A Dynamic Model of Weak and Strong Ties in the
Labor Market∗
Yves Zenou†
May 5, 2012 (Revised March 20, 2014)
Abstract: We develop a simple and tractable social interaction model where workers
can obtain a job through either their strong or weak ties. We show that increasing the time
spent with weak ties raises the employment rate of workers. We also show that when the
job-destruction rate or the job-information rate increases, workers choose to rely more on
their weak ties in order to find a job . Finally, we extend the model so that the unemployed
workers can learn of a vacancy directly from an employer rather than via a social interaction.
We show that the equilibrium employment and time spent with weak ties are sometimes,
but not in all cases, positively related.
Keywords: Weak ties, labor market, social networks.
JEL Classification: A14, J6, Z13.
∗
I would like to thank the editor Jonathan Guryan as well as two anonymous referees for very helpful
comments.
†
Stockholm University and Research Institute of Industrial Economics (IFN), Sweden, and GAINS. Email: [email protected].
1
1
Introduction
A number of studies for a range of countries have emphasized the popularity of using friends
and family as a job search mechanism and indicate that they are an effective mechanism for
obtaining job offers (Rees, 1966; Granovetter, 1974, 1979; Blau and Robins, 1990; Topa, 2001;
Wahba and Zenou, 2005; Bayer et al., 2008; Bentolila et. al, 2010; Pellizzari, 2010). The
empirical evidence reveals that around 50 percent of individuals obtain or hear about jobs
through friends and family (Holzer 1987, 1988; Montgomery, 1991; Addison and Portugal,
2001). Such methods have the advantage that they are relatively less costly and may provide
more reliable information about jobs compared to other methods. Little is known, however,
about the way people search and what kind of mechanisms lead to a job finding.
Building on Granovetter (1973, 1974, 1983)’s idea that weak ties are superior to strong
ties for providing support in getting a job,1 we develop a dynamic model of the labor market
in which dyad members do not change over time so that two individuals belonging to the
same dyad hold a strong tie with each other. However, each dyad partner can meet other
individuals outside the dyad partnership, referred to as weak ties or random encounters.
By definition, weak ties are transitory and only last for one period. The process through
which individuals learn about jobs results from a combination of a socialization process that
takes place inside the family (in the case of strong ties) and a socialization process outside
the family (in the case of weak ties).2 Thus, information about jobs is essentially obtained
through strong and weak ties and thus word-of-mouth communication. In particular, we
assume that an unemployed (employed) individual meets either a strong or weak tie during
a given period, potentially gaining (providing) information on a job vacancy from the tie
they meet only if that tie is employed (unemployed).3
1
Granovetter (1973, 1974, 1983) defines weak ties 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.
In other words, for Granovetter, weak ties are useful because they essentially increase the information set
relative to strong ties by providing access to different information on job openings than can be provided by
strong ties.
2
This idea was first put forward by Bisin and Verdier (2000, 2001) in the context of the transmission of
a trait like, for example, religion or identity.
3
More recent empirical evidence lends some support to the strength of weak ties in finding a job.
Yakubovich (2005) uses a large scale survey of hires made in 1998 in a major Russian metropolitan area and
finds that a worker is more likely to find a job through weak ties than through strong ones. These results
come from a within-agent fixed effect analysis, so are independent of workers’ individual characteristics.
Using data from a survey of male workers from the Albany NY area in 1975, Lin et al. (1981) find similar
results. Lai et al. (1998) and Marsden and Hurlbert (1988) also find that weak ties facilitate the reach to a
contact person with higher occupational status, who in turn leads to better jobs, on average. Finally, Giulietti et al. (2014) show that weak ties are important for rural-urban migration in China because they help
2
We first characterize the steady-state equilibrium employment rate and determine the
time spent in a 0 dyad (both workers are unemployed), in a 1 dyad (one worker is employed
and the other one is unemployed) and in a 2 dyad (both workers are employed). We then
show that the higher is the job-destruction rate or the lower is the job-information rate, the
lower is the employment rate and more workers spend time in a 0 dyad. We also show that
by increasing the probability  of meeting new workers (i.e. weak ties), the steady-state
employment rate increases, formally demonstrating the Granovetter’s informal idea of the
strength of weak ties in finding a job. This is not a trivial result, since, by increasing , we
have different and opposite effects on the job formation/destruction process. On the one
hand, we increase the probability of getting out of unemployed dyads, while, on the other
hand, we potentially give up the information of an employed partner in favor of a link with
an unemployed one. We obtain this result because, in our model, it is indeed better to meet
weak ties because a strong tie does you no good in state 0 since your best friends are all
unemployed. But a weak tie can do you good in any state because that person might be
employed. So there is an asymmetry that is key to the model and that can explain why some
workers may be stuck in unemployment traps (i.e. 0 dyads) having little contact with weak
ties that can help them leaving the 0 dyad.
Observe that our definition of weak ties is slightly different than that of Granovetter. As
underscored by Granovetter, in a close network where everyone knows each other, information
is shared and so potential sources of information are quickly shaken down so that the network
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. In his 1983 paper, Granovetter reviews some of the empirical literature
up to that time, including his own work, where the definition of strong and weak ties varies
depending on the paper (and on data availability). As a result, what constitutes a strong
and weak tie is somewhat unclear in this literature. Here, we adopt a definition of weak
and strong ties based on the time spent together so that weak ties are more transient and
changing than strong ties. This enables us to show that weak ties provide better job access
in many cases because they provide an unemployed worker essentially with more draws at
the underlying probability of becoming employed.
We then extend this framework by endogeneizing social interactions. When workers
decide how much time they spend with their weak and strong ties, we show that, when the
job-destruction rate or the job-information rate increases, workers spend more with their
weak ties. This means that, in downturn periods where jobs are destroyed at a faster rate,
workers tend to spend more time with their weak ties because they know they will help them
migrants obtaining an urban job. See also Patacchini and Zenou (2008) who find evidence of the strength
of weak ties in crime.
3
exit unemployment. In an economy where the flow of job information is faster, the same
results occur.
Finally, we allow for the possibility that the unemployed workers can learn of a vacancy
directly from an employer rather than via a social interaction. In the new model, employed
and unemployed workers can hear directly from a job and the unemployed workers can, as
before, find a job through their social network. We show that the equilibrium employment
and time spent with weak ties are sometimes, but not in all cases, positively related.
Related literature 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; CalvóArmengol et al., 2009; Goyal, 2007; Jackson, 2008, 2014; Ioannides, 2012; Jackson and Zenou,
2014; Jackson et al., 2014), especially in the labor market. However, few models of social
networks in the labor market are dynamic. Montgomery (1994) and Calvó-Armengol et al.
(2007) propose a dynamic model of weak and strong ties but the former focuses on inequality
while the latter on the interaction between crime and labor markets. Calvó-Armengol and
Jackson (2004, 2007) have a richer network analysis (since they can encompass any network
structure) by modeling strong ties in a more general way so that workers who hear about
a job vacancy does not always pass along this information to their strong ties. This will
depend on the number of their friends. On the other hand, we model weak ties in a richer
way since, in our framework, weak ties can directly provide information about jobs while
they can’t in Calvó-Armengol and Jackson (2004) and workers have to decide how much
time they spend with their weak and strong ties.4
There is also a theoretical literature on job search and social networks. See, in particular,
Diamond (1981), Montgomery (1991), Mortensen and Vishwanath (1994), Calvó-Armengol
(2004), Calvó-Armengol and Zenou (2005), Calvó-Armengol and Jackson (2004, 2007),
Cahuc and Fontaine (2009), Galeotti and Merlino (2014), and the literature surveyed by
Ioannides and Loury (2004) and Topa (2011).
Our model is simpler than these models in the sense that we do not model the whole
network structure but only focus on dyads. However, our model is richer since it explicitly
analyzes the relationship between weak and strong ties and their impact on the labor-market
outcomes of workers. Because of its simplicity, our model is tractable and all our equilibrium
variables can be written in closed-form solutions. This enables us to extent our model in
different directions. In particular, we allow workers to choose the time spent with weak and
strong ties. We also look at a model where jobs can be found either directly or through
strong and weak ties.
The paper unfolds as follows. In the next section, we present the basic model of weak and
4
See Section 5 where we have a more thorough discussion on the comparison of our model and that of
Calvó-Armengol and Jackson (2004).
4
strong ties. In Section 3, we characterize the steady-state equilibrium, derive some important
comparative statics results, calculate the correlation in employment status between workers
in the same dyad and endogeneize the choice of social interactions. In Section 4, we allow
unemployed workers to directly hear from a job. In Section 5, we compare our model to
that of Calvó-Armengol and Jackson (2004). Finally, Section 6 concludes. All proofs can be
found in the Appendix.
2
The model
Consider a population of individuals of size one.
Dyads As in Montgomery (1994), 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 ever and can never be broken. Thus, we can think
of strong ties as links between members of the same family, or between very close friends.
Individuals can be in either of two different states: employed or unemployed. Dyads,
which consist of paired individuals, can thus be in three different states,5 which are the
following:
() both members are employed −we denote the number of such dyads by 2 ;
() one member is employed and the other is unemployed (1 );
() both members are unemployed (0 ).
Aggregate state By denoting the employment rate and the unemployment rate at time
 by () and (), where () () ∈ [0 1], we have:
½
() = 22 () + 1 ()
(1)
() = 20 () + 1 ()
The population normalization condition can then be written as
() + () = 1
(2)
or, alternatively,
2 () + 1 () + 0 () =
5
The inner ordering of dyad members does not matter.
5
1
2
(3)
Social interactions Time is continuous and individuals live for ever. We assume repeated
random pairwise meetings over time. Matching can take place between dyad partners or not.
At time , each individual can meet a weak tie with probability () (thus 1 − () is the
probability of meeting her strong-tie partner at time ).6 In Sections 2 and 4, we assume
these probabilities to be constant and exogenous, not to vary over time and thus, they can
be written as  and 1 − . We endogeneize  in Section 3.5 below.
We refer to matchings inside the dyad partnership as strong ties, and to matchings
outside the dyad partnership as weak ties or random encounters. Within each matched pair,
information is exchanged, as explained below. Observe that we assume symmetry within
each dyad, that is if I meet a strong (or a weak) tie, then my strong (or weak) tie has to
meet me. In the language of graph theory, this means that the network of relationships is
undirected (Jackson, 2008).
Information transmission Each job offer is taken to arrive only to employed workers,
who can then direct it to one of their contacts (through either strong or weak ties).7 This is a
convenient modelling assumption, which stresses the importance of on-the-job information.8
To be more precise, employed workers hear of job vacancies at the exogenous rate  while
they lose their job at the exogenous rate . All jobs and all workers are identical so that
all employed workers obtain the same wage. Therefore, employed workers, who hear about
a job, pass this information on to their current matched partner, who can be a strong or a
weak tie. Thus, information about jobs is essentially obtained through social networks.
This information transmission protocol defines a 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 . Because of the continuous time Markov
process, the probability of a two-state change is zero (small order) during a small interval of
time  and  + . This means, in particular, that both members of a dyad cannot change
their status at the same time.
6
If each individual has one unit of time to spend with his friends, then () can also be interpreted as
the percentage of time spent with weak ties.
7
There is strong evidence that firms rely on referral recruitment (Bartram et al. 1995; Barber et al., 1999;
Mencken and Winfield, 1998; Pellizzari, 2010) and it is even common and encouraged strategy for firms to
pay bonuses to employees who refer candidates who are successfully recruited to the firm (Berthiaume and
Parsons, 2006).
8
We allow unemployed workers to hear directly about jobs in Section 4 below.
6
Flows of dyads between states It is readily checked that the net flow of dyads from
each state between  and  +  is given by:
⎧ •
⎪
⎪
⎨ •2 () = [1 −  +  ()] 1 () − 22 ()
(4)
1 () = 2 ()0 () − ( + [1 −  +  ()] ) 1 () + 22 ()
⎪
⎪
•
⎩
0 () = 1 () − 2 ()0 ()
Let us explain in details these equations. Take the first one. Then, the variation of dyads
•
composed of two employed workers (2 ()) is equal to the number of 1 −dyads in which the
unemployed worker has found a job (through either her strong tie with probability (1 − )
or her weak tie with probability ()) minus the number of 2 −dyads in which one of
the two employed workers has lost her job. In the second equation, the variation of dyads
•
composed of one employed and one unemployed worker (1 ()) is equal to the number of
0 −dyads in which one of the unemployed workers has found a job (only through her weak
tie with probability  () since her strong tie is unemployed and cannot therefore transmit
any job information) minus the number of 1 −dyads in which either the employed worker
has lost her job (with probability ) or the unemployed worker has found a job with the help
of her strong or weak tie (with probability [1 −  +  ()] ) plus the number of 2 −dyads
in which one the two employed has lost her job. Finally, in the last equation, the variation
•
of dyads composed of two unemployed workers (0 ()) is equal to the number of 1 −dyads
in which the employed worker has lost her job minus the number of 0 −dyads in which one
of the unemployed workers has found a job (only through her weak tie, with probability
 ()) These dynamic equations reflect the flows across dyads. Graphically,

