Sep 13, 2002 - hiring and of the structure of social networks on three labor market ... If social networks are organized as a random graph each agent in.
Social Network Structure, Equality and Segregation in a Labor Market with Referral Hiring Troy Tassier∗ Department of Economics Filippo Menczer Department of Management Sciences The University of Iowa Iowa City, Iowa 52242 {troy-tassier,filippo-menczer}@uiowa.edu September 13, 2002
Abstract We construct an agent-based model to examine the effects of referral based hiring and of the structure of social networks on three labor market phenomena: the steady state level of equality between groups, measures of segregation at steady state, and the rate of convergence to steady state. When firms hire by referral the relative structure of social networks in majority and minority groups determines the level of equality; groups prefer to have more regular (less random) social networks relative to other groups in the population. While regular networks are beneficial in regard to the steady state level of equality, they slow the rate of convergence to steady state. Thus there is a tension in that the preferred social network structure for the minority (disadvantaged) group depends on whether the labor market has converged to steady state.
∗
Current address: Center for the Study of Complex Systems, University of Michigan
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1
Introduction
When comparing the labor market outcomes of ethnic groups in the United States one notices substantial differences across groups. For instance, the difference between black and white wages is approximately 25–40% (Smith and Welch 1989). The differences extend to many other ethnic groups as well (Farley 1990; Darity Jr., Guilkey, and Winfrey 1996). Some, but not all, of these differences can be explained by varying levels of education and other job market skills across groups (Farley 1990; Card and Krueger 1992; Darity Jr., Guilkey, and Winfrey 1996). One clear candidate for the remaining differences is discrimination. Another is more subtle: differences in social networks may cause un-equal information flows about jobs for some groups since approximately 50 percent of workers find their jobs through friends, relatives, and other social contacts (Granovetter 1995). Some believe this finding to prove the old adage “it’s not what you know but who you know” (Montgomery 1991). From this belief a line of research has developed that argues referral hiring may be one cause of the persistence of inequality for labor force minorities.1 Since social networks tend to consist of members of the same ethnic group, race, or social class (Marsden 1988) and large amounts of job information travel through social networks, a member of an under-represented group in the labor market may find out about fewer job opportunities relative to a member of a more widely represented group. Thus a potentially self-perpetuating poverty trap can be created where members of a group cannot get good jobs because their friends do not have good jobs, and so on. Thereby some groups may be “locked-out” of portions of the labor market. In this paper we investigate this possibility by systematically studying measures of inequality between majority and minority groups in a model labor market. We pri1
See the appendix to Granovetter (1995) for a review of the referral hiring literature through 1995. More recent work includes Arrow and Borzekowski (2000), Calvo-Armengol (2001), Calvo-Armengol and Jackson (2002), and Mouw (1999).
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marily focus on how the structure of social networks influences labor market outcomes both at steady state and on the path to steady state. We capture social network structure by varying the amount of randomness and regularity in social networks. Social networks tend to contain cliques where friends of a given individual are more likely to be friends with each other than they are with other randomly chosen members of the population. Such a clustering property of social networks can be modeled by creating regular graphs (Tassier and Menczer 2001; Watts and Strogatz 1998; Watts 1999). However, each member of a given clique also tends to have a small amount of random friends who are not part of the clique. The random friends are generally not as close (in social terms) as friends who belong to the clique. Thus they are sometimes referred to as weak ties (Granovetter 1973). Here we model the structure of social networks by varying the amount of social connections that are regular vs. random — strong vs. weak ties.2 The structure of social networks affects the flow of information through the distance of network members. Random graphs approximate the shortest average path length between all pairs of nodes in a graph for a given number of edges and nodes (Bollobas 2001). If social networks are organized as a random graph each agent in the network is closer to more job information than if the networks are organized as a regular graph. This proximity to information may appear to be an advantage for any given agent. But there is also a disadvantage to being close to other individuals: Since any job information held by a given agent or group of agents in a random graph is closer to all other members of the population, keeping the information inside a given subset of a social network becomes increasingly difficult. Thus, the preferred structure of social networks for a group is not immediately clear. In one respect 2
Note two things here. First, as we will describe below, random networks do not imply that groups are integrated; the social networks of a given group can be completely random within the group and still maintain complete segregation from other groups. Second, our networks are directed; thus we also may vary the amount of segregation between groups.
