Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
www.elsevier.com/locate/econbase
A dynamic model of job networking and social
in'uences on employment
Brian V. Krauth∗
Department of Economics, Simon Fraser University, 8888 University Dr, Burnaby,
B.C., Canada V5A 1S6
Received 23 October 2001; accepted 19 February 2003
Abstract
This paper explores an economy in which personal connections facilitate job search. In the
model, a 7rm receives information on the productivity of those job applicants with social ties to
its current employees. In addition to providing a theory of networking, the model endogenously
generates two classic theories in economic sociology. First, there is a highly non-linear relationship between average human capital in a group of socially connected individuals and the group’s
employment rate. Small changes in group composition may cause large changes in employment,
as suggested in Wilson’s ‘social isolation’ explanation for high unemployment rates among poor
African-Americans. The model also supports Granovetter’s ‘strength of weak ties’ hypothesis,
which holds that acquaintances are more valuable job contacts than close friends.
? 2003 Elsevier B.V. All rights reserved.
JEL classi$cation: E24; J64; Z13
Keywords: Social interactions; Networks; Search; Non-linearity
1. Introduction
A line of empirical research going back to the 1930s indicates that social networks
play an important role in job search. Bewley (1999) lists 24 studies between 1932 and
1990 that estimate the fraction of jobs obtained through friends or relatives, with most
This paper bene7ts from comments by the editor and anonymous referees, William Brock, Lorne
Carmichael, Kim-Sau Chung, Steven Durlauf, and Jonathan Parker, as well as workshop participants at
University of Wisconsin, the Santa Fe Institute Graduate Economics Workshop, the 2001 CEA meetings,
and the 2001 WEHIA conference. All errors are mine.
∗ Tel.: +1-604-291-4438; fax: +1-604-291-5944.
E-mail address: [email protected] (B.V. Krauth).
0165-1889/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0165-1889(03)00079-4
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
estimates between 30 and 60 percent. Despite this, the theory of job networking remains
relatively underdeveloped. This paper analyzes a model economy in which networking
arises because 7rms have limited information on the skills of job applicants. A 7rm’s
current employees provide information on the job-speci7c skills of their friends, thus
improving the likelihood of a productive match. As a result of these information 'ows,
a job searcher’s employment probability and expected productivity are increasing in
the number of employed workers among his or her friends.
In addition to providing a simple theory of networking’s role in the labor market,
the model has a number of implications for the behavior of employment rates within
social groups (groups of individuals who share social ties, potentially de7ned by neighborhood, ethnicity, or other characteristics). First, the long-run employment rate of an
individual is weakly increasing in the initial employment status, number of social ties,
and human capital of any other member of his or her social group, and strictly increasing in his or her own initial employment status, social ties, and human capital. A
second set of results characterizes the group-level relationship between human capital
and long-run employment rates. There exists a critical level of human capital such that
no group member is employed in the long run if all members are below that level, but
the long-run employment rate is strictly positive if all members are above that level.
The relationship is highly non-linear; near the critical value, small changes in group
composition lead to large changes in employment. The third set of results describes the
relationship between the social network itself and the long-run behavior of the employment rate. First, small groups without ties to the rest of society are at risk of falling
into a zero-employment trap. Second, the structure of the social ties within the group
matter. All else being equal, employment is increasing in the proportion of social ties
that are ‘weak’ ties.
In addition to being of independent interest, these results formalize in an economic
model two classic sociological theories: Granovetter’s (1973) ‘strength of weak ties’
hypothesis and Wilson’s (1987, 1996) ‘social isolation’ theory of inner-city unemployment. The strength of weak ties hypothesis holds that acquaintances are more valuable
in job search than close friends and family, because they provide a more diversi7ed
source of information. There is a sizable literature on sociology on the strength of weak
ties. Montgomery (1992) uses a formal model to analyze its empirical implications, but
assumes rather than derives a bene7t to weak ties. In contrast, an aggregate version of
the strength of weak ties arises endogenously in the model presented here.
The social isolation theory holds that poor access to job networks plays a key role
in explaining high unemployment rates among low income African-Americans. Like
the strength of weak ties, it has a long history in social science. In an early contribution, Kain (1968) argues that racial segregation leads to poorer job prospects for
African-Americans, in part through their isolation from the job network. Almost three
decades later, Wilson (1996) argues that growing isolation from job networks explains
the signi7cant decline in employment in low income African-American communities
from the late 1960s to mid-1990s, and that this decline explains much of their contemporaneous growth in social problems. Wilson suggests that this is an unintended
consequence of desegregation: with the end of formal segregation, the black middle
class left traditional ghettos, depriving the remaining residents of critical contacts with
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1187
employers. This small change in neighborhood composition led to a large deterioration in employment conditions through a vicious cycle of increased unemployment and
further weakening of job networks. These dynamics arise endogenously in the model
developed in this paper. Finally, I provide evidence supporting the social isolation theory: the cross-sectional relationship between average human capital and employment
rates across US Census tracts exhibits a non-linearity very similar to that predicted by
the model and by Wilson. While this empirical result does not in itself prove the social
isolation theory is correct, the match between model and data is striking.
Taken as a whole, the model provides new insight into a number of ideas raised in
other social sciences but previously unaddressed in economic theory.