d0
2
d1
d2
1     e 
2 e
Figure 1: Flows in the labor market
Observe that the assumption stated above that both members of a dyad cannot lose their
status at the same time is reflected in the flows described by (4). What is crucial in our
analysis is that members of the same dyad (strong ties) always remain together throughout
their life. So, for example, if a 2 −dyad becomes a 0 −dyad, the members of this dyad are
exactly the same; they have just changed their employment status.
7
3
Steady-state equilibrium and comparative statics analysis
3.1
Steady-state equilibrium
In a steady-state (∗2  ∗1  ∗0 ), each of the net flows in (4) is equal to zero. Setting these net
flows equal to zero leads to the following relationships:
∗2
(1 −  + ∗ ) ∗
=
1
2
(5)
2∗  ∗
0

(6)
1
− ∗2 − ∗1
2
(7)
∗1 =
where
∗0 =
∗ = 2∗2 + ∗1
(8)
∗ = 1 − ∗
(9)
Definition 1 A steady-state labor market equilibrium is a four-tuple (∗2  ∗1  ∗0  ∗  ∗ ) such
that equations (5), (6), (7), (8) and (9) are satisfied.
Define  = (1 − ) ,  =  (). We have the following result.
Proposition 1
() There always exists a steady-state equilibrium U where all individuals are unemployed
and only 0 −dyads exist, that is ∗2 = ∗1 = ∗ = 0, ∗0 = 12 and ∗ = 1.
() If
+



p
 (4 − 3)
2
(10)
there exists a steady-state equilibrium I where 0  ∗  1 is defined by
∗ =
2
−  −   0
2∗0
8
(11)
0  ∗  1 by (9), and 0  ∗0  12 is the unique (feasible) solution of the following
equation:
µ ¶2
 ∗2 (1 + ) ∗

− 0 −
=0
(12)
0 +

2
2
Also, the other dyads are given by:
2∗ ∗

 0
(13)
( + ∗ ) ∗ ∗
0
2
(14)
∗1 =
∗2 =
If condition (10) holds, then an interior equilibrium always exists. Indeed, if the jobdestruction rate  is sufficiently low and/or the job-contact rate  is sufficiently high, then
an interior equilibrium exists. As a result, we are in a “reasonable” economy where jobs
are not destroyed too fast and jobs are created at the sufficient high rate (otherwise we will
end up with the steady-state equilibrium U where all workers are unemployed). Following
Shimer (2005) and Pissarides (2009), the monthly transitions data in the United States give
a mean value for 1960−2004 of 0594 for the job finding rate and 0036 for the job separation
rate, i.e.  = 0594 and  = 0036. In that case, condition (10) is always satisfies even for
very low values of , like e.g.  = 001.
The steady-state equilibrium U is obviously uninteresting and, from now on, we only
focus on the labor market equilibrium I. For this equilibrium, we can, in fact, calculate
explicitly ∗ and ∗0 . We obtain:9
p
 [ + 4 (1 − )] − 2 + 2 − 
(17)
∗ =
2
∗0 =
2
p
2  +   [ + 4 (1 − )]
(18)
To obtain (17) and (18), we proceed as follows. First, we plug the values of ∗2 and ∗1 from (5) and (6)
into ∗ = 2∗2 + ∗1 to obtain:
∙
¸
(1 −  + ∗ ) 
2∗  ∗
∗
(15)
 =
+1
0


9
Then, we plug the values of ∗2 and ∗1 from (5) and (6) into ∗0 = 12 − ∗2 − ∗1 to obtain:
∙
¸
(1 −  + ∗ ) 
2∗  ∗ 1
∗
0 +
+1
0 =
2