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agents prefer more random networks in order to be proximate to information they do not hold but potentially would like to distance themselves from other agents in order to protect information they do hold. The effect that the structure of social networks has on employment outcomes is especially important since many minority groups tend to have more structured networks. Using data from the General Social Survey, Patterson (1998) finds blacks to have more overlap in their social connections than whites. Portes and Sensenbrenner (1993) cite numerous reasons related to social capital theory and provide examples that suggest recent immigrants are likely to live in tightly knit social networks. Additionally, the breadth of social networks is believed to be positively correlated with social status (Homans 1950; Patterson 1998). Given that social networks differ across groups we aim to investigate how varying network structures affect employment outcomes when referral hiring occurs in a labor market. We uncover two main findings in regard to the structure of social networks. First, as the social networks of a given group become more random, members of this group do worse in terms of job attainment and unemployment rates. The effect of protecting information appears to dominate the effect of being close to the information of others. A group with relatively more structure or less randomness in their network is able to contain job information inside the group. In addition this creates an insurance structure by which individuals who become unemployed can more easily find new employment opportunities (Tassier and Menczer 2001). Second, if a minority group starts in a position of inequality and has the same social network structure as the majority group, random networks increase the rate at which the minority group achieves steady state and thus equality. In this case random networks limit the ability of an advantaged group to hold job information inside their group and places the disadvantaged group closer to the job information contained
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in the social network. Random networks allow job information to leak out to a disadvantaged group more quickly than if networks are organized in a more regular structure. Thus these two findings create a tension for policy makers interested in improving the labor market conditions of minority groups. If a minority group is in a position of inequality and the labor market has not reached steady state, methods to increase the randomness of the social network of the minority group (such as student bussing programs) may prove helpful. But if the labor market is already at steady state, increasing the randomness of the social network may place the minority group in a disadvantaged position.
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Model Specification
We proceed by specifying an agent-based model of the referral hiring process. The main advantage of using an agent-based framework in this context is the ability to easily model networks of various topologies. Previous models that study the effects of referral hiring having been limited by analytical tractability to studying random matching processes (Mailath, Samuelson, and Shaked 2000), random networks (Tassier 2002) or simplified networks such as dyads (Montgomery 1991; Montgomery 1994). Our model of the agents’ social network is flexible both in terms of size and structure. Thus we are able to define a series of experiments to test how labor market outcomes are influenced by social networks over a wide set of structures.
2.1
Agents and Networks
Our model consists of two networks, one for jobs and one for social connections, and a hiring process related to the networks. The are two levels of jobs, high and low. Each level consists of M jobs. Number the low level jobs L1 , L2 , . . . , LM and the high 5
level jobs H1 , H2 , . . . , HM . We arrange the jobs into a network such that each set forms a one dimensional toroidal lattice (a circle) where consecutively labeled jobs are neighbors on the lattice. The reader may think of this being a specification of how the jobs are related. A job is located spatially close in the job network to similar jobs. Thus occupants of a given job are able to both learn of available open jobs similar to theirs and provide referrals for these jobs. There are N agents with γN belonging to group A and (1 − γ)N belonging to group B. Both γ and the group to which an agent belongs are exogenously chosen in the model. Thus there is a minority and a majority group for any γ 6= 1/2. Without loss of generality we consider γ < 1/2 and call group A the minority group. Agents have positive utility UH for a high level job and a lower positive utility UL for a low level job (UH > UL ). Agents have a reservation utility of 0 if unemployed. Thus agents always prefer employment to unemployment. We assume agents are equally able to search when employed and unemployed. All workers are equally qualified for both job levels. Agent networks are initialized by forming a torus of dimension d for each group consisting exclusively of all members of the group. Each agent is connected to d neighbors in each direction. Thus we initially form two regular and segregated networks, one for group A and one for group B. The edges of the network are directed. As an example of the two networks consider Figure 1. In the example each agent is employed at one of the jobs and all jobs are filled. There are eight members of each group and eight jobs of each type. The social network of group A is shown by the nodes in the two left columns of the graph. The social network of group B is shown in the two columns on the right of the graph. High level jobs are the upper two rows of nodes in the graph and low level jobs are the lower two rows of nodes in the graph. In the example, four members of each group hold a high level job and
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A
B
Figure 1: An example social network with N = 16 agents and γ = 1/2. The agents of group A are shown as white nodes and the agents of group B as gray nodes. The social network connections (d = 1) are shown by white and black arrows for group A and B, respectively. Dotted lines represent the network of high level jobs and dashed lines represent the network of low level jobs (M = 8).