2. Related literature
The theoretical literature in economics on job networking owes much to the contribution of Montgomery (1991). In his two-period model, 7rms lack information on the
productivity of job candidates, but positive correlation in productivity between friends
leads 7rms to oMer higher wages to the friends of productive period-one employees. In
more recent work (CalvNo-Armengol and Zenou, 2002; CalvNo-Armengol and Jackson,
2002), individuals randomly discover job vacancies, and currently employed workers
pass vacancies to unemployed friends. These papers provide additional insight into
the long-run dynamics of job networking that are absent from Montgomery’s simple
two-period model. The model developed here is like that of Montgomery in that networks facilitate the transmission of information on productivity. However the model
is richer in that it is explicitly dynamic and allows for long-run analysis, as well as
analysis of the relationship between network structure and long run outcomes.
As previously mentioned, the empirical literature implies that networking is extremely
common. Its economic importance is somewhat less well established. Networking may
play a critical role in job matching, or it may be merely a slightly cheaper substitute
for formal search, with little impact on the resulting outcomes. Researchers have found
that obtaining a job through networking is associated with higher acceptance rates of
job oMers (Holzer, 1987), higher reported job satisfaction (Granovetter, 1995, p. 13),
and lower quit rates (Datcher, 1983), though not necessarily higher wages (Granovetter,
1995, p. 147). Studies by Ludwig et al. (2000) and Topa (2001) suggest that a person’s social environment has a signi7cant impact on subsequent employment. Taken
as a whole, the evidence suggests that social networks do aMect the eventual assignment of workers to jobs. However, there are substantial selection issues in interpreting
the results these studies, so the economic importance of networking remains an open
empirical question.
3. Description of the model
The model features a population of workers and 7rms. Productivity varies across
worker–7rm matches, and 7rms have limited information on the productivity of a
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
particular match. Exogenous social ties between workers facilitate the transmission
of additional information to 7rms. A 7rm’s current employees provide references, or
reports on the quality of the worker–7rm match, for each of their friends. Social ties
are thus valuable in job search.
3.1. Demography and social network
The economy is populated by a number of in7nitely lived workers, indexed by
i ∈ W = {1; 2; : : : ; N }. The set W can also be countably in7nite, in which case W = Z+ .
There is also a set of 7rms, indexed by f ∈ F.
Workers’ social ties with one another are represented by a time-varying directed
graph or network. Each node in the network represents one worker. The edges in the
network are described by a stochastic process {(t)}. The random variable (t) is an
‘adjacency matrix’, an N × N matrix such that
ij (t) ∈ {0; 1};
(1)
ii (t) = 0:
(2)
Direct connections between nodes in the network represent social ties between workers.
ij (t) equals one if worker i is friends with (can provide a personal reference for) j
and zero otherwise.
The network follows an exogenous stochastic process in which the number of people
that know any individual is bounded:
ij (t) 6 D for all j
for some D ¡ ∞
(3)
i∈W
and the process is Markov:
Pr((t)|(t − 1); (t − 2); : : :) = Pr((t)|(t − 1)):
(4)
Otherwise, no structure is imposed on the network. In particular, it may be deterministic
or random, may be 7xed or changing over time, and may have a 7nite or countably
in7nite set of nodes. 1 The network is directed, so friendships can be one sided. Actual
social networks are unlikely to be simple or unchanging, so it is important to know
the impact of network structure on the model’s properties. This 'exible treatment of
the social network is thus a desirable model feature.
3.2. Workers, $rms, and referrals
Labor is the only factor of production, and the production process exhibits constant
returns to scale. A worker’s productivity is determined by his or her general human
1 Formally, the network is de7ned as the pair (W; (t)), though the rest of the paper often refers informally
to the ‘network’ (t).
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1189
capital as well as a match-speci7c component which diMers across worker-7rm pairs.
Let ai be the worker’s general human capital, which is exogenous and observable. Let
mfi (t) be the quality of the match between worker i and 7rm f, which is exogenous
but not always observable. The worker’s output at the 7rm is
yif (t) = ai mfi (t)nfi (t);
(5)
where nfi (t) is the amount of labor supplied to 7rm f. Each worker has one unit
of labor per period. Although there is no restriction on dividing labor among several
7rms, in equilibrium each agent strictly prefers not to do so.
Match quality is exogenous and stochastic. The quality of a match between a particular worker and 7rm changes over time, as in models of endogenous job destruction
(Mortensen and Pissarides, 1994). The random variable mfi (t) is IID across workers,
7rms, and time, with continuous CDF Fm :
Fm (x) ≡ Pr(mfi (t) 6 x):
(6)
Without loss of generality, mfi is normalized to have a mean of 1 so that E(mfi (t)) = 1
and E(yif (t)=nfi (t)) = ai . In words, the human capital term ai describes the worker’s
average productivity across a variety of jobs, while the match-speci7c term mfi (t) gives
the worker’s relative productivity in a particular job.
Match quality is not generally observable prior to employment. In addition, output
and match quality are not veri7able after employment, so workers and 7rms cannot
write contingent contracts. The 7rm can acquire more information on a worker by either
direct observation (if the worker is already a current employee), or referral (if the
worker has a social tie to a current employee). Only direct social ties lead to referrals,
and the information received is complete and costless. Formally, let Ef (mfi (t)) be the
expected value of mfi (t) based on 7rm f’s information at the beginning of period t.
