 dyads over timeis 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 thed1 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, Ec (t), is given by Ec (t) = 2d2 (t)+d1 (t)whereas the number of unemployed workers, Uc (t), is given by Uc (t) = 2d0 (t) + d1 (t). As a result,•••Nc (t) = Ec (t) + U (t) = 2 [d0 (t) + d1 (t) + d2 (t)]. In steady state, d2 (t) = 0, d1 (t) = 0, d0 (t) = 0 and•

Nc (t) = 0, and the flows in the labor market can be described by the following figure:[Insert F igure 1 here]
By solving (6) in steady state, we obtain:d∗0 =d∗1 = d∗2
d∗2 =

∙

δNc∗
2(δ + 2λ)

(8)

¸
δ
∗
− γφ(Nc )2λ(9)2λ2 Nc∗(δ + 2λ)[δ + 2λ(1 − γφ(Nc∗ ))]8(10)

For d∗1 to be strictly positive, we assume that

2λφγ < δ(11)Moreover, for d∗2 to be positive, we assume that δ + 2λ(1 − γφ) > 0, which can be rearranged as
2λφγ < δ + 2λ(12)2λφγ < δ(13)

Combining (11) and (12) leads to

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 arural worker, we have:rΩ = wr − δ Ω(14)where Ω is lifetime expected utility of a rural worker, wr is the rural wage defined by (4), and r is thediscount rate. Observe that, 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 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 thecity, they switch from a d2 dyad from the rural area to a d0 dyad in the city.
Let us now write Wce , the lifetime expected utility of an urban worker who is employed in the city. It
is given by:
rWce = wc − δWce

(15)

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 exist the economy(at rate δ) either because they retire or they die.10 This is, however, not true for the unemployed workersu,because their chance to escape unemployment depends on which dyad they belong to. Denote by W0c

the lifetime expected utility of an urban unemployed worker in a d0 dyad. Then, we have:
u
u
u
u
u
= b + λ(Wce − W0c) + λ(W1c− W0c) − δW0c
rW0c

(16)

Indeed, when someone is unemployed in a d0 dyad, she obtains an unemployment benefit of 0 < b < wc .
Then, she can hear directly about a job at rate λ and become employed and thus obtains a surplus of
10This is clearly a simplified assumption, which helps us solve the model. If we relax it, the Bellman equations will becomemuch more complicated and the resolution of the model more cumbersome but the results will be qualitatively unchanged.

9u > 0 or her partner in the dyad, who is also unemployed, finds a job at rate λ, and thus
Wce − W0c
u − W u > 0. Finally, the dyad can die at rate δ andswitches to a d1 dyad so that the surplus is now W1c0c
u < 0.
the negative surplus is 0 − W0c

Finally, the lifetime expected utility of an urban unemployed worker in a d1 dyad is equal to:uuu= b − βγ + g ∗ (Wce − W1c) − δW1c
rW1c

(17)

where g ∗ is the steady-state rate at which the unemployed worker from a d1 dyad finds a job and is equal
to:11d∗g ∗ = 2λ + γφ (Nc∗ ) 2λ 2∗
d1

(18)

Indeed, an unemployed worker in a d1 dyad earns an unemployment benefit of b and pays a cost of βγ forinteracting with weak ties in the city where β > 0 is the marginal cost (remember that γ is the worker’su > 0 or theeffort). This worker can leave unemployment at rate g ∗ and obtain a surplus of Wce − W1cdyad can die at rate δ. We assume that γ ≤ b/β, which ensures that b − βγ ≥ 0. Since b/β is the natural
upper bound of γ, we set γ = b/β without loss of generality. As a result, condition (13) can be written
as:
2λφ

b
<δ
β

(19)

Solving for the Bellman equations, we easily obtain
u
W0c
=

u)
b + λ(Wce + W1c
r + δ + 2λ

(20)

∗

wc
b − βγ + gr+δu
W1c =r + δ + g∗
wc
Wce =
r+δ
wr
Ω=
r+δ

(21)
(22)
(23)

4 Steady-state equilibrium
4.1 Definition
Definition 1 A steady-state equilibrium (Nc∗ , γ ∗ ) is such that the equilibrium migration condition is•satisfied Nc (t) = 0, i.e.
u
=Ω
W0c

(24)

u.
which determines Nc∗ , and the unemployed workers in a d∗1 dyad optimally choose γ that maximizes W1c
11

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

104.2 Optimal choice of social interactions
u , taking
Let us determine the optimal γ. Unemployed workers in a d∗1 dyad choose γ that maximizes W1c

as given Nc∗ , d∗1 and d∗2 . Define−1

Nc ≡ φµδβ(r + δ + 2λ)2
4λ (wc − b) + 2λγβ(r + δ + 2λ)

¶

(25)

We have the following result:
Proposition 1 Assume (19).
(i) If the city population is sufficiently small, i.e. Nc < N c , then unemployed workers in a d∗1 dyad
choose not to interact with weak ties, i.e. γ ∗ = 0.
(ii) If the city population is sufficiently large, i.e. Nc ≥ N c , then unemployed workers choose tointeract as much as possible with weak ties, i.e. γ ∗ = γ.
This is an interesting result that links urban population to social interactions. The intuition of thisresult is as follows. When choosing their optimal social-interaction effort γ, workers trade off the benefits of increasing γ, which is finding a job more quickly, and the costs, which is simply the effort cost
βγ. However, the benefits from increasing γ is captured by g ∗ , the rate at which the workers leave unemployment, which increases with γ but depends on the size of the urban population Nc (agglomeration
effect). When this 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 provide full socialization effort with weak ties.
As stated in the Introduction, there is a literature in sociology that supports this result (see e.g. Wirth,
1938, Coleman, 1988, and Putman, 1993, 2001).12 Fisher (1982) found that urban dwellers had moredispersed 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 theneighborhood. 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.
12

This, in fact, goes back to Tonnies (1957) and Simmel (1995) with the idea of rural gemeinschaft (or community) andurban gesellchaft (or association).

11

However, although individuals may have fewer strong relationships in cities than in villages, they
have more random encounters (weak ties), 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 forproviding support in getting a job. He criticized the assumption that strong ties in close networks werestrong 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 sourcesof 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 ofassistance received and provided by friends and family. He finds that urbanites receive more help fromfriends than do rural dwellers, give more help to friends, expect more help from friends, and expect lesshelp 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 arethose Solidarity Networks that combine positive aspects of dense and loose networks. They consist of awide 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 informalorganizations on the other. As Cattell concludes “the more varied the network, the greater the range ofresources accessible, and the greater the potential benefits to health.” (Cattell 2001: 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 weakties than in less dense cities.