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four members of each group hold a low level job. Consider the node labeled A in the picture. This node represents an agent of group A who is employed in a high level job. The node labeled B represents an agent of group B who is employed in a low level job. Note that the picture actually contains four separate networks: the social networks of groups A and group B and the job networks of high and low level jobs. In most instances the mapping between social and job networks will be much more complicated than in this example. Once we create the initial social networks we modify them in two ways to create randomness both in the structure of the social network and in the amount of overlap (integration) in the social connections between the two groups. We create random networks by defining a randomness parameter ρ. With probability ρ we replace a given edge (i, j) with edge (i, k) where k is chosen uniformly over all members of the population not currently connected to i with one restriction. We define the restriction by a segregation parameter ψ. With probability ψ the choice of k must come from the same group as i. Thus if ψ = 1 the network retains complete segregation even for completely random networks. As ψ decreases the social networks of the two groups become more integrated. Thus we are able to compare results of the model for social networks that span from strictly regular to strictly random by adjusting ρ; and from completely segregated to completely integrated by adjusting ψ. The example in Figure 1 shows social networks that are completely regular (ρ = 0) and segregated (ψ = 1). Again note that because the edges are directed the groups can have different levels of segregation and randomness.
2.2
Job Search and Dynamics
We now discuss the dynamic process by which agents find and lose jobs and how they interact in the labor market. Psuedocode is provided in Figure 2. Agents may be 8
Initialize M high level and M low level jobs arranged on a lattice Initialize N agents Assign agents to groups A and B Initialize a regular social graph for each group Permutate social graph edges according to ψ, ρA , and ρB Assign initial jobs to agents according to experiment structure for each period for each agent i Fire i with probability δ Update open job list endfor for each job j for each agent i Notify i of available jobs based on search intensity, sA or sB Notify i of available jobs based on social connections endfor Agents apply for the job Randomly choose an applicant to fill job j endfor Measure statistics endfor
Figure 2: Psuedocode of the dynamics used to simulate our job market. either employed or unemployed. Employed agents are subject to removal from their jobs with probability δ each period. This creates open jobs in the economy. A list of open jobs is maintained throughout the simulation. In each period all open jobs are chosen from the list in a random order and given the opportunity to be filled. When a job is chosen to be filled each worker is notified of the job with probability sA or sB , si ∈ [0, 1], as determined by the group affiliation of the agent. In addition, each member of the population may receive information about job openings from any of its 2d neighbors. Each agent who has a social network connection who knows of the opening is notified about the opening with probability 1. An agent knows of an open job j if she is employed at job k and k is connected in the job network to j. After notification all workers learning of the opening who are unemployed or who are employed at a level lower than the opening apply for the job. A firm makes a hiring decision by choosing randomly from the applicants (since the workers are all equally qualified the firm is indifferent as to the specific worker hired). It may be that no one applies for a given open job. In this case the job remains open until the following period, at which time another attempt will be made to fill it. 9
2.3
Measures
Before moving forward we need to define a few measures in order to compare simulation sets across parameter values. Def ’n Inequality:
I=
! PM A 1−γ i=1 i · PM − 1 × 100%, γ B j j=1
where Ai = 1 if job Hi is occupied by a member of Group A and 0 otherwise. Bj is defined similarly. Thus the two groups are equal in terms of the proportion of high level jobs held if each group holds the same number of high level jobs relative to population size. If I > 0 group A is advantaged ; If I < 0 group A is dis-advantaged. Def ’n Unemployment rate is simply defined as the proportion of agents in each group who do not currently occupy a job. Def ’n Job level autocorrelation, or job segregation, is the likelihood that job k and job k + 1 are occupied by a member of the same agent group. If agents are placed randomly at jobs and I = 0, job segregation should be γ 2 + (1 − γ)2 . This value represents the baseline for comparing the segregation of workers across jobs. Def ’n Agent level autocorrelation, or agent segregation, is the likelihood that agent i and agent j are in the same employment state given i is connected to j in the social network. Assuming random placement of workers and I = 0 agent segregation 2 N −2M 2 should be 2 M + . This value represents the baseline for comparing how N N the job status of agent i depends on his social network.