2
By solving simultaneously equations (15) and (16), we get (17) and (18).
9
(16)
3.2
Comparative statics results
We have the following comparative-statics results:
Proposition 2 Consider the steady-state equilibrium I and assume (10). Then, an increase
in the job-destruction rate  decreases the employment rate ∗ and the time spent in dyad ∗2
but increases the time spent in a ∗0 dyad. If the elasticity of ∗0 with respect to  is lower in
absolute value than the elasticity of ∗ with respect to , i.e.,
¯ ∗ ¯ ¯ ∗ ¯
¯ 0  ¯ ¯   ¯
¯ ¯
¯
¯
(19)
¯  ∗ ¯  ¯  ∗ ¯
0
then an increase in  reduces the time spent in a ∗1 dyad .
The effect of , the job destruction rate, on the different endogenous variables is interesting and not straightforward. Consider first the effect of  on the different dyads. When
 increases, workers lose their job at a faster rate and thus spend more time in the ∗0 dyad
(where both workers are unemployed) and less time in the ∗2 dyad (where both workers are
employed). The effect of  on the time spent in the ∗1 dyad is, however, less clear. Indeed,
when  increases, this can either increase or decrease the time spent in a ∗1 dyad because
there will be flow in 1 from 2 and flow out 1 to 0 .
Let us now focus on , the job-information rate. We have:
Proposition 3 Consider the steady-state equilibrium I and assume and assume (10). An
increase in the job-information rate  increases the employment rate ∗ but decreases the time
spent in a ∗0 dyad. Moreover, if the elasticity of ∗ with respect to  is higher in absolute
value than the elasticity of ∗0 with respect to , i.e.,
¯ ∗ ¯ ¯ ∗ ¯
¯   ¯ ¯ 0  ¯
¯
¯ ¯
¯
¯  ∗ ¯  ¯  ∗ ¯
0
then an increase in  increases the time spent in dyad ∗1 . Finally, if
3
(1 − )  ( − 2) ( 2 − 42  2 )
an increase in  raises the time spent in dyad ∗2 .
∗0 
(20)
Interestingly, the effect of the job-information rate  on the different endogenous variables
is quite different than that of the job-destruction rate . When  increases, people spend less
time in a ∗0 dyad since both unemployed workers have a higher chance of meeting a weak
tie with a job information. This increases employment as long as  is not too small. The
effects on the time spent in a ∗1 dyad is less clear because unemployed workers tend to find
a job at a faster rate, which both increases (from a ∗0 to a ∗1 dyad) and decreases (from a
∗1 to a ∗2 dyad) the time spent in a ∗1 dyad. Finally, the effect of  on the time spent in a
∗2 dyad is positive only if the size of the ∗0 dyads is sufficiently large because there should
be enough individuals moving first from a ∗0 to a ∗1 dyad and then from a ∗1 to a ∗2 dyad.
10
3.3
Social interactions
Let us study the impact of social interactions (captured by ) on the different endogenous
variables. We have the following important result.
Proposition 4 Consider steady-state equilibrium I and assume (10). Then, increasing the
percentage of weak ties  decreases the number of 0 −dyads and increases the employment
rate ∗ in the economy, i.e.
∗0
∗
0

0


The effects of  on ∗1 and on ∗2 are, however, ambiguous.
We show here that by increasing the probability of meeting new workers (i.e. weak ties),
the steady-state employment rate increases. This is not a trivial result, since, by increasing
, we have different and opposite effects on the job formation/destruction process. On the
one hand, we increase the probability of getting out of unemployed dyads, while, on the other
hand, we potentially give up the information of an employed partner in favor of a link with
an unemployed one. In our model, it is indeed better to meet weak ties because a strong tie
does you no good in state 0 since your best friends are all unemployed. But a weak tie can
do you good in any state because that person might be employed.
Observe that, if we take as given  and assume that you spend the same amount of
time with weak and strong ties, then it is clear that, in a 1 dyad, a strong tie is more
valuable than a weak tie in order to find a job because the strong tie is employed (with
probability 1) while the weak tie that you meet will be employed with probability   1
and thus (1 − )   . This is not, however, enough to compensate the fact that in a
0 dyad strong ties play no role at all while only the weak ties help the unemployed workers
to leave a 0 dyad. Indeed, since ∗ is the fraction of time you spend employed over your
lifetime (or equivalently the probability of being employed in steady state), what is shown
in Proposition 4, is that, increasing marginally the time spent with weak ties  increases the
total the time you spend employed over your lifetime. In other words, since ∗ = 2∗1 + ∗2
and ∗ = 1 − ∗ = 2∗0 + ∗1 , this means that, when  increases, you spend relatively more
time in the 2 dyads over your lifetime and relatively less time in the 0 dyad. This result
is also interesting, since it solves a trade off between status-quo relations and new relations.
It also formally demonstrates the Granovetter’s informal idea of the strength of weak ties in
finding a job.
Observe that our model does not directly incorporate the idea that a strong tie indicates
a stronger personal relationship than a weak tie, and therefore a strong tie might be more
willing to help you out than a weak tie. Indeed, whenever a weak tie has information about
a job she is always willing to help you, exactly like a strong tie. A simple way to model
the fact that a strong tie might be more willing to help you out than a weak tie is to add a
11
parameter   1 so that the probability of finding a job via a weak tie is always multiplied
by  while that of finding a job via a strong tie is not. In that case, the probability of leaving
a 0 dyad or 1 dyad for an unemployed worker is now  instead of , which means
that, to find a job, you need to meet a weak tie (probability ), who is employed (probability
), has heard about a job (probability ) and who is willing to help you (probability ). In
that case, strong ties are more willing to help you than weak ties. It is readily checked that
the net flow of dyads from each state between  and  +  is then given by:
⎧ •
⎪
⎪
⎨ •2 () = [1 −  +  ()] 1 () − 22 ()
1 () = 2 ()0 () − ( + [1 −  +  ()] ) 1 () + 22 ()
⎪
⎪
⎩ •
0 () = 1 () − 2 ()0 ()
We can derive the steady-equilibrium as in Proposition 1 and show that, if
p
 +  (4 − 4 + )



2
there exists an interior steady-state equilibrium I where 0  ∗  1 is defined by
q £
¤
 4 (1 − ) +  (1 − )2 +  (2 − 2 + ) − 2 −  (1 − ) + 
∗ =
2
and the dyads are equal to:
∗0 =
2
q £
¤
2  (1 −  + ) +   4 (1 − ) +  (1 − )2 +  (2 − 2 + )
2∗  ∗
0

2  (1 −  + ∗ ) ∗ ∗
∗2 =
0
2
In that case, however, the effect of  on ∗ is now ambiguous and depends on . At the
extreme, if  is very small (close to zero), strong ties would be more useful in finding a
job than weak ties and there will be a positive effect of strong ties (and not weak ties) on
employment. This is an extreme case where most job information stemming from weak ties
is unreliable. On the contrary, if  is close to 1, then, by continuity, the result of Proposition
∗
4, i.e. 
 0, should hold. For intermediate values of , the effect of  on ∗ is clearly