4.3 Characterization of equilibriumu , increases with
Equation (20) implies that the asset value of an unemployed worker in the d0 dyad, W0c
u . Because W e does not depend on γ, we know that W u |uW1cc0c γ=γ is larger than W0c |γ=0 if and only ifu|u
W1cγ=γ is larger than W1c |γ=0 , i.e., the city population is sufficiently large so that Nc > N c .

12

¢
¡
Definition 2 A No-Interaction Equilibrium NcNI∗ , γ NI∗ is when all workers in the city choose γ N I∗ =¡¢0 while a Full-Interaction Equilibrium NcF I∗ , γ F I∗ is when all workers in the city choose γ F I∗ = γ.
The city population is determined by the decision of the new entrants to migrate to the city. Thisrequires that the asset value of a rural employed worker to be equal to that of an urban unemployedu . The following proposition characterizes the equilibrium.
worker in the d0 dyad, i.e. Ω = W0c

Proposition 2 Assume (19).
¢¡(i) If Nc < N c , then there exists a No-Interaction Equilibrium NcNI∗ , γ NI∗ such that all urban
workers provide effort γ NI∗ = 0.

¡¢(ii) If Nc ≥ N c , then there exist a Full-Interaction Equilibrium NcF I∗ , γ F I∗ such that all urban
workers exert effort γ F I∗ = γ.

In the proof of Proposition 2, we provide, for each equilibrium, the equilibrium values of the numbersof dyads, d∗0 , d∗1 and d∗2 , the equilibrium urban population size Nc∗ , the level of urban unemployment Uc∗ ,the job arrival rate of an unemployed worker in a d1 dyad, g ∗ , and all the equilibrium value functions
for all workers. This proposition confirms our previous result showing how urbanization (capture by Nc )affects the social behavior of workers.

4.4 Existence and uniqueness of equilibrium
Since, in Proposition 2, the condition for equilibrium depends on Nc , an endogenous variable, we needto examine under which condition there exists an equilibrium and if it is unique. The determination of
the city population Nc is described in Figure 2(a) and Figure 2(b).

[Insert F igures 2(a) and 2(b) here]
In Figure 2(a), the horizontal axis represents the urban population Nc from the right. The rural
population Nr is represented by the difference between N and Nc . The vertical axis shows the assetu . Ω is represented by a upward sloping curve (with respect to the urban population
values Ω and W0c

Nc ) because dΩ/dNc = −dΩ/dNr = −f 00 /(r + δ) > 0. From (32), (33) and (35), we know that
u|
u
u
W0c
γ=0 is independent of Nc whereas W0c |γ=γ is increasing in Nc . Observe first that Ω and W0c |γ=0
u|0
has a unique interaction if W0c
γ=0 > f (N )/(r + δ) because we assumed the Inada conditions for f (·).
u|u
Observe second that W0c
γ=γ > W0c |γ=0 as long as Figure 2(b) is relevant. The Inada condition foru|uf (·) combined with the fact that W0cγ=γ,φ=φ(N ) has a finite value, ensures that Ω and W0c |γ=γ has atleast one intersection.
13

In Figure 2(a), when the total population N increases, the urban population, Nc , also increases. Thisu|
can be confirmed by totally differentiating Ω = W0c
γ=0 since we have: dNc /dN = 1 > 0. Whenbc , the regime switches from the no-social interaction case (γ ∗ = 0) to
the city population reaches N

the full social interaction case (γ ∗ = γ) and the equilibrium population distribution is determined byu|Ω = W0cγ=γ . We have the following proposition:

Proposition 3 Assume (19).
(i) If Nc∗ is small enough, i.e. Nc∗ < N c , then the city is characterized by no social interaction¡¢(γ ∗ = 0) and there is therefore a No-Interaction Equilibrium NcN I∗ , γ NI∗ . Furthermore, if¢¡
Condition (52) holds, then the No-Interaction Equilibrium NcNI∗ , γ NI∗ is unique and given by(37)−(38) in Proposition 2 (i) (see Figure 2(a)).

(ii) If Nc∗ is large enough, i.e. Nc∗ > N c , then the city is characterized by full social interactions¡¢(γ ∗ = γ) and there is therefore a Full-Interaction Equilibrium NcF I∗ , γ F I∗ . Furthermore, if¢¡
Conditions (52) and (53) hold, then the Full-Interaction Equilibrium NcF I∗ , γ F I∗ is unique andgiven by (45)−(47) in Proposition 2 (ii) (see Figure 2(b)).

When the level of urbanization is low, the contact rate with weak ties, φ(Nc ), is also low so that
it is not worth devoting much effort to interact with weak ties. As the urban population grows, theagglomeration effect regarding the contact rate gets larger and it becomes rewarding to social interact
with weak ties. As will be shown in the next section, the urban population size, Nc∗ , increases with thetotal population size (rural plus urban), N , in the No-Interaction equilibrium. Moreover, Nc∗ , determinedby (44), has the property that limN →∞ Nc∗ = ∞. Hence, there exists a threshold value of N underwhich we observe the No-Interaction equilibrium and beyond which we observe the Full-Interaction
equilibrium. Thus, if we consider a continuous growth process of total population size, there will be aregime change from the No-Interaction equilibrium to the Full-Interaction equilibrium in the city.

5 Comparative statics
In this section, we provide the results of basic comparative statics with respect to the urban population
size, Nc∗ , and the unemployment rate, u∗c = U ∗ /N .