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2.4
Parameters
In all of the experiments below we initialize the model with N = 1, 000 agents of which 25% belong to group A, γ = 0.25. M = 480 jobs.3 Thus there is at least 4% unemployment in the population.4 Each of the jobs is connected to one job in each direction. Each agent has two social connections, d = 1. Specific parameters of the networks (ψ, ρA , and ρB ) and search intensity (sA and sB ) vary across experiments. We specify these parameters in the introduction to each experiment.
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Steady State Characteristics
In a related model with strictly random directed networks Tassier (2002) shows that the only asymptotically stable steady states of the referral hiring market have equality (there defined as equal unemployment rates). Thus one of the questions of this work is whether such a result depends on the assumption of random networks. In this section we run four experiments to study the group level equality at steady state. In the first experiment we maintain equal network structures between the two groups. In the second we allow the network structures to differ across groups. In the third search intensities differ and in the fourth we increase the amount of connections held by agents of the majority group. We initialize our experiments by randomly assigning agents to jobs in the population. Because of the random initial assingment the experiments in this section begin in a state with no inequality I = 0, and baseline values for the other measures on average. We then run the experiment for 200 periods and compare the measures described 3
Sample results do not show qualitative dependence on the choice of these parameters. Of course changing the population makeup, γ, will change absolute values of measures such as job-level autocorrelation but the results are not qualitatively different. 4 There is more than 4% unemployment in the experiments because some jobs will be un-filled in any given period.
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Table 1: Inequality is 0.0 for ρA = ρB . ρ 0.0 .25 .50 .75 1.0
s=.0001 s=.001 s=.01 s=.1 −8.4∗∗ 1.3 -3.8 -0.1 ∗ 0.8 6.4 -1.7 -0.6 -2.4 0.9 0.1 -3.6 -0.9 5.5∗ 0.2 -0.3 -0.5 -1.6 1.1 1.0
(*) significantly different from 0.0 at the 95% level. (**) significantly different from 0.0 at the 99% level.
above. The results presented are averages over 50 replications of the experiment.
3.1
Segregation with Equivalent Social Networks
In this experiment we leave the level of randomness in the networks equal between the two groups, ρA = ρB = ρ, and we also leave the level of search intensity equal between the two groups, sA = sB = s. We then vary ρ and s. Social networks are completely segregated, ψ = 1. We first show the level of inequality in Table 1. With three exceptions all of the parameter combinations display average inequality not significantly different from 0.0 at the 95% confidence level. Another measure of the equality between the two groups is the average length of unemployment duration between the two groups. Let DA be the average number of periods that a member of Group A remains unemployed after being fired. Define DB in the same way for Group B. We present results for the ratio DA /DB in Table 2. The duration of unemployment spells between the two groups are equal except for three cases noted in the table. The results presented in Tables 1 and 2 suggest that equality between groups results if the groups have the same network structure, even in the presence of segregation. 12
Table 2: The ratio of average duration of unemployment spells between Group A and Group B for ρA = ρB . ρ 0.00 0.25 0.50 0.75 1.00
s=.0001 s=.001 s=.01 s=.1 1.45∗∗ 0.96 1.00 0.98 0.99 1.04∗ 0.97 0.97 0.96 1.00 0.98 1.01 1.03 0.98 1.01 1.04 1.01 0.97∗ 0.99 1.00
(*) significantly different from 1.0 at the 95% level. (**) significantly different from 1.0 at the 99% level.
