A Social Network Analysis of Occupational Segregation
aa r X i v : . [ ec on . T H ] A p r A Social Network Analysis of Occupational Segregation ∗ I. Sebastian Buhai † and Marco J. van der Leij ‡ This version: March 2020
Abstract
We develop a network model of occupational segregation between social groups dividedalong gender or racial dimensions, generated by the existence of positive assortative match-ing among individuals from the same group. If referrals are important for job search, thenexpected homophily in the structure of job contact networks induces different career choicesfor individuals from different social groups. This further translates into stable occupationalsegregation equilibria in the labor market. We derive conditions for wage and unemploymentinequality in the segregation equilibria and characterize both the first and the second bestsocial welfare optima. We find that utilitarian socially optimal policies always involve seg-regation, but that integration policies are justifiable by additional distributional concerns.Our analysis suggests that social interaction through homophilous job referral networks is animportant channel for the propagation and persistence of gender and racial inequalities in thelabour market, complementary to classical theories such as taste or statistical discrimination.
JEL codes : D85, J15, J16, J24, J31
Keywords : Social Networks, Homophily, Occupational Segregation, Labour Market In-equality, Social Welfare ∗ For useful discussions and suggestions at previous stages of this paper, we are grateful to Willemien Kets,Mich`ele Belot, Sanjeev Goyal, Wilbert Grevers, Maarten Janssen, Joan de Mart´ı, Friederike Mengel, JamesMontgomery, Jos´e Luis Moraga-Gonz´alez, Wojciech Olszewski, Gilles Saint-Paul, Ott Toomet, Jan van Ours,Fernando Vega-Redondo, and Yves Zenou. We also thank audiences from seminars at Stockholm University,Babes-Bolyai University, Northwestern University, CPB Netherlands Bureau for Economic Policy Analysis, AarhusUniversity, Tartu University, Tinbergen Institute, University College London, and from numerous workshops andconferences. † SOFI at Stockholm University, NIPE at Minho University, and CEPREMAP Paris, email: [email protected] ‡ University of Amsterdam and Tinbergen Institute; email: [email protected] Introduction
Occupational segregation between various social groups is an enduring and pervasive phe-nomenon, with important implications for the labour market. There have been countless empiri-cal studies within sociology and economics documenting the extent and the shape of occupationalsegregation; Richard Posner summed them up concisely: “a glance of the composition of differ-ent occupations shows that in many of them, particularly racial, ethnic, and religious groups,along with one or the other sex and even groups defined by sexual orientation (heterosexual vs.homosexual), are disproportionately present or absent”. Most studies investigating the causesof labour market inequality agree that ’classical’ theories such as taste or statistical discrimina-tion by employers cannot, alone, explain pay, employment, and occupational disparities betweengenders or races, and their remarkable persistence over time. While several meritorious comple-mentary theories have been advanced, some leading social scientists have suggested that socialinteractions could also be an important, yet relatively little explored channel in this context, seefor instance Arrow (1998). In this paper, we investigate a potential network channel leading to occupational segregationand wage inequality in the labour market, by developing and analysing an intuitive, parsimonioussocial interactions model. We construct a four-stage model of occupational segregation betweentwo homogeneous, exogenously given, mutually exclusive social groups (e.g., genders or races)acting in a two-job labour market. In the first stage each individual chooses one of two specializededucations to become a worker. In the second stage individuals randomly form ”friendship” tieswith other individuals, with a tendency to form relatively more ties with members of the same For more recent overviews of potential channels explaining observed labour market inequality between genders,see for instance Bertrand (2011), Goldin and Katz (2011, 2016), or Blau and Kahn (2017). The role of informal personal networks for inter-gender labour market inequality had been emphasized insociology, often as part of the gender-specific ”social capital”, at least since Burt (1992,1998). In economics theinterest took up more slowly, but recently there has been a wave of studies on the role of personal networks forgender disparities on the labour market, using a diverse set of approaches and methodologies, see, e.g., Zeltzer(2020), Mengel (2020), or Lindenlaub and Prummer (2020). (inbreeding) homophily ”, “ inbreeding bias ” or” assortative matching ”. In the third stage workers use their networks of friendship contactsto search for jobs. In the fourth stage workers earn a wage and spend their income on a singleconsumption good.We obtain the following results. First, unsurprisingly, we show that with inbreeding ho-mophily within social groups, a complete polarization in terms of occupations across the twogroups arises as a stable equilibrium outcome. This result follows from standard arguments onnetwork effects. If a group is completely segregated and specialized in one type of job, theneach individual in the group has many more job contacts if she ”sticks” to her specialization.Hence, sticking to one specialization ensures good job opportunities to group members, andthese incentives stabilize segregation.We next extend the basic model allowing for “good” and “bad” jobs, in order to analyzeequilibrium wage and unemployment inequality between the two social groups. We show thatwith large differences in job attraction (i.e., wages at equal labour supply), the main outcomeof the model is that one social group ”fully specializes” in the good job, while the other group”mixes” over the two jobs. In this partial segregation equilibrium, the group that specializes inthe good job always has a higher payoff and a lower unemployment rate. Furthermore, with asufficiently large intra-group homophily, the fully-specializing group also has a higher equilibriumemployment rate and a higher wage rate than the ”mixing” group, thus being twice advantaged.Hence, our model is able to explain typical empirical patterns of gender, race, or ethnic labourmarket inequality. The driving force behind our result is the fact that the group that fullyspecializes, being homogenous occupationally, is able to create a denser job contact networkthan the mixing group. We emphasize at this point that we do not intend in any way to implythat there is no more taste or statistical discrimination by employers in the labour market. Onthe contrary, we regard our network interaction model, classical discrimination theories, as wellas other theories such as, e.g., gender-specific work amenity preferences, as complementary —allnecessary bits if we aim to fully explain the observed patterns of labour market inequality.We finally consider whether society benefits from an integration policy, in the sense thatlabour inequality between the social groups would be attenuated. To this aim, we analyze a socialplanner’s first and second-best policy choices. Surprisingly, segregation is the preferred outcomein the first-best analysis, while a laissez-faire policy leading to segregation shaped by individual Homophily measures the relative frequency of within-group versus between-group friendships. There existsinbreeding homophily or an inbreeding bias if the group’s homophily is higher than what would have been expectedif friendships are formed randomly. See, e.g., Currarini et al (2009) for formal definitions. This work points out thatindividual performance on the labour market crucially depends on the position individuals takein the social network structure. However, these studies typically do not focus on the role thatnetworks play in accounting for persistent patterns of occupational segregation and inequalitybetween races, genders or ethnicities. A recent exception is Bolte et al (2020) who discusshow the distribution of job referrals could lead to persistent inequality and intergenerationalimmobility. Here, instead of focusing on the network structure, we take a simpler reduced formapproach, which allows us to highlight the mechanism relating the role of the job networks inthe labour market to occupational segregation and inequality between social groups.The paper is organized as follows. The next section shortly overviews empirical findings onoccupational segregation. We review empirical evidence on the relevance of job contact networksand the extent of social group homophily in Section 2; we set up our model of occupational segre-gation in Section 3; and we discuss key results on the segregation equilibria in Section 4. Section5 analyses the social welfare outcome. We summarize and conclude the paper in Section 6.
In this section we present the empirical background that motivates the building blocks of ourmodel. We first discuss evidence on occupational segregation, and the relation to gender andrace wage gaps. Next we overview some empirical literature on the role of job contact networksand on homophily.
Although labour markets have become more open to traditionally disadvantaged groups, wagedifferentials by race and gender remain stubbornly persistent, see, e.g., Altonji and Blank (1999),Blau and Kahn (2000, 2006, 2017), or England et al (2020). Altonji and Blank (1999) note for The seminal paper on the role of networks in the labor markets is Montgomery (1991). Other well citedpapers include, e.g., Calv´o-Armengol and Jackson (2004, 2007), Ioannides and Soutevent (2006), or Bramoull´eand Saint-Paul (2010); for more general reviews on the economic analysis of social networks see,e.g., Goyal (2007),Jackson (2008), Goyal (2016). Calv´o-Armengol and Jackson (2004) find that two groups with two different networks may have differentemployment rates due to the endogenous decision to drop out of the labor market. However, their finding drawsheavily on an example that already assumes a large amount of inequality; in particular, the groups are initiallyunconnected and the initial employment state of the two groups is unequal. direct statistics on the occupational segregation and wage inequality patterns by gender, raceor ethnicity, see, e.g., Beller (1982), Albelda (1986), King (1992), Padavic and Reskin (2002),Charles and Grusky (2004), Cotter et al (2004), Blau and Kahn (2017), England et al (2020).They all agree that, despite substantial expansion in the labour market participation of womenand affirmative action programs aimed at labour integration of racial and ethnic minorities,women typically remain clustered in female-dominated occupations, while blacks and severalother races and ethnic groups are over-represented in some occupations and under-represented Some of these papers, e.g. Sørensen (2004), discuss in detail the extent of labor market segregation betweensocial groups, at the workplace, industry and occupation levels. Here we shall be concerned with modelingsegregation by occupation alone (known also as ”horizontal segregation”), which appears to be dominant at leastrelative to segregation by industry. Weeden and Sørensen (2004) convincingly show that occupational segregationin the USA is much stronger than segregation by industries and that if one wishes to focus on one single dimension,“occupation is a good choice, at least relative to industry”.
