A dynamic model of job networking and social influences on employment

Journal of Economic Dynamics & Control 28 (2004) 1185–1204www.elsevier.com/locate/econbase

A dynamic model of job networking and socialin'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 themodel, a 7rm receives information on the productivity of those job applicants with social ties toits current employees. In addition to providing a theory of networking, the model endogenouslygenerates two classic theories in economic sociology. First, there is a highly non-linear relation-ship between average human capital in a group of socially connected individuals and the group’semployment 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 poorAfrican-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 networksplay an important role in job search. Bewley (1999) lists 24 studies between 1932 and1990 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, LorneCarmichael, Kim-Sau Chung, Steven Durlauf, and Jonathan Parker, as well as workshop participants atUniversity 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

mailto:[email protected]

1186 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 remainsrelatively underdeveloped. This paper analyzes a model economy in which networkingarises because 7rms have limited information on the skills of job applicants. A 7rm’scurrent employees provide information on the job-speci7c skills of their friends, thusimproving the likelihood of a productive match. As a result of these information 'ows,a job searcher’s employment probability and expected productivity are increasing inthe 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 withinsocial groups (groups of individuals who share social ties, potentially de7ned by neigh-borhood, ethnicity, or other characteristics). First, the long-run employment rate of anindividual 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 in-creasing in his or her own initial employment status, social ties, and human capital. Asecond set of results characterizes the group-level relationship between human capitaland long-run employment rates. There exists a critical level of human capital such thatno group member is employed in the long run if all members are below that level, butthe 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 groupcomposition lead to large changes in employment. The third set of results describes therelationship between the social network itself and the long-run behavior of the employ-ment rate. First, small groups without ties to the rest of society are at risk of fallinginto a zero-employment trap. Second, the structure of the social ties within the groupmatter. All else being equal, employment is increasing in the proportion of social tiesthat are ‘weak’ ties.

In addition to being of independent interest, these results formalize in an economicmodel two classic sociological theories: Granovetter’s (1973) ‘strength of weak ties’hypothesis and Wilson’s (1987, 1996) ‘social isolation’ theory of inner-city unemploy-ment. The strength of weak ties hypothesis holds that acquaintances are more valuablein job search than close friends and family, because they provide a more diversi7edsource of information. There is a sizable literature on sociology on the strength of weakties. Montgomery (1992) uses a formal model to analyze its empirical implications, butassumes rather than derives a bene7t to weak ties. In contrast, an aggregate version ofthe 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 rolein explaining high unemployment rates among low income African-Americans. Likethe strength of weak ties, it has a long history in social science. In an early contri-bution, Kain (1968) argues that racial segregation leads to poorer job prospects forAfrican-Americans, in part through their isolation from the job network. Almost threedecades later, Wilson (1996) argues that growing isolation from job networks explainsthe signi7cant decline in employment in low income African-American communitiesfrom the late 1960s to mid-1990s, and that this decline explains much of their con-temporaneous growth in social problems. Wilson suggests that this is an unintendedconsequence of desegregation: with the end of formal segregation, the black middleclass 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 deteriora-tion in employment conditions through a vicious cycle of increased unemployment andfurther weakening of job networks. These dynamics arise endogenously in the modeldeveloped in this paper. Finally, I provide evidence supporting the social isolation the-ory: the cross-sectional relationship between average human capital and employmentrates across US Census tracts exhibits a non-linearity very similar to that predicted bythe model and by Wilson. While this empirical result does not in itself prove the socialisolation 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 inother social sciences but previously unaddressed in economic theory.

2. Related literature

The theoretical literature in economics on job networking owes much to the contri-bution of Montgomery (1991). In his two-period model, 7rms lack information on theproductivity of job candidates, but positive correlation in productivity between friendsleads 7rms to oMer higher wages to the friends of productive period-one employees. Inmore recent work (CalvNo-Armengol and Zenou, 2002; CalvNo-Armengol and Jackson,2002), individuals randomly discover job vacancies, and currently employed workerspass vacancies to unemployed friends. These papers provide additional insight intothe long-run dynamics of job networking that are absent from Montgomery’s simpletwo-period model. The model developed here is like that of Montgomery in that net-works facilitate the transmission of information on productivity. However the modelis richer in that it is explicitly dynamic and allows for long-run analysis, as well asanalysis of the relationship between network structure and long run outcomes.