We now turn to population level characteristics of the results. The unemployment rate is shown in Figure 3 and the average duration of unemployment is shown in Figure 4. Note that the unemployment rate increases as s decreases. For s = 0.10 the unemployment rate is just above 4%. For s = 0.0001 the unemployment rate increases to approximately 14%. As s increases more people find out about available open jobs. The unemployment rate also increases as networks become more random although the effect is secondary to that of changes in s. For s = .0001 moving from completely regular to completely random networks increases the unemployment rate from 10.9% to 14.2%. Regular networks propagate job information more effectively than random networks. As we will see below, agents closely linked in the social network partition the job network such that agents near each other socially work near each other. This creates more structure in the way agents find jobs. These results are directly tied to the unemployment duration results. Qualitatively the same relationships are observed but the effect of network structure is very small here. Figure 5 displays the level of job autocorrelation in the job network across the replications. Since γ = 0.25 the baseline comparison is 62.5%. Note that for relatively high levels of s job segregation is close to random but increases sharply for low levels
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Unemployment Rate (%) 14 12 10 8 6
16 14 12 10 8 6 4
1 0.8 0.6
0.0001 0.4
0.001 0.01 search intensity
0.1
randomness
0.2 1 0
Figure 3: The unemployment rate increases as s decreases or as networks become more random. Unemployment Duration (periods) 9 8 7 6 5 4 3 10 8 6 4 2
1 0.8 0.6
0.0001 0.4
0.001 0.01 search intensity
0.1
randomness
0.2 1 0
Figure 4: The duration of unemployment spells decreases with s. Increasing the randomness of networks increases the average unemployment duration but the effect appears small. 14
Job Level Autocorrelation (%) 80 75 70 65
85 80 75 70 65 60
1 0.8 0.6
0.0001 0.4
0.001 0.01 search intensity
0.1
randomness
0.2 1 0
Figure 5: Job segregation increases as s decreases. The effect of network structure on job segregation appears minimal. of s. The network structure does not appear to significantly affect the level of job segregation. Agent autocorrelation across employment states is shown in Figure 6. The baseline level is 46.25% for our parameter configuration. We observe levels close to the baseline for high levels of s and highly random networks. But for high levels of regularity and low levels of s agent segregation increases as we would expect. As networks become more regular the job status of the social connections of an agent become increasingly important.
3.2
Changing Relative Social Network Structure
We now ask what in our model could account for the qualitative differences in employment outcomes observed in the US data mentioned earlier? The analysis is broken down into two parts: Differences in the social structure of the groups and differences 15
Social Group Autocorrelation (%) 60 55 50
65 60 55 50 45
0 0.2 0.4
0.0001 0.6
0.001 0.01 search intensity
randomness
0.8
0.1
1 1
Figure 6: Agent segregation increases as s decreases and as networks become less random.
Table 3: Ratio of average unemployment spells of Group A to Group B.
ρB 1.0 .75 .50 .25 0.0
1.0 0.97∗ 0.98∗ 1.06∗ 1.12∗ 1.31∗
.75 1.00 0.98 1.09∗ 1.10∗ 1.23∗
ρA .50 0.91∗ 0.94∗ 1.00 1.12∗ 1.23∗
.25 0.92∗ 0.89∗ 0.98 1.04∗ 1.15∗
0.0 0.80∗ 0.79∗ 0.86∗ 0.86∗ 0.96
(*) significantly different from 1.0 at the 95% level.
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Table 4: The Level of inequality favors networks that are less random relative to the other group.
ρB 1.0 .75 .50 .25 0.0
1.0 -0.02 0.00 -0.01 -0.04 −0.07∗
.75 0.03 0.06∗ -0.01 0.00 -0.06
ρA .50 .25 -0.01 0.08∗ 0.03 0.03 0.01 0.02 -0.03 0.06∗ -0.07 -0.02
0.0 0.07∗ 0.10∗ 0.07 0.06∗ 0.01
(*) significantly different from 0.0 at the 95% level.