5.1 No-Interaction Equilibrium¡¢
Let us start with the No-Interaction equilibrium NcNI∗ , γ NI∗ , where Nc∗ is determined by (44) and is
described in Figure 2(a). In order to focus on the unique No-Interaction equilibrium, we assume that14

condition (52) holds. We provide here explanations based on the figure and relegate the formal proofs
to the Appendix. As shown in Figure 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 au|d0 dyad, W0c
γ=0 , whereas it doesn’t affect the asset value of a rural employed worker, Ω. Hence, byincreasing these variables, cities will attract more people and the urban population size, Nc∗ , will becomelarger. 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, Nc∗ , decreases. An increase in the total
population, size N , reduces Ω, leading to a larger Nc∗ .
The effects on unemployment rate can be readily examined. Because the number of unemployedworkers is given by (38), the effect of an increase of wc , b, or r on the unemployment rate, u∗c = U ∗ /N ,is proportional to that of Nc∗ . An increase in N affects both the numerator, U ∗ , and the denominator, N ,of u∗c = U ∗ /N . Still, we can see from the Appendix that ∂Nc∗ /∂N = 1, implying that ∂(Nc∗ /N )/∂N =(N − Nc∗ )/N 2 > 0. Hence, we know that an increase in N raises u∗c . However, an increase in δ or λhas additional impacts on u∗c , making its total effect on unemployment rate ambiguous. The following
proposition summarizes our findings:¢¡
Proposition 4 Suppose that Condition (52) holds. In the No-Interaction equilibrium NcN I∗ , γ NI∗ , anincrease in wc , b, N , or λ, or a decrease in δ or r increases the urban population size, Nc∗ . Moreover,an increase in wc , b, or N , or a decrease in r increases the unemployment rate, u∗c . An increase in δ or
λ has an ambiguous effect on u∗c .

5.2 Full-Interaction Equilibrium
We next examine the Full-interaction equilibrium, where the urban population size is determined by (51)and is described in Figure 2(b). Here, we assume that conditions (52) and (53) hold so that the model has¢¡
the unique Full-Interaction equilibrium NcF I∗ , γ F I∗ . As shown in Figure 2(b), a higher value of wc

u|shifts the asset value of an urban unemployed worker in a d0 dyad, W0c
γ=γ , upwards whereas it doesn’tchange the asset value of a rural employed worker, Ω. Thus, it increases Nc∗ . Furthermore, an increaseu|∗in N lowers Ω whereas it doesn’t change W0c
γ=γ , resulting in a larger Nc . Unfortunately, the effects

of a change in other parameters are ambiguous.
The effects of unemployment rate, u∗c , are somewhat more complicated in this case because now theurban population size (and hence density) has agglomeration effects on the contact rate with weak ties
described by φ(Nc ). From (47), we can see that, on the one hand, a larger Nc∗ has an direct effect ofraising u∗c . On the other hand, it indirectly lowers u∗c by raising φ(Nc ). Such an indirect effect is not15

sufficient to dominate the direct effect in the case of an increase in wc , but it is so in the case of anincrease in N .
¡¢Proposition 5 Suppose that conditions (52) and (53) hold. In the Full-Interaction equilibrium NcF I∗ , γ F I∗ ,an increase in wc or N increases the urban population size, Nc∗ . Moreover, an increase in wc increases
the unemployment rate, u∗c , whereas an increase in N decreases it. An increase in b, δ, λ, or r has
ambiguous effects on Nc∗ and u∗c .

The following table summarizes all our comparative statics results:
No-Interaction equilibrium (γ ∗ = 0)
wc

b

N

δ

λ

r

Nc∗

+

+

+

−

+

−

u∗c

+

+

+

?

?

−

Full-Interaction equilibrium (γ ∗ = γ)

Nc∗

wc

b

N

δ

λ

r

+

?

+

?

?

?

+ ? − ? ?
?
u∗cTable 1: Comparative statics for both equilibria

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 ∗ , efficient? and (ii) is the degree of urbanization, Nc∗ ,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.

SW =

Z ∞
0

e−rt [f (N − Nc (t)) + Ec (t)wc + Uc (t)b − βd1 (t)γ(t)] dt(26)

The dynamics of the number of dyads are given by (6). Note that
Ec (t) = 2d2 (t) + d1 (t)(27)

Uc (t) = 2d0 (t) + d1 (t)(28)

Nc (t) = 2 [d0 (t) + d1 (t) + d2 (t)]

(29)

and

16

The planner chooses γ(t) and Nc (t) that maximize (26) under the flow constraint (6). We obtain thepresent-value Hamiltonian:
Ht = e−rt {f (N − Nc (t)) + Ec (t)wc + Uc (t)b − βd1 (t)γ(t)}+ μ1 (t)[2λd0 (t) − (δ + g(t))d1 (t)] + μ2 (t) [g(t)d1 (t) − δd2 (t)]•where μ1 (t) and μ2 (t) are the Lagrangian multipliers (or co-state variables) corresponding to d1 (t) and••d2 (t). We don’t need to write the constraint corresponding to d0 (t) because we use (29). The control
variables of the social planner are γ(t) and Nc (t) and the state variables are d1 (t) and d2 (t). We havethe following result:
Proposition 6 When the urban population is sufficiently small (Nc ≤ N oc ), there is no social interactions
in equilibrium and it is optimal from a social welfare viewpoint, i.e. γ ∗ = γ o = 0. When the urban
population is sufficiently large (Nc > N ∗c ), there is full social interactions in equilibrium and and it
is optimal from a social welfare viewpoint, i.e. γ ∗ = γ o = γ. When the urban population takes
intermediate values (N oc < Nc ≤ N ∗c ), there is no social interaction in equilibrium (γ ∗ = 0) while fullsocial interactions are optimal (γ o = γ).
Define φbo as

φbo ≡βδ(δ + 2λ).
2λ(bδ + 2wc λ)(30)

As shown in the proof of Proposition 6, γ = γ is optimal if φ > φbo and γ = 0 is optimal if φ < φbo .
Define next N oc as³ ´
N oc ≡ φ−1 φbo .

Then, we obtain that γ = γ is optimal if Nc > N oc and γ = 0 is optimal if Nc < N oc . Because (34) andih(30) result in sgn φb∗ − φbo = sgn [wc − b], we know that N ∗c > N oc , implying Proposition 6. Figure 3
describes when the equilibrium decision on social interactions is optimal and when it is not.

[Insert F igure 3 here]
From the comparative statics results (Table 1), we know that the urban population size becomes larger
when the total population size increases. Thus, the results of Proposition 6 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 ofurbanization. The source of this inefficiency is due to (positive) externalities related to the decision on

17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 ahigher 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 g ∗ ) for d1 dyads.
However, an unemployed worker in a d1 dyad ignores this positive externality on other unemployedworkers when making her social-interaction decision γ, resulting in inefficiency of this decision.
Next, we move to the efficiency analysis of urban population size, Nc∗ . In order to focus on theproperties of urbanization, we mainly restrict our attention to the cases where the equilibrium decision
on social interactions is efficient (i.e., γ ∗ = γ o , or equivalently, Nc ≤ N oc or Nc > N ∗c ). In such cases,
the equilibrium condition for Nc∗ , (24), is equal to½ u
W0c |γ=0 for γ = 0
Ω=
u|
W0c
γ=γ for γ = γ
whereas the optimal condition is given byΩ=½

Θot |γ=0 for γ = 0
Θot |γ=γ for γ = γ

where Θot |γ=0 and Θot |γ=γ are defined in (65). As shown in proof of the propositions 6 and 7 in theu|ouoAppendix, we have that: W0c
γ=0 > Θt |γ=0 and W0c |γ=γ > Θt |γ=γ , which implies that the curves of
u|
u
o
o
W0c
γ=0 and W0c |γ=γ are above the curves of Θt |γ=0 and Θt |γ=γ , as can be seen in Figures 4(a) and

4(b), respectively.