As previously mentioned, the empirical literature implies that networking is extremelycommon. Its economic importance is somewhat less well established. Networking mayplay a critical role in job matching, or it may be merely a slightly cheaper substitutefor formal search, with little impact on the resulting outcomes. Researchers have foundthat obtaining a job through networking is associated with higher acceptance rates ofjob 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 per-son’s social environment has a signi7cant impact on subsequent employment. Takenas a whole, the evidence suggests that social networks do aMect the eventual assign-ment of workers to jobs. However, there are substantial selection issues in interpretingthe results these studies, so the economic importance of networking remains an openempirical question.

3. Description of the model

The model features a population of workers and 7rms. Productivity varies acrossworker–7rm matches, and 7rms have limited information on the productivity of a


particular match. Exogenous social ties between workers facilitate the transmissionof additional information to 7rms. A 7rm’s current employees provide references, orreports on the quality of the worker–7rm match, for each of their friends. Social tiesare thus valuable in job search.

3.1. Demography and social network

The economy is populated by a number of in7nitely lived workers, indexed byi∈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 directedgraph or network. Each node in the network represents one worker. The edges in thenetwork 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) jand zero otherwise.

The network follows an exogenous stochastic process in which the number of peoplethat know any individual is bounded:(∑

i∈W

ij(t)6D for all j

)for some D¡∞ (3)

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 deterministicor random, may be 7xed or changing over time, and may have a 7nite or countablyin7nite set of nodes. 1 The network is directed, so friendships can be one sided. Actualsocial networks are unlikely to be simple or unchanging, so it is important to knowthe impact of network structure on the model’s properties. This 'exible treatment ofthe 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 constantreturns 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 informallyto the ‘network’ (t).


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. Letmf

i (t) be the quality of the match between worker i and 7rm f, which is exogenousbut not always observable. The worker’s output at the 7rm is

yfi (t) = aim

fi (t)nfi (t); (5)

where nfi (t) is the amount of labor supplied to 7rm f. Each worker has one unitof labor per period. Although there is no restriction on dividing labor among several7rms, in equilibrium each agent strictly prefers not to do so.

Match quality is exogenous and stochastic. The quality of a match between a partic-ular worker and 7rm changes over time, as in models of endogenous job destruction(Mortensen and Pissarides, 1994). The random variable mf

i (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(mf

i (t))=1and E(yf

i (t)=nfi (t)) = ai. In words, the human capital term ai describes the worker’saverage productivity across a variety of jobs, while the match-speci7c term mf

i (t) givesthe worker’s relative productivity in a particular job.

Match quality is not generally observable prior to employment. In addition, outputand match quality are not veri7able after employment, so workers and 7rms cannotwrite contingent contracts. The 7rm can acquire more information on a worker by eitherdirect observation (if the worker is already a current employee), or referral (if theworker 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(mf

i (t)) be theexpected value of mf

i (t) based on 7rm f’s information at the beginning of period t.Then

Ef(mfi (t)) =

mfi (t) if current employee (nfi (t − 1)¿ 0) or

friend of employee (∃j : nfj ¿ 0; ji = 1);

1 otherwise:

(7)

The expected productivity of worker i at 7rm f is then aimfi (t) with a reference, and

ai without a reference. References thus have no impact on the average productivity ofapplicants, but enable 7rms to select from the upper tail of the productivity distribu-tion. Firms may hire multiple workers without decreasing returns to scale, but eachmultiple-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 economytoward a single employer. This assumption is just a convenient shortcut to model theeconomic 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 onlybene7t from a given match is its current output, and the equilibrium is simple tocharacterize. This allows analysis of the behavior of the economy for a wide class of


networks. In contrast, if agents do not discount the future, each job match providesboth current output and future matches to the 7rm. The number of these future matchesand their impact on output depends on the exact state of the economy—not only theaggregate employment state but the state of the social network and the identity of eachemployed 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, Iimpose the restriction that the worker’s productivity in home production is higher thanhis or her expected productivity at a typical 7rm:

hi ¿ai: (8)