in direct access to information (search intensities and social network size). We will show through a series of experiments that differences in the breadth of the social structure of individuals does not yield the same outcomes as differences in direct access to information. We first look at how levels of equality change as we vary ρA and ρB with sA = sB = 0.001. Table 3 shows ratio of the average length of unemployment spells for group A to group B. As the network of one group becomes more random relative to the other group the average duration of unemployment spells increase. This finding has direct relevance to the level of equality for different network structures shown in in Table 4. Note that as the randomness of the minority group increases (to the left of the table) the minority group becomes worse off. Its workers hold fewer of the high level jobs relative to the majority group. The result follows from the ability of a social network to retain job information. This is the standard argument for why referral hiring may cause persistence of inequality. If social networks are sufficiently regular, closely connected agents tend to work at jobs near each other in the job network. When an opening occurs it is relatively more likely an agent within the social network of the nearby job occupants gets
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the job than when networks are random. Thus regular social networks tend to hold job information inside cliques of agents; it is difficult for outsiders to gain access to information about these jobs. As networks become more random it becomes harder for agents in a clique to exclude other agents from the jobs they currently occupy, since all agents are closer in the social network. Given a choice, a given partition of a social network prefers to be organized more regularly than the social network of other agents in the population. Note that this is a group level effect. Other research has shown that more random connections or weak ties are benefical for the individual (Calvo-Armengol and Jackson 2002). If the amount of connections that are random increase for one individual agent, leaving other agents highly clustered, that agent would gain the advantage of having access to information in a number of groups. But as we mentioned before, in doing so the agent also acts as a conduit to the other clusters as well. CalvoArmengol and Jackson (2002) show that at the individual level access to more clusters is beneficial; but our experiment suggests that if all agents in a group increase the amount of connections that are random the group is worse off. The results of this section highlight that the incentives for the individual and the group differ.
3.3
Changing Access to Information
We now look at how changes in the direct availability of information compare to changes in social structure. We model differential access to information in two ways: changing the relative search intensities between groups and changing the relative size of the social networks within groups. We first look at differences in search intensity, sA and sB . Differences in search intensity may occur in two ways: First, one group may use formal means in a job search more or less frequently than another (Holzer 1988). Second, if some firms 18
Table 5: Effect of changing the relative search intensity across groups. sB /sA I DA DB
0.5 0.26 2.53 4.33
0.75 1∗ 1.25 1.5 0.13 0.01 -0.09 -0.18 3.03 3.45 3.84 4.18 3.82 3.52 3.26 3.05
(*) Excepting the sA = sB case, all levels of inequality and differences between DA and DB are significantly different from 0.00 at the 95% level.
prefer to hire members of one group over another these firms may target their formal job search toward the preferred group. For example if a firm prefers blue workers, the firm could advertise the job opening in newspapers where primarily blue people live. This may occur as a result of statistical or overt discrimination. We run 5 sets of simulations. We set sA = 0.01 and vary sB between 0.005 and 0.015. The results are shown in Table 5. As one can see, increases in search intensity relative to the other group directly lead to better jobs and shorter duration of unemployment spells. We also perform a series of experiments where we increase the number of friends for each member of group B from 2 to 4 (d = 2). In this case the level of inequality is I = −0.07. Regarding duration of unemployment spells we find DA = 3.80 and DB = 3.30. Both the level of inequality and the difference in unemployment duration are significantly different from 0.0 at the 95% level. This result shows that increasing the size of a group’s social network improves the position of the workers in that group. The most important point to take from the two sets of experiments in this subsection is that changing access to information through social structure is much more subtle than if the changes are made through direct means. Making the social structure of one group more random shortens the average distance between members of the group; it increases the breadth of the social structure of individual agents in the
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group. Increasing the number of connections held by each agent has the same effect. But the results are different. In the case of social structure, reducing the distance hurts the group; in the case of the number of friends it helps the group. Thus, analyzing the effects of social structure is more difficult than simply looking at the expected amount of information that flows to an average agent. Increasing the network size of a group of agents is more closely related to increasing the search intensities of those agents. And it appears neither can be directly used to model social network structure without accounting for the actual network structure.
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Attainment of Equality
Even though equality was attained in the experiments described in the previous section (where network structures are equal), we started the experiment with labor market equality between the two groups through our random assignment of jobs. We now ask three questions: First, if we start the minority group in a position of stark disadvantage, will equality be attained? Second, if equality is attained, how long does it take? And third, does the length of time to achieve equality depend on the structure of social networks? We proceed to address these questions by starting the minority group in the worst possible situation. In the next two experiments we initialize each replication with all members of group A either being unemployed or having a low level job. All members of group B have a high level job or a low level job. We then view how long it takes to reach 95% equality as defined above, I = −0.05.5 All results are averaged over 10 runs of each parameter setting. 5
To account for noise we keep a running average of I over the last ten periods and use such an average as our measure of inequality.