Then
 f
m (t) if current employee (nfi (t − 1) ¿ 0) or


 i
Ef (mfi (t)) =
(7)
friend of employee (∃j : nfj ¿ 0; ji = 1);



1
otherwise:
The expected productivity of worker i at 7rm f is then ai mfi (t) with a reference, and
ai without a reference. References thus have no impact on the average productivity of
applicants, but enable 7rms to select from the upper tail of the productivity distribution. Firms may hire multiple workers without decreasing returns to scale, but each
multiple-worker 7rm divides into a set of one-worker 7rms at the end of each period.
Otherwise, networking provides a strong tendency for the evolution of the economy
toward a single employer. This assumption is just a convenient shortcut to model the
economic forces that counteract such a tendency.
Workers are risk neutral and maximize current expected income in each period, i.e.,
they do not value future income. When agents discount the future entirely, the only
bene7t from a given match is its current output, and the equilibrium is simple to
characterize. This allows analysis of the behavior of the economy for a wide class of
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
networks. In contrast, if agents do not discount the future, each job match provides
both current output and future matches to the 7rm. The number of these future matches
and their impact on output depends on the exact state of the economy—not only the
aggregate employment state but the state of the social network and the identity of each
employed worker. 2
Instead of market work, a worker can choose instead to engage in home production,
with expected productivity hi . In order for the model to have positive unemployment, I
impose the restriction that the worker’s productivity in home production is higher than
his or her expected productivity at a typical 7rm:
hi ¿ ai :
(8)
Under this condition, and the assumptions about the labor market outlined in the next
section, the unemployment rate is positive and all jobs are obtained through networking.
Section 5.1 adds non-networked job search to the model.
3.3. The labor market
In characterizing the labor market, I look at the wide class of market structures
that generate short-run eQcient assignment, i.e., the allocation of workers to jobs that
maximizes current income. Alternatively, one might make speci7c assumptions about
the wage setting mechanism and characterize the equilibrium allocation. This section
de7nes the eQcient allocation, considers three wage setting mechanisms – competitive
equilibrium, bidding, and bargaining – and demonstrates that all lead to the eQcient
allocation. The short-run eQcient allocation is de7ned as follows:
Denition 3.1 (EQcient allocation): For each i; t, the (short-run) eQcient allocation of
labor {nfi (t)}f∈F solves:




ai Ef (mfi (t))nfi (t) + hi 1 −
nfi (t) ;
(9)
yi (t) = max 
{nfi (t)¿0}
f
f
where Ef (mfi (t)) is described by (7) and taken as given, and
f
nfi (t) 6 1.
Competitive equilibrium: A competitive equilibrium is a set of match-speci7c wages
{wif } and allocations {nfi } such that, taking wages as given, the allocations solve each
7rm’s pro7t-maximization problem
f
f
f
f
wi (t)ni (t) −
ai Ef (mi (t))ni (t)
(10)
max
{nfi (t)¿0}
i
i
2 It may be possible to apply suQcient restrictions on the social network process such that the identity of
each employed worker is not needed to characterize the state, though that is left to future research.
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
and the worker’s income-maximization problem:




f

wi (t)nfi (t) + hi 1 −
nfi (t) :
max
{nfi (t)¿0;
f
nfi (t)61}
f
1191
(11)
f
In order for 7rm f to have 7nite and positive labor demand, it must be that wif =
ai Ef (mfi ). This implies that the worker’s income-maximization problem corresponds
to the output-maximization problem (9), so any competitive equilibrium allocation is
also an eQcient allocation.
Bargaining: There are positive economic (bilateral monopoly) rents associated with
a worker’s highest-productivity match because its expected output can be strictly higher
than any alternative. This is similar to the case in the search and matching literature
(see Mortensen and Pissarides (1999) for a recent survey). In these models, formation
of a match requires costly search, so an existing match has economic rents. With bilateral monopoly rents, price taking is suboptimal behavior and the competitive model
may be inappropriate. The search and matching literature generally models wage setting as a bilateral bargaining problem, speci7cally one whose outcome corresponds to
the axiomatic Nash bargaining solution. The Nash bargaining solution has two basic
properties: (1) a match occurs if it is eQcient (produces a positive surplus over all
alternatives), (2) each participant receives income from a match at least equal to his
or her opportunity cost, and (3) the worker receives an exogenous fraction of the
surplus and the 7rm receives 1 − , where is usually interpreted as the worker’s
bargaining power. Though de7ned axiomatically, Nash bargaining corresponds to the
equilibrium outcome of a wide variety of bargaining games (Mortensen and Pissarides,
1999, p. 1188). Returning to the job networking model, any bilateral bargaining process corresponding to the Nash solution will have each eQcient match occurring, so
the equilibrium allocation will correspond to the eQcient allocation.
Bidding: Alternatively, the worker could take binding oMers (bids) from potential
employers and select among them. This may be a more appropriate model than bargaining if the worker does not observe the 7rm’s information. If a worker can credibly
report oMers made by one 7rm to others and solicit counter-oMers, the labor market
corresponds to a set of independent English (open ascending-price) auctions. If offers are strictly private, the worker cannot solicit counter-oMers and the labor market
corresponds to a set of independent sealed-bid 7rst-price auctions. In either case, a
7rm’s bid is increasing in Ef (mfi ), so the equilibrium allocation will correspond to the
eQcient allocation.