6n others; these occupations are usually of lower ’quality’, meaning inter alia that they are payingless on average, which explains partly the male-female and white-black wage differentials.King (1992) offers, for instance, detailed evidence that throughout 1940-1988 there was apersistent and remarkable level of occupational segregation by race and sex, such that “approx-imately two-thirds of men or women would have to change jobs to achieve complete genderintegration”, with some changes in time for some subgroups. Whereas occupational segregationbetween white and black women appears to have diminished during the 60’s and the 70’s, oc-cupational segregation between white and black males or between males and females remainedremarkable stable. Several studies by Barbara Reskin and her co-authors, c.f. the discussion andreferences in Padavic and Reskin (2002), document the extent of occupational segregation bynarrow race-sex-ethnic cells and find that segregation by gender remained extremely prevalentand that within occupations segregated by gender, racial and ethnic groups are also alignedalong stable segregation paths. England et al (2020) summarizes the trends in the segregationof occupations by means of an occupational segregation index D, ranging from 0 (complete in-tegration) to 1 (complete segregation), that ”has fallen steadily since 1970, with D moving from0.60 to 0.42. However, it moved much faster in the 1970s and 1980s than it has since 1990;segregation dropped by 0.12 in the 20-y period after 1970, but by a much smaller 0.05 in thelonger 26-y period after 1990.” Though most of these studies are for the USA, there is also in-ternational evidence confirming that, with some variations, similar patterns of segregation hold,e.g. Pettit and Hook (2005).The recent study by England et al (2020) on US exhaustive CPS data is relevant in severalrespects for our purpose at hand. Next to demonstrating, as already mentioned above, thatdespite some progress, the convergence on gender pay gaps, employment gaps and occupationaldesegregation has clearly slowed down, it also shows that the gender desegregation by fields ofstudy has stalled — and that despite the fact that women by now overtook men in terms of bothbachelor and doctoral degrees. Using a desegregation index for fields of study across genders,similar to the one for occupations, England et al (2020) show that ”[for bachelor degrees] it fellfrom 0.47 in 1970 to 0.28 in 1998, and has not gone down since, but rather, segregation has risenslightly. For doctoral degrees, segregation went from 0.35 in 1970 to a low of 0.18 in 1987 andhas hovered slightly higher since. In neither case has there been any net reduction in segregationfor over 20 years.” As the authors of that study also recognize, this is extremely important, as”segregated fields of study contribute to occupational gender segregation.” In the main part ofour paper, we actually provide a mechanism of exactly how that can happen and how that canthen lead to persistent inequality in wages and employment.7 .2 Job contact networks
There is by now an established set of facts showing the importance of the informal job networksin matching job seekers to vacancies. For instance, on average about 50 percent of the workersobtain jobs through their personal contacts, e.g. Rees (1966), Granovetter (1995), Holzer (1987),Montgomery (1991), Topa (2001); Bewley (1999) enumerates several studies published beforethe 90’s, where the fraction of jobs obtained via friends or relatives ranges between 30 and 60percent. It is also established that on average 40-50 percent of the employers actively use socialnetworks of their current employees to fill their job openings, e.g. Holzer (1987). Furthermore,employer-employee matches obtained via contacts appear to have some common characteristics.Those who found jobs through personal contacts were on average more satisfied with their job,e.g. Granovetter (1995), and were less likely to quit, e.g. Datcher (1983), Devine and Kiefer(1991), Simon and Warner (1992), Datcher Loury (2006). For more detailed overviews of studieson job information networks, see Ioannides and Datcher Loury (2004) or Topa (2011); for morerecent empirical research on the influence and value of job referral networks, see, e.g., Bayer etal (2008), Hellerstein et al (2011), or Burks et al (2015).
There is considerable evidence on the existence of social “homophily”, also labeled “assortativematching” or “inbreeding social bias”, that is, there is a higher probability of establishing linksamong people with similar characteristics. Extensive research shows that people tend to befriends with similar others, see McPherson et al. (2001) for a good review, with characteristicssuch as race, ethnicity or gender being essential dimensions of homophily. It has also beendocumented that friendship patterns are more homophilous than would be expected by chanceor availability constraints, even after controlling for the unequal distribution of races or sexesthrough social structure, e.g. Shrum, Cheek and Hunter (1988). There are also studies pointingtowards ”pure” same race preferences in marrying or dating, see, e.g., Wong (2003), Fishman et The difference in the use of informal job networks among professions is also documented. Granovetter (1995)pointed out that although personal ties seem to be relevant in job search-match for all professions, their incidenceis higher for blue-collar workers (50 to 65 percent) than for white-collar categories such as accountants or typists(20 to 40 percent). However, for certain other white-collar categories, the use of social connection in job findingis even higher than for blue-collars, e.g. as high as 77 percent for academics. The ”homophily theory” of friendship was first introduced and popularized by sociologists Lazarsfeld andMerton (1954), with Coleman (1958) introducing ”inbreeding homophily” indices, and the notion extensivelyused in sociology ever since. In economics, the notion got popular much later, in the second half of the 2000s.
8l (2006); among young kids, see, e.g., Hraba and Grant (1970), or among television audiences,see, e.g., Dates (1980) or Lee (2006).In our ”job information network” context, early studies by Rees (1966) and Doeringer andPiore (1971) showed that workers who had been asked for references concerning new hires werein general very likely to refer people ”similar” to themselves. While these similar featurescould be anything, such as ability, education, age, race and so on, the focus here is on groupsstratified along exogenous characteristics (i.e. one is born in such a group and cannot alterher group membership) such as those divided along gender, race or ethnicity lines. Indeed,most subsequent evidence on homophily was in the context of such ’exogenously given’ socialgroups. For instance, Marsden (1987) finds using the U.S. General Social Survey that personalcontact networks tend to be highly segregated by race, while other studies such as Brass (1985)or Ibarra (1992), using cross-sectional single firm data, find significant gender segregation inpersonal networks. More recent evidence on various homophilous social networks is also givenby Mayer and Puller (2008), Currarini et al. (2009), or Zeltzer (2020).Direct evidence of large gender homophily within job contact networks comes from tabula-tions in Montgomery (1992). Over all occupations in a US sample from the National Longi-tudinal Study of Youth, 87 percent of the jobs men obtained through contacts were based oninformation received from other men and 70 percent of the jobs obtained informally by womenwere as result of information from other women. Montgomery shows that these outcomes holdeven when looking at each narrowly defined occupation categories or one-digit industries, in-cluding traditionally male or female dominated occupations, where job referrals for the minoritygroup members were obtained still with a very strong assortative matching via their own gendergroup. For example, in male-dominated occupations such as machine operators, 81 percent ofthe women who found their job through a referral, had a female reference. Such figures aresurprisingly large and are likely to be only lower bounds for magnitudes of inbreeding biaseswithin other social groups. Weeden and Sørensen (2004) estimate a two-dimensional model of gender segregation, by industry and occu-pation: they find much stronger segregation across occupations than across industries. 86% of the total associationin the data is explained by the segregation along the occupational dimension; this increases to about 93% onceindustry segregation is also accounted for. The gender homophily is likely to be smaller than race or ethnic homophily, given frequent close-knit rela-tionships between men and women. This is confirmed for instance by Marsden (1988), who finds strong inbreedingbiases in contacts between individuals of the same race or ethnicity, but less pronounced homophily within gendercategories. recruitment and the hiring stages for an entry-level job ata call center of a large US bank. This study also finds that contact networks contribute to thegender skewing of jobs, in addition documenting directly that there is strong evidence of genderhomophily in the refereeing process: referees of both genders tend to strongly produce same sexreferrals.Finally, we briefly address the relative importance of homophily within ”exogenously given”versus ”endogenously created” social groups. As mentioned above, assortative matching takesplace along a great variety of dimensions. However, there is empirical literature suggesting thathomophily within exogenous groups such as those divided by race, ethnicity, gender, and- to acertain extent- religion, typically outweighs assortative matching within endogenously formedgroups such as those stratified by educational, political or economic lines. E.g., Marsden (1988)finds for US strong inbreeding bias in contacts between individuals of the same race or ethnicityand less pronounced homophily by education level. Another study by Tampubolon (2005),using UK data, documents the dynamics of friendship as strongly affected by gender, maritalstatus and age, but not by education, and only marginally by social class. These facts motivatewhy we focus here on ”naturally” arising social groups, such as gender, racial or ethnic ones;nevertheless, as will become clear in the modeling, assuming assortative matching by education, in addition to gender, racial or ethnic homophily, does not matter for our conclusions.