[Insert F igures 4(a) and 4(b) here]
We have the following proposition.
Proposition 7 When the equilibrium level of social interactions is optimal (γ ∗ = γ o ), the equilibrium
urbanization level Nc∗ is always higher than the optimal level one Nco .
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 Full-Interaction equilibrium, there exists agglomeration(positive) externalities and congestion (negative) externalities from finding jobs via weak ties. When aworker decides to migrate to the city, it increases the size of the urban population and thus the possibilityof contacting a weak tie, which is represented by the term φ(Nc ) in g(t). This creates positive agglomeration externalities. At the same time, the decision to migrate increases the number of workers in d0
18

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, 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 whichγ ∗ 6= γ o (i.e., N oc < Nc ≤ N ∗c ). In this case, although we cannot determine analytically the efficiencyu|o13
properties of Nc∗ , simple numerical examples indicate that W0c
γ=γ > Θt |γ=0 . Therefore, it would besafe 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 urban population) so that there are too few social interactions in equilibrium. The second
comes from urbanization. When social interactions are optimal, which corresponds to either small orlarge 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 weakor 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 urbanizationlevels, 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 withfull 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 thecity. We show that these equilibria are usually not socially efficient when the urban population has anintermediate size because there are too few social interactions compared to the social optimum. Whenthe 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 enhances the development of person-specific humancapital and increases the rate at which new ideas are formed. We believe that the role of social interac13

We tried two specifications of φ(Nc ) (i.e., φ(Nc ) = Nc /(1 + Nc ) and φ(Nc ) = 1 − exp[−Nc ]). We also tried variousu|γ=γ > Θot |γ=0 .
sets of parameter values. For any combination, we obtained that W0c

19

tions and the fact that people tend to extend their social networks by meeting more weak ties in bigger
cities that help them finding jobs are crucial in explaining agglomeration. We also believe that this canlead to over-urbanization, which would imply that cities are oversized. These issues certainly need morethorough empirical investigations.

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23

8 Appendix: Proofs
Proof of Proposition 1: Let us determine the optimal γ. Unemployed workers in a d∗1 dyad choose γ
u , taking as given N ∗ , d∗ and d∗ . Using (21) and (18), we obtainthat maximize W1c
c
1
2
h
i
´
³
∗
∗
∗ ) d2 wc (r + δ + g ∗ ) − b − βγ + g ∗ wc∗ ) d2
−β
+
2λφ
(N
2λφ
(N
u
∗
c
c
d1 r+δ
r+δ
d∗1
∂W1c
=
2
∂γ
(r + δ + g ∗ )

=

=

=

d∗

d∗

d∗

1

1

1

wx
wc
−β(r + δ + g ∗ ) + r+δ
(r + δ + g ∗ )2λφ (Nc∗ ) d2∗ − (b − βγ)2λφ (Nc∗ ) d2∗ − g ∗ r+δ
2λφ (Nc∗ ) d2∗(r + δ + g ∗ )2

i
h
d∗
d∗
d∗
wc
−β r + δ + 2λ + γ2λφ (Nc∗ ) d2∗ + r+δ
(r + δ)2λφ (Nc∗ ) d2∗ − (b − βγ)2λφ (Nc∗ ) d2∗
1

1

1(r + δ + g ∗ )2

d∗

d∗

1

1

−β(r + δ + 2λ) + wc 2λφ (Nc∗ ) d2∗ − b2λφ (Nc∗ ) d2∗(r + δ + g ∗ )2d∗1∗

d∗ 2λφ (Nc ) (wc − b) − β d∗2 (r + δ + 2λ)= 2∗
d1(r + δ + g ∗ )2This implies that
¸
∙
¸
u
∂W1c
d∗1
∗
= sgn 2λφ(Nc )(wc − b) − β ∗ (r + δ + 2λ)
sgn
∂γ
d2
∙

Using (9) and (10), we see thatδd∗1
− γφ(Nc∗ )
=
∗
d2
2λ
d∗

Plugging d1∗ into the above equation, we can see that
2

sgn

∙

¸
∙
¶
¸
µ
u
∂W1c
δ
= sgn 2λφ(Nc∗ )(wc − b) + β γφ(Nc∗ ) −(r + δ + 2λ) .
∂γ
2λThis leads tou∂W1cδ2λ(wc − b)
R0⇔γR
−
∗
∂γ
2λφ(Nc ) β(r + δ + 2λ)This condition only depends on Nc∗ . We therefore know that an unemployed worker in the d∗1 dyad
choose either γ = 0 (no interaction at all with weak ties) or γ = γ (maximum interaction with weakties), i.e. we have only corner solutions.
Let us characterize all the solutions of this maximization problem.
(i) If2λ(wc − b)δ<,∗
2λφ(Nc )β(r + δ + 2λ)which is equivalent toφ(Nc∗ ) >δβ(r + δ + 2λ)
4λ2 (wc − b)24then γ ∗ = γ. Since φ0 (.) > 0, this means that, if Nc∗ > N c , then γ ∗ = γ, whereµ¶−1 δβ(r + δ + 2λ)Nc ≡ φ4λ2 (wc − b)(ii) Ifδ2λ(wc − b)−> γ,∗
2λφ(Nc ) β(r + δ + 2λ)which is equivalent toφ(Nc∗ ) <δβ(r + δ + 2λ)
4λ2 (wc − b) + 2λγβ(r + δ + 2λ)then γ ∗ = 0. Since φ0 (.) > 0, this means that, if Nc∗ < N c , then γ ∗ = 0, where¶µδβ(r + δ + 2λ)−1
Nc ≡ φ4λ2 (wc − b) + 2λγβ(r + δ + 2λ)(31)(iii) If0<δ2λ(wc − b)−<γ∗2λφ(Nc ) β(r + δ + 2λ)then both cases γ ∗ = 0 and γ ∗ = γ can arise.
Figure A1 illustrates these different cases.