Under this condition, and the assumptions about the labor market outlined in the nextsection, 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 structuresthat generate short-run eQcient assignment, i.e., the allocation of workers to jobs thatmaximizes current income. Alternatively, one might make speci7c assumptions aboutthe wage setting mechanism and characterize the equilibrium allocation. This sectionde7nes the eQcient allocation, considers three wage setting mechanisms – competitiveequilibrium, bidding, and bargaining – and demonstrates that all lead to the eQcientallocation. The short-run eQcient allocation is de7ned as follows:

De�nition 3.1 (EQcient allocation): For each i; t, the (short-run) eQcient allocation oflabor {nfi (t)}f∈F solves:

yi(t) = max{nfi (t)¿0}

∑

f

aiEf(mfi (t))nfi (t)

+ hi

1 −

∑f

nfi (t)

; (9)

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{wf

i } and allocations {nfi } such that, taking wages as given, the allocations solve each7rm’s pro7t-maximization problem

max{nfi (t)¿0}

(∑i

wfi (t)nfi (t)

)−(∑

i

aiEf(mfi (t))nfi (t)

)(10)

2 It may be possible to apply suQcient restrictions on the social network process such that the identity ofeach employed worker is not needed to characterize the state, though that is left to future research.


and the worker’s income-maximization problem:

max{nfi (t)¿0;

∑f nfi (t)61}

∑

f

wfi (t)nfi (t)

+ hi

1 −

∑f

nfi (t)

: (11)

In order for 7rm f to have 7nite and positive labor demand, it must be that wfi =

aiEf(mfi ). This implies that the worker’s income-maximization problem corresponds

to the output-maximization problem (9), so any competitive equilibrium allocation isalso 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 higherthan 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, formationof a match requires costly search, so an existing match has economic rents. With bi-lateral monopoly rents, price taking is suboptimal behavior and the competitive modelmay be inappropriate. The search and matching literature generally models wage set-ting as a bilateral bargaining problem, speci7cally one whose outcome corresponds tothe axiomatic Nash bargaining solution. The Nash bargaining solution has two basicproperties: (1) a match occurs if it is eQcient (produces a positive surplus over allalternatives), (2) each participant receives income from a match at least equal to hisor her opportunity cost, and (3) the worker receives an exogenous fraction � of thesurplus and the 7rm receives 1 − �, where � is usually interpreted as the worker’sbargaining power. Though de7ned axiomatically, Nash bargaining corresponds to theequilibrium outcome of a wide variety of bargaining games (Mortensen and Pissarides,1999, p. 1188). Returning to the job networking model, any bilateral bargaining pro-cess corresponding to the Nash solution will have each eQcient match occurring, sothe 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 bar-gaining if the worker does not observe the 7rm’s information. If a worker can crediblyreport oMers made by one 7rm to others and solicit counter-oMers, the labor marketcorresponds to a set of independent English (open ascending-price) auctions. If of-fers are strictly private, the worker cannot solicit counter-oMers and the labor marketcorresponds to a set of independent sealed-bid 7rst-price auctions. In either case, a7rm’s bid is increasing in Ef(mf

i ), so the equilibrium allocation will correspond to theeQcient allocation.

These results indicate that the allocation of workers to 7rms is the same for a varietyof wage setting mechanisms. The wage setting mechanism will only aMect the divisionof rents among the worker and 7rm. Given these results, the remainder of the paperfocuses directly on the dynamic behavior of the short-run eQcient allocation.

4. Implications

This section describes the dynamic behavior of the eQcient allocation. To facilitatethis discussion, I introduce some additional notation. Let ni(t) be the employment status


of worker i in period t:

ni(t) ≡{

1 if nfi (t)¿ 0 for some f;

0 otherwise(12)

and let n(t) ≡ (n1(t); n2(t); : : : ; nN (t)). For a 7nite number of workers, let Rn(t) be theemployment rate:

Rn(t) ≡ 1N

N∑i=1

ni(t): (13)

For an in7nite number of workers Rn(t) 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 dividedinto short-run (properties of the transition function) and long-run (properties of theassociated stationary distributions). Section 4.1 describes short-run properties, whileSections 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) ormarket production (ni(t) = 1), depending on expected productivity:

ni(t) =

{1 if aiEf(mf

i (t))¿hi for some f;

0 otherwise:(14)