ambiguous. This is because there is still an asymmetry between weak and strong ties since
only weak ties help leaving a 0 dyad but it is compensated by the fact that strong ties have
even a bigger role in helping unemployed workers leaving a 1 dyad. Indeed, when  = 1, for
someone spending the same amount of time with his weak and strong ties, the rate at which
workers left a 1 dyad via a weak tie was (1 − )  while, via a strong tie, it was , with
(1 − )   . When 0    1, this inequality becomes (1 − )   , which means
that the difference in rates at which unemployed workers leave unemployment has increased.
∗1 =
12
3.4
Correlation in employment status
In order to better understand the model, let us determine the correlation in employment
status between workers in the same dyad. Denote by
p
(21)
∆ ≡  [ + 4 (1 − )]
We have seen that the employment rate was given by:
∗ = 2∗2 + ∗1 =
and ∗0 by:
∗0 =
∆ − 2 + 2 − 
2
(22)
2
 ( + ∆)
We can easily calculate ∗1 and ∗2 as follows:
∗1 =
∗2 =
 (∆ − 2 + 2 − )
 ( + ∆)
(∆ − 2 + ) (∆ − 2 + 2 − )
4 ( + ∆)
Denote by  = (1  2 ) the state of the dyad and observe that ∗0 corresponds to {1 = 0 2 = 0},
∗2 to {1 = 1 2 = 1} and ∗1 to either {1 = 0 2 = 1} or {1 = 1 2 = 0}. The latter implies that, in steady state, the fraction of people who will be in state {1 = 0 2 = 1} is
∗1 2, and the fraction of people who will be in state {1 = 1 2 = 0} is ∗1 2. As a result,
the steady-state joint distribution is given by:
2 = 0
1 = 0
1 = 1
2
(+∆)
(∆−2+2−)
2(+∆)
2 = 1
(∆−2+2−)
2(+∆)
(−2+∆)(∆−2+2−)
4(+∆)
(23)
From this table, we can calculate the marginal probability of being employed. Indeed, to be
employed one needs either to belong to a 1 −dyad or a 2 −dyad. This probability is thus
equal to
 (∆ − 2 + 2 − )  (∆ − 2 + 2 − ) 2 ( − 2 + ∆) (∆ − 2 + 2 − )
+
+
2 ( + ∆)
2 ( + ∆)
4 ( + ∆)
∆ − 2 + 2 − 
=
2
which is (22).
13
Let us now calculate the correlation in employment status between two employed individuals in a dyad.10 We have:
where
 (1 = 1 2 = 1)
 (1 = 1 2 = 1) = p
  (1 = 1)   (2 = 1)
 (1 = 1 2 = 1) = E (1 = 1 2 = 1) − [E (1 = 1)] [E (2 = 1)]
= 2∗2 − (∗ )2
(∆ − 2 + 2 − ) [( + ∆) (2 +  − ∆) − 4]
=
42  2 ( + ∆)
and
  (1 = 1) = E (1 = 1) − [E (1 = 1)]2 = ∗ − (∗ )2
(∆ − 2 + 2 − ) (2 +  − ∆)
=
42  2
=  (2 = 1) = E (2 = 1) − [E(2 = 1)]2
Using the value of ∆ in (21), we obtain:
p
 [ + 4 (1 − )] − 
 (1 = 1 2 = 1) = p
0
 [ + 4 (1 − )] −  + 2
The employment status of two employed individuals belonging to the same dyad is thus positively correlated: if one agent is employed, the probability that the other agent is employed
is positive. This is because, in this model, workers obtain a job through social interactions
only and two workers belonging to the same dyad have a strong tie relationship with each
other and thus share information about jobs 1 −  percent of their time. As a result, since
they help each other finding a job, their employment statuses are correlated.
To better understand this result, consider the case when  = 0, i.e. employed workers are
not informed about jobs. In that case, the correlation is equal to zero since all workers end
up in a 0 dyad and stay there for ever. Similarly, when  = 0, i.e. there is no job-destruction
rate, the correlation is also equal to zero since, as soon as   0, everybody will end up
employed independently of her partner in the dyad. When  = 0, workers only interact
with their strong ties and their employment statuses are perfectly correlated. Finally, the
correlation in employment status between two employed workers is decreasing with  and
 and increasing in . Indeed, when the job-destruction rate  increases, workers tend to
be more often unemployed and thus help each other more to find a job. As a result, their
10
The other correlations between different employment statuses can be calculated in a similar way.
14
correlation in employment status increases. The same reasoning applies for a decrease in .
When  increases so that social interactions with weak ties rise at the expense of the social
interactions with strong ties, workers in the same dyad help each other less and thus their
correlation in employment status decreases.
3.5
Choosing social interactions
We would like now to extend the model so that  is chosen by individuals and not exogenously
defined as in the previous sections. The timing is as in the previous section. We assume that
there is some cost of interacting with weak ties. Let  denotes the marginal cost of these
interactions. The expected utility is now given by:
 () = ∗ () + [1 − ∗ ()]  −  
where ∗ () is defined by (11) or (17). Each individual optimally chooses  that maximizes
 (). The first-order condition yields:
∗ ()
 ()
=
( − ) −  = 0


(24)
∗ ()

 0. We have the following
We have seen (see Proposition 4) that if (10) holds, then
result:
Proposition 5 Assume (10) and consider steady-state equilibrium I. Then there exists a
unique interior 0   ∗  1 that maximizes  (). Higher wages or lower unemployment
benefits or lower interaction costs will increase the interactions with weak ties, i.e.
 ∗
0

 ∗
0

∗
0

Furthermore, if  ≥ 2, an increase in , the job-destruction rate or an increase in , the
job-information rate induces workers to spend more time with their weak ties, i.e.,
∗
0

 ∗
0

There is clear trade-off between the benefits of interacting with weak ties and the costs
associated with it (see (24)). Indeed, workers want to interact with weak ties because it
increases their probability of being employed (or, equivalently, the time they spend employed
∗
during their lifetime), i.e. ()  0. Concerning the wage  and the unemployment benefit
, a higher  or  increases the value of employment and, since ∗ () and  are positively
related, workers will interact more with weak ties. Quite naturally, increasing the cost  of
social interactions reduces the time spent with weak ties.
15
What is interesting and new here is the effect of the aggregate labor-market variables
on social interactions. When  or  increases, workers spend more time with their weak
∗2
∗2
ties because the cross effect of  or  on employment is positive, i.e. 
 0 and 

∗
0. Indeed, when  or  increases, the positive effect of weak ties  on employment  is
even stronger and thus workers rely more on their weak ties. This is an interesting result
since it shows that, in downturn periods where jobs are destroyed at a faster rate, workers
tend to spend more time with their weak ties because they know they will help them exit
unemployment. In an economy where the flow of job information is faster, the same results
occur.
4
Finding a job outside the social network
While we do know that lots of jobs are obtained through networks, lots of jobs are not, and,
as a result, one limitation of our model is that there is no role for job finding outside of the
social network. Let us now consider the possibility of an unemployed worker learning of a
vacancy directly from an employer as well as via a social interaction. To be more precise, we
now assume that the unemployed workers can also hear directly about a job at rate .11 For
example, employers put help-wanted signs on their windows or have adds in newspapers. In
that case, anybody (employed or unemployed) can have access to this information. As in the
previous section, the job information that only goes through employed workers is when the
employers do not advertise their vacancies publicly but only tell their own employed workers
about them.
In that case, the unemployed workers can now find a job either directly or through their
social network. It is readily verified that the net flow of dyads from each state between 
and  +  is now given by:
⎧ •
⎪
⎪
⎨ •2 () = [2 −  +  ()] 1 () − 22 ()
1 () = 2 [ () + 1] 0 () − ( + [2 −  +  ()] ) 1 () + 22 ()
⎪
⎪
⎩ •
0 () = 1 () − 2 [ () + 1] 0 ()
(25)
Indeed, the only difference with the previous section where only the employed workers could
hear about a job (see (4)) is that now the unemployed workers can also hear about a job at
rate . For example, the rate at which a dyad 0 becomes a dyad 1 is not anymore 2 but
2 + 2 since now each individual in the dyad 0 can also hear directly about a job. As
a result, a person in a dyad 0 will find a job either directly at rate  or through her social
11
This is for simplicity. Below, we relax this assumption and assume that the rate at which the employed
and the unemployed workers hear about a job is not the same.
16
network at rate  since her weak tie has to be employed and has to hear about a job.
Observe that jobs obtained through the social network are only transmitted by employed
workers because if an unemployed worker hears directly about a job she will not give this
information to someone in her network but will take it.
In a steady-state (∗2  ∗1  ∗0 ), each of the net flows in (25) is equal to zero and we obtain:
∗2 =
 (2 −  +  ∗ ) ∗1
2
(26)
2 ( ∗ + 1) ∗0

(27)
1
− ∗2 − ∗1
2
(28)
∗1 =
where
∗0 =
∗ = 1 − ∗
(29)
∗ = 2∗2 + ∗1
(30)
We have a first result.
Lemma 1 When the unemployed workers can directly hear from a job, neither a full-unemployment
equilibrium U for which ∗ = 0 and ∗ = 1 nor a full-employment equilibrium E for which
∗ = 1 and ∗ = 0 can exist.
This is a first interesting result which says that a full-unemployment equilibrium U such
that ∗ = 0 and ∗ = 1 is not anymore possible here while it always existed in the benchmark
model. This is because when workers are stuck in a 0 dyad, they can always leave it without
using their network since there is always a positive probability that one of the unemployed
worker belonging to a 0 dyad hears directly about a job. As a result, nobody gets “stuck”
in a 0 dyad.
Proposition 6 When the unemployed workers can directly hear from a job, there exists a
unique steady-state interior equilibrium I where 0  ∗  1 is (implicitly) given by
 (∗ + 1) [ (2 −  +  ∗ ) + ]

 2 +  ( ∗ + 1) [ (2 −  +  ∗ ) + 2]
(31)
 2 2

= 2
 +  ( ∗ + 1) [ (2 −  +  ∗ ) + 2]
(32)
∗ =
0  ∗0  12 by
∗0
and ∗2 and ∗1 by (26) and (27), respectively.
17
Even though the model is much more complicated, we are able to prove that there exists
a unique steady-state interior equilibrium and give the equilibrium values of all variables.
Their values are, however, more cumbersome and only implicitly defined. We would like now
to see if one of our main results, namely the positive relationship between employment and
weak ties, still holds in this more general model.
Define
Φ() ≡ −23 − (3 − 4 + 2) 2 + [ + 2 (2 − )]  − 
(33)
We have the following result.
Proposition 7 When the unemployed workers can directly hear from a job, an increase in
the percentage of weak ties  increases the employment rate ∗ in the economy only when ∗
takes intermediate values. On the contrary, when ∗ is low or high, then an increase in 
reduces ∗ . Formally, if  and  are the positive real roots of Φ(), then
() When   ∗  , then
∗