[Insert F igure A1 here]
Let us now deal with case (iii) only since the other cases are straightforward. Then, using the fact
u can be written as:that γ = b/β, W1cu|γ=0 =
W1cu|γ=γ =
W1ccb + 2λwr+δr + δ + 2λ

2λδwc(r + δ)[βδ(r + δ + 2λ) − 2bλ(r + δ)φ(Nc∗ )]

(32)
(33)

Comparing the asset values under no social interaction (γ = 0) to that under full social interactionu|u(γ = γ), we can see that W1cγ=0 = W1c |γ=γ if and only if

Furthermore, we have that
³
´
u|
u|
−
W
∂ W1c
1c γ=0γ=γ∂φ(Nc )

φ(Nc∗ ) = φb ≡
=

³
´
u|
∂ W1c
γ=γ
∂φ(Nc )βδ(r + δ + 2λ)2λ[b(r + δ) + 2wc λ]=4bβwc δλ2
>0
[βδ(r + δ + 2λ) − 2bλφ(Nc )(r + δ)]2

(34)

(35)

As a result, an unemployed worker in the d∗1 dyad always chooses γ ∗ = γ if and only if φ(Nc ) > φb andb From the assumption that φ0 (Nc ) > 0, we have a thresholdchooses γ ∗ = 0 if and only if φ(Nc ) < φ.

b i.e.
bc ≡ φ−1 (φ),
regarding the city population: N¶µβδ(r + δ + 2λ)−1bNc ≡ φ2λ[b(r + δ) + 2wc λ]
25

(36)

bc = N c , given by (25). Observe also that
Since γ = b/β, then it is easily verified that N
bc = N c < N cNwhich implies that an unemployed worker in the d∗1 dyad always chooses γ = γ in case (iii). Proposition
1 summarizes these results.
Proof of Proposition 2: Using (23), the steady-state migration equilibrium condition (24) can be written
as
u
=
W0c

f 0 (N − Nc )(r + δ)Hence, the equilibrium conditions given in Proposition 1 (see the proof of Proposition 1) becomeu|b + λ(Wce + W1c
f 0 (N − Nc )
γ=0 )
u
= W0c
|γ=0 =r+δ
r + δ + 2λu|0b + λ(Wce + W1c
f (N − Nc )
γ=γ )
u
= W0c
|γ=γ =r+δ
r + δ + 2λ

if and only if Nc < N c ,if and only if Nc ≥ N c .

From these conditions, we obtain Proposition 2.
Let us first characterize the No-Interaction Equilibrium for which all workers provide zero effort, i.e.
γ ∗ = 0. Using (8), (9) and (10) and the fact that γ = 0, we easily obtain:
d∗0 =

δNc∗
λδNc∗
2λ2 Nc∗∗, d∗1 =,d=22(δ + 2λ)(δ + 2λ)2(δ + 2λ)2(37)Since the urban unemployment level is: Uc∗ = 2d∗0 + d∗1 , we obtain:U∗ =δ (δ + 3λ) Nc∗(δ + 2λ)2(38)

g ∗ = 2λ(39)

From (18), we obtain:

From (20), (21), (22) and (23), we have:u∗W0c|γ=0 =(r + δ) (r + δ + 3λ) b + λ (r + δ + 4λ) wc(r + δ) (r + δ + 2λ)2(r + δ)b + 2λwc
(r + δ) (r + δ + 2λ)wc
Wce∗ =r+δ

u∗|γ=0 =W1cΩ=

f 0 (N − Nc∗ )
r+δ

26

(40)

(41)
(42)

(43)

u = f 0 (N − N ) / (r + δ). By using
Finally, the migration equilibrium condition (24) is given by W0cc(40), this equation (24) can be written as
f 0 (N − Nc∗ ) =(r + δ) (r + δ + 3λ) b + λ (r + δ + 4λ) wc(r + δ + 2λ)2(44)

which implicitly defined Nc∗ .
Let us now characterize the Full-Interaction Equilibrium for which all workers provide maximal
effort, i.e. γ ∗ = γ. Proceeding exactly as above, we easily obtain:The equilibrium values are given by:
∙
¸
δ
δNc∗
2λ2 Nc∗∗d∗0 =, d∗1 =
−
γφ(N
)
(45)
c
2(δ + 2λ)(δ + 2λ)[δ + 2λ(1 − γφ(Nc∗ ))] 2λ
d∗2 =
U∗ =

u
W0c
=

2λ2 Nc∗(δ + 2λ)[δ + 2λ(1 − γφ(Nc∗ ))](46)

δ (δ + 3λ) − 2λγφ(Nc∗ ) (δ + λ) ∗N(δ + 2λ)[δ + 2λ(1 − γφ(Nc∗ ))] c2δλg∗ =δ − 2λγφ(Nc∗ )

(47)
(48)

βδ [b (r + δ) (r + δ + 2λ) + λwc (r + δ + 4λ)] − 2bλφ(Nc∗ ) (r + δ) [b (r + δ) + λwc ](49)(r + δ) (r + δ + 2λ) [δβ (r + δ + 2λ) − 2λbφ(Nc∗ ) (r + δ)]

u
W1c
=

f 0 (N − Nc∗ )2βδλwc
wc
e
,
W
,
Ω
=
=
c
(r + δ) [βδ (r + δ + 2λ) − 2bλφ(Nc∗ ) (r + δ)]r+δr+δ

f 0 (N − Nc∗ ) =(50)βδ [b (r + δ) (r + δ + 2λ) + λwc (r + δ + 4λ)] − 2bλφ(Nc∗ ) (r + δ) [b (r + δ) + λwc ](r + δ + 2λ) [δβ (r + δ + 2λ) − 2λbφ(Nc∗ ) (r + δ)](51)This completes the proof.
Proof of Proposition 3: To show that there is a unique equilibrium with γ ∗ = 0, we need to show that¯¯u|
> f 0 (N )/ (r + δ). This is equivalent to:
W0c
γ=0 ¯
Nc =0

(r + δ) (r + δ + 3λ) b + λ (r + δ + 4λ) wc > (r + δ + 2λ)2 f 0 (N )(52)

To show that there is a unique equilibrium with γ ∗ = γ, we further need to show that Ω0 (Nc )|Nc =Nc∗ >
¯
¯
u|
/∂N
. This is equivalent to:
∂ W0c
c¯
γ=γ
∗
Nc =Nc4λ3 wc (2λ + δ) (r + δ) γφ0 (Nc∗ )− (r + δ + 2λ) [1 + γφ (Nc∗ )]2 f 00 (N − Nc∗ ) > n
h
i
o2
∗
c)
2
(r + δ) δ−2λγφ(N+4λ1+γφ(N ∗ )

(53)

c

This completes the proof.