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’scurrent friends who were employed at the end of the previous period:

ki(t) ≡∑j∈W

ji(t)nj(t − 1): (15)

Also let

qi ≡(

1 − Fm

(hiai

))(16)

and let q ≡ (q1; q2; q3; : : : ; qN ). If employed in the previous period, worker i will receivean acceptable oMer (one which exceeds the reservation wage hi) to stay at that job withprobability qi. In addition, each employed friend will generate an acceptable job oMerfor i with probability qi. The value of qi, which I call the worker’s ‘oMer rate’, sum-marizes the contribution of an individual’s personal characteristics (reservation wageand human capital) to his or her employment probability, and can be interpreted as anindex of employability. It is often convenient to work directly with qi rather than itscomponents.


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 oMerrate. Output is also increasing (in the sense of 7rst-order stochastic dominance) in thenumber of employed friends.

4.2. Long-run dynamics: a simple example with IID networks

Having described the Markov process that governs employment, I next describethe model’s long-run behavior, i.e., the stationary distribution(s) of that process. Thisanalysis begins with a particular special case of the model for which the behavior ofemployment can be characterized analytically. In this special case, all agents are exante identical and the social network is IID across time periods.

Proposition 4.1 (The employment rate with IID networks). Suppose that qi = q for alli∈W , and that (t) is an IID random draw from the set of r-regular graphs (graphssuch that each node has exactly r edges). Then:

(1) The employment rate in the economy follows the stochastic di:erence equation

Et−1( Rn(t)) = 1 − (1 − q Rn(t − 1))r+1 (18)

which implies that E0( Rn(t)) is strictly increasing in q, r and Rn(0) whenever q¿ 0and Rn(0)¿ 0.

(2) If Rn(t) = 0, then Rn(s) = 0 with probability one for all s¿ t.(3) As the number of workers approaches in$nity, then the employment rate con-

verges in probability to a deterministic variable n(t) that obeys the di:erenceequation:

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 ≡ 1r + 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 Rn(t − 1))r . If i is unemployed in period t − 1, then Et−1(ni(t)) =1 − (1 − qi Rn(t − 1))r . Averaging over all workers yields Eq. (18). Given some initialcondition, Eq. (19) follows directly from Eq. (18) and the Law of Large Numbers. Onecan verify by inspection that zero is a steady state of (19). One can verify directly thatthe 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)@n

∣∣∣∣n=0

= q(r + 1):


Fig. 1. Relationship between oMer rate (q) and long-run employment rate for an economy with an IID socialnetwork 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 andis stable. If it is greater than one, then (by continuity) there is a positive and stablesteady state.

Although there is no closed form solution for the stable steady state of (19) itcan 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 employmentrate is zero when q6 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 smallincrease or decrease in each worker’s human capital or reservation wage would leadto a large change in the employment rate. Note that the employment probability ofworker i changes smoothly in qi when the employment status of his or her friends isheld constant. The non-linearity in the model is of particular interest because it arisesonly in the aggregate.

The qualitative features of this special case apply much more generally. Section 4.3provides analytic results, while Section 4.4 uses simulations of a wide variety of modelvariations 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 generalsetting. The results are similar to those found in the IID networks case. The 7rst resultconsiders an economy in which the social network can be broken up into cohesive socialgroups which do not connect with one another, and 7nds that these subnetworks canbe analyzed independently. This suggests that social groups can be analyzed separatelyas a 7rst approximation if there are only a few intergroup connections.


Proposition 4.2 (Independence of unconnected groups). Suppose that there is someW1 ⊂ W such that, for any a∈W1, b∈W c

1 , and t, Pr(ab(t) = 1) = 0. Then forany a∈W1, b∈W c

1 , t, and t′, na(t) is independent of nb(t′).

Proof. The network Markov process {(t)}, the set of individual oMer rates q, andthe 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 construc-tion there is no path of open edges leading from a to b or from b to a, so changingthe state of b has no eMect on the state of a. Since this is true for all elements of thesample space, the two are independent.