These results indicate that the allocation of workers to 7rms is the same for a variety
of wage setting mechanisms. The wage setting mechanism will only aMect the division
of rents among the worker and 7rm. Given these results, the remainder of the paper
focuses directly on the dynamic behavior of the short-run eQcient allocation.
4. Implications
This section describes the dynamic behavior of the eQcient allocation. To facilitate
this discussion, I introduce some additional notation. Let ni (t) be the employment status
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
of worker i in period t:
1 if nfi (t) ¿ 0 for some f;
ni (t) ≡
0 otherwise
(12)
and let n(t) ≡ (n1 (t); n2 (t); : : : ; nN (t)). For a 7nite number of workers, let n(t)
R
be the
employment rate:
N
1 ni (t):
n(t)
R ≡
N
(13)
i=1
For an in7nite number of workers n(t)
R
must be de7ned as a probability limit.
The state of the economy at time t can be described by the pair ((t); n(t)),
which follows a Markov process. The model’s properties can be fruitfully divided
into short-run (properties of the transition function) and long-run (properties of the
associated stationary distributions). Section 4.1 describes short-run properties, while
Sections 4.2–4.4 describe long-run properties.
4.1. Short-run dynamics
In the eQcient allocation, worker i engages in either home production (ni (t) = 0) or
market production (ni (t) = 1), depending on expected productivity:
1 if ai Ef (mfi (t)) ¿ hi for some f;
(14)
ni (t) =
0 otherwise:
If employed, i works at the 7rm f with the highest Ef (mfi (t)). As Fm is continuous,
there is a unique such 7rm with probability one.
A worker’s probability of employment is a function of his or her human capital,
reservation wage, and number of job contacts. Let ki (t) be the number of worker i’s
current friends who were employed at the end of the previous period:
ji (t)nj (t − 1):
(15)
ki (t) ≡
j∈W
Also let
qi ≡
1 − Fm
hi
ai
(16)
and let q ≡ (q1 ; q2 ; q3 ; : : : ; qN ). If employed in the previous period, worker i will receive
an acceptable oMer (one which exceeds the reservation wage hi ) to stay at that job with
probability qi . In addition, each employed friend will generate an acceptable job oMer
for i with probability qi . The value of qi , which I call the worker’s ‘oMer rate’, summarizes the contribution of an individual’s personal characteristics (reservation wage
and human capital) to his or her employment probability, and can be interpreted as an
index of employability. It is often convenient to work directly with qi rather than its
components.
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1193
The probability that i will be employed in period t is
Pr(ni (t) = 1) = 1 − (1 − qi )ki (t)+ni (t−1) :
(17)
This probability is increasing in the number of employed friends and in the oMer
rate. Output is also increasing (in the sense of 7rst-order stochastic dominance) in the
number of employed friends.
4.2. Long-run dynamics: a simple example with IID networks
Having described the Markov process that governs employment, I next describe
the model’s long-run behavior, i.e., the stationary distribution(s) of that process. This
analysis begins with a particular special case of the model for which the behavior of
employment can be characterized analytically. In this special case, all agents are ex
ante identical and the social network is IID across time periods.
Proposition 4.1 (The employment rate with IID networks). Suppose that qi = q for all
i ∈ W , and that (t) is an IID random draw from the set of r-regular graphs (graphs
such that each node has exactly r edges). Then:
(1) The employment rate in the economy follows the stochastic di:erence equation
Et−1 (n(t))
R
= 1 − (1 − qn(t
R − 1))r+1
(18)
which implies that E0 (n(t))
R
is strictly increasing in q, r and n(0)
R
whenever q ¿ 0
and n(0)
R ¿ 0.
(2) If n(t)
R = 0, then n(s)
R = 0 with probability one for all s ¿ t.
(3) As the number of workers approaches in$nity, then the employment rate converges in probability to a deterministic variable n(t) that obeys the di:erence
equation:
n(t) = 1 − (1 − qn(t − 1))r+1 :
(19)
Let n∗ be the stable steady state of Eq. (19). n∗ is positive if and only if
q ¿ qc ≡
1
:
r+1
(20)
Proof. If worker i is employed in period t − 1, then by Eq. (17), Et−1 (ni (t)) =
1 − (1 − qi )(1 − qi n(t
R − 1))r . If i is unemployed in period t − 1, then Et−1 (ni (t)) =
1 − (1 − qi n(t
R − 1))r . Averaging over all workers yields Eq. (18). Given some initial
condition, Eq. (19) follows directly from Eq. (18) and the Law of Large Numbers. One
can verify by inspection that zero is a steady state of (19). One can verify directly that
the right-hand side of (19) is bounded between zero and one, as well as continuous,
increasing and concave in n(t − 1). It can also be shown that
@(1 − (1 − qn)r+1 ) = q(r + 1):
@n
n=0
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
Fig. 1. Relationship between oMer rate (q) and long-run employment rate for an economy with an IID social
network and 4 contacts/worker. This non-linear pattern appears across a variety of network structures.
