Occupational mismatch and social networks

0 downloads 34 Views 947KB Size Report
Aug 23, 2014 - Further, Moscarini (2001) shows in a two-sector model with heterogeneous workers that employees whose productivity does not differ much.

Journal of Economic Behavior & Organization 106 (2014) 442–468

Contents lists available at ScienceDirect

Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Occupational mismatch and social networks Gergely Horváth ∗ Southwestern University of Finance and Economics, China

a r t i c l e

i n f o

Article history: Received 18 July 2013 Received in revised form 25 June 2014 Accepted 17 July 2014 Available online 23 August 2014 JEL classification: E24 J62 J64 Keywords: Social networks Labor market search Occupational mismatch Homophily

a b s t r a c t A labor market model with heterogeneous workers and jobs is provided to investigate the effects of social networks as a job information channel regarding the level of mismatch between workers and firms. The efficiency in producing good matches of the formal market is compared to that of social networks. It is assumed that links between workers represent favoring relationships: workers recommend each other for any kinds of jobs, regardless of the quality of the resulting match. This study shows that as the fraction of ties connecting similar agents (homophily) increases, the level of mismatch decreases. If this fraction is sufficiently high, networks provide good matches at a higher rate than the formal market, for any efficiency level of the market. In this case, the mismatch level is lower in economies with social networks than it would be if workers did not use social contacts for job search. Hence, the presence of social networks can reduce mismatch despite favoritism. Implications of mismatch creation for the expected wages of jobs obtainable through different search methods are also discussed. © 2014 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Motivation Workers often use social contacts while searching for a job, in addition to formal methods such as newspaper ads or direct application to employers. Research shows that 30–60 percent of workers obtain employment through informal methods (see, for example, Granovetter, 1995 [1974]; Holzer, 1987; Bentolila et al., 2010; Pellizzari, 2010). The extensive use of social networks originates from the important roles such networks play in mitigating two primary informational problems prevalent in the labor market: (1) job referrals provide information about the unobserved characteristics of workers for firms (see, for example, Montgomery, 1991; Galenianos, 2013); and (2) employed workers transmit information about vacancies to their unemployed social contacts and in this way reduce search frictions (Calvo-Armengol and Zenou, 2005). While most papers analyze the signaling function of referrals, this paper considers the role of social networks in reducing search frictions. The few articles studying social networks in the search and matching framework assume that workers are homogeneous and investigate the impact of social contacts and network connectivity on the unemployment rate (CalvoArmengol and Zenou, 2005; Ioannides and Soetevent, 2006). This paper analyzes two new aspects of the job search via

∗ Correspondence to: School of Public Administration, Southwestern University of Finance and Economics, 227, Zhizhi Building, 555 Liutai Road, Wenjiang District, 611130 Chengdu, Sichuan, China. Tel.: +86 18628332756. E-mail address: [email protected] URL: http://sites.google.com/site/horvathgergely/. http://dx.doi.org/10.1016/j.jebo.2014.07.017 0167-2681/© 2014 Elsevier B.V. All rights reserved.