Proposition 4.3 generalizes part (1) of Proposition 4.1. It indicates that higher initialemployment, a higher oMer rate, or a denser social network all increase the probabilityof 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 twoidentical percolation graphs from an arbitrary element of the support of {(t)}. Assigna random number z drawn independently from the standard uniform distribution to eachedge in the 7rst graph, and assign the same number to the corresponding edge in thesecond 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 percolationtheory. See Grimmett (1989) for an introduction to percolation. An understanding of percolation is notnecessary to understand the substance of any propositions.

4 Percolation graphs are constructed from the original model as follows. The node set of the percolationgraph 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 edgebetween 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 employedin period t (in the language of percolation, node i; t is ‘wet’) if and only if there is a path of open edgesfrom 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 thepurpose of comparing probabilities.


the second graph, mark all edges as open if z¡q′i , closed otherwise. By construction,these two random graphs represent the two percolation processes. Because q¿ q′, everyedge that is open in the second graph is also open in the 7rst graph. If node i; t is wetin the second graph, it must also be wet in the 7rst graph. Therefore, the probabilitythat a given node will be wet in the 7rst graph must be greater than or equal to theprobability that it will be wet in the second graph. The probability of a given workerbeing employed at a given point in time is therefore (weakly) increasing in q. Thesame 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 theeconomy (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 subnetworkwill eventually become trapped in the zero-employment state. Although Proposition 4.4indicates that all 7nite economies converge to zero employment, simulation results inSection 4.4 indicate that, over reasonable time horizons, only very small economies arelikely to become trapped in the zero-employment state. Instead, the employment ratein 7nite economies tends to 'uctuate around an apparent long-run average for a verylong 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)

limt→∞ Pr(ni(t) = 0|q; {(t)}; n(0)) = 1: (23)

Proof. Eq. (22) follows directly from Eq. (17). To prove (23), note that Eq. (17)implies that Pr(n(t) = 0|n(t − 1))¿ (1 − q)N

2¿ 0. Since the zero-employment state

is an absorbing state and there is a positive probability of reaching it from any otherstate, (23) follows.

Proposition 4.5 generalizes part (3) of Proposition 4.1. It shows that a large (in-7nite) 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 highlynon-linear behavior near the critical value.

Proposition 4.5 (Existence of critical value). For any {(t)} there exists qc ∈ (0; 1]such that

limt→∞ Pr(ni(t) = 1|q; ; n(0) = 1) = 0 if q¡qc

and

limt→∞ Pr(ni(t) = 1|q; ; n(0) = 1)¿ 0 if q¿qc: (24)


In addition, if W is countably in$nite and there is a connected network over Wsuch 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 expectednumber of such paths, as the number of open paths is no less than one for any elementof the sample space in which there is an open path. By (3), the number of possibleopen 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 equalto Dt ∗ qt . Therefore if q¡ 1=D, the probability of an open path leading to i; t goesto zero as t goes to in7nity. This implies that (24) holds for qc¿ 1=D¿ 0. Next, weprove that, under the assumptions that the set of workers is in7nite and the network isconnected, qc ¡ 1. If is the one line (i.e., each i is connected to i + 1 and no oneelse), then Durrett (1984) shows that qc ¡ 1, so it remains to show that the one lineis a subgraph of . Now suppose that is an arbitrary connected graph with no morethan D connections into and out of each node. Pick a node i. Let �i(j) be the distanceof the shortest path (sequence of nodes connected by edges in the graph) from node ito some node j. Since is connected, �i(j) is always 7nite. Let Sn be the set of allpoints j such that �i(j)6 n. By earlier assumption there are no more than D edgesto or from any node, so the size of Sn is always no more than Dn, and thus always7nite. Therefore, for any n there is always a node j such that �i(j)¿n. Clearly, theshortest path does not go through the same point twice, or else one can construct ashorter path. We can use the same argument to show that, for any n, there is a nodek such that �k(i)¿n and that the shortest path from k to i does not cross the shortestpath from i to j. We can thus construct a subgraph ′ which has an in7nite number ofpoints 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 ¡ 1for the original graph as well.