Based on the stylized facts mentioned in Section 2.2, we build a parsimonious theoretical model ofsocial network interaction able to explain stable occupational segregation, and employment andwage gaps, that can complement existing theories and fill the remaining unexplained disparitiesin inter-genders, or inter-races, or even inter-ethnicities, labour market outcomes.Let us consider the following setup. A continuum of individuals with measure 1 is equallydivided into two social groups, Reds ( R ) and Greens ( G ). The individuals are ex ante homoge-neous apart from their social color. They can work in two occupations, A or B . Each occupationrequires a corresponding thorough specialized education (career track), such that a worker can-not work in it unless she followed that education track. We assume that it is too costly for10ndividuals to follow both educational tracks. Hence, individuals have to choose their educationtrack before they enter the labour market. Consider now the following order of events:1. Individuals choose one education in order to specialize either in occupation A or in occu-pation B ;2. Individuals randomly establish “friendship” relationships, thus forming a network of con-tacts;3. Individuals participate in the labour market. Individual i obtains a job with probability s i .4. Individuals produce a single good for their firms and earn a wage w i . They obtain utilityfrom consuming goods that they buy with their wage.We proceed with an elaboration of these steps. The choice of education in the first stage involves strategic behavior. Workers choose the educa-tion that maximizes their expected payoff given the choices of other workers, and we thereforelook for a Nash equilibrium in this stage. This can be formalized as follows.Denote by µ R and µ G the fractions of Reds and respectively Greens that choose education A . It follows that 1 − µ X of group X ∈ { R, G } chooses education B . The payoffs will dependon these strategies: the payoff of a worker of group X that chooses education A is given byΠ XA ( µ R , µ G ), and mutatis mutandis, Π XB ( µ R , µ G ). Define ∆Π X ≡ Π XA − Π XB . The functionalform of the payoffs is made more specific later, in Subsection 3.4.In a Nash equilibrium each worker chooses the education that gives her the highest payoff,given the education choices of all other workers. Since workers of the same social group arehomogenous, a Nash equilibrium implies that if some worker in a group chooses education A For example, graduating high school students may face the choice of pursuing a medical career or a career intechnology. Both choices require several years of expensive specialized training, and this makes it unfeasible tofollow both career tracks. B ), then no other worker in the same group should prefer education B ( A ). This implies thata pair ( µ R , µ G ) is an equilibrium if and only if, for X ∈ { R, G } , the following hold: ∆Π X ( µ R , µ G ) ≤ µ X = 0 (1)∆Π X ( µ R , µ G ) = 0 if 0 < µ X < X ( µ R , µ G ) ≥ µ X = 1 . (3)To strengthen the equilibrium concept, we restrict ourselves to stable equilibria . We use asimple stability concept based on a standard myopic adjustment process of strategies, whichtakes place before the education decision is made. That is, we think of the equilibrium asthe outcome of an adjustment process. In this process, individuals repeatedly announce theirpreferred education choice, and more and more workers revise their education choice if it isprofitable to do so, given the choice of the other workers. Concretely, we consider stationarypoints of a dynamic system guided by the differential equation ˙ µ X = k ∆Π X ( µ R , µ G ). Thisimplies that µ ≡ ( µ R , µ G ) is a stable equilibrium if it is an equilibrium and (i) for X ∈ { R, G } : ∂ ∆Π X /∂µ X < X = 0; (ii) det( D ∆Π( µ )) > R = 0 and ∆Π G = 0, where D ∆Π( µ )is the Jacobian of (∆Π R , ∆Π G ) with respect to µ .We assumed that individuals first choose an education, and then form a network of jobcontacts (see next subsection). As a consequence, individuals have to make expectations aboutthe network they could form, and base their education decisions on these expectations. This isin contrast to some of the earlier work on the role of networks in the labour market. In thatresearch, the network was supposed to be already in place, or the network was formed in thefirst stage (Montgomery 1991, Calv´o-Armengol 2004, Calv´o-Armengol and Jackson 2004).Our departure from the earlier frameworks raises legitimate questions about the assumedtiming of the education choice. Are crucial career decisions made before or after job contactsare formed? One might be tempted to answer: both. Of course everyone is born with family The question whether the equilibrium is in pure or mixed strategies is not relevant, because the player set is ameasure of identical infinitesimal individuals (except for group membership). Our equilibrium could be interpretedas a Nash equilibrium in pure strategies; then µ X is the measure of players in group X choosing pure strategy A . The equilibrium could also be interpreted as a symmetric Nash equilibrium in mixed strategies; in that casethe common strategy of all players in group X is to play A with probability µ X . A hybrid interpretation is alsopossible. One could think of such a process as the discussions students have before the end of the high school abouttheir preferred career. An alternative with a longer horizon is an overlapping generations model, in which theeducation choice of each new generation partly depends on the choice of the previous generation. p + κ + λ p + λ group different p + κ p ties, and in early school and in the neighborhood children form more ties. It is also known thatpeer-group pressure among children has a strong effect on decisions to, for instance, smoke orengage in criminal activities and, no doubt, family and early friends do form a non-negligiblesource of influence when making crucial career decisions. However, we argue that most job-relevant contacts (the so called ’instrumental ties’) are made later, for instance at the university,or early at the workplace, hence after a specialized career track had been chosen. In spite ofthe fact that those ties are typically not as strong as family ties, they are more likely to providerelevant information on vacancies to job seekers; Granovetter (1973, 1985) provides convincingevidence that job seekers more often receive crucial job information from acquaintances (”weakties”), rather than from family or very close friends (”strong ties”). If the vast majority of suchinstrumental ties are formed after the individual embarked on a (irreversible) career, then it isjustified to consider a model in which the job contact network is formed after making a careerchoice. In the second stage the workers form a network of contacts. We assume this network to berandom, but with social color homophily. That is, we assume that the probability for twoworkers to create a tie is p ≥ λ >
0. Similarly, if two workers choose the sameeducation, then the probability of creating a tie increases with κ ≥
0. Hence, we allow forassortative matching by education, in addition to the one by social color. We do not imposeany further restrictions on these parameters, other than securing p + λ + κ ≤ . This leads tothe tie formation probabilities from Table 1. We shall refer to two workers that create a tie as“friends” 13e assume the probability that an individual i forms a tie with individual j to be exogenouslygiven and constant. In practice, establishing a friendship between two individuals typically in-volves rational decision making. It is therefore plausible that individuals try to optimize theirjob contact network in order to maximize their chances on the labour market. In particular,individuals from the disadvantaged social groups should have an incentive to form ties with indi-viduals from the advantaged group. While this argument is probably true, we do not incorporatethis aspect of network formation in our model. The reality is that strategic network formationdoes not appear to dampen the inbreeding bias in social networks significantly; in Section 2.2we provided evidence that strong homophily exists even within groups that have strong labourmarket incentives not to preserve such homophily in forming their ties. The reason could be thatthe payoff of forming a tie is mainly determined by various social and cultural factors, and onlyfor a smaller part by benefits from the potential transmission of valuable job information. Ontop of that, studies such as, for instance, Granovetter (2002), also note that many people wouldfeel exploited if they find out that someone befriends them for the selfish reason of obtaining jobinformation. These elements might hinder the role of labour market incentives when formingties. Hence, while we do not doubt that incentives do play a role when forming ties, we believethese incentives are not sufficient to undo the effects of the social color homophily. We thereforeassume network formation exogenous in this paper.
The third stage we envision for this model is that of a dynamic labour process, in which infor-mation on vacancies is propagated through the social network, as in, e.g., Calv´o-Armengol andJackson (2004), Calv´o-Armengol and Zenou (2005), Ioannides and Soetevent (2006) or Bramoull´eand Saint-Paul (2006). Workers who randomly lose their job are initially unemployed because ittakes time to find information on new jobs. The unemployed worker receives such informationeither directly, through formal search, or indirectly, through employed friends who receive theinformation and pass it on to her (in the particular case where all her friends are unemployed,only the formal search method works). As the specific details of such a process are not importantfor our purposes, we do not consider these dynamic models explicitly, but take a ”reduced form”approach. See Calv´o-Armengol (2004) for a model of strategic network formation in the labor market. Currarini et al.(2009) discuss a model of network formation in which individuals form preferences on thenumber and mix of same-group and other-group friends. In this model inbreeding homophily arises endogenously.