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

443

social ties. First, it introduces heterogeneous workers and vacancies to the model and studies how social networks influence the matching of workers to the jobs suitable for their skills. It defines “mismatch” as the disagreement between the skills required by a job and those possessed by the worker occupying the position.1 Second, this paper emphasizes that labor market outcomes are affected not only by the number of contacts, but also by the characteristics of those contacts, in this case the types of the skills they possess. The results of this study show that the impact of social networks on matching efficiency depends on the degree of homophily, defined as the tendency of workers with identical skill background to be connected in the network. This study finds that when the homophily level is sufficiently high, the mismatch level in the economy with social networks is lower than it would be if workers did not use social contacts for their job search. This paper also compares social networks and the formal market with regard to the likelihood of creating good matches. It assumes that social networks consist of favoring relationships: workers recommend each other to vacancies even if the referred worker lacks the required skills for the job.2 The formal market is modeled as a random arrival process of jobs to unemployed workers, where an efficiency parameter captures the likelihood that the market produces good matches. This study finds that, despite favoritism, social contacts are less likely to create mismatch than the formal market when the homophily level is large enough; this result holds true for any efficiency level of the market. Three factors lead to the above-mentioned results. First, similar contacts are more likely to provide information on good matches than dissimilar ones; therefore, for higher degrees of homophily the network is more efficient in terms of matching. Second, it is assumed that employed workers hear about the new openings within the sector of their current job. It follows that a worker can transmit a good job offer to those contacts who possess similar skills, as long as s/he is employed in a good match. In this way, when the homophily level is high, the network becomes more efficient when there are more workers employed in good matches. The fraction of workers employed in good matches increases when the market efficiency parameter rises; consequently, the efficiencies of the two search methods are interrelated: a more efficient market implies a more efficient network. Because of this interconnection, for high levels of homophily, there is no level at which the market is more efficient at matching than the social network. Third, it is assumed that job tenure is shorter in bad matches than in good matches, which increases the fraction of workers employed in good matches in the equilibrium; it also increases the likelihood of hearing about a good job via social contacts. Previous literature also focused on the impact of social networks on mismatch. Bentolila et al. (2010) show that the social network always increases the mismatch level in society, and that the formal market is more efficient in terms of matching than the network even for high values of homophily. There are two important differences between Bentolila et al. (2010) and this model. First, in Bentolila et al. (2010), agents perfectly direct their search on the formal market and, consequently, the market does not create mismatch, only the social network. In the model presented here, the directed search on the formal market is imperfect, and a parameter is introduced which captures how often this search method provides a good match. Second, their model is static in the sense that the information access of social contacts is exogenously given and does not depend on their employment status. Therefore, the market efficiency does not affect the arrival rate of good offers via social contacts. In the dynamic model presented here, every agent moves between three states: unemployment, employment in a bad match, or employment in a good match. This means that contacts can transmit different types of job information depending on their actual sector of employment, and the market efficiency increases the efficiency of the social network. These important differences give rise to the possibility that the social network is more efficient in terms of matching than the formal market when the homophily level is large enough. Montgomery (1991) analyzes the impact of social networks on match quality in a context where workers differ in unobservable ability and homogeneous firms seek to employ high-ability workers. He finds that the average match quality is higher through social ties than on the market whenever more than half of the links connect similar workers. In contrast, in this study’s model, the required homophily level is always higher and depends on other parameters of the model. The first difference between his paper and this study is that this study considers a model with heterogeneous firms and the problem of assignment of workers to sectors according to their observable skills, rather than the role of social networks in signaling unobservable ability. Moreover, in Montgomery’s model, firms choose to hire through referrals only when the referral’s (already observed) productivity is high and the firm can exploit the homophilous nature of social ties to find high-ability workers. In this study’s model, the arrival of workers to firms through the network is random, as workers pass job information to a randomly chosen neighbor of theirs and firms do not choose the network channel for hiring based on information about the referral’s type or his/her (possibly unknown) network characteristics. Despite this random arrival process, the network can be more efficient than the formal market.

1 Mismatch has alternative definitions as well. Some papers define mismatch based on the quantity of education: a worker is mismatched if s/he is overor undereducated compared to the job’s required quantity of education (Leuven and Oosterbeek, 2011; Korpi and Tahlin, 2008). Another literature defines mismatch as the difference in the occupational or geographical distribution of labor between demand and supply (Thisse and Zenou, 2000; Shimer, 2007; Sahin et al., 2011) and investigates the impact of this difference on the unemployment rate. 2 Agents might have good reasons to recommend someone with inappropriate skills for a job despite possibly suffering reputation loss with the employer. One such reason is that social ties are used in many contexts other than the labor market, for example risk sharing, and the reputation loss might be compensated by benefits along these other dimensions (Beaman and Magruder, 2012). Forwarding job offers contributes to the maintenance of such beneficial links. Another reason for passing along unsuitable offers is that such behavior is reciprocated in the future, resulting in a shorter unemployment period for the individual (Bramoulle and Goyal, 2013).