Proof of Propositions 4: In the No-interaction equilibrium, the urban population size, Nc∗ , is determinedby (51). By totally differentiating (51), we obtain, after straightforward but tedious calculations, thefollowing results of basic comparative statics:27£¤b(r + δ)2 (r + δ + 4λ) + 2wc λ r2 + δ 2 + 6δλ + 4λ2 + 2r(δ + 3λ)
∂Nc∗
=
<0
∂δ
f 00 (N − Nc∗ )(r + δ)(r + δ + 2λ)3∂Nc∗
(wc − b)(r + δ)(r + δ + 6λ)= − 00>0∂λf (N − Nc∗ )(r + δ + 2λ)3(r + δ)(r + δ + 3λ)∂Nc∗= − 00>0∂bf (N − Nc∗ )(r + δ + 2λ)2∂Nc∗
λ(r + δ + 4λ)= − 00>0∂wc
f (N − Nc∗ )(r + δ + 2λ)2
∂Nc∗
=1>0
∂N
£
¤
b(r + δ)2 (r + δ + 4λ) + 2wc λ r2 + δ 2 + 6δλ + 4λ2 + 2r(δ + 3λ)
∂Nc∗
=
<0
∂r
f 00 (N − Nc∗ )(r + δ)(r + δ + 2λ)3The equilibrium unemployment rate is defined by
u∗ =

U∗
N

where U ∗ is given by (38). Hence, we readily obtain∂N ∗N ∗ λ(δ + 6λ) + δ(δ + 2λ)(δ + 3λ) ∂δc
∂u∗
= c
∂δ
N (δ + 2λ)3
∂u∗
∂λ

=

h
¢ c∗ i¡δ −Nc∗ (δ + 6λ) + δ 2 + 5δλ + 6λ2 ∂N∂λN (δ + 2λ)3∂u∗
δ(δ + 3λ) ∂Nc∗=>0∂b
N (δ + 2λ)2 ∂b∂u∗δ(δ + 3λ) ∂Nc∗=
>0∂wc
N (δ + 2λ)2 ∂wc
∂u∗
∂N

=

³
´
∗
c
δ(δ + 3λ) −Nc∗ + N ∂N∂N
N 2 (δ + 2λ)2=δ(δ + 3λ) (N − Nc∗ )>0
N 2 (δ + 2λ)2δ(δ + 3λ) ∂Nc∗
∂u∗
=
<0
∂r
N (δ + 2λ)2 ∂r
This completes the proof.
28

Proof of Propositions 5: Here, we assume that (53) holds, i.e.,
¯
u|
¯
¯
∂ W0c
¯
γ=γ
0
Ω (Nc )¯Nc =N ∗ −
¯
c
∂Nc ¯

>0

(54)

Nc =Nc∗to ensure that the model has the unique Full-interaction equilibrium. In the Full-interaction equilibrium,
the urban population size is determined by (51). By totally differentiating (51), we can obtain

∂Nc∗
∂δ

⎡

⎤−1

¯
u|
¯
∂ W0c
γ=γ ¯
−
¯
Nc =Nc∗∂Nc ¯¯= ⎣ Ω0 (Nc )¯
−
+

½
−

b(r + δ)2
(r + δ)2 (r + δ + 2λ)2Nc =Nc∗£¤2wc λβ 2 δ 2 (r + δ + 2λ) (r + δ)2 + 6(r + δ)λ + 4λ2⎦(r + δ)2 (r + δ + 2λ)2 [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]2¤£
4wc λ2 bβ(r + δ) 2δ(r + δ)2 − (r − 8δ)(r + δ)λ − 2(r − 3δ)λ2 φ(Nc∗ )(r + δ)2 (r + δ + 2λ)2 [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]2¾
8wc λ3 b2 (r + δ)2 (r + δ + λ)φ(Nc∗ )2−(r + δ)2 (r + δ + 2λ)2 [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]2

⎡
⎤−1
¯
u|
¯
¯
∂ W0c
∂Nc∗ ⎣ 0γ=γ ¯⎦= Ω (Nc )¯Nc =N ∗ −
¯
c
∂λ
∂Nc ¯
∗
Nc =Nc½ 2 2
β δ (r + δ + 2λ) [wc (r + δ + 6λ) − 2b(r + δ + 2λ)]×(r + δ + 2λ)2 [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]24bβδ(r + δ)λ [2b(r + δ + 2λ) − wc (r + δ + 3λ)] φ(Nc∗ )+(r + δ + 2λ)2 (βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ ))2¾
4b2 (2b − wc )(r + δ)2 λ2 φ(Nc∗ )2−(r + δ + 2λ)2 (βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ ))2
∂Nc∗
∂b

⎡

¯
= ⎣ Ω0 (Nc )¯Nc =N ∗ −
c

×

(

¯

u|
¯
∂ W0c
γ=γ ¯

∂Nc

¯
¯

Nc =Nc∗⎤−1⎦£¤β 2 δ 2 (r + δ + 2λ)2 + 4βδλ wc λ2 − b(r + δ)(r + δ + 2λ) φ(Nc∗ )(r + δ + 2λ) [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]2¾
4b2 (r + δ)2 λ2 φ(Nc∗ )2+(r + δ + 2λ) [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]2

∂Nc∗
∂wc

⎡

¯
= ⎣ Ω0 (Nc )¯
×

¯
u|
¯
∂ W0c
γ=γ ¯
−
¯
Nc =Nc∗∂Nc ¯

Nc =Nc∗⎤−1⎦

λ [βδ(r + δ + 4λ) − 2b(r + δ)λφ(Nc∗ )](r + δ)(r + δ + 2λ) [βδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc∗ )]

>0
29

(55)