14n particular, we assume that unemployed workers have a higher propensity to receive jobinformation when they have more friends with the same job background , that is, with the samechoice of education . On the one hand, this assumption is based on the result of Ioannides andSoetevent (2006) that in a random network setting the individuals with more friends have alower unemployment rate. On the other hand, this assumption is based on the conjecturethat workers are more likely to receive information about jobs in their own occupation. Forexample, when a vacancy is opened in a team, the other team members are the first to knowthis information, and are also the ones that have the highest incentives to spread this informationaround.Formally, denote the probability that individual i becomes employed by s i = s ( x i ), where x i is the measure of friends of i with the same education as i has. We thus assume that s ( x )is differentiable, 0 < s (0) < s ′ ( x ) > x > s i depends on the education choices of i and the choices of allother workers. Remember that µ R and µ G are the fractions of Reds and respectively Greensthat choose education A . Given the tie formation probabilities from Table 1 and some algebra,the employment rate s XA of A -workers in group X ∈ { R, G } will be given by: s XA ( µ R , µ G ) = s (( p + κ )¯ µ + λµ X /
2) (4)and likewise, the employment rate s XB of B -workers in group X will be s XB ( µ R , µ G ) = s (( p + κ )(1 − ¯ µ ) + λ (1 − µ X ) /
2) (5)where ¯ µ ≡ ( µ R + µ G ) / s XA > s YA and s XB < s YB for X, Y ∈ {
R, G } , X = Y , if and only if µ X > µ Y and λ >
0. We will see in Section 4.1 that the ranking of the employment rates is crucial, asit creates a group-specific network effect. That is, keeping this ordering, if only employmentmatters (jobs are equally attractive), then individuals have an incentive to choose the sameeducation as other individuals in their social group. Importantly, it is straightforward to see This result is nontrivial, as the unemployed friends of employed individuals tend to compete with each otherfor job information. Thus, if a friend of a jobseeker has more friends, the probability that this friend passesinformation to the jobseeker decreases. In fact, in a setting in which everyone has the same number of friends,Calv´o-Armengol and Zenou (2005) show that the unemployment rate is non-monotonic in the (common) numberof friends. λ , but it does not depend on κ . Therefore,only the homophily among members of the same social group- and not the eventual assortativematching by education- is relevant to our results. The eventual payoff of the workers depends on the wage they receive, the goods they buy withthat wage, and the utility they derive from consumption. Without loss of generality we assumethat an unemployed worker receives zero wage. However, the wages of employed workers are notexogenously given, but they are determined by supply and demand.When firms offer wages, they take into account that there are labour market frictions andthat it is impossible to employ all workers simultaneously. Thus what matters is the effectivesupply of labour as determined by the labour market process in stage 3. Let L A be the totalmeasure of employed A -workers and L B be the total measure of employed B -workers. Hence, L A ( µ R , µ G ) = µ R s RA ( µ R , µ G ) / µ G s GA ( µ R , µ G ) / L B ( µ R , µ G ) = (1 − µ R ) s RB ( µ R , µ G ) / − µ G ) s GB ( µ R , µ G ) / . (7)Given (4) and (5) from above, it is easy to check that L A is increasing with µ R and µ G , whereas L B is decreasing with µ R , µ G .As in Benabou (1993), consumption, prices, utility, the demand for labour and the impliedwages are determined in a 1-good, 2-factor general equilibrium model. All individuals havethe same utility function U : R + → R , which is strictly increasing and strictly concave with U (0) = 0. The single consumer good sells at unit price, such that consumption of this goodequals wage and indirect utility is given by U i = U ( w i ).Firms put A -workers and B -workers together to produce the single good at constant returnsto scale. Wages are then determined by the production function F ( L A , L B ). As usually, weassume that F is strictly increasing and strictly concave in L A and L B and ∂ F/∂L A ∂L B > A -workersand B -workers, w A and w B , are given by w A ( µ R , µ G ) = ∂F∂L A ( L A ( µ R , µ G ) , L B ( µ R , µ G )) , and w B ( µ R , µ G ) = ∂F∂L B ( L A ( µ R , µ G ) , L B ( µ R , µ G )) .
16t is easy to check that w A is strictly decreasing with µ R and µ G , and mutatis mutandis, w B .We can now define the payoff of a worker as her expected utility at the time of decision-making. The payoff function of an A -educated worker from social group X ∈ { R, G } is thusΠ XA ( µ R , µ G ) = s XA ( µ R , µ G ) U ( w A ( µ R , µ G )) . (8)Similarly, Π XB ( µ R , µ G ) = s XB ( µ R , µ G ) U ( w B ( µ R , µ G )) . (9)If we do not impose further restrictions, then there could be multiple equilibria, most ofthem uninteresting. To ensure a unique equilibrium in our model (actually: two symmetricequilibria), we make the following two assumptions. Assumption 1
For the wage functions w A and w B lim x ↓ U ( w A ( x, x )) = lim x ↓ U ( w B (1 − x, − x )) = ∞ . Assumption 2
For X ∈ { R, G } , and for all µ R , µ G ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂s XA /s XA ∂µ X /µ X (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) ∂U/U∂w A /w A (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂w A /w A ∂µ X /µ X (cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12) ∂s XB /s XB ∂µ X /µ X (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) ∂U/U∂w B /w B (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂w B /w B ∂µ X /µ X (cid:12)(cid:12)(cid:12)(cid:12) . Assumptions 1 and 2 guarantee the uniqueness of our results. Assumption 1 implies thatthe wage for scarce labour is so high that at least some workers always find it attractive tochoose education A or respectively B ; everyone going for one of the two educations cannot bean equilibrium. In Assumption 2 we assume that the education choice of an individual has asmaller marginal effect on the employment probability within a group than on the wages andoverall utility. Note that the assumption implies that for X ∈ { R, G } ∂ Π XA ∂µ X < < ∂ Π XB ∂µ X , and it is this feature that guarantees the uniqueness of our results. The assumption is notrestrictive as long as there is sufficient direct job search, because the employment probabilityof each individual in our model is bounded between s (0) > s (0) capturing the17mployment probability in the absence of any ties and thus induced only by the exogenouslygiven direct job finding rate. Hence, a higher s (0) implies less of an impact of the network effecton the employment rate.It should be noted that we make these assumptions above only in order to focus our analysison segregation outcomes, for the sake of clarity and brevity. These assumptions are not necessary.For instance, in the calibration exercise of Section 5.2.1, Assumption 2 is violated, but there arestill (two) unique equilibria. We now present the equilibrium analysis of our model. The formal proofs of all subsequentpropositions are relegated to the Appendix. Without loss of generality we assume throughoutthe section that w A (1 , ≥ w B (1 , A -occupation is weakly more attractive thanthe B -occupation when effective labour supply is equal. We call A the “good” job, and B the“bad” job. We are in particular interested in those equilibria in which there is segregation. We define complete segregation if µ R = 0 and µ G = 1, or, vice versa, µ R = 1 and µ G = 0. On the otherhand, we say that there is partial segregation if for X ∈ { R, G } and Y ∈ { R, G } , Y = X : µ X = 0but µ Y <
1, or, vice versa, µ X = 1 but µ Y > Proposition 1
Suppose Assumptions 1 and 2 hold. Define s H ≡ s (( p + κ + λ ) / and s L ≡ s (( p + κ ) / .(i) If ≤ U ( w A (1 , U ( w B (1 , ≤ s H s L , (10) then there are exactly two stable equilibria, both with complete segregation.(ii) If U ( w A (1 , U ( w B (1 , > s H s L , (11) then there are exactly two stable equilibria, both with partial segregation, in which either µ R = 1 or µ G = 1 .
18e first note that a non-segregation equilibrium cannot exist, even in the case of a tinyamount of homophily λ . The intuition is that homophily in the social network among membersof the same social group creates a group-dependent network effect. Thus, if slightly more Redworkers choose A than Greens do, then the value of an A -education is higher for the Reds thanfor the Greens, while the value of a B -education is lower in the Reds’ group. Positive feedbackthen ensures that the initially small differences in education choices between the two groupswiden and widen, until at least one group segregates completely into one type of education.Second, if the wage differential between the two jobs (for equal numbers of A-educated andB-educated workers) is not ”too large” vis-`a-vis the social network effect (condition 10), completesegregation is the only stable equilibrium outcome, given a positive inbreeding bias in the socialgroup. Thus one social group specializes in one occupation, and the other group in the otheroccupation. On the other hand, the proposition makes clear that complete segregation cannotbe sustained if the wage differential is ”too large” vis-`a-vis the social network effect (condition11). Starting from complete segregation, a large wage differential gives incentives to the groupspecialized in B -jobs to switch to A -jobs.Interestingly, the ”unsustainable” complete segregation equilibrium is then replaced by apartial equilibrium in which one group specializes in the “good” job A , while the other group hasboth A and B -workers. Partial segregation in which one group, say the Greens, fully specializesin the “bad” job B is unsustainable, as that would lead to an oversupply of B -workers and aneven larger wage differential. This would provide the Red B -workers with strong incentives toswitch en masse to the A -occupation. The discussion so far ignored eventual equilibrium differentials in wages and unemploymentbetween the two types of jobs. We now tackle that case. We continue to assume that w A (1 , ≥ w B (1 ,
0) and, in light of the results of Proposition 1, we focus without loss of generality on theequilibrium in which µ R = 1. Thus, the Reds specialize in the “good” job A , while the “bad”job B is only performed by Green workers.We first consider the case in which wage differentials are small enough so that completesegregation is an equilibrium ( µ R = 1 and µ G = 0). In this case the implications are straightfor-ward. Since both groups specialize in equal amounts, the network effects are equally strong, andthe employment rates are equal. Given that employment rates are equal, the effective laboursupply is also equal, and therefore the wage of the “good” job is weakly higher. We thus havethe following result: 19 roposition 2 Suppose Assumptions 1 and 2 hold. Define s H ≡ s (( p + κ + λ ) / and s L ≡ s (( p + κ ) / and suppose that ≤ w A (1 , w B (1 , ≤ s H s L . Suppose ( µ R , µ G ) = (1 , is a stable equilibrium.In that equilibrium w A ≥ w B ,s RA = s GB > s RB = s GA , and Π RA ≥ Π GB ≥ Π GA ≥ Π RB . (12)Next, we turn to the analysis of the more interesting case in which wage differentials arelarge. In that case there is a partial equilibrium in which ( µ R , µ G ) = (1 , µ ∗ ) where µ ∗ ∈ (0 , GA (1 , µ ∗ ) = Π GB (1 , µ ∗ ) , or equivalently s GA (1 , µ ∗ ) U ( w A (1 , µ ∗ )) = s GB (1 , µ ∗ ) U ( w B (1 , µ ∗ )) . Thus, whereas workers in group R prefer the A -job, the workers in group G make an individualtrade-off: lower wages should be exactly compensated by higher employment probabilities andvice versa.We are particularly interested in whether this individual trade-off between unemploymentand wages translates into a similar trade-off at the ’macro-level’, in which an inter-group wagegap is compensated by a reversed employment gap. We have the following proposition. Proposition 3