444

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

Similar to Montgomery (1991), Galenianos (2012) also studies the firm’s challenge of finding high-productivity workers in a one-sector model and finds that the average productivity of workers hired through referrals is larger than of those contracted through the market when at least half of the links in the network connect similar workers.3 The implication is that high-productivity workers will be more likely to be employed and, through homophily, can recommend other highly productive workers to the firms. The main difference between Galenianos (2012) and this study is that while he assumes that the arrival of workers of different types to firms is uniformly random in the formal market, this study introduces a parameter that captures the frequency of the formation of good matches on the market. It shows that the social network can provide good matches at a higher rate than the market for any efficiency level of the market provided that the homophily level is sufficiently high. In other words, in this study’s model, market efficiency has a positive impact on the arrival rate of highly productive workers through social contacts; this mechanism is not pointed out by Galenianos (2012).4 The interest in match quality is motivated by the fact that mismatch to a large extent affects the wages earned; consequently, it influences the risks embedded in the investment in human capital (see Berkhout et al., 2010). Recent literature investigates the wage penalty for workers who do not work in the occupation related to their college major. For example, Robst (2007) finds that in the U.S., mismatched workers earn 11 percent less on average than workers whose major fits their current occupation. Using Swedish data, Nordin et al. (2010) obtain that the wage penalty amounts to 34 percent for men and to 32 percent for women, and that this penalty decreases but does not entirely disappear over time. As match quality influences the wages earned, based on mismatch creation, it is possible to compare the two search methods with respect to the expected wages of jobs they provide. The theoretical and empirical literature is divided regarding the wage differences between search methods (Pellizzari, 2010): some articles suggest that social networks pay a wage premium over the formal market (see, for example, Dustmann et al., 2011; Galenianos, 2013; Kugler, 2003), while others draw opposite conclusions (see, for example, Bentolila et al., 2010; Ponzo and Scoppa, 2010). For different values of the homophily level, the model presented here can explain both a wage premium and a discount for social contacts. Since mismatched employees earn less than workers employed in a good match, job finding through the search method that creates more mismatch leads to a wage discount. For a sufficiently high degree of homophily, the job search via social contacts creates less mismatch and pays higher wages than the job search in the formal market, while for low homophily levels the opposite holds true. These predictions are in accordance with the empirical findings. First, the wages of jobs obtained through social networks increase with the number of social contacts possessing similar education as the individual (see Mouw, 2003). Second, the network of professional contacts, which mostly comprises workers of similar backgrounds as the individual, pays a wage premium over the market. In contrast, jobs obtained through family members, who are more heterogeneous regarding educational background, pay a wage discount (see Antoninis, 2006). This paper is organized as follows. Section 2 describes the model and deduces the equilibrium conditions. Section 3 presents the main results in two steps: (1) the impact of social networks on mismatch; and (2) the comparison of search methods regarding expected wages. Section 4 considers an extension of the model, where both the vacancy rate and wages are endogenously determined. Section 5 discusses the limitations and possible extensions of the model, and Section 6 concludes. Appendix A contains some robustness checks of the results, as well as the proofs and figures.

2. Model This paper describes a labor market characterized by search frictions. Workers can identify vacancies through two simultaneously present channels: the formal market and social networks. The model herein is based on a matching function derived from assumptions on the job information transmission between workers connected to each other by a social network. This section assumes that both the number of vacancies and wages are exogenous; it focuses on the implications of the matching process for the mismatch level in society and the expected wages of jobs obtainable through the two search channels. Section 4 provides a standard Diamond–Mortensen–Pissarides search model with endogenous vacancy posting and wages, and shows that the results hold true in that version of the model as well. There are two sectors s ∈ {X, Y} and two types of workers j ∈ {X, Y} in the economy, each group of workers having unit mass. The worker’s type means the skills possessed (or education acquired) by the worker; the firm’s type means the types of workers it prefers to employ. These types are observable. Every worker is able to fill a job of any sector; type i workers constitute a good (bad) match for vacancies of sector i (j), i = / j, i, j ∈ {X, Y}. Workers accept the first arriving job when unemployed, and firms employ the first arriving worker when they have a vacancy. Therefore, partners accept bad matches, which is rational if there are search frictions on the labor market and agents would have to wait too long for a

3

Note that this model version with heterogeneous workers has been removed from the new version of the paper (Galenianos, 2014). van der Leij and Buhai (2008) also study the impact of homophily on the labor market. They show that homophily in the social network providing job information causes occupational segregation. This is because workers in the same group choose the same occupation to exploit the efficiency of the job arrival process, which rises with the homophily level. This paper instead focuses on the impact of homophily on the mismatch level and the expected wages of job search via social contacts. The consequences of homophily are also analyzed by Bramoulle and Saint-Paul (2010). They model a dynamically evolving network and assume that two employed agents are more likely to connect than an employed agent and an unemployed agent. Therefore, homophily in their model means being connected to workers in the same employment state, not having the same skills as defined here. 4