By concavity, if this derivative is less than one, then zero is the only steady state and
is stable. If it is greater than one, then (by continuity) there is a positive and stable
steady state.
Although there is no closed form solution for the stable steady state of (19) it
can be found numerically. Fig. 1 shows n∗ as a function of q for the case r = 4.
Proposition 4.1 implies that qc = 0:2, and the 7gure shows the long-run employment
rate is zero when q 6 0:2 and positive when q ¿ 0:2, as implied by Proposition 4.1.
In addition, the long-run employment rate is highly non-linear in q near 0.2. A small
increase or decrease in each worker’s human capital or reservation wage would lead
to a large change in the employment rate. Note that the employment probability of
worker i changes smoothly in qi when the employment status of his or her friends is
held constant. The non-linearity in the model is of particular interest because it arises
only in the aggregate.
The qualitative features of this special case apply much more generally. Section 4.3
provides analytic results, while Section 4.4 uses simulations of a wide variety of model
variations to further characterize the model’s properties.
4.3. Long-run employment: general results
This section describes the long-run behavior of the economy in a more general
setting. The results are similar to those found in the IID networks case. The 7rst result
considers an economy in which the social network can be broken up into cohesive social
groups which do not connect with one another, and 7nds that these subnetworks can
be analyzed independently. This suggests that social groups can be analyzed separately
as a 7rst approximation if there are only a few intergroup connections.
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1195
Proposition 4.2 (Independence of unconnected groups). Suppose that there is some
W1 ⊂ W such that, for any a ∈ W1 , b ∈ W1c , and t, Pr(ab (t) = 1) = 0. Then for
any a ∈ W1 , b ∈ W1c , t, and t , na (t) is independent of nb (t ).
Proof. The network Markov process {(t)}, the set of individual oMer rates q, and
the initial condition n(0) can be used to de7ne an oriented bond percolation process. 3
Construct a percolation graph 4 for any element of the support for {(t)}. By construction there is no path of open edges leading from a to b or from b to a, so changing
the state of b has no eMect on the state of a. Since this is true for all elements of the
sample space, the two are independent.
Proposition 4.3 generalizes part (1) of Proposition 4.1. It indicates that higher initial
employment, a higher oMer rate, or a denser social network all increase the probability
of employment over any time horizon.
Proposition 4.3 (Monotonicity of employment). For all i and t ¿ 0,
Pr(ni (t) = 1|q; {(t)}; n(0))
(21)
is weakly increasing in all three arguments, i.e.,
(1) q ¿ q implies
Pr(ni (t) = 1|q; {(t)}; n(0)) ¿ Pr(ni (t) = 1|q ; {(t)}; n(0)):
(2) {(t)} ¿ {(t)} implies
Pr(ni (t) = 1|q; {(t)}; n(0)) ¿ Pr(ni (t) = 1|q; {(t)} ; n(0)):
(3) n(0) ¿ n(0) implies
Pr(ni (t) = 1|q; {(t)}; n(0)) ¿ Pr(ni (t) = 1|q; {(t)}; n(0) ):
Proof. First, I show that the probability of employment is increasing in q. Create two
identical percolation graphs from an arbitrary element of the support of {(t)}. Assign
a random number z drawn independently from the standard uniform distribution to each
edge in the 7rst graph, and assign the same number to the corresponding edge in the
second graph. In the 7rst graph, mark each edge as open if z ¡ qi , closed otherwise. In
3 As a technical aside, several of the proofs in this section use mathematical tools related to percolation
theory. See Grimmett (1989) for an introduction to percolation. An understanding of percolation is not
necessary to understand the substance of any propositions.
4 Percolation graphs are constructed from the original model as follows. The node set of the percolation
graph is W × Z, and a node is indexed by i; t. For each i add edges from i; t − 1 to i; t. For each edge
between i and j in (t), add a corresponding edge from i; t − 1 to j; t in the percolation graph. Finally,
designate each edge to worker i as ‘open’ with probability qi and ‘closed’ otherwise. Worker i is employed
in period t (in the language of percolation, node i; t is ‘wet’) if and only if there is a path of open edges
from a period-zero employed worker to i; t. The most useful tool for the purposes of this paper is coupling,
which is simply the construction of two or more stochastic processes on the same probability space for the
purpose of comparing probabilities.
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
the second graph, mark all edges as open if z ¡ qi , closed otherwise. By construction,
these two random graphs represent the two percolation processes. Because q ¿ q , every
edge that is open in the second graph is also open in the 7rst graph. If node i; t is wet
in the second graph, it must also be wet in the 7rst graph. Therefore, the probability
that a given node will be wet in the 7rst graph must be greater than or equal to the
probability that it will be wet in the second graph. The probability of a given worker
being employed at a given point in time is therefore (weakly) increasing in q. The
same argument, with the obvious substitutions, can be used for {(t)} and n(0).
Proposition 4.4 generalizes part (2) of Proposition 4.1. It demonstrates that if the
economy (or any isolated subnetwork) is ever in a state where no agent is employed,
it will stay there permanently. In addition, any 7nite economy or isolated subnetwork
will eventually become trapped in the zero-employment state. Although Proposition 4.4
indicates that all 7nite economies converge to zero employment, simulation results in
Section 4.4 indicate that, over reasonable time horizons, only very small economies are
likely to become trapped in the zero-employment state. Instead, the employment rate
in 7nite economies tends to 'uctuate around an apparent long-run average for a very
long time.