∂Nc∗
∂N

=−

⎡

f 00 (N − Nc∗ ) ⎣
r+δ

⎡

¯
u|
¯
¯
∂ W0c
¯
γ=γ
0
¯
Ω (Nc ) Nc =N ∗ −
¯
c
∂Nc ¯

Nc =Nc∗

¯
u|
¯
∂ W0c
f 00 (N − Nc∗ ) ⎣ f 00 (N − Nc∗ )
γ=γ ¯
−
=−
−
¯
r+δ
r+δ
∂Nc ¯
>1

⎤−1
⎦

Nc =Nc∗

⎤−1
⎦

⎡

⎤−1
¯
u|
¯
¯
∂
W
0c γ=γ ¯⎦= ⎣ Ω0 (Nc )¯Nc =N ∗ −
¯
c
∂r
∂Nc ¯
Nc =Nc∗½(r + δ) [b(r + δ) + wc λ] + wc λ(r + δ + 2λ)× −(r + δ)2 (r + δ + 2λ)2⎫2
2
34wc β δ λ (r+δ+2λ)
2wc βδλ2 (3(r+δ)+4λ) ⎬−βδ(r+δ+2λ)−2b(r+δ)λφ(βδ(r+δ+2λ)−2b(r+δ)λφ)2+2⎭(r + δ) (r + δ + 2λ)2

∂Nc∗

(56)

where the inequality in (55) comes from assumptions (54) and (19) which states that δ > 2γφλ, and theinequality in (56) comes from (54) and the assumption that f 00 (·) ≤ 0.
In the Full-interaction equilibrium, U ∗ is given by (47). Hence, we can see that

∂u∗
∂δ

=

n
¤o
∗ £
3 N ∗ φ0 (N ∗ )
c
βδ(δ
+
2λ)(δ
+
3λ)
−
4bλ
β(δ + 2λ) βλNc∗ (δ + 6λ) + ∂N
c
c
∂δ
−
+

∂u∗
=
∂λ

N (δ + 2λ)2 [β(δ + 2λ) − 2bλφ(Nc∗ )]2
h
¡ 2
¢i
∗
2
c
4bβλφ(Nc∗ ) λNc∗ (δ + 3λ) + ∂N
(δ
+
2λ)
δ
+
3δλ
+
λ
∂δ
N (δ + 2λ)2 [β(δ + 2λ) − 2bλφ(Nc∗ )]2hi∗c4b2 λ2 φ(Nc∗ )2 λNc∗ + ∂N
(δ
+
λ)(δ
+
2λ)
∂δ
N (δ + 2λ)2 [β(δ + 2λ) − 2bλφ(Nc∗ )]2

∂Nc∗
∗
∗
∂λ (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc )] [βδ(δ + 3λ) − 2bλ(δ + λ)φ(Nc )]N (δ + 2λ)2 [β(δ + 2λ) − 2bλφ(Nc∗ )]2hi∗
0 (N ∗ ) + βδ(δ + 6λ)cNc∗ β 4bλ3 ∂N
φc∂λ−N(δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]24bδλNc∗ φ(Nc∗ ) [β(δ + 3λ) − bλφ(Nc∗ )]+N (δ + 2λ)2 [β(δ + 2λ) − 2bλφ(Nc∗ )]2∗ ©
c−4Nc∗ βλ3 φ + ∂N
β 2 δ(δ + 2λ)(δ + 3λ) + 4b2 λ2 (δ + λ)φ(Nc∗ )2
∂u∗
∂b
=
∂b
N (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]2¢¡¤ª£ 2 ∗ 0 ∗4bβλ λ Nc φ (Nc ) + δ 2 + 3δλ + λ2 φ(Nc∗ )−N (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]2

30

∂u∗
∂Nc∗
=
∂wc
∂wc
+

(

¢
¡
β 2 δ(δ 2 + 5δλ + 4λ2 ) − 4bβλ δ 2 + 3δλ + λ2 φ(Nc∗ )N (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]2¤)£
2λ2 2b2 (δ + λ)φ(Nc∗ )2 + β 2 δ − 2bβλNc∗ φ0 (Nc∗ )N (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]2

(57)

>0

½
∂Nc∗ δβ(δ + 3λ) − (δ + λ)2bλφ(Nc∗ ))(Nc∗ − N∂Nβ(δ + 2λ) − 2bλφ(Nc∗ )¾∂Nc∗4bβλ3 N Nc∗ φ0 (Nc∗ )+∗2(β(δ + 2λ) − 2bλφ(Nc )) ∂N

∂u∗
1
=− 2
∂N
N (δ + 2λ)

(58)

<0

∂u∗
=
∂r

∂Nc∗ © 2β δ(δ + 2λ)(δ + 3λ) + 4b2 λ2 (δ + λ)φ(Nc∗ )2∂r
N (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]2¡£¢¤ª4bβλ λ2 Nc∗ φ0 (Nc∗ ) + φ(Nc∗ ) δ 2 + 3δλ + λ2−
N (δ + 2λ) [β(δ + 2λ) − 2bλφ(Nc∗ )]2

where the inequality in (57) comes from assumption of concavity of φ(·) and assumption (19) whichstates that δ > 2γφλ, and the inequality in (58) comes from the facts that ∂Nc∗ /∂N > 1 and N > Nc∗ ,and assumption (19).
Proof of Propositions 6 and 7: Using (27) and (28), the Hamiltonian can be written as:
Ht = e−rt {f (N − Nc (t)) + [2d2 (t) + d1 (t)] wc + [Nc (t) − 2d2 (t) − d1 (t)] b − βd1 (t)γ(t)}
½ µ
¶ ∙
µ
¶¸
¾
Nc (t)
γd2 (t)− d1 (t) − d2 (t) − δ + 2λ 1 +φ(Nc (t)) d1 (t)]+ μ1 (t) 2λ2
d1 (t)∙ µ¶¸γd2 (t)φ(Nc (t)) d1 (t) − δd2 (t)+ μ2 (t) 2λ 1 +d1 (t)(59)

Moreover, we need to consider the inequality constraints γ ≥ 0 and γ ≥ γ. The Lagrangian for the
maximization problem becomes
Lt = Ht + ξ0 γ + ξ1 (γ − γ).