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

445

consecutive good match.5 The literature on horizontal skill mismatch documents that a significant fraction of workers are indeed employed in bad matches; for example, in Robst (2007), the mismatch incidence is 20–25 percent. This paper describes the matching process between workers and vacancies. Unemployed workers might become aware of vacancies either directly in the market or indirectly through their social contacts. The assumptions about the direct matching of workers to vacancies are presented below. 2.1. Matching on the formal market The formal market is modeled as a stochastic arrival process, where the number of vacancies is assumed to be exogenous. At a rate ωi , a sector i job offer arrives and the receiver of job information is randomly chosen from the population of workers, assuming that 1. unemployed workers of type i have a higher chance to hear about the vacancy than others – since workers and firms direct their search toward good matches, it is plausible to think that good matches are more frequently informed by the market than bad matches; 2. among employed workers only those employed in sector i can hear about the vacancy – this assumption captures the idea that employed workers hear about the new openings within their own sector. Assumption 2 establishes a connection between the types of contacts and the types of jobs arriving via social ties to unemployed workers, and hence makes it possible to study the impact of homophily. Since under Assumption 1, workers are more likely to be employed in a good match than in a bad one, a contact with the same skills is more likely to transmit a good offer than a neighbor with different skills. If offers were uniform randomly distributed between employed workers, the degree of homophily would have no impact on the types of offers obtained through contacts, as any employed contact would be able to transmit any kind of offers.6 In the following notation, the fraction of unemployed is denoted among type i ∈ {X, Y} agents by ui , the fraction of type i j j workers employed in sector i by eii , and the fraction of type i agents employed in sector j( = / i) by ei (hence, ui + eii + ei = 1). Using this notation, the market (direct) arrival rate of sector i offers to unemployed agents of type i is written as follows: Pr M G,i =

ui ui + uj + eii + eji

ωi

Parameter  represents the extent of directed search between firms and workers in the market (Assumption 1); it is referred to as the market efficiency parameter in the sequel.7 If  = 1, the direct arrival is uniform random. As  rises, the number of good matches formed in the market increases. ui + uj + eii + eji is the size of the group of workers from which the receiver of job information is randomly drawn. Note that workers employed in sector j are not included in Assumption 2. In the same way, the market arrival rate of sector i offers to unemployed workers of type j = / i is the following: Pr M B,j =

uj ui + uj + eii + eji

ωi

This expression is decreasing in parameter : as the market becomes more efficient, bad matches are less likely to be formed through this channel. Finally, the market arrival rate of sector i offers to type l ∈ {X, Y} workers employed in sector i is as follows: eli ui + uj + eii + eji

ωi

2.2. Network structure and information transmission Workers are connected by an underlying (undirected) network. Every worker has k neighbors. The neighbors are randomly chosen from the population in such a way that a neighbor of a worker is of the same type with probability  and is of a

5 Appendix A provides explicit conditions that guarantee that the acceptance of bad matches is optimal for the firm. Both workers and firms incur costs when waiting: the firm incurs some costs to maintain a vacancy, while a worker is better off earning the wage associated with a bad match than earning unemployment benefits. In addition, firms observe the type of worker and can condition wages on the match quality, which compensates them for accepting a bad match. Further, Moscarini (2001) shows in a two-sector model with heterogeneous workers that employees whose productivity does not differ much between the two sectors search for and accept offers from both sectors. 6 Another way of connecting job arrival to contact types would be to assume that type i workers can hear exclusively about the offers of sector i. This assumption is less plausible since there is no obvious explanation why offers would be sorted among employed workers in this way, while Assumption 2 captures the idea that workers hear about the new vacancies within their own employment sector. 7 Another interpretation can be that the institutions of the formal market (for example, agencies) facilitate the encounters of firms and suitable workers and thus the formation of good matches.