Proposition 4.4 (Zero-employment absorbing state). For any q and {(t)}:
Pr(ni (t) = 0|q; {(t)}; n(0) = 0) = 1;
t ¿ 0:
(22)
If W is $nite and q ¡ 1, then for any initial condition n(0)
lim Pr(ni (t) = 0|q; {(t)}; n(0)) = 1:
(23)
t→∞
Proof. Eq. (22) follows directly from Eq. (17). To prove (23), note that Eq. (17)
2
implies that Pr(n(t) = 0|n(t − 1)) ¿ (1 − q)N ¿ 0. Since the zero-employment state
is an absorbing state and there is a positive probability of reaching it from any other
state, (23) follows.
Proposition 4.5 generalizes part (3) of Proposition 4.1. It shows that a large (in7nite) economy does not necessarily converge to the zero-employment state. Instead,
the long-run behavior depends on q being above some network-speci7c critical value.
The simulations in Section 4.4 show that the long-run employment rate exhibits highly
non-linear behavior near the critical value.
Proposition 4.5 (Existence of critical value). For any {(t)} there exists qc ∈ (0; 1]
such that
lim Pr(ni (t) = 1|q; ; n(0) = 1) = 0
if q ¡ qc
lim Pr(ni (t) = 1|q; ; n(0) = 1) ¿ 0
if q ¿ qc :
t→∞
and
t→∞
(24)
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1197
In addition, if W is countably in$nite and there is a connected network over W
such that Pr((t) ¿ ) = 1 for all t, then qc ∈ (0; 1).
Proof. The probability of an open path of length t is less than or equal to the expected
number of such paths, as the number of open paths is no less than one for any element
of the sample space in which there is an open path. By (3), the number of possible
open paths of length t is less than or equal to Dt . If edges are open with probability q,
then the expected number of open paths of length t leading to i is less than or equal
to Dt ∗ qt . Therefore if q ¡ 1=D, the probability of an open path leading to i; t goes
to zero as t goes to in7nity. This implies that (24) holds for qc ¿ 1=D ¿ 0. Next, we
prove that, under the assumptions that the set of workers is in7nite and the network is
connected, qc ¡ 1. If is the one line (i.e., each i is connected to i + 1 and no one
else), then Durrett (1984) shows that qc ¡ 1, so it remains to show that the one line
is a subgraph of . Now suppose that is an arbitrary connected graph with no more
than D connections into and out of each node. Pick a node i. Let i (j) be the distance
of the shortest path (sequence of nodes connected by edges in the graph) from node i
to some node j. Since is connected, i (j) is always 7nite. Let Sn be the set of all
points j such that i (j) 6 n. By earlier assumption there are no more than D edges
to or from any node, so the size of Sn is always no more than Dn , and thus always
7nite. Therefore, for any n there is always a node j such that i (j) ¿ n. Clearly, the
shortest path does not go through the same point twice, or else one can construct a
shorter path. We can use the same argument to show that, for any n, there is a node
k such that k (i) ¿ n and that the shortest path from k to i does not cross the shortest
path from i to j. We can thus construct a subgraph which has an in7nite number of
points on either side of i and is equivalent to a one line (with a relabeling of nodes).
Durrett’s result shows that qc ¡ 1 for this subgraph. Proposition 4.3 implies that qc ¡ 1
for the original graph as well.
4.4. Numerical results
This section describes a set of simulation experiments which further characterize
the behavior of the long-run employment rate in the model. The simulations generally
follow 1000 agents over many time periods, with an initial condition of full employment
(n(0) = 1). The typical simulation features homogeneous agents (qi = q) and a 7xed
number r of connections per worker arranged in a simple nearest-neighbor loop. An
r-nearest-neighbor loop connects agent i to r nearest neighbors (for example, a two-loop
connects i to i − 1 and i + 1) and wraps around so that agent N and agent 1 are
‘neighbors’. Fig. 2 shows the 7rst 100 periods of a representative simulation run with
q = 0:35 and r = 3. As the 7gure shows, the employment rate quickly settles down to
'uctuations around some apparent long-run average. This behavior is typical.
Although Proposition 4.4 implies that any 7nite economy eventually reaches zero
employment permanently, simulation results like those in Fig. 2 suggest that a large
economy can have a stable and positive employment rate for many periods. To gain a
more systematic understanding of this model property, I perform a simple simulation
experiment. The parameterization of the model used to generate Fig. 2 is simulated
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
Fig. 2. The time series of the employment rate in a representative simulation run. Network is a loop with
1000 workers, 3 contacts/worker and oMer rate of 0.35.
Table 1
Number of periods before zero-employment state reached
No. agents
in network
10
100
1,000
Periods elapsed before nRt = 0
Min
Max
Median
% ¿1 m
7
741,592
¿1 m
1028
¿1 m
¿1 m
134
¿1 m
¿1 m
0
99
100
Network is loop with r = 3, q = 0:35.
for 1 million periods, and the period in which employment 7rst reaches zero is noted.