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Number of Periods to Reach Equality 2.5e+03 2e+03 1.5e+03 1e+03 500
3000 2500 2000 1500 1000 500 0
0 0.2 0.4
0.0001 0.6
0.001 0.01 search intensity
0.1
randomness
0.8 1 1
Figure 7: The number of periods to reach equality increases as s decreases and as networks become more regular.
4.1
Effect of Network Structure on Attainment of Equality
In this experiment we maintain segregation in the population, ψ = 1, and let the amount of randomness in the networks and the search intensity be the same for the two groups (ρA = ρB = ρ, sA = sB = s) since this is the case where we expect equality to be attained. We vary ρ and s and view how long it takes the minority group to reach equality as defined above. Figure 7 present the results. Addressing the question of whether the minority group would reach equality if starting out disadvantaged, all runs reach equality. Now note that the number of periods to reach equality is strictly decreasing in s. As more information about jobs is made available outside of referrals the minority group reaches equality faster. Additionally, as networks become more random the minority group reaches equality faster as long as s is relatively low. Notice that when s = .10 the structure of the network does not appear to matter. But as s decreases the 21
structure of the network becomes important for minorities to obtain high level jobs. This is because when s is low referrals are very important for finding jobs; When s is high referrals matter less and the structure of networks is not important. Essentially increasing s acts as a substitute for the access to information provided by networks when s is low. For high enough levels of s access to information through networks is dominated by traditional search. For s = .0001 it takes the minority group three times as long to reach equality in a strictly regular network than it does in a strictly random network. Additionally, as s decreases from .10 to .0001 the number of periods to reach equality increases just over four times in a random network. But in a regular network, the increase is more than 16 times as many periods. The result can be understood from our discussion of the previous experiment. Random networks have short characteristic path lengths between nodes. Thus information can easily spread from any one agent to any other agent in a small number of steps. Unless networks are sufficiently regular, it is very difficult to hold information inside a social network. And once some foothold is gained by the minority group the advantage quickly spreads to many other members of the group.
4.2
Effect of Segregation on Attainment of Equality
It is likely that the level of segregation also has an important effect on the time it takes the minority group to reach equality. Here we fix ρ at .50 and vary s and ψ. Results are reported in Figure 8. This experiment shows two things: First, the level of segregation appears to only matter when s is small. Second, it appears that the effects of segregation are only important when the level of segregation is very high.This result can be understood in terms of the small world phenomenon. It takes very few random connections between the minority and majority group to greatly shorten the characteristic path length 22
Number of Periods to Reach Equality 900 800 700 600 500 400 300 200
1000 800 600 400 200 0
1 0.8 0.6
0.0001 0.4
0.001 0.01 search intensity
0.1
social segregation
0.2 1 0
Figure 8: The number of periods to reach equality increases significantly if both s is low and there are extreme levels of segregation. For moderate levels of segregation changes in s have much smaller effects. between all members of the population.6
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Policy Discussion
In steady state we observe that the structure of social networks does not have implications for the level of inequality as long as groups have equivalent network structures. But if the social network structures differ between groups inequality may result. Having less random networks relative to other groups allows a group to contain job information inside the group, a result which leads to better jobs and lower unemployment rates for the group. However, the results observed in the steady state experiments contrast with those in the attainment of equality experiments. When a minority 6
See Watts (1999), and Watts and Strogatz (1998) for a general discussion of the small world phenomenon or Tassier and Menczer (2001) for a discussion related to labor networks.
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group attempts to catch up to the status-quo the short characteristic path length of random networks allow the job information to diffuse more quickly through the minority network. In contrasting the results of these two experiments, we present one difficulty facing policy makers; they must ascertain whether the labor market is at steady state or on the path to steady state in order to make the best public policy prescriptions. In order to do so, difficult questions must be addressed about the causes of the inequality we observe for some minority groups: Are the differences in labor market outcomes primarily the result of differences in educational attainment or other labor market skills? If so are these differences stable? If labor market opportunities improve will educational attainment also improve? Depending on answers to questions such as these, policy makers should recommend different policies with respect to social networks and referral hiring. If at steady state, our results suggest that policies to create tight bonds in minority social groups along with the introduction of some good jobs into these groups may be most beneficial. If on the path to steady state, policies that introduce random connections extending beyond the minority group would be best. This would include programs such as student bussing programs, peer mentoring, or internships at institutions with large numbers of status-quo members. Each of these items would accelerate the attainment of equality by creating additional randomness in the minority social networks and/or decreasing the segregation between social networks. Note however that these programs also may be dangerous for minority groups. As mentioned above it has been shown that minority groups tend to have more structured social networks with more overlap of connections (Patterson 1998). Minority groups tend to live in social networks with tighter cliques. Our results suggest that this is the preferred structure in regard to the flow of information in steady state. If social
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policies create too much randomness in the social networks of minority groups in order to get them to equality faster this may put them in a long term position of disadvantage. One way to circumvent some of these problems is to concentrate on social policies that create random connections from the status-quo group to the minority group. As examples consider bussing status-quo members to minority schools, or sending peer mentors from the suburbs to the inner city instead of sending individuals out of the inner city to the suburbs. This would allow the minority groups to maintain their tight, local social networks but provide the random links to the status-quo that would allow for information to diffuse more easily. One difficulty in interpreting our results is the conception of time. While our model indicates that a minority group will always catch up to the status-quo we cannot estimate from our model how long it will take. We can only speak to the relative time between various parameter settings. While the model indicates it may take up to 16 times as long to reach equality with regular as opposed to random networks, the concern of society over this result depends on the units of time being considered. Whether the correct units of time are months, years, or decades greatly determines how strongly society should consider acting. Regarding social segregation, our model suggests that the time for a minority group to attain equality is only affected by social segregation if very high levels of segregation exist. The result can be interpreted in light of the small world phenomenon. It takes but a few cross group connections to greatly shorten the distance from the members of the minority group to the job information held by the statusquo. This leads one to believe that the benefits to public policies that create even minimal amounts of integration could be extremely beneficial if extreme levels of segregation exist. But if segregation is only moderate, public policies that create more
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integration may have minimal effects at least in terms of creating access to jobs. Of course this is only one dimension policy makers may consider for wanting to increase the level of integration in society. Finally consider the results of the auto-correlation of agent employment states in our first experiment. Our results indicate that there may be strong relationships between the employment states of agents linked in the social network if networks are sufficiently non-random and referrals are important in the hiring process. But on average, since I = 0, any subset of agents in a social network will spend the same amount of time as all others in each employment state. There are two points to consider here. First, viewing correlations across agent networks is not sufficient to claim that there will be long term inequality. Second, the correct interpretation of time again becomes important. Even if these correlations are not long term, if they persist for generations we should be much more concerned about them than if they only persist for, say, a few years. If they appear to persist for a “long time,” as a society we may still want to attempt to limit these affects of referral hiring even if the effects are transitory.
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Summary and Future Work
The results of this paper may be divided into the effects of referral hiring and the effects of the social structure in the presence of referral hiring. In isolation, referral hiring does not create inequality by itself as long as groups have equivalent network structures. However, referral hiring does slow the rate of convergence to equality if a given group begins in a state of inequality. In this case random networks provide an advantage with respect to the speed at which a disadvantaged group attains equality. This is because random networks allow for information to diffuse faster. But if the
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groups do not have equivalent network structures, members of groups with more random social networks are at a disadvantage relative to members in groups with less random social networks. The difference in preferred network structure between models that look at steady state or equilibrium behavior versus models that look at the dynamic process of convergence to steady state highlights an important feature of agent-based models, as the former models leave out the dynamic story. One additional item which we do not consider in this work is the evolution of social norms and in particular education choices in a referral hiring labor market context. The opportunity cost and advantages of investing in education depend on how likely one is to find employment in an industry where education is valued. For instance, if a given person has poor labor market connections for gaining employment in an industry that values education highly she may choose to forgo investing in education. Once these co-evolutionary effects are considered a much richer understanding of the effects of referral hiring and social networks begins to develop. But, these effects also become much more complex and difficult to model and analyze. We have recently begun a study where agents evolve strategies for investing in education as a function of the labor market outcomes of their parents and social neighbors and hope this study will shed even more light on the labor market effects of referral hiring and social networks.
Acknowledgements We thank Ted Temzelides for helpful suggestions early in this research project and Scott Page for helpful comments on a previous draft.
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