446

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

different type with the complementary probability.  measures the homophily level in the society, i.e. the extent of workers with the same skill background to be connected in the social network. When  = 1, the society consists of two separate groups, while when  = 0.5, the two groups are completely mixed. The homophily level and the number of neighbors take the same values for the two groups of workers. It is assumed that the network consists of favoring relationships: workers would like to help each other find a job and recommend each other to jobs without considering the resulting match quality. In concrete terms, it is assumed that an employed worker having job information recommends a randomly chosen unemployed direct neighbor of his/hers. This behavior creates a mismatch to the extent that neighbors have different skills than the referral, which makes the degree of homophily a crucial parameter regarding the effectiveness of social contacts in creating good matches. In reality, favoritism characterizes some relationships, but surely not all relationships are favoring.8 From a theoretical point of view, however, the random information transmission describes a case in which the effectiveness of the network in creating good matches is at its lower bound. That is, if the referrals direct offers to good matches, the network will create less mismatch. It is thus expected that real-world networks are even more effective than these results suggest. Furthermore, it is supposed that when an employed agent with knowledge of an offer does not have an unemployed neighbor, the offer is lost (i.e. the vacancy remains unfilled).9 To derive the matching function, the so-called homogeneous mixing assumption is used, which means that the network connections of a given individual are randomly redrawn at each instant of time. This assumption implies that the state of a neighbor is independent of the state of an individual connected to him/her. Moreover, the probability that a neighbor can be found in a particular state is equal to the population frequency of that state. For example, the probability that a type i neighbor is unemployed is given by the unemployment rate of type i workers in the economy.10 Given these assumptions about the information transmission via social contacts, the arrival rate of job offers through social contacts is derived.

2.2.1. Bad matches A bad match occurs when a type i unemployed worker receives an offer of sector j where i, j ∈ {X, Y}, i = / j. The arrival rate of bad offers to type i unemployed workers through the network channel is expressed as follows:

 Pr N B,i

= ui



j kei

ωj

j Q + k(1 − )ej j j i uj + ui + ei + ej uj

ωj j

j

+ ui + ei + ej

(1)

Qj

The fraction of unemployed among type i workers is ui . Each unemployed worker has k neighbors of type i who are employed ωj j in sector j with probability ei . Each employed neighbor has an available offer with probability and passes it to a j j uj +ui +e +e i

j

given unemployed contact with probability Qi when picking a random individual from the pool of unemployed neighbors. Qi is expressed as follows:

Qi =

 k−1   k−1 s=0

s

s

ui (1 − ui )

k−1−s

1 1 − (1 − ui ) = s+1 ui k

k

(2)

where ui = ui + (1 − )uj is the average probability that a neighbor of a type i individual is unemployed: each of his/her contacts is either of type i or type j; in both cases s/he has to be unemployed, which happens with probability ui or uj , respectively. s is the number of competitors for the same information: the probability that exactly s other neighbors are s k−1−s unemployed apart from the given worker is ui (1 − ui ) . The probability that among (s + 1) unemployed contacts a given 11 worker is randomly picked is 1/(1 + s). The second term can be similarly explained. A type i unemployed worker has k(1 − ) neighbors of type j who are employed ωj j and passes it to a in sector j with probability ej . Each employed neighbor has an available offer with probability j j uj +ui +e +e i

8

j

In fact, the referral may choose to recommend someone whose education fits the job instead randomly picking someone. The information travels only one step in the network since it cannot reach unemployed workers at more distant positions in the network. This assumption largely facilitates the analysis, as only the state of direct neighbors needs to be modeled but not the state of more distant agents. The same assumption was used in Calvo-Armengol and Zenou (2005) and in Calvo-Armengol and Jackson (2004). 10 In reality, as shown by Calvo-Armengol and Jackson (2004, 2007), the states of connected agents in a fixed network are not independent, but rather are correlated in the long run. Section A.3 presents some simulation results on fixed networks, which show that the findings obtained using the homogeneous mixing assumption also hold true on fixed networks. 11 Note that similar derivations were presented in Boorman (1975) and Calvo-Armengol and Zenou (2005), except that the workers are homogenous in those models. 9

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

447

given unemployed contact with probability Qj when picking a random individual from the pool of unemployed neighbors. Qj is defined as: Qj =

 k−1   k−1 s

s=0

s

uj (1 − uj )

k−1−s

1 − (1 − uj ) 1 = s+1 uj k

k

where uj = uj + (1 − )ui . The derivation of this formula is completely analogous to the previous case. 2.2.2. Good matches A good match is formed when an unemployed worker of type i gets an offer of sector i (i ∈ {X, Y}). The arrival rate of good offers via social contacts to type i unemployed workers can be constructed in a similar way to the case of the arrival rate of bad offers. It can be expressed as follows:



Pr N G,i

= ui



keii

ωi

Qi + k(1 − )eji ui + uj + eii + eji ui

ωi + uj + eii + eji

Qj

(3)