The simulation is then repeated 100 times to get a distribution of the 7rst passage
time to zero employment, then the entire experiment is repeated for a diMerent number
of agents. The results, summarized in Table 1, indicate that the economy becomes
trapped in the zero-employment state very quickly if the number of connected agents
is quite small (10) but has a positive employment rate for a very long time even if
the number of connected agents is moderate (100). This experiment has been repeated
with various speci7cations of the network and oMer rate, and the results are similar.
Only extremely isolated social groups or social groups with low oMer rates tend to
approach zero employment.
Given these results, additional simulation experiments are used to characterize the
relationship between the oMer rate and long-run employment. These simulations are
run for 2000 periods and the overall employment rate is averaged over the last 500
time periods to get the ‘long-run employment rate’. Fig. 3 summarizes the results of a
baseline experiment which features simple loop networks and homogeneous agents. As
the 7gure shows, the relationship between the oMer rate and the long run employment
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1199
Fig. 3. Relationship between oMer rate and long-run average employment rate for several diMerent loop
networks. Number of contacts per worker (r) indicated on graph.
rate exhibits many of the patterns described in Section 4.2. Long-run employment is
zero if q is below some critical value and the long-run employment rate is highly
non-linear in q near that critical value. Additional simulation experiments 5 featuring a
heterogeneous distribution of qi across individuals, variations in the number of agents,
and a variety of alternative network structures yield similar results.
5. Applications and extensions
5.1. Incorporating non-networked job search
The basic model presented in previous sections is highly stylized. In particular,
the assumptions guarantee that all jobs are obtained through networking and none
are obtained through formal search. However, empirical research indicates that about
half of jobs are obtained through networking and the other half are obtained through
more formal methods. This section adds a formal job search component to the model.
Because there is already a vast literature on formal search, the modeling here is very
basic.
In addition to those matches gained through networking, each worker is matched
up with !i (t) 7rms, each of which gets to observe mfi . The exact number of 7rms
is stochastic, with probability distribution S. Therefore the probability of receiving an
acceptable job oMer through formal search is
!
hi
1 − Fm
si ≡
S(!)
(25)
ai
!
5
The GAUSS code to run these experiments is available from the author on request.
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
and the overall probability of employment is
Pr(ni (t) = 1) = 1 − (1 − si )(1 − qi )ki (t)+ni (t−1) :
(26)
Rather than specifying S, the remainder of the paper treats si as primitive.
When si ¿ 0, many but not all of the previous results remain. In particular, Propositions 4.2 and 4.3 continue to hold but Propositions 4.4 and 4.5 do not. Furthermore, the
relationship between the oMer rate and the employment rate remains highly non-linear,
with an appearance similar to that in Fig. 3. Fig. 5 shows an example, the details of
which are described in Section 5.3.
5.2. The strength of weak ties
The modeling approach in this paper allows for a wide class of social networks. This
'exibility provides generality and allows for the consideration of the relationship between network structure and long-run outcomes. One particular hypothesis that appears
frequently in sociology is the ‘strength of weak ties’:
[There is] a structural tendency for those to whom one is only weakly tied to have
better access to job information one does not already have. Acquaintances, as compared
to close friends, are more prone to move in diMerent circles than one’s self. Those to
whom one is closest are likely to have the greatest overlap in contact with those one
already knows, so that the information to which they are privy is likely to be much
the same as that which one already has (Granovetter, 1995, pp. 52–53).
This section demonstrates that the model in this paper exhibits an aggregate strengthof-weak-ties property. All else equal, the proportion of a group’s social ties that are
weak is positively associated with its long-run employment rate.
In the context of this model, a social tie from agent i to agent j is de7ned as
strong if j has a social tie to one or more of i’s other friends, and weak otherwise.
This de7nition corresponds to the key features of weak and strong ties described in
the above quote. A systematic way of constructing a family of networks with a given
proportion of weak ties is given by Watts and Strogatz (1998). Their algorithm starts
with a loop network with r ¿ 2 social ties per agent. Such a network has only strong
ties, as each agent’s friends are also socially tied to another of his or her friends.
Then, for some p ∈ [0; 1], each edge is switched to a diMerent (randomly selected)
destination node with probability p. If the number of agents is large, the probability
that a randomly generated tie is a strong tie is close to zero. The result is a family of
random networks, indexed by p, in which the fraction of weak ties is approximately p.
For various values of p, the model is simulated with 1000 agents for 2000 periods, and
the average employment over the last 500 is taken to give the long-run employment
rate. Because the networks are random, the exercise is repeated 10 times for each value
of p to average over the distribution of networks with a fraction p of weak ties.
Fig. 4 shows the results from a representative simulation with q = 0:3 and r = 3. The
horizontal axis is the fraction of weak ties (p) and the vertical axis is the long-run
average employment rate. The endpoints of the curve correspond to the loop network
(p = 0) and a fully random regular network (p = 1). As the 7gure shows, the long-run
employment rate is increasing in the proportion of weak ties. Simulations with other
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1201
Fig. 4. The strength of weak ties, i.e., the relationship between fraction of weak ties and long run employment.
Parameter values are r = 3, q = 0:3.
values of q and r indicate this is a general property of the model as long as q is
suQciently high for the long-run employment rate to be positive. An increase in the
relative prevalence of weak ties has a similar eMect to that of a moderate increase in
the oMer rate.