Suppose Assumptions 1 and 2 hold. Define s H ≡ s (( p + κ + λ ) / and s L ≡ s (( p + κ ) / and suppose that w A (1 , w B (1 , > s H s L . Define ˆ µ ∈ (0 , , such that w A (1 , ˆ µ ) = w B (1 , ˆ µ ) , (13) and let ( µ R , µ G ) = (1 , µ ∗ ) be a stable equilibrium. In that equilibrium Π RA > Π GB = Π GA > Π RB . (14) Moreover, i) if ˆ µ < λ p + κ + λ ) , then s RA > s GB > s GA > s RB , and w A (1 , µ ∗ ) > w B (1 , µ ∗ ); (ii) if ˆ µ > λ p + κ + λ ) , then s RA > s GA > s GB > s RB , and w B (1 , µ ∗ ) > w A (1 , µ ∗ ) . The main implication of this proposition is that an inter-group wage gap is not compensatedby a reversed employment gap. On the contrary, it is possible that the group specializing inthe good job, here the Reds, both earns a higher wage and has higher employment probabilitiesthan the Greens group. This is especially clear when the group homophily bias λ is large relativeto p and κ (in fact p + κ ) and there is a big difference in attractiveness between the good andthe bad jobs (case (i) above).This result can be understood by the following observation: the workers in the ’specializing’group R have a higher employment probability than all workers in group G . This is always thecase, regardless of whether the individual in G is an A or a B worker, and whether s GB > s GA or not. As all members of group R choose the same occupation, the Reds remain a stronghomogenous social group. Network formation with homophily then implies that they are ableto create a lot of ties, and hence, that they benefit most from their social network. On the otherhand, the Greens are dispersed between two occupations. This weakens their social network andthis decreases their chances on the labour market, both for A and B -workers in group G .Whether the wage differential between the workers in the two groups is positive or negativedepends on the relative size of λ relative to p + κ , in the term λ p + κ + λ ) from the inequalityconditions in Proposition 3. This can be roughly assessed in light of the empirical evidence onhomophily discussed earlier in this paper. First, as seen from the stylized facts from Section 2.2,the assortative matching by education, κ , is typically found to be lower relative to racial, ethnicalor gender homophily. The second interesting situation is a scenario where the probability ofmaking contacts in general, p , were already extremely high relative to the intra-group homophilybias. However, given the surprisingly large size of intra-group inbreeding biases in personalnetworks of contacts found empirically, this is also unlikely. Hence, the likelihood is very highthat in practice λ would dominate the other parameters in the cutoff term λ p + κ + λ ) .21et us summarize the implications of this last proposition. The fully specializing group isalways better off in terms of unemployment rate and payoff, independent of either relative orabsolute sizes of λ , p and κ (as long as λ > λ dominant relativeto p and κ is likely to be met. This ensures that the group fully specializing in the good jobalways has a higher wage in the equilibrium than the group mixing over the two jobs, as provenin Proposition 3. Note that this partial segregation equilibrium is in remarkable agreement withobserved occupational, wage and unemployment disparities in the labour market between, forinstance, males-females or blacks-whites. This suggests that our simple model offers a plausibleexplanation for major empirical patterns of labour market inequality. In the previous section we observed that individual incentives lead to occupational segregationand wage and unemployment inequality. This suggests that a policy targeting integration mayreduce inequality as well, and in fact may just be socially beneficial. This is an argument oftenused for instance by proponents of positive discrimination. We set out here to analyze theimplications of our model from a social planner’s point of view.Consider a utilitarian social welfare function: W ( µ R , µ G ) = µ R Π RA / − µ R )Π RB / µ G Π GA / − µ G )Π GB / , (15)where Π XA ≡ Π XA ( µ R , µ G ) and Π XB ≡ Π XB ( µ R , µ G ) are given by equations (8) and (9). Sinceunemployed workers obtain zero utility, we can also write the welfare function as W ( µ R , µ G ) = L A U (cid:18) ∂F∂L A ( L A , L B ) (cid:19) + L B U (cid:18) ∂F∂L B ( L A , L B ) (cid:19) , (16)where L A ≡ L A ( µ R , µ G ) and L B ≡ L B ( µ R , µ G ) were introduced by (6) and (7). The formulationin (16) is useful, because it shows that what matters for social welfare is the effect of a policyon the society’s effective labour supply.We consider a first-best social optimum, that is, the social planner is able to fully manage µ R ∈ [0 ,
1] and µ G ∈ [0 ,
1] and therefore the social optimum µ S = ( µ SR , µ SG ) is defined as µ S = argmax µ R ∈ [0 , ,µ G ∈ [0 , W ( µ R , µ G ) . We obtain the following result: 22 roposition 4
If for all x ∈ [0 , ( p + κ + λ ) /
2] : s ′′ ( x ) > − λ s ′ ( x ) (17) then any social optima involves complete or partial segregation. Thus a segregation policy is socially preferred, as long as s ( x ), the employment probabilityof having x friends with the same education, is ”not too concave”. This proposition can beintuitively understood as follows. Suppose that there is no segregation, and 0 < µ G < µ R < A -occupation, s RA > s RB ,whereas the Greens have a higher employment rate as B -workers, s GB > s GA . Now consider theeffect on segregation, wages and employment when a social planner forces a Red individualinitially choosing a B -occupation and respectively, a Green individual initially choosing an A -occupation, into switching their occupation choice . In that case µ R slightly increases, whereas µ G slightly decreases. The result of this event is, first, that segregation increases; the gap between µ R and µ G becomes larger. Second, the total fraction of individuals that choose occupation A , µ R + µ G , does not change. So the ratio of A -workers versus B -workers does not change much, andtherefore the ratio of wages does not change much either. Thus the effect on wage inequalityis only marginal. Third, by switching occupations, the Red worker can now benefit from adenser network, and have an employment probability s RA instead of s RB . The same is true for theGreen worker switching from B to A . Thus, the combined payoff of the two workers increases,as they are both more likely to become employed. We also need to consider the externalityon the employment rates of the workers not involved in the occupation switch. In particular,the switch of occupations increases the network effects of the other Red A -workers and Green B -workers, whereas it decreases the network effects of Red B -workers and Green A -workers.The restriction on the concavity of s ( x ) ensures that the switch of occupations puts on averagea positive externality on the employment probabilities of other workers. We conclude that theswitch of occupations of the two workers hardly affects wage inequality, while it increases thelabour supply of both A and B . Therefore, social welfare increases.The general message of this result is that an integration policy might also have unintendeddetrimental effects on employment. Under our model’s assumptions, integration might weakenthe employment chances of individuals, because the network effects are weaker in mixed net-works. In the case of complete segregation, individuals are surrounded by similar individualsduring their education. Therefore, it is easier for them to make many friends they can rely onwhen searching on the job market. Consequently, employment probabilities are high. On theother hand, if educations are mixed, then individuals have more difficulties in creating useful job23ontacts, and therefore their employment probabilities are lower. It is worth emphasizing thatthe result that integration weakens network effects and decreases labor market opportunitieshas empirical support in related literature on segregation. For example, Currarini et al. (2009)find clear evidence that larger (racial) minorities create more friendships, and Marsden (1987)finds a similar pattern in his network of advice. Therefore, it is more beneficial for a worker tochoose an education in which she is only surrounded by similar others, instead of an education inwhich racial groups are mixed, let alone one in which she is a small minority. In a different butrelated context, Alesina and La Ferrara (2000, 2002) find that participation in social activitiesis lower in racially mixed communities and so is the level of trust. These and our results suggestthat possible negative impacts of integration on social network effects should also be taken intoaccount.Our outcome on the first-best social optimum hinges for a large part on the fact that the socialplanner is able to increase employment by increasing segregation, while still controlling wageinequality. In reality however, a social planner may not have this amount of control. Perhapsa more feasible policy is a policy in which the social planner enforces and stabilizes integration,but where the exact allocation of workers to occupations is determined by individual incentives.In the case of segregation there would be a potentially large inequality in payoffs between thesocial groups, whereas in the case of integration there may be complete payoff equality, butemployment may be lower. This suggests a second-best analysis of social welfare, in which thereis a potential trade-off of segregation between network benefits and inequality. Such an analysisis unfeasible without further specification of the parameters, hence we will perform that analysissubsequent to calibrating the model for suitable parameters and functional forms. We calibrate the parameters, in order to perform a small numerical simulation of our model.The purpose of this simulation is to get a better feeling on the mechanisms of the model, therestrictiveness of our assumptions, and the magnitude of the wage gap that can be generated.This straightforward simulation also allows us to get some key insights about a second-bestwelfare policy. A detailed analysis would require an extension of the model and is beyond thescope of this paper.We first specify functional forms for s ( x ), the employment probability as function of thenumber of friends with the same education, F ( L A , L B ), the production function and thus thederived wage functions, and U ( x ), the utility function. Regarding the employment probability,24e consider a function that follows from a dynamic labour process, in which employed individualsbecome unemployed at rate 1, and in which unemployed individuals become employed at rate c + c x , where c is the rate at which unemployed workers directly obtain information on jobvacancies, and c measures the strength of having friends. This leads to the following employmentfunction: s ( x ) = c + c x c + c x . Since we have defined s = s (0) as the employment probability when only direct search is used,it follows that s = c / (1 + c ).For the production function we assume the commonly used Cobb-Douglas function withconstant returns to scale, F ( L A , L B ) = θL αA L − αB . For the utility function we consider a function with constant absolute risk aversion, where ρ isthe coefficient of absolute risk aversion. That is U ( x ) = 1 − e − ρx . We calibrate the parameters s , c ( p + κ ) , c λ, p and θ , leaving α as a free parameter. First,we calibrate s , c ( p + κ ), and c λ from three equations that are motivated by the empiricalevidence given in Section 2 and 3. This parameterization is sufficient to perform the simulation,and it is thus not necessary to separately specify c , p , κ and λ . The first equation is obtainedby imposing the restriction that about 50% of the workers find their job through friends, assuggested in Section 2. This restriction implies that the direct job arrival rate c should equalthe indirect job arrival rate through friends c x . The indirect job arrival rate differs, dependingon the choices of the individuals, but if we focus on the case complete segregation, in which µ R = 1 and µ G = 0, then we can impose the following restriction: c = c ( p + κ + λ ) / . Next, we calibrate the amount of inbreeding homophily in the social group. This amounttypically differs depending on the group defining characteristic. For example, analyzing data onFacebook participants at Texas A&M, Mayer and Puller (2008) find that two students living inthe same dorm are 13 more likely to be friends than two random students, two black students 17more likely, but two Asian students 5 times more likely and two Hispanic students twice as likelyto be friends. In light of this evidence, we chose to keep the amount of inbreeding homophily inthe simulation modest, imposing λ = 3( p + κ ).25able 2: Chosen parameter values in the simulation and the sensitivity with respect to ˆ α andthe maximum wage gap.Parameter Value Elasticity of ˆ α Elasticity of wage gapˆ α = . G (1 ,
0) = . s .9048 -1.71 -9.47 c ( p + κ ) 4.75 -.04 -.23 c λ ρ . × − .38 2.09 θ c c = 0 . , and this implies that s = c c = 1921 ≈ . . and further that c ( p + κ ) = 4 .