 0.
() When 0  ∗   or   ∗  1, then
∗

 0.
This is an interesting result which shows that the positive relationship between ∗ and 
only holds when employment is neither too low nor too high. Indeed, when the employment
rate ∗ is very low, then weak ties do not help a lot because there is little chance that they
are employed. This is because, when someone is stuck in a 0 −dyad, she can leave it without
the help of a weak tie since she can directly hear from a job at rate . In the previous model,
only a weak tie could help her finding a job because neither her strong tie nor herself could
find a job. When the employment rate ∗ is very high, then weak ties are less important
because strong ties are also more likely to be employed.
This new result provides a new perspective on the notion of “weak ties” proposed by
Granovetter. As highlighted in the Introduction, Granovetter was postulating that weak
ties were always superior to strong ties in helping workers finding a job. Proposition 4 says
that when the unemployed workers can only find jobs through their social contacts (which
is the case in developing countries for unskilled workers who are often illiterate and thus
have no access to newspapers12 or in developed countries for very specific jobs which are not
advertised formally), then weak ties are superior to strong ties in finding a job. However,
when the unemployed workers can find a job both directly and through their social contacts,
Proposition 7 states that strong ties can be superior to weak ties when the employment rate
in the economy is either low or high.
12
See for example Wahba and Zenou (2005, 2012) for Egypt.
18
To illustrate this result and to have some sense of what is a high and low employment
rate, let us perform some simple numerical simulations. As in Section 3.3, following Shimer
(2005) and Pissarides (2009), we take  = 0594 and  = 0036. For the time spent with
weak ties, we take  = 02. As stated in Proposition 6, there is a unique solution, which is
here given by ∗ = 09699 and ∗0 = 000073. The following table gives all the equilibrium
values.
Table 1: Steady-state equilibrium
∗ (%)
∗ (%)
∗0
2∗0 (%)
∗1
2∗1 (%)
∗2
2∗2 (%)
9699
301
000073
015
00286
572
04707
9413
The unemployment rate is equal to 3 percent. Workers do not stay that much in a 0
dyad (only 015 percent of their time)13 because they hear directly of a job and have their
social network. On the contrary, they spend most of their time in a 2 dyad (9413 percent
of their time).
We can then determine the relationship between ∗ and . In the proof of Proposition
∗
7, we showed that the sign of 
is the same as that of Φ(), which is given by (33). It is

easily verified that Φ() has three real solutions, one being negative and equals to  = −256,
and two positive and equal to  = 03609 and  = 09709. Here, since ∗ = 09699, it is
between  = 03609 and  = 09709. Then, following Proposition 7, an increase in  always
∗
increases ∗ , i.e. 
 0. It is interesting to notice that, when we say that ∗ has to take an

∗
intermediate value for 
 0, it is relative to the values of  and . In the present example,

∗
 0.
the equilibrium employment rate is quite high (9699 percent) and we still have that 

Let us now assume that the rate at which the employed (denoted by 1 ) and the unemployed workers (denoted by 0 ) hear about a job is not the same. In that case, the flows in
the labor market are now given by:
For the interpretation of the results, it is better to use 2∗0 than ∗0 since the former is normalized and
gives the time spent in a 0 dyad. The same applies for ∗1 and ∗2 .
13
19
⎧ •
⎪
⎪
⎨ •2 () = (0 + [1 −  +  ()] 1 ) 1 () − 22 ()
1 () = 2 [ ()1 + 0 ] 0 () − ( + 0 + [1 −  +  ()] 1 ) 1 () + 22 ()
⎪
⎪
⎩ •
0 () = 1 () − 2 [ ()1 + 0 ] 0 ()
(34)
In a steady-state (∗2  ∗1  ∗0 ), each of the net flows in (34) is equal to zero and we obtain:
∗2 =
0 + (1 −  +  ∗ ) 1 ∗
1
2
(35)
2 (0 +  ∗ 1 ) ∗0

(36)
∗1 =
Since ∗ = 2∗2 + ∗1 , we have:
∗ = [0 + (1 −  +  ∗ ) 1 + ]
2 (0 +  ∗ 1 ) ∗0
2
(37)
Since ∗0 = 12 − ∗1 − ∗2 , we easily obtain:
2∗0 =
2
 2 + (0 +  ∗ 1 ) [0 + (1 −  +  ∗ ) 1 + 2]
(38)
Using the same arguments as in the proof of Proposition 6, it is straightforward to show that
there exists a unique interior steady-state equilibrium I where ∗ and ∗0 are given by (37)
and (38) and ∗2 and ∗1 by (35) and (36). Plugging the value of 0 from (38) into (37), we
obtain:
(0 +  ∗ 1 ) [0 + (1 −  +  ∗ ) 1 + ]
∗ = 2
 + (0 +  ∗ 1 ) [0 + (1 −  +  ∗ ) 1 + 2]
Proceeding as in the homogenous case (when 0 = 1 = ), we have:
where
∗
e
R 0 ⇔ Φ()
R0

e
Φ()
= −21 3 − [20 + 2 + 1 (1 − 4)] 2 + [ + 30 + 1 (1 − 2)]  − 0
We obtain exactly Proposition 7 with the only difference that  and  are now the positive real
e
roots of Φ()
and not of Φ(). Let us now run some numerical simulations by assuming that
the average rate is the same as in the homogeneous case so that (0 + 1 ) 2 =  = 0594.
Let us take: 0 = 05 and 1 = 0688 so that 1 − 0 is relatively small. In that case,
20
the results are nearly the same as in the homogenous case since the employment rate is
∗ = 09698 and  = 03286 and  = 09709.
Let us keep the same average (0 + 1 ) 2 = 0594, but let us now increase the difference
between 0 and 1 . We take: 0 = 02 and 1 = 0988. In that case, we obtain a similar
employment rate, i.e. ∗ = 09687, but the lower bound in employment rate is much smaller
since  = 01830 and  = 09710. So we see that the lower is 0 , the lower is  and the larger
is the interval between  and  and therefore the more likely there is a positive relationship
between  and ∗ .
5
Discussion
In order to highlight the contribution of our paper, let us compare it with that of CalvóArmengol and Jackson (2004) (CAJ hereafter), a prominent paper in the theoretical literature
on social networks in the labor market.14 Contrary to the present model where only a
very specific network structure (i.e. the dyad) is assumed, CAJ explicitly model a social
network (which can have any possible structure) using graph theory (Jackson, 2008) where
information flows between individuals having a link with each other. They show that an
equilibrium with a clustering of workers with the same status is likely to emerge since, in the
long run (i.e. steady state), employed workers tend to be friends with employed workers.
The main differences between our model and that of CAJ are as follows. First, CAJ
model strong ties in a richer way since each individual may have many strong ties (not only
one as in our model) and thus must be in competition with friends of friends. This leads
to the fact that, if my (employed) strong tie hears about a job, I will not necessarily obtain
this formation since my strong tie will share it with all her unemployed strong ties. This is
not the case in our model. Second, we model weak ties in a richer way than CAJ because
the unemployed workers meet directly them at every period of time while, in CAJ, workers
never meet directly weak ties. The latter seems to be a strong assumption and not consistent
with evidence on weak ties as a direct of source of information about jobs, as suggested by
Granovetter. In CAJ, my weak ties will never directly provide me with information about
jobs. They will help my strong ties getting a job, which eventually will help me finding a
job. Third, CAJ assume that workers spend all their time with strong ties (in the language
of our model, this means that  = 0) while, in our model, workers choose how much time to
spend with weak and strong ties. This helps us to formally demonstrate Granovetter (1973,
1974, 1983)’s idea that weak ties are superior to strong ties for providing support in getting
14
As stated in the Introduction, there are other theoretical papers dealing with social networks in the labor
market. They are usually not dynamic and, if they are, they are just a particular case of Calvó-Armengol
and Jackson (2004). That is why here we mainly compare our model to that of Calvó-Armengol and Jackson
(2004).
21
a job by showing that  has a positive impact on the employment rate in the economy.
This result cannot be shown in CAJ because it is based on the asymmetric role of weak and
strong ties in helping the unemployed workers finding a job. Finally, the main advantage of
our approach is that it leads to a very tractable model.15 We obtain closed-form solutions
for the equilibrium values and it is easy to incorporate new aspects (for example, explaining
racial differences in employment) and still obtain clear-cut results.
6
Conclusion
In this paper, we have developed a simple and tractable model of weak and strong ties in
the labor market. We have formally demonstrated the Granovetter’s informal idea of the
strength of weak ties in finding a job. In our model, it is better to meet weak ties because
a strong tie does not help in the state where all best friends are unemployed. But a weak
tie can help leaving unemployment in any state because that person might be employed. So
there is an asymmetry that is key to the model and that explains why weak ties are superior
to strong ties in helping out unemployed workers finding a job. This can also explain why
some workers may be stuck in unemployment traps having little contact with weak ties
that can help them escape unemployment. When we allow unemployed workers to learn of
a vacancy directly from an employer as well as via a social interaction, we show that the
previous result is not always true. It is only valid when the employment rate in the economy
is not too high because now the unemployed workers can escape from the unemployment
traps by finding a job by themselves.
One of the virtues of our model is that it is simple and tractable. In particular, it is
easy to extend it and to incorporate new aspects. For example, Zenou (2013) uses this
model to explain racial differences in employment. Indeed, if minority workers have little
interaction with weak ties, especially white workers, they will end up experiencing high
unemployment rate. This is because, as highlighted in this paper, weak ties are an important
source of job information and when ethnic minorities miss it, they end up having a higher
unemployment rate than whites. There is indeed a vicious circle because, if ethnic minorities
are discriminated against in the labor market, they will experience a higher unemployment
rate than whites and will mostly rely on other ethnic minority workers (strong ties) who also
experience a high unemployment rate. Since jobs are mainly found through social networks
via employed friends,16 ethnic minorities will then be stuck in a “bad” labor market situation.
15
For example, in Section 3.4, we were able to explicitly calculate the correlation in employment status
between different workers in a dyad and to see how this correlation varies with the different parameters of
the model.
16
See, for example, Battu et al. (2011) who show that networks are a popular method for the ethnic
minorities in the UK, even though they are not necessarily the most effective in terms of gaining employment.
22
This result will be true even if unemployed workers can find a job directly as in Section 4.
This may be because minority workers are more likely to reside in segregated areas (Ihlanfeldt
and Sjoquist, 1998). As a result, if black workers do not have access to weak ties (especially
whites), in particular because they are segregated and separated from them, then their main
source of information about jobs will be provided by their strong ties. But if the latter are
themselves unemployed, this might partially explain unemployment differences by race.
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26
Appendix
Proof of Proposition 1
We establish the proof in two steps. First, Lemma 1 characterizes all steady-state dyad
flows. Lemma 2 then provides conditions for their existence.
Lemma 2 There exists at most two different steady-state equilibria: () a full-unemployment
equilibrium U such that ∗ = 0 and ∗ = 1, () an interior equilibrium I such that 0  ∗  1
and 0  ∗  1.
Proof.
By combining (5) to (8), we easily obtain:
∗ = [(1 −  + ∗ ) + ]
2∗  ∗