31The first-order conditions for the maximization are
Nc (t) :γ(t) :
d1 :
d2 :

∂Lt
,
∂Nc (t)
∂Lt
0=
,
∂γ(t)
∂Lt
•
,
μ1 (t) = −∂d1 (t)∂Lt•,
μ2 (t) = −∂d2 (t)0=This is equivalent to
0=
0=

¤
£
∂Ht
= e−rt −f 0 (N − Nc (t)) + b + μ1 λ∂Nc (t)∂Ht+ ξ0 − ξ1 = −e−rt βd1 (t) + 2λφ (Nc (t)) d2 (t) [μ2 (t) − μ1 (t)] + ξ0 − ξ1
∂γ(t)

(60)
(61)

•

μ1 (t) = −e−rt (wc − b − βγ) + μ1 (t)(δ + 4λ) − 2μ2 (t)λ•μ2 (t) = −2e−rt (wc − b) + 2μ1 (t)λ [1 + γφ(Nc (t))] + μ2 (t) [δ − 2γλφ(Nc (t))]
Solving the last two differential equations, we obtain∙¸
e−rt
2λ(wc − b + βγ)
wc − b − βγ +μ1 (t) =r + δ + 2λr + δ + 2λ − 2γλφ(Nc (t))μ2 (t) = 2e−rt(62)

rwc + βγλ [1 + γφ(Nc (t))] + wc [δ + λ [3 − γφ(Nc (t))]] − b[r + δ + λ [3 − γφ(Nc (t))]](r + δ + 2λ) [r + δ + 2λ − 2γλφ(Nc (t))](63)

Equation (61) can be written as∂Ht= ξ1 − ξ0 ,∂γ(t)which implies that the optimal γ is a corner solution. If ξ0 > 0 and ξ1 = 0, then γ o = 0, whereas if
ξ0 = 0 and ξ1 > 0, then γ o = γ. Substituting (62) and (63) into (61), we obtain:
¸
∙ ∙
¸
¸
2λφ(Nc (t)) (wc − b − βγ)δ∂Ht= sgn β γφ(Nc (t)) −
+
.
sgn
∂γ(t)
2λ
r + δ + 2λ − 2γλφ(Nc (t))∙

Because we have assumed (19), i.e. δ > 2γφλ, then ∂Ht /∂γ(t) > 0 can be possible only whenwc − b − βγ = wc − 2b > 0. Moreover, plugging the steady state conditions (8), (9) and (10), into the
Hamiltonian (59), we obtain
Ht |γ=γ̄ − Ht |γ=0 =bNc (t)λ[−βδ(δ + 2λ) + 2λ(bδ + 2wc λ)φ(Nc (t))](δ + 2λ)2 [β(δ + 2λ) − 2bλφ(Nc (t))]

Defineφbo ≡βδ(δ + 2λ)
2λ(bδ + 2wc λ)
32

(64)

From (64) and using (19), we know that if
Ht |γ=γ̄ R Ht |γ=0 ⇔ φ R φbo

Let N oc denote the urban population that satisfiesφ(N oc ) = φbowhich means that

N oc = φ−1µβδ(δ + 2λ)
2λ(bδ + 2wc λ)¶Remember that φb∗ was the equilibrium threshold defined by (34). Simple comparison yields

implying that N ∗c > N oc .

φb∗ − φbo =rβδ(wc − b)
> 0,(bδ + 2wc λ) [b(r + δ) + 2wc λ]Next, we examine whether the urbanization level is optimal when the equilibrium level of social
interaction is efficient. Substituting μ1 (t) and μ2 (t) in (62) and (63) and using (60), we obtain that½ oΘt |γ=0 for γ = 0,Ω=Θot |γ=γ for γ = γ
where Θot |γ=0 and Θot |γ=γ are defined byλ(wc − b)(r + δ + 4λ)
b
−
,
r+δ
(r + δ) (r + δ + 2λ)2∙¸
λb
2βwc λo−wc − 2b +Θt |γ=γ ≡r + δ (r + δ) (r + δ + 2λ)c(r + δ + 2λ) − 2bλφ(Nc (t))Θot |γ=0 ≡(65)and Ω is given by (23) (Ω ≡ f 0 (N − Nc (t))/ (r + δ)). In contrast, the equilibrium urbanization level isdetermined byΩ=u|uwhere W0c
γ=0 and W0c |γ=γ are given byu|γ=0 =
W0c

½

u|
W0c
γ=0 for γ = 0,uW0c |γ=γ for γ = γb(r + δ)(r + δ + 3λ) + wc λ(r + δ + 4λ)(r + δ)(r + δ + 2λ)2andu|γ=γ =
W0c

βwc δλ(r + δ + 4λ) − 2b2 λφ(Nc (t))(r + δ)2 + b(r + δ)[βδ(r + δ + 2λ) − 2wc λ2 φ(Nc (t)](r + δ)(r + δ + 2λ)[βδ(r + δ + 2λ) − 2bλφ(Nc (t))(r + δ)]

Taking the difference, we have:uW0c|γ=0 − Θot |γ=0 =2(wc − b)λ(r + δ + 4λ)>0(r + δ)(r + δ + 2λ)2
33

and
u
|γ=γ − Θot |γ=γ
W0c

2λ
=
(r + δ) (r + δ + 2λ)+∙½−2b + wc 1 +βλβ(r + δ + 2λ) − 2bλφ(Nc (t))¸¾βλδβδ(r + δ + 2λ) − 2b(r + δ)λφ(Nc (t))>0where the last inequality comes from the fact that γ = γ can be optimal only when wc − 2b > 0, andassumption (19) which states that δ > 2γφλ. From figures 4(a) and 4(b), we obtain the results of the
proposition.

34



d0

d1

2    ( N c ) 2  d 2 / d1

2Figure 1: Flows in the labor market in steady stated2

W0uc , W0uc



with an increasein, ,or with a decreasein,with an increasein , ,

 0

0

N c*

N Nc
N r*

Figure 2(a): Equilibrium population distribution when
there are no social interactions ( *  0 )W0uc , W0uc



with an increasein  ,  ( N )W0uc

with an increase
in

W0uc

 

 0

0

Nc

N c*

N Nc

N r*
Figure 2(b): Equilibrium population distribution
when there are full social interactions (    )* 0
ˆo

o 0

* 

o 



ˆ*

Too few social interactions
* 

* 0
N

 0
o

o
c

N

 

*
c

Nc

o

Figure 3: Equilibrium versus optimal social interactions

W0uc , 


W0uc
o

 0
 0

0

N co

N c*

N

Nc

Figure 4(a): Equilibrium versus optimal population
distribution when there are no social interactions(  0 )W0uc , 

W0uc
o



 
 

0

N co

N c*

N Nc

Figure 4(b): Equilibrium versus optimal population
distribution when there are full social interactions(   )W0uc
0


0

W0uc
0


 ( b /  )2 ( wc  b)
2 ( N c )  (r    2 )Figure A1: Optimal choice of social interactions