75 and c λ = 14 . θ and the coefficient of absolute risk aversion ρ . The coefficient of absolute risk aversion has been estimated between 6 . × − and 3 . × − (Gertner 1993, Metrick 1995, Cohen and Einav 2007 ). We set the risk aversion at 1 . × − ,which means a coefficient of relative risk aversion of 4 at a wealth level of $ 40,000, or indifferenceat participating in a lottery of getting $ 100.00 or losing $ 99.01 with equal probability.The productivity parameter, θ , is chosen such that average income equal $ 40,000 in thecase of complete segregation, ( µ R , µ G ) = (1 , α = .
5. Since in that situation w A (1 ,
0) = w B (1 ,
0) = θ/
2, we have θ = 80 , α with s , c ( p + κ ), c λ , ρ and θ as summarized in Table 2, and in which µ R and µ G are determined by equilibriumconditions (1)-(3). Given the result of Proposition 1 that there is either a complete equilibriumor a partial equilibrium, in which one group specializes in the good job, we concentrate ourattention to the parameter space in which α ∈ [1 / , µ R = 1 and µ G ∈ [0 , A is “good”, and group R specializes in A .We first show a plot of ∆Π G (1 , µ G ) as a function of µ G for different values of α . This functionillustrates the payoff evaluation that a Green individual makes when deciding on its occupation.If ∆Π G (1 , µ G ) > ( < )0, then the Green individual prefers A ( B ) if she beliefs that all Reds26 elta Pi^G0,40,30,20,10-0,1-0,2-0,3 mu_G 0,80,60,40,20 alpha=.5 alpha=.6 alpha=.7 alpha=.8 alpha=.9 Figure 1: ∆Π G (1 , µ G ) as a function of µ G for different values of α .choose A and fraction µ G of Greens choose A . Clearly, in an equilibrium it should hold thateither ∆Π G (1 , < G (1 , µ G ) = 0.The plot is displayed in Figure 1. This plot nicely illustrates the workings of the model.First, note that for α = .
5, ∆Π G (1 , µ G ) is clearly negative, so given that the Reds choose A, theGreens prefer B and complete segregation is an equilibrium. However, ∆Π G (1 , µ G ) increaseswith α , such that for α > . ≡ ˆ α , we have that ∆Π G (1 , > If α < . s RA = s GB = .
95 and s RB = s GA = . . Wages have a particular simple form in the case of complete segregation, being w A (1 ,
0) = θα and w B (1 ,
0) = θ (1 − α ). Therefore, if we define the wage gap as G ( µ R , µ G ) = 1 − w B ( µ R , µ G ) /w A ( µ R , µ G ), then the wage gap under complete segregation is G (1 ,
0) = 2 − /α .Note that at α = ˆ α = . w A (1 ,
0) = 47 ,
233 and w B (1 ,
0) = 32 , ∆Π G (1 , µ G ) is not monotonically decreasing for very large α , which implies that Assumption 2 is violated.Nonetheless, there is still a unique equilibrium for all values of α . G (1 ,
0) = . R is com-pletely specialized in education A . Therefore the wage and unemployment gap are determined bythe trade off that workers from group G are making. Choosing education A gives Green workersa higher wage than education B , but in education B there would be few Green colleagues, andtherefore fewer job contacts. Therefore choosing A would result in a lower employment rate forGreen workers. What is surprising is that this unemployment gap may be quite small comparedto the wage gap that compensates the unemployment gap. In particular, in our simulation, at α = ˆ α ≡ . We would like to know whether an even larger wage gap can be sustained in a partial segre-gation equilibrium when α > ˆ α = . w A (1 , µ ∗ ) and w B (1 , µ ∗ ), and equilibrium employments, s RA (1 , µ ∗ ), s RB (1 , µ ∗ ), s GA (1 , µ ∗ ) and s GB (1 , µ ∗ ), as func-tion of α . Remember that the equilibrium µ ∗ equals zero when α ≤ ˆ α , and solves ∆Π G (1 , µ ∗ ) = 0when α > ˆ α . These plots are shown in Figures 2 and 3.The pictures clearly confirm Propositions 2 and 3. Moreover, for the chosen parameters wealso observe that the wage gap G (1 , µ ∗ ) is maximized at α = ˆ α . When α becomes larger thanˆ α , the wage of A declines and the wage of B increases until the wage gap is reversed.We next look at the sensitivity of ˆ α with respect to the parameter choices, as we saw thatat α = ˆ α the wage gap is maximized. We do this by computing the elasticities of ˆ α and of theimplied wage gap G (1 ,
0) at the chosen parameters. That is, we look at the percentage increaseof ˆ α and the maximum wage gap change when a parameter increases by 1% . The elasticitiesare shown in columns 2 and 3 of Table 2. We note that ˆ α and the implied maximum wage gapare most sensitive to ρθ , the coefficient of relative risk aversion. A 1% increase in this coefficientleads to a 2% increase in the maximum wage gap. On the other hand, our calibration seemsleast sensitive to the network parameters c ( p + κ ) and c λ . The maximum wage gap seems to The risk aversion effect, and thus the wage gap, may be smaller if unemployment is only temporary, andindividuals only care about permanent income, or if agents get unemployment benefits/ social support. Onthe other hand, from prospect theory it is known that individual agents tend to emphasize small probabilities(Kahneman and Tversky 1979), and thus the small probability of becoming unemployed may get excessive weightin the education decision. lpha 0,90,80,70,60,5wage 1E58E46E44E42E40 w_A(1,mu*) w_B(1,mu*) Figure 2: Equilibrium wages as function of α . alphaemployment0,94 0,90,80,60,95 0,70,50,91 Figure 3: Equilibrium employment rates as function of α .29e close to linear with respect to 1 − s , the unemployment rate if a worker only consider directsearch techniques. That is, if we chose s = .
95 instead of s = .
90, it would roughly halve themaximum wage gap.
We now consider the analysis of a second-best optimum. Namely, we suppose that the govern-ment (social planner) does not have the institutions to completely control µ R and µ G , but that itis able to stabilize a symmetric equilibrium, such that µ R = µ G = µ S . Should the governmentdo this? In case the government stabilizes integration, we still impose the equilibrium condition,which is in this case symmetric. ThereforeΠ RA ( µ S , µ S ) = Π RB ( µ S , µ S ) = Π GA ( µ S , µ S ) = Π GB ( µ S , µ S ) . Hence, in the symmetric case there is complete equality. On the other hand, in the case ofsegregation, we consider the equilibrium allocation ( µ R , µ G ) = (1 , µ ∗ ), such that Reds obtain ahigher payoff than Greens. Therefore, we might face a tradeoff when assessing an integrationpolicy. It enforces equality, but it might decrease employment.To this purpose we plot the increase in social welfare from such an integration policy, I = W ( µ S , µ S ) /W (1 , µ ∗ ) −
1, as function of α . Figure 4 shows this plot. alpha 0.90.80.70.60.50-0.004-0.008-0.012-0.016 Figure 4: Percentage increase in welfare of a policy that enforces perfect integration. In the proof of Lemma 1 we show that there exists a symmetric equilibrium, but that it is unstable; that is,after a small deviation from the equilibrium individual incentives drive education choices to segregation.