The intuition behind this result is that weak ties are a way of diversifying one’s
social resources. Eq. (17) implies that an individual faces a form of decreasing returns
to the number of employed friends. When a person’s friends are also friends with
one another, this increases variance in the number of employed friends by increasing
the correlation in employment status between the person’s friends. In the aggregate, a
network with a higher proportion of weak ties reduces inequality in the distribution of
employed friends, leading to a higher overall employment rate.
5.3. Social networks and neighborhoods
In two in'uential books, Wilson (1987, 1996) identi7es the departure of the black
middle class as the key shock to inner-city communities which led to the gradual ‘disappearance’ of regular employment in these communities, and eventually an increase
in various social pathologies. In his explanation this seemingly small shock to neighborhood composition had a disproportionately large eMect on community employment
rates because of the self-reinforcing process of employment decline, job network deterioration, and further employment decline. If social networks are primarily determined
by neighborhoods, the model presented in this paper exhibits this property. This section
explores this idea in more detail.
There is some empirical support for Wilson’s hypothesis. Holzer (1987) 7nds that
black youth in the NLSY have a lower success rate than white youth in obtaining
job oMers through networking, despite a similar rate of success in more formal means.
Krauth (2000) 7nds strong evidence that the relationship between neighborhood
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B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
Fig. 5. Scatter plot of employment rate versus percentage of adults with university (Bachelor) degrees,
across Chicago CMSA census tracts. Dark line is non-parametric regression (supersmoother), lighter line is
job networking model calibrated to match.
composition and neighborhood employment is highly non-linear. A representative example of the results in that paper appears in Fig. 5. The scatter plot is of Chicago
Census tract-level data from 1990. The horizontal axis is the fraction of tract residents
with college degrees and the vertical axis is the employment rate in the tract. The
thick line is a non-parametric regression line. It indicates that, as Wilson suggests, as
the fraction of workers in a community with high human capital falls below a critical
level the predicted employment rate falls dramatically. Away from this critical level
the relationship is much weaker. Krauth (2000) repeats this exercise for the 20 largest
US cities and 7nds that the pattern is similar in all 20 cities. Although these results
may be consistent with alternative models, they are certainly consistent with Wilson’s
hypothesis.
Figs. 1 and 3 suggest that the model here also implies a non-linear relationship
between average human capital and employment rates. However, the parameterizations
used to generate those 7gures may not be realistic, which raises the question of how
closely a reasonable parameterization can match the empirical relationship in Fig. 5. I
address this question by ‘calibrating’ the model, though the microeconomic evidence
on parameter values may be too weak to merit the term. Because empirical results
indicate that half of jobs are obtained through formal means and half are obtained
through networking, the model extension described in Section 5.1 is used. The success
rate in formal search is set to give equal probability
of obtaining a job through formal
√
and informal means, implying that s = 1 − 1 − n.
R With a 1990 employment rate
in Chicago of approximately 90%, this implies that s = 0:68. The social network is
simple in structure: each period an individual has r randomly generated social ties to
other members of the community. There is very limited and controversial evidence on a
reasonable value for r, in part because there is no clear dividing line between friend and
stranger. Glaeser (2000, p. 128) estimates that the average resident of a large city has 5
B.V. Krauth / Journal of Economic Dynamics & Control 28 (2004) 1185 – 1204
1203
close friends. A school of thought in evolutionary psychology associated with Dunbar
(1996) argues that human brains can ‘know’ approximately 150 other individuals at
any one time. Knowledge of a person in this sense means suQcient knowledge of
a person’s past behavior to predict future behavior or enforce social norms against
predation. The number of social ties which are potentially useful in job search likely
falls between these two extremes, so r is set to 50. There are two types of agents, those
with college degrees (qi =qH ) and those without (qi =qL ). The fraction of workers with
college degrees varies across communities. The rate of success in networking for each
type of worker is set at (qH = 0:2) and (qL = 0:0051) to produce a close match to Fig.
5. The model is then simulated with the percentage of workers with a college degree
varying from 0 to 100. The results are depicted in the thinner line in Fig. 5. As the
7gure shows, the model can generate a relationship between neighborhood composition
and neighborhood employment which is quite similar to both the observed relationship
and to Wilson’s hypothesis.
6. Conclusion and further directions
This paper has developed a model in which personal connections help transmit important information about the quality of a worker-job match. The resulting dynamics
of the model have several interesting features. Long run employment is aMected by the
number of ties in the social network and the proportion of ties that are weak, as well
as individual characteristics such as human capital and reservation wages. The relationship between any of these factors and long-run employment is highly non-linear. Both
the non-linearity of employment in group characteristics and the value of weak ties
are commonly discussed sociological theories which are given more detailed economic
foundations here.
In many ways, the model presented here is just an early step toward understanding
the aggregate behavior of labor markets with job networking. Although simplifying assumptions have been made explicitly with the view that there is an unavoidable tradeoM
between the complexity of a model’s agents and the complexity of their interactions,
a full understanding of the issues here requires a model with more intelligent agents.
Desirable extensions include incorporating forward-looking behavior and risk aversion,
including a human capital accumulation decision, and allowing the network itself to
respond to economic incentives.
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