4.4. Numerical results

This section describes a set of simulation experiments which further characterizethe behavior of the long-run employment rate in the model. The simulations generallyfollow 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 7xednumber r of connections per worker arranged in a simple nearest-neighbor loop. Anr-nearest-neighbor loop connects agent i to r nearest neighbors (for example, a two-loopconnects 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 withq = 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 zeroemployment permanently, simulation results like those in Fig. 2 suggest that a largeeconomy can have a stable and positive employment rate for many periods. To gain amore systematic understanding of this model property, I perform a simple simulationexperiment. The parameterization of the model used to generate Fig. 2 is simulated


Fig. 2. The time series of the employment rate in a representative simulation run. Network is a loop with1000 workers, 3 contacts/worker and oMer rate of 0.35.

Table 1Number of periods before zero-employment state reached

No. agents Periods elapsed before Rnt = 0in network

Min Max Median % ¿ 1 m

10 7 1028 134 0100 741,592 ¿ 1 m ¿ 1 m 99

1,000 ¿ 1 m ¿ 1 m ¿ 1 m 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 passagetime to zero employment, then the entire experiment is repeated for a diMerent numberof agents. The results, summarized in Table 1, indicate that the economy becomestrapped in the zero-employment state very quickly if the number of connected agentsis quite small (10) but has a positive employment rate for a very long time even ifthe number of connected agents is moderate (100). This experiment has been repeatedwith 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 toapproach zero employment.

Given these results, additional simulation experiments are used to characterize therelationship between the oMer rate and long-run employment. These simulations arerun for 2000 periods and the overall employment rate is averaged over the last 500time periods to get the ‘long-run employment rate’. Fig. 3 summarizes the results of abaseline experiment which features simple loop networks and homogeneous agents. Asthe 7gure shows, the relationship between the oMer rate and the long run employment


Fig. 3. Relationship between oMer rate and long-run average employment rate for several diMerent loopnetworks. Number of contacts per worker (r) indicated on graph.

rate exhibits many of the patterns described in Section 4.2. Long-run employment iszero if q is below some critical value and the long-run employment rate is highlynon-linear in q near that critical value. Additional simulation experiments 5 featuring aheterogeneous 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 noneare obtained through formal search. However, empirical research indicates that abouthalf of jobs are obtained through networking and the other half are obtained throughmore 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 verybasic.

In addition to those matches gained through networking, each worker is matchedup with !i(t) 7rms, each of which gets to observe mf

i . The exact number of 7rmsis stochastic, with probability distribution S. Therefore the probability of receiving anacceptable job oMer through formal search is

si ≡∑!

(1 − Fm

(hiai

))!S(!) (25)

5 The GAUSS code to run these experiments is available from the author on request.


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, Proposi-

tions 4.2 and 4.3 continue to hold but Propositions 4.4 and 4.5 do not. Furthermore, therelationship 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 ofwhich 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 be-tween network structure and long-run outcomes. One particular hypothesis that appearsfrequently in sociology is the ‘strength of weak ties’:

[There is] a structural tendency for those to whom one is only weakly tied to havebetter access to job information one does not already have. Acquaintances, as comparedto close friends, are more prone to move in diMerent circles than one’s self. Those towhom one is closest are likely to have the greatest overlap in contact with those onealready knows, so that the information to which they are privy is likely to be muchthe same as that which one already has (Granovetter, 1995, pp. 52–53).

This section demonstrates that the model in this paper exhibits an aggregate strength-of-weak-ties property. All else equal, the proportion of a group’s social ties that areweak 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 asstrong 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 inthe above quote. A systematic way of constructing a family of networks with a givenproportion of weak ties is given by Watts and Strogatz (1998). Their algorithm startswith a loop network with r¿ 2 social ties per agent. Such a network has only strongties, 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 probabilitythat a randomly generated tie is a strong tie is close to zero. The result is a family ofrandom 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, andthe average employment over the last 500 is taken to give the long-run employmentrate. Because the networks are random, the exercise is repeated 10 times for each valueof 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. Thehorizontal axis is the fraction of weak ties (p) and the vertical axis is the long-runaverage 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-runemployment rate is increasing in the proportion of weak ties. Simulations with other


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 issuQciently high for the long-run employment rate to be positive. An increase in therelative prevalence of weak ties has a similar eMect to that of a moderate increase inthe oMer rate.