30e observe that I is negative for all values of α . So for the chosen parameters the integrationpolicy is never preferred. People are better off segregated.Our results are very clear; with a utilitarian welfare function, a second best policy involvesa “laissez-faire” policy, such that society becomes segregated. The intuition behind this resultis twofold. First, in the case of partial segregation the equilibrium is determined by the Greenworkers. They trade off a benefit in wage against a loss in employment. Their individualincentives therefore already put a limit on the amount of wage inequality that can be sustainedin equilibrium. Second, an integration policy would lead to lower employment rates. In a societywith risk-averse individuals, society puts large emphasis on unemployment, and therefore prefersto allow for some inequality in order to obtain these higher employment rates.We finally remark that an integration policy is only beneficial when society has additional distributional concerns that are not captured by the concavity of the individual utility function.For example, consider the case of a maximin social welfare function: W min = min i Π i . In theintegrated case, µ R = µ G = µ S , everyone obtains the same payoff, whereas in the segregated caseworkers from group G are worse off. Therefore, W min (1 , µ ∗ ) = Π GB (1 , µ ∗ ) and W min ( µ S , µ S ) =Π GB ( µ S , µ S ). We show a comparison of these two payoffs, Π GB ( µ S , µ S ) / Π GB (1 , µ ∗ ) −
1, in Figure 5. -0.005-0.0100.005 alpha 0.90.80.70.60.5
Figure 5: Percentage increase in Green workers’ payoffs of a policy enforcing perfect integration.We observe that the Green workers would benefit from an integration policy for values of α around ˆ α , where the wage gap is particularly large. In such a case, strong distributional concernswould justify integration . Graham et al (2010) provide a general conceptual framework to test for a potential equity-efficiency tradeoff,focusing to that extent on a ”local segregation inequality effect”. One can rule segregation-increasing efficiencygains unacceptable if they increase inequality across groups. Conclusion
We have proposed and investigated a simple social network framework where jobs are obtainedthrough a network of contacts formed stochastically, after career decisions had been made. Wehave established that even with a very small amount of homophily within each social group,stable occupational segregation equilibria arise. If the wage differential across the occupationsis not too large, complete segregation is always sustainable. If the wage differential is large,complete segregation cannot be sustained, but a partial segregation equilibrium in which oneof the group fully specializes in one education while the other group mixes over the careertracks, is sustainable. Furthermore, our model is able to explain sustained employment andwage differences between the social groups.We also analyze the implications of our model from a social planner’s point of view. In the firstbest social welfare optimum, we find that segregation is the socially preferred outcome. Subjectto proper calibration of our model parameters, a second best social welfare analysis supports alaissez-faire policy, where society also becomes segregated, shaped by individual incentives. Boththese conclusions are valid in light of ’reasonable’ concavity features of the individual utilityfunction. Our social welfare conclusions cast some doubts on the usual ”always integration”policy advocacy; if job referrals through contact networks are relevant in matching workersto vacancies, and if the crucial mechanisms of our model are the correct ones, an integrationapproach would only be justified under strong additional distributional concerns, not reflected inthe individual utility functions. We highlight therefore that these distributional concerns shouldtake center stage in typical debates on social integration.While our job referral social interaction model can relate empirical patterns of educationaland occupational segregation to wage and employment inequality between gender, racial orethnical groups, other factors are also documented to play a significant role in this context. Thismodel should thus be seen as complement to a number of alternatives, including here the classicaltheories of taste discrimination or rational bias by employers—which are still documented to bepresent in the market, despite their predicted erosion over time given competitive pressure andinstitutional instruments. It is therefore pertinent, in future empirical research on suitable data,to assess the relative strength of our model in explaining observed labour market disparitiesbetween genders, races or ethnicities, vis-`a-vis other proposed channels.Our model easily allows for interesting extensions. One such avenue for future research is toextend with an analysis of minority versus majority groups, by modeling the interaction betweensocial groups of unequal sizes. Another avenue is to consider heterogeneity in productivity. Thelatter would allow to analyze the mismatch of workers to firms due to network effects.32
Proofs
The proof of Proposition 1 uses the following lemma:
Lemma 1
Suppose Assumptions 1 and 2 hold. A weakly stable equilibrium ( µ ∗ R , µ ∗ G ) , in which < µ ∗ R < and < µ ∗ G < , does not exist. Proof.
Suppose ( µ ∗ R , µ ∗ G ) is a stable equilibrium, and µ ∗ R ∈ (0 ,
1) and µ ∗ G ∈ (0 , RA ( µ ∗ R , µ ∗ G ) = Π RB ( µ ∗ R , µ ∗ G ) and Π GA ( µ ∗ R , µ ∗ G ) = Π GB ( µ ∗ R , µ ∗ G ) (18)Substituting (8)-(9) into (18) and rewriting, these equations become U ( w A ( µ ∗ R , µ ∗ G )) U ( w B (( µ ∗ R , µ ∗ G )) = s RB ( µ ∗ R , µ ∗ G ) s RA ( µ ∗ R , µ ∗ G ) = s GB ( µ ∗ R , µ ∗ G ) s GA ( µ ∗ R , µ ∗ G ) . (19)Since λ > µ ∗ R > µ ∗ G implies s RA > s GA and s RB < s GB . But this means that if µ ∗ R > µ ∗ G , then s RB ( µ ∗ R , µ ∗ G ) s RA ( µ ∗ R , µ ∗ G ) < s GB ( µ ∗ R , µ ∗ G ) s GA ( µ ∗ R , µ ∗ G ) . which contradicts (19). The same reasoning holds for µ ∗ R < µ ∗ G . Hence, it must be that µ ∗ R = µ ∗ G .However ( µ ∗ R , µ ∗ G ) with µ ∗ R = µ ∗ G cannot be a stable equilibrium. To see this, suppose that( µ ∗ , µ ∗ ) with µ ∗ ∈ (0 ,
1) is a stable equilibrium. Hence Π XA ( µ ∗ , µ ∗ ) = Π XB ( µ ∗ , µ ∗ ) for X ∈ { R, G } and ∂ ∆Π X ∂µ X < µ R = µ G = µ ∗ , and det( G ( µ ∗ , µ ∗ ) >
0, where G ( µ ) = D ∆Π( µ ) is the Jacobianof ∆Π ≡ (∆Π R , ∆Π G ) with respect to µ ≡ ( µ R , µ G ).Since λ >
0, it must be that ∂s XA ∂µ X > ∂s XA ∂µ Y > ∂s XB ∂µ X < ∂s XB ∂µ Y < X, Y ∈ {
R, G } and Y = X . Furthermore, if µ R = µ G = µ ∗ , then s XA = s YA , ∂L A ∂µ X = ∂L A ∂µ Y , ∂L B ∂µ X = ∂L B ∂µ Y , and therefore, ∂w A ∂µ X = ∂w A ∂µ Y (22)and ∂w B ∂µ X = ∂w B ∂µ Y . (23)33rom (20)-(23) and Assumption 2, it follows that, at µ R = µ G = µ ∗ , ∂ ∆Π X ∂µ Y < ∂ ∆Π X ∂µ X < . for X, Y ∈ {
R, G } , X = Y . But then it is straightforward to see that det( G ( µ ∗ , µ ∗ )) <
0. Thiscontradicts stability.
Proof of Proposition 1. (i) If (10) holds, thenΠ RA (1 , > Π RB (1 ,
0) and Π GA (1 , < Π GB (1 , . Hence, ( µ R , µ G ) = (1 ,
0) is clearly a stable equilibrium. The same is true for ( µ R , µ G ) = (0 , GA (1 , > Π GB (1 , . (24)Furthermore, from Assumption 1 we know that ∂ ∆Π G (1 ,µ G ) ∂µ G < µ G ∈ [0 , F , U and s , that there must be a unique µ ∗ , such that Π GA (1 , µ ∗ ) = Π GB (1 , µ ∗ ) . Moreover, s RA (1 , µ ∗ ) > s GA (1 , µ ∗ ) and s GB (1 , µ ∗ ) > s RB (1 , µ ∗ ), so we have at ( µ R , µ G ) = (1 , µ ∗ )Π XA > Π YB = Π YA > Π XB . (25)It is therefore clear that ( µ R , µ G ) = (1 , µ ∗ ) is a stable equilibrium. The same is true for( µ R , µ G ) = ( µ ∗ , RA (1 , > Π RB (1 , . Assump-tion 2 then implies that Π RA ( µ, > Π RB ( µ,
0) for all µ ∈ [0 , µ,
0) and, similarly, (0 , µ )cannot be an equilibrium. By Lemma 1 we also know that there is no mixed equilibrium.
Proof of Proposition 2.