2 0
(39)
We consider two different cases.
() If ∗ = 0, then equation (39) is satisfied. Furthermore, using (5) and (6), this implies
that ∗1 = ∗2 = 0 and, using (7) and (9), we have ∗0 = 12 and ∗ = 1. This is referred to as
steady-state U (full unemployment).
() If ∗  0, then solving equation (39) yields:
∙ 2
¸

1
(1 − )
∗
− −
 =
∗
 20

Define  = (1 − ) ,  =  (). This equation can now be written as:
∗ =
2
−−
2∗0
(40)
Moreover, by combining (5) and (6), we obtain:
2∗ ∗
( + ∗ ) ∗ ∗
0 ,
∗2 =
0
(41)

2
• Let us first focus on the case where ∗ = 1. In that case, it has to be that only
2 −dyads exist and thus ∗0 = ∗1 = 0, which, using (41) implies that: ∗2 = 0. So this case is
not possible.
• Let us now thus focus on the case: 0  ∗  1 (which implies that 0  ∗  1)
By plugging (40) and (41) in (7) and after some algebra, we obtain that ∗0 solves Φ(∗0 ) =
0 where Φ() is the following second-order polynomial:
µ ¶2

 2 (1 + )
∗
Φ(0 ) = −  −
+
(42)

2
2
∗1 =
27
Lemma 3
() The steady-state equilibrium U always exists.
() The steady-state equilibrium I exists when   [ +
p
(4 − 3)]2.
Proof.
() In this equilibrium ∗ = 0, which implies that () = (1 − )  and () = 0. There
are only 0 −dyads so all workers are unemployed and will never receive a job offer since
() = 0. So when a 0 −dyad is formed it is never destroyed and thus this equilibrium is
always sustainable.
() We know from Lemma 2 that a steady-state I exists and that ∗ 6= 1. We now
have to check that ∗  0 and 0  ∗0  12. Let us thus verify whether there exists some
0  ∗0  12 such that Φ(∗0 ) = 0, where Φ(·) is given by (42). We have Φ(0) = (2)2  0
and Φ0 (0) = − (1 + ) 2  0. Therefore, (42) has a unique positive root smaller than 12
if and only if
∙
¸
1
1
1

2
Φ(12) =
 − (1 + ) −
= (1 + )( 2 −  − )  0
4

4

h
i
p
2
The unique positive solution to  −  −  = 0 is 1 + (4 − 3) 2. Then, ∗0  12 if
i
h
p
and only if   1 + (4 − 3) 2, equivalent to:
+



p
(4 − 3)
2
Observe that ∗0  12 guarantees that ∗  0.
Proof of Proposition 2
By differentiating (17), we obtain:
We have:
which is equivalent to:
∗
1−
1
= p
−

  [ + 4 (1 − )] 
∗
1−
1
0⇔ p


  [ + 4 (1 − )] 
(1 − )2    + 4 (1 − )
Simplifying further this inequality leads to:
 ( − 2)  4 (1 − )
28
(43)
which is always true since   1.
By differentiating (12), we get:
∙
¸
∗0
1
2
=
0

 4∗0 + (1 + ) 
(44)
By differentiating (13), we get:
∙
¸
∗
∗1
2 ∗ ∗
∗ 0
∗ ∗
= 2
 +
 −  0


 0

(45)
Using (43), (44) and the fact that  ≡ (), we obtain:
"
#
∗
3 ∗
∗1
(1 − ) ∗0 
2
2∗2
∗0 
0  + 
∗ ∗
p
= 2
+
−  0
−


  [ + 4 (1 − )]  4∗0  + (1 + ) 
This is clearly ambiguous. However, if we go back to (45), observe that if
∗
then 1  0. This is equivalent to:
∗

∗ ∗
  +∗ 0 
 0
 0,
∗0 
∗ 
+
0
 ∗
 ∗0
∗0 
∗ 

−
 ∗0
 ∗
¯ ∗ ¯ ¯ ∗ ¯
¯   ¯ ¯   ¯
¯
⇔ ¯¯ 0 ∗ ¯¯  ¯¯
 
 ∗ ¯
⇔
0
Finally, by differentiating (14), we get:
½
¾
∗
¡ ∗
¡ ∗
¢ ∗
¢ ∗
∗2
()2
∗  ∗
∗2 0
∗2
( + 2 )
=
  +  + 
 − 2  +  0

3
 0

∗
∗
∗   0, let us show that (∗ + ∗2 ) 0  − 2 (∗ + ∗2 ) ∗0  0. Using
Since ( + 2∗ ) 
 0
(44) and the fact that  ≡ (), this last inequality is equivalent to:
3
2∗2
0 +
− 2∗0  0
4∗0  + (1 + ) 
which is equivalent to:
3  ∗2 (1 + ) ∗
−
 −
0 +
2 0
2
29
µ ¶2

0
2
Since ∗0 is defined as (see (12)):
(1 + ) ∗

0 +
− ∗2
0 −

2
µ ¶2

=0
2
the inequality above is thus equivalent to:
−
1  ∗2
 0
2 0
which is always true.
Proof of Proposition 3
Differentiate first (12). We obtain:
2
 ∗2
 + 23 2
∗0
0
= −   0∗ 1+

2  0 + 2
Now, by differentiating (11), we have:
By differentiating (17), we have:
22
As a result,
p
∗
2 + 2 (1 − )
= p
−  [ + 4 (1 − )] + 2

 [ + 4 (1 − )]
2 (1 − )
= 2 − p
 [ + 4 (1 − )]
2 (1 − )
∗
 0 ⇔ 2  p

 [ + 4 (1 − )]
⇔ 4 (1 − )   ( − 2)
which is always true since   2.
By differentiating (14), we obtain:
3
¡
¡
¢ ∗0
¢ 
∗2
∗
=
( + 2∗ ) ∗0 + ∗ + ∗2
 + 2 ∗ + ∗2 2 ∗0




¡
¡
¢
¢  ∗0
∗


( + 2∗ )
∗0 + 2 ∗ + ∗2 2 ∗0 + ∗ + ∗2
=



 
30
(46)
Using (46), we have:

∗
3 2

3

¡ ∗
¢  ∗ (1 − ) ∗2
¢
 ∗
∗
0 + 23  2 ¡
∗
∗2
∗
∗2
−
+

=
( + 2 )
0 + 2  + 


2
0


2 
2  (1−)
∗0 + 2

#
"
4
∗2
∗
∗
¡
¢
+
2
(1
−
)

 ∗

20
0
3 2
( + 2∗ )
0 + ∗ + ∗2
−
=
∗
2
2



4  (1 − ) 0 + 
#
"
4
2
∗2
2
∗
∗
¡
¢

(1
−
)

(4
−
)
+
2

−
2
 ∗

0
0

( + 2∗ )
0 + ∗ + ∗2
=
∗
2
2


  [4  (1 − ) 0 + ]
We know from (12) that
−
1 ∗
2
(1 − )  ∗2
0 −
0 + 2 2 = 0

2
4 
and thus we have:
3
∗2
 ∗ 2 (∗ + ∗2 ) (1 − )  ( − 2) ∗2
∗
0
=
( + 2∗ )
 −


 0
 [42  (1 − ) ∗0 + ]
Using (??), we obtain:
∗

 3 2 = 2 − 3 ( + 2∗ )


Ã
∗
2 (1 − ) ∗2
0 (4 − ) + 20  −
44 3 [4 (1 − ) ∗0 + ]
3
2 
!
2 (∗ + ∗2 ) (1 − )  ( − 2) ∗2
0
−
∗
2
 [4  (1 − ) 0 + ]
Using (12), we get:
∗
3 2


= 2 +


⇔ 3
4 ( + 2∗ )
³
(1−)∗2
0 (−2)

−
∗0

´
− 43 2 (∗ + ∗2 ) (1 − )  ( − 2) ∗2
0
24 2 [4 (1 − ) ∗0 + ]
4
∗
∗
∗2

(1 − )  ( − 2) [ 3 ( + 2∗ ) − 42 2 ( + ) ∗ ] ∗2
0 −  ( + 2 ) 0
= 2 +


24  3 [4 (1 − ) ∗0 + ]
A sufficient condition for
∗2

 0 is:
£
¤
4
∗
∗
(1 − )  ( − 2)  3 ( + 2∗ ) − 42 2 ( + ) ∗ ∗2
0 −  ( + 2 ) 0  0
which is equivalent to:
∗0 
3
£
¡ + ¢ ¤
(1 − )  ( − 2)  2 − 42  2 +2
∗
∗
31
An upper bound for
¡
+
+2∗
¢
∗ is 1 and thus this condition can be written as:
∗0 
3
(1 − )  ( − 2) ( 2 − 42  2 )
which is condition (20).
Finally, by differentiating (13), we get:
¸
∙
∗
∗1
2 ∗ ∗
∗ 0
∗ ∗
=
 +
 +  0

  0

(47)
This is clearly ambiguous. However, since ∗ ∗0  0, a sufficient condition for
∗ ∗
∗
0  + ∗ 0   0


⇔
∗0 
∗ 
+
0
 ∗
 ∗0
∗ 
∗0 

−
 ∗
 ∗0
¯ ∗ ¯ ¯ ∗ ¯
¯   ¯ ¯ 0  ¯
¯¯
¯
⇔ ¯¯
 ∗ ¯ ¯  ∗ ¯
⇔
0
Proof of Proposition 4
() By totally differentiating (12), we obtain:
 2
 + 21 2 0 −
∗0
=  0 (1−)

2  0 +
and thus
∗
sgn 0 = sgn

Let us study
∙
2
22 3
1
2
 2
1
2
0 + 2 0 − 2 3

2
2 
¸
 2
1
2
0 + 2 0 − 2 3

2
2 
2
Φ(0) = − 2 3  0
2 

1
Φ0 (0 ) = 2 0 + 2  0 when 0 ≥ 0

2

Φ00 (0 ) = 2  0

Φ(0 ) ≡
32
∗1

 0 is:
We have a quadratic function that crosses only once the positive orthant. Let us calculate
b0  0 the value for which Φ(0 ) crosses the 0 −axis. For that, we have to solve: Φ(b0 ) = 0.
It is easy to verify that:
Ãr
!

8
b0 =
−1 0
1+
4 2

It should be clear that if b0  12, then Φ(0 )  0 for 0  0  12 and thus
us thus check that b0  12, which is equivalent to:
∗0

 0. Let
µ ¶3
µ ¶



≡2
−  − 3  0
Ω



We have:
Ω (0) = −3  0
µ ¶
µ ¶2


0
=6
−
Ω


with
r
µ ¶



Ω
0⇔ 


6
0
p
∗
As a result, when  
, b0  12 and thus 0  0. Since we are in equilibrium I,
6
condition (10) has to hold, i.e.
p
 + (4 − 3)



2
Let us show that
This inequality is equivalent to:
r
+


6
p
(4 − 3)
2
p
2  3 2 + 3(4 − 3) + 6 (4 − 3)
p
⇔ 3 2 + 9(1 − ) +  + 6 (4 − 3)  0
which is always true since   1.
() By totally differentiating (17), we obtain:
p
 [ + 4 (1 − )] − 2 + 2 − 
∗
 =
2
33
(48)
"
#
∗
2

+
4
−
2

= 2 +  − p
2 2

 [ + 4 (1 − )]
Thus
(2 + )2 ( + 4 − 4)   ( + 4 − 2)2
(2 + )2   (1 − ) +  ( + 2)
2 ( + 2)   (1 − )
∗

 0
⇔ (2 + )2 [ + 4 (1 − )]   ( + 4 − 2)2
⇔  + 4  −
which is always true.
Finally, from (13) and (14), it is easy to see that
∗1

and
∗2

cannot be signed.
Proof of Proposition 5
First, it is easily verified that, using (17),
p
⎤
(+4−2)
√ ⎡ 2  + 4 (1 − ) − √
4
2
2 +4(1−)

 ⎣
⎦0
=
2
23
 + 4 (1 − )
(49)
so that second order condition is always satisfied. As a result,  ∗ , the solution to (24), is
unique.
Let us show that 0  ∗  1. The optimal  ∗ is given by:
∗ ()

=

−
By differentiating (17), we obtain:
( + 2)

2
=

2
Using (50), we have
p
 [ + 4 (1 − )] −  [ + 2 (2 − )]
p
 [ + 4 (1 − )]
"
#
√
()
1  + 2 − 2 + 4
lim
= 2
= +∞
→0 

2
since  + 2 
√
2 + 4. We also have
34
(50)
()
=0
→1 
lim
Furthermore, since
2
 2
 0, we know that
()

is a decreasing function and since

−
is a

and −
lies between 0 and 1. As
constant, it has to be that the intersection between ()

∗
a result, there is a unique  , which is between 0 and 1.
Second, by differentiating (24), it is straightforward to show that
∗
0

 ∗
0

∗
0

Third, by differentiating (24), we have
 ∗
=−

∗2
 2 ∗ ()

∗2
2
 ∗
=−
and

 2 ∗ ()

∗2
2
2 ∗
∗
 ()
Since 
 0 (see (49)), the sign of 
is the same as of  
and the sign of
2

 2 ∗ ()
same as of  . Let us now calculate these cross-derivatives.
By differentiating (17), we have:
"
#
∗


+
2
(2
−
)
2 2
= 2 +  −  p

 [ + 4 (1 − )]
 ∗

Thus, by differentiating again this equation with respect to , we obtain:
 2 ∗
22 + 4 (2 − ) (1 − )
p
=2−

[ + 4 (1 − )]  [ + 4 (1 − )]
p
2 ∗
[ + 4 (1 − )]  [ + 4 (1 − )] + 2 (2 − ) (1 − ) − 2
2  
p
⇔ 
=

[ + 4 (1 − )]  [ + 4 (1 − )]
2 2
Let us show that
 2
 2 ∗
0
 2
This is equivalent to
p
 [ + 4 (1 − )] + 2 (2 − ) (1 − )  2
r
r
 + 4 (1 − )
 + 4 (1 − )
⇔
+ 4 (1 − )
+ 2 (2 − ) (1 − )  


r
r
 + 4 (1 − )
4 (1 − )

⇔ 1+
1


[ + 4 (1 − )]
Since
35
is the
(51)
2 ∗
2 ∗
 
 
is always true, then 2 
 0 and thus 
 0. As a result,
Now, by differentiating (51) with respect to , we obtain:
 ∗