An offer of sector i may reach an unemployed individual of type i through his/her k type i or k(1 − ) type j contacts. In both cases, these contacts must be employed in sector i, which happens with probability eii and eji , respectively. Any employed contact is aware of a vacancy with probability

ωi

ui +uj +ei +ei i

. S/he passes the vacancy to a randomly chosen unemployed

j

neighbor. The probability that a given unemployed neighbor receives the information is Qi when the employed contact is of type i and Qj when s/he is of type j. 2.3. Aggregate matching function The aggregate matching function describes the encounters between workers and vacancies through the two job search channels. The model is written in continuous time, which implies that one channel, at most, provides information at the same time. The aggregate arrival rate of offers is the sum of the arrival rates in the formal market and via social contacts. The aggregate matching function for bad matches is expressed as follows: ui ωj

N ui qB,i = Pr M B,i + Pr B,i =

= ui

j

j

uj + ui + ei + ej

ωj

+ ui

ωj

j

j

j

uj + ui + ei + ej

j

j

j

uj + ui + ei + ej

j

(kQi ei + k(1 − )Qj ej )

j

(1 + kQi ei + k(1 − )Qj ej )

(4)

The aggregate matching function for good matches is expressed as follows:

N ui qG,i = Pr M G,i + Pr G,i =

=

ui ωi ui + uj + eii + eji

ui ui + uj + eii + eji

ωi + ui

ωi ui + uj + eii + eji

( + kQi eii + k(1 − )eji Qj )

(kQi eii + k(1 − )eji Qj ) (5)

2.4. Equilibrium Workers move between three states: unemployment, employment in a bad match, or employment in a good match. Unemployed agents find a bad job at rate qB,i and a good job at rate qG,i . It is assumed that job separation takes place according to a Poisson process. Bad matches are separated at a higher rate than good matches: the Poisson parameter for good matches is , while for bad matches it is ˛ with ˛ ≥ 1. ˛ thus measures how much longer good matches last than bad ones.12 The model is analyzed for the case in which the economic environment is similar in the two sectors: the arrival rate of offers, the efficiency parameter of the direct arrival and the separation rate difference between bad and good matches

12 It is assumed that there is no explicit on-the-job search. However, the higher separation rate for bad matches captures some of the consequences of on-the-job search (e.g. good matches last longer than bad matches and the fraction of workers employed in good matches increases relative to bad matches).

448

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

take the same value in the two sectors (ωX = ωY ,  X =  Y , ˛X = ˛Y ). Every worker has the same number of neighbors and is homophilous to the same extent. These assumptions give rise to a symmetric equilibrium13 where • uX = uY ≡ u • eY = eX ≡ eB X Y • eX = eY ≡ eG X Y Note that 1 − u = eB + eG . For convenience, the subscripts i and j are dropped from the notation of every parameter and variable. The equilibrium of the model is a steady state in which the number of workers finding a job is equal to the number of employed workers losing a job, for both bad and good matches. It is defined as follows: ∗ ) satisfying the following two equations: The equilibrium of the model is expressed by the pair (eB∗ , eG

Definition.

uqB (u, eB , eG ) = ˛eB

(6)

uqG (u, eB , eG ) = eG

(7)

where 1 − u = eB + eG

uqB (u, eB , eG ) =

uω u + 1

uqG (u, eB , eG ) =

uω u + 1

 1 + (eB + (1 − )eG )

  + (eG + (1 − )eB )

1 − (1 − u)k u 1 − (1 − u)k u

 (8)

 (9)

The first two equations of the definition provide the steady state conditions. Eqs. (8) and (9) are the aggregate matching functions in the symmetric case. The following proposition establishes the existence and uniqueness of the equilibrium: Proposition 1.

A unique equilibrium of the model exists when

1. ˛(1 + (1 − 2eB∗ )) + ω(1 − ) + (eB∗ )k (1 + k)ω > 0 where eB∗ is given by:

eB∗

=

⎧ ⎨ 1 ⎩

2˛ k(1 + k)ω

1/(k−1)

if − 2˛ + kω(1 + k) ≤ 0 if − 2˛ + kω(1 + k) > 0

and 2. −˛eB∗∗ + ω( + eB∗∗k (1 + (−1 + k))) > 0 where eB∗∗ is expressed