The intuition behind this result is that weak ties are a way of diversifying one’ssocial resources. Eq. (17) implies that an individual faces a form of decreasing returnsto the number of employed friends. When a person’s friends are also friends withone another, this increases variance in the number of employed friends by increasingthe correlation in employment status between the person’s friends. In the aggregate, anetwork with a higher proportion of weak ties reduces inequality in the distribution ofemployed 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 blackmiddle class as the key shock to inner-city communities which led to the gradual ‘dis-appearance’ of regular employment in these communities, and eventually an increasein various social pathologies. In his explanation this seemingly small shock to neigh-borhood composition had a disproportionately large eMect on community employmentrates because of the self-reinforcing process of employment decline, job network dete-rioration, and further employment decline. If social networks are primarily determinedby neighborhoods, the model presented in this paper exhibits this property. This sectionexplores this idea in more detail.

There is some empirical support for Wilson’s hypothesis. Holzer (1987) 7nds thatblack youth in the NLSY have a lower success rate than white youth in obtainingjob oMers through networking, despite a similar rate of success in more formal means.Krauth (2000) 7nds strong evidence that the relationship between neighborhood


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 isjob networking model calibrated to match.

composition and neighborhood employment is highly non-linear. A representative ex-ample of the results in that paper appears in Fig. 5. The scatter plot is of ChicagoCensus tract-level data from 1990. The horizontal axis is the fraction of tract residentswith college degrees and the vertical axis is the employment rate in the tract. Thethick line is a non-parametric regression line. It indicates that, as Wilson suggests, asthe fraction of workers in a community with high human capital falls below a criticallevel the predicted employment rate falls dramatically. Away from this critical levelthe relationship is much weaker. Krauth (2000) repeats this exercise for the 20 largestUS cities and 7nds that the pattern is similar in all 20 cities. Although these resultsmay be consistent with alternative models, they are certainly consistent with Wilson’shypothesis.

Figs. 1 and 3 suggest that the model here also implies a non-linear relationshipbetween average human capital and employment rates. However, the parameterizationsused to generate those 7gures may not be realistic, which raises the question of howclosely a reasonable parameterization can match the empirical relationship in Fig. 5. Iaddress this question by ‘calibrating’ the model, though the microeconomic evidenceon parameter values may be too weak to merit the term. Because empirical resultsindicate that half of jobs are obtained through formal means and half are obtainedthrough networking, the model extension described in Section 5.1 is used. The successrate in formal search is set to give equal probability of obtaining a job through formaland informal means, implying that s = 1 − √

1 − Rn. With a 1990 employment ratein Chicago of approximately 90%, this implies that s = 0:68. The social network issimple in structure: each period an individual has r randomly generated social ties toother members of the community. There is very limited and controversial evidence on areasonable value for r, in part because there is no clear dividing line between friend andstranger. Glaeser (2000, p. 128) estimates that the average resident of a large city has 5


close friends. A school of thought in evolutionary psychology associated with Dunbar(1996) argues that human brains can ‘know’ approximately 150 other individuals atany one time. Knowledge of a person in this sense means suQcient knowledge ofa person’s past behavior to predict future behavior or enforce social norms againstpredation. The number of social ties which are potentially useful in job search likelyfalls between these two extremes, so r is set to 50. There are two types of agents, thosewith college degrees (qi=qH) and those without (qi=qL). The fraction of workers withcollege degrees varies across communities. The rate of success in networking for eachtype 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 degreevarying from 0 to 100. The results are depicted in the thinner line in Fig. 5. As the7gure shows, the model can generate a relationship between neighborhood compositionand neighborhood employment which is quite similar to both the observed relationshipand to Wilson’s hypothesis.

6. Conclusion and further directions

This paper has developed a model in which personal connections help transmit im-portant information about the quality of a worker-job match. The resulting dynamicsof the model have several interesting features. Long run employment is aMected by thenumber of ties in the social network and the proportion of ties that are weak, as wellas individual characteristics such as human capital and reservation wages. The relation-ship between any of these factors and long-run employment is highly non-linear. Boththe non-linearity of employment in group characteristics and the value of weak tiesare commonly discussed sociological theories which are given more detailed economicfoundations here.

In many ways, the model presented here is just an early step toward understandingthe aggregate behavior of labor markets with job networking. Although simplifying as-sumptions have been made explicitly with the view that there is an unavoidable tradeoMbetween 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 torespond to economic incentives.

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A dynamic model of job networking and social influences on employment