 0.
 2 ∗

4 + 8 2 (2 − ) (1 − )
p
=− 3 +


[ + 4 (1 − )]  [ + 4 (1 − )]
p
2 ∗
44 + 8 2 3 (2 − ) (1 − ) −  [ + 4 (1 − )]  [ + 4 (1 − )]
2  
p
=
⇔ 2

3 [ + 4 (1 − )]  [ + 4 (1 − )]
22
Let us show that:
44 + 8 2 3 (2 − ) (1 − )   [ + 4 (1 − )]
This is equivalent to:
p
 [ + 4 (1 − )]
167 + 64 2 5 (2 − )2 (1 − )2 + 646  (2 − ) (1 − )  [ + 4 (1 − )]3
⇔ 167 + 645  2 (2 − )2 (1 − )2 + 646  (2 − ) (1 − )
 3 + 122  (1 − ) + 48 2 (1 − )2 + 64 3 (1 − )3
¢
£
¤
¡
⇔ 3 4 − 1 + 12 (1 − ) (2 − ) 4 − 1 [ + 4 (1 − )]
+157 + 52 (1 − ) 6 (2 − ) + 16 (1 − )2 5  (2 − )2 − 64 3 (1 − )3  0
If  ≥ 2, then all terms are positive but the last one. Observe that (10), i.e.

i
p
h
 +  (4 − 3)
2
implies that (by taking the upper bound of  on the right-hand side):
"
√ #
 1+ 4
3

= 
2
2
2
As a result,
µ ¶3
3
3 = 2163
64 (1 − )  64
2
3
Since  ≥ 2,
3
157 = 154 3 ≥ 15 × 24 3 = 2403
2 ∗
 
and thus the inequality above is always true. As a result, 2 2 
 0, thus
∗

 0.

36
 2 ∗

 0 and
Proof of Lemma 1. By combining (26) to (30), we easily obtain:
∗ = [(2 −  +  ∗ )  + ]
2 (∗ + 1) ∗0
2
(52)
First, a full-unemployment equilibrium U such that ∗ = 0 and ∗ = 1 is not anymore
possible here. Indeed, if ∗ = 0, then for equation (39) to be satisfied it has to be that
∗0 = 0. Using (26) and (27), this implies that ∗1 = ∗2 = 0. But, in that case, equation
(28) can never be satisfied since it implies that 0 = 12, which is impossible. As a result, a
full-unemployment equilibrium U cannot exist.
Second, let us see if full-employment equilibrium E such that ∗ = 1 and ∗ = 0 can exist.
In that case, it has to be that only 2 −dyads exist and thus ∗0 = ∗1 = 0, which, using (26)
implies that: ∗2 = 0. So this case is also not possible.
Proof of Proposition 6. In the proof of Lemma 1, we have shown that the equilibrium
employment rate is now given by (52), which we rewrite for the ease of exposition:
∗ = [(2 −  +  ∗ )  + ]
2 (∗ + 1) ∗0
2
(53)
By combining (26), (27) and (28), we easily obtain the equilibrium value of ∗0 as follows:
2∗0 =
2
 2 +  ( ∗ + 1) [ (2 −  +  ∗ ) + 2]
(54)
As a result, the equilibrium values of ∗ and ∗0 are determined by equations (53) and (54).
Let us first study (54) that we write as:
0 () =
1
2
2
2  +  (  + 1) [ (2 −  +  ) + 2]
We have:
0
0

Let us now study equation (53), which can be written as:
0 (0)  0,
0 () =
 2
2 ( + 1) [(2 −  +  )  + ]
We have
0 (0) = 0
We also have:
( + 1) [(2 −  +  )  + ] −  [(2 −  +  )  + ] −   ( + 1)
2 0
=
2
 
[( + 1) [(2 −  +  )  + ]]2
37
Since the denominator is strictly positive, let us focus on the numerator and show that it is
also positive. We want to show that:
( + 1) [(2 −  +  )  + ]   [(2 −  + )  + ] +  ( + 1)
⇔ ( + 1) [(2 −  +  )  + ]   [(2 − )  +  + 2 + ]
⇔  [(2 −  +  )  + ] + (2 −  +  )  +    [(2 − )  +  + 2 + ]
⇔ (2 −  +  )  +    ( + 1)
(2 − )  +    2 2
Since  2 2   (both  and  are less than 1) and since (2 − )   , this inequality is
0
always true. As a result, 
 0.

Since the first function always decreases starting from a strictly positive number while
the other always increases starting from zero, there is a unique intersection between these
two curves and thus a unique equilibrium where ∗  0 and ∗0  0. We need now to show
that ∗0  12 and ∗  1. From (54), we see that straightaway that ∗0  12 whenever
∗  0. Let us now show that ∗  1. Plugging the value of 0 from (54) into (53), we
obtain:
 (∗ + 1) [ (2 −  +  ∗ ) + ]
∗ = 2
 +  ( ∗ + 1) [ (2 −  +  ∗ ) + 2]
It is clear from this last equation that, for any   0, the equilibrium value of  has to be
less than 1. We can also show it in the following way. By developing this equality, we easily
obtain:
£
¤
2  2 3 + [ (3 − 2) + 2] 2 +  2 +  (2 − ) + 2 (2 − ) (1 − ) − 2  
= 2 (2 − ) + 
If we denote by
£
¤
Θ() ≡ 2  2 3 + [ (3 − 2) + 2] 2 +  2 +  (2 − ) + 2 (2 − ) (1 − ) − 2  
− (2 − ) 2 − 
then, it is easily verified that Θ(0)  0. Let us show that Θ0 ()  0. We have:
Θ0 () = 32  2 2 + 2 [ (3 − 2) + 2]  +  2 +  (2 − ) + 2 (2 − ) (1 − ) − 2 
The only negative term is −2 . Let us show that
2 [ (3 − 2)]   2 
38
This is equivalent to

1
2 (3 − 2)
1
is increasing in , let us take the upper bound of , which is  = 1. In that
Since 2(3−2)
case, this inequality rewrites as:
  05
which is always true for reasonable value of employment . Thus Θ0 ()  0. This confirms
the uniqueness and positivity of  (because Θ(0)  0 and Θ0 ()  0). To show that the
solution of this equation is less than 1, i.e. ∗  1, it suffices to show that Θ(1)  0 (since
we know that Θ(0)  0 and Θ0 ()  0). This is equivalent to
2 2 + [ (3 − 2) + 2]  +  2 +  (2 − ) + 2 (2 − ) (1 − ) − 2  − (2 − ) 2 −   0
 [ (1 − ) + 2] + 2 +  (1 − )  0
which is always true. Since there is a unique 0  ∗  1 and a unique 0  ∗0  1, ∗2 is
uniquely determined by (26) and ∗1 by (27).
Proof of Proposition 7. The equilibrium employment rate is given by (31) or equivalently by17
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Θ() ≡ 2  2 3 + [ (3 − 2) + 2] 2 +  2 +  (2 − ) + 2 (2 − ) (1 − ) − 2  
− (2 − ) 2 − 
By totally differentiating this equation, we obtain:
∗
23 + (3 − 4 + 2) 2 − [ + 2 (2 − )]  + 
= −

Θ0 ()
We have shown in the proof of Proposition 6 that Θ0 ()  0. Thus the sign of
as the sign as
∗

Φ() ≡ −23 − (3 − 4 + 2) 2 + [ + 2 (2 − )]  − 
so that
∗
R 0 ⇔ Φ() R 0

Let us study the cubic function Φ(). We have Φ(0)  0 and
Φ(1) = −2 − (3 − 4 + 2) + [ + 2 (2 − )] −  = −  0
17
See the proof of Proposition 6.
39
is the same
Let us now calculate Φ0 () since its roots will provide the critical points where the slope of
the cubic function is zero. We have:
Φ0 () = −62 − 2 (3 − 4 + 2)  + [2 (2 − ) + ]
The discriminant of Φ0 () is equal to:
ª
©
∆ = 4 (3 − 4 + 2)2 + 6 [2 (2 − ) + ]  0
Since the discriminant ∆  0, then the cubic function Φ() has a local maximum and a local
minimum. It is readily verified that one root of Φ0 () is negative and the other one positive
and given by:
q
− (3 − 4 + 2) + (3 − 4 + 2)2 + 6 [2 (2 − ) + ]
e =
0
6
Let us show that e  1. This is equivalent to
q
(3 − 4 + 2)2 + 6 [2 (2 − ) + ]  6 + (3 − 4 + 2)
⇔ 2 + 3  0
which is always true. The following figure plots the curve of Φ() where  and  are the
positive real roots of Φ().
F(e)
1
0
F(1)
F(0)
40
e
∗
As a result, since Φ() determines the sign of 
, we have that:

∗
∗
() When     , then   0.
∗
() When 0  ∗   or   ∗  1, then 
 0.

41