The equations follow almost directly. We have s RA (1 ,
0) = s GB (1 ,
0) = s H > s L = s RB (1 ,
0) = s GA (1 , . Further, by assumption w A ≥ w B at ( µ R , µ G ) = (1 , µ R , µ G ) = (1 , U ( w A ) s RA ≥ U ( w B ) s GB ≥ U ( w A ) s GA ≥ U ( w B ) s RB , and this is equivalent to (12). 34 roof of Proposition 3. Consider the stable equilibrium at (1 , µ ∗ ). Since it is an equilibriumwe know that Π GA (1 , µ ∗ ) = Π GB (1 , µ ∗ ) . In the proof of Proposition 1, equation (25), we already demonstrated the inequality (14) Further,by Assumption 2 we know that ∆Π G (1 , µ G ) is strictly monotonically decreasing in µ G .(i) If ˆ µ < λ p + κ + λ ) , then s GA (1 , ˆ µ ) < s GB (1 , ˆ µ ). As w A (1 , ˆ µ ) = w B (1 , ˆ µ ) it must be thatΠ GA (1 , ˆ µ ) < Π GB (1 , ˆ µ ) . But then it also must be that µ ∗ < ˆ µ . As we consider a partial equilibrium, we know that µ ∗ >
0. Hence, 0 < µ ∗ < ˆ µ and w A (1 , ˆ µ ∗ ) > w B (1 , ˆ µ ∗ ), as w A ( µ R , µ G ) is a decreasing function,whereas w B ( µ R , µ G ) is increasing.(ii) If ˆ µ > λ p + κ + λ ) , then s GA (1 , ˆ µ ) > s GB (1 , ˆ µ ) and Π GA (1 , ˆ µ ) < Π GB (1 , ˆ µ ) . But then µ ∗ > ˆ µ . ByAssumption 1 we know that µ ∗ <
1. Hence, ˆ µ < µ ∗ <
1, and therefore w A (1 , ˆ µ ∗ ) < w B (1 , ˆ µ ∗ )We next continue with the proof of Proposition 4. This proof uses the following lemma: Lemma 2
Suppose that for all x ∈ [0 , ( p + κ + λ ) / s ′′ ( x ) > − λ s ′ ( x ) . (26) (i) If µ X > µ Y for X, Y ∈ {
R, G } , then ∂L A ∂µ X ( µ R , µ G ) > ∂L A ∂µ Y ( µ R , µ G ) > , (27) and ∂L B ∂µ Y ( µ R , µ G ) < ∂L B ∂µ X ( µ R , µ G ) < . (28) (ii) If µ R = µ G = µ , then ∂ L A ( ∂µ X ) ( µ, µ ) > ∂ L A ∂µ X ∂µ Y ( µ, µ ) , (29) and ∂ L B ( ∂µ X ) ( µ, µ ) > ∂ L B ∂µ X ∂µ Y ( µ, µ ) . (30)35 roof. (i) It is easy to derive that for X ∈ { R, G } : ∂L A ∂µ X = 12 (cid:18) s XA + µ R ∂s RA ∂µ X + µ G ∂s GA ∂µ X (cid:19) > ∂L B ∂µ X = 12 (cid:18) − s XB + (1 − µ R ) ∂s RB ∂µ X + (1 − µ G ) ∂s GB ∂µ X (cid:19) < µ R , µ G ). From (31) and (32), we find that for all X, Y ∈ {
R, G } : ∂L A /∂µ X > ∂L A /∂µ Y isequivalent to s XA + µ X (cid:18) ∂s XA ∂µ X − ∂s XA ∂µ Y (cid:19) > s YA + µ Y (cid:18) ∂s YA ∂µ Y − ∂s YA ∂µ X (cid:19) . (33)With the definition of s XA in (4) we can write out s XA + µ X (cid:18) ∂s XA ∂µ X − ∂s XA ∂µ Y (cid:19) = s (( p + κ )¯ µ + λµ X /
2) + µ X λ s ′ (( p + κ )¯ µ + λµ X /
2) (34)when X = Y . Therefore µ X > µ Y is equivalent to (33), whenever (34) is strictly monotoneincreasing with µ X , where we can treat ¯ µ = ( µ X + µ Y ) / µ X > µ Y . With a similar derivation one can show that condition (26) implies (28) as well.(ii) The second derivatives of L A and L B with respect to µ X and µ Y are ∂ L A ∂µ X ∂µ Y = 12 (cid:18) ∂s XA ∂µ Y + ∂s YA ∂µ X + µ R ∂ s RA ∂µ X ∂µ Y + µ G ∂ s GA ∂µ X ∂µ Y (cid:19) (35) ∂ L B ∂µ X ∂µ Y = 12 (cid:18) − ∂s XB ∂µ Y − ∂s YB ∂µ X + (1 − µ R ) ∂ s RB ∂µ X ∂µ Y + (1 − µ G ) ∂ s GB ∂µ X ∂µ Y (cid:19) . (36)Taking the second derivatives of s XA , evaluating at µ R = µ G = µ and reordering, we get that(29) is equivalent to s ′′ (( p + κ + λ ) µ/ < − λµ s ′ (( p + κ + λ ) µ/ . (37)Inequality (37) clearly holds if condition (26) holds, which proves (29). In a similar fashion, (26)implies (30) Proof of Proposition 4.
Suppose that W ( µ R , µ G ) is maximized at ( µ R , µ G ) = (˜ µ R , ˜ µ G ), where˜ µ R ∈ (0 ,
1) and ˜ µ G ∈ (0 , c ≡ L A (˜ µ R , ˜ µ G ) /L B (˜ µ R , ˜ µ G ), and consider the constrainedmaximization problem: max µ R ∈ [0 , ,µ G ∈ [0 , W ( µ R , µ G ) s.t. L A ( µ R , µ G ) = cL B ( µ R , µ G ) . (38)36ecause by definition of c , the solution (˜ µ R , ˜ µ G ) satisfies the restriction g ( µ R , µ G ) = cL B ( µ R , µ G ) − L A ( µ R , µ G ) = 0 , (39)it actually solves the maximization problem (38).Define the feasible set C = { µ R ∈ [0 , , µ G ∈ [0 , | g ( µ R , µ G ) = 0 } . By the assumption ofconstant returns to scale, we have that for all ( µ R , µ G ) ∈ C : w A ( µ R , µ G ) and w B ( µ R , µ G ) areconstant, and therefore, at all ( µ R , µ G ) ∈ C , the welfare function (16) can be written as W ( µ R , µ G ) = L A ( µ R , µ G )( U ( w A ) + U ( w B ) /c ) , which is monotone increasing with L A ( µ R , µ G ). Therefore, the solution (˜ µ R , ˜ µ G ) also solves thefollowing maximization problem:max µ R ∈ [0 , ,µ G ∈ [0 , L A ( µ R , µ G ) s.t. L A ( µ R , µ G ) = cL B ( µ R , µ G ) . (40)We verify that (˜ µ R , ˜ µ G ) indeed satisfy the first- and second-order conditions of problem (40).The Lagrangian is given by L ( µ R , µ G , ψ ) = (1 − ψ ) L A ( µ R , µ G ) + ψcL B ( µ R , µ G ) . Since (˜ µ R , ˜ µ G ) is supposed to be interior, the following first order constraints should hold: ∂ L ∂µ R (˜ µ R , ˜ µ G , ψ ) = (1 − ψ ) ∂L A ∂µ R (˜ µ R , ˜ µ G ) + ψ ∂L B ∂µ R (˜ µ R , ˜ µ G ) = 0 (41) ∂ L ∂µ G (˜ µ R , ˜ µ G , ψ ) = (1 − ψ ) ∂L A ∂µ G (˜ µ R , ˜ µ G ) + ψ ∂L B ∂µ G (˜ µ R , ˜ µ G ) = 0 . (42)The first part of Lemma 2 implies that ψ ∈ (0 ,
1) and that under condition (26): µ R > µ G ifand only if ∂ L /∂µ R > ∂ L /∂µ G . Therefore, condition (26) and the first-order conditions implythat ˜ µ R = ˜ µ G ≡ ˜ µ .Since ˜ µ R = ˜ µ G defines a unique point in C , the second-order condition should hold at˜ µ R = ˜ µ G , which says that the Hessian of the Lagrangian with respect to ( µ R , µ G ) evaluated atthe social optimum, D µ R ,µ G L (˜ µ, ˜ µ, ψ ), is negative definite on the subspace { z R , z G | z R ( ∂g/∂µ R )+ z G ( ∂g/∂µ G ) = 0 } . The second order condition is thus that at ( µ R , µ G ) = (˜ µ, ˜ µ ):2 ∂g∂µ R ∂g∂µ G ∂ L ∂µ R ∂µ G − (cid:18) ∂g∂µ R (cid:19) ∂ L ( ∂µ G ) − (cid:18) ∂g∂µ G (cid:19) ∂ L ( ∂µ R ) > . (43)Because ∂g∂µ R (˜ µ, ˜ µ ) = ∂g∂µ G (˜ µ, ˜ µ ) , and ∂ L ( ∂µ G ) (˜ µ, ˜ µ ) = ∂ L ( ∂µ R ) (˜ µ, ˜ µ ) , the second order condition (43)simplifies to ∂ L ∂µ R ∂µ G (˜ µ, ˜ µ ) > ∂ L ( ∂µ R ) (˜ µ, ˜ µ ) , or equivalently(1 − ψ ) ∂ L A ∂µ R ∂µ G (˜ µ, ˜ µ ) + ψ ∂ L B ∂µ R ∂µ G (˜ µ, ˜ µ ) > (1 − ψ ) ∂ L A ( ∂µ R ) (˜ µ, ˜ µ ) + ψ ∂ L B ( ∂µ R ) (˜ µ, ˜ µ ) . (44)37y the second part of Lemma 2, inequality (44) cannot hold under condition (26). Therefore wehave a contradiction and the non-segregation allocation (˜ µ R , ˜ µ G ) cannot be a social optimum.Since a social optimum exists by continuity of W and compactness of [0 , , the social optimumnecessarily has to involve complete or partial segregation. References [1] Albelda, R.P., ”Occupational Segregation by Race and Gender: 1958–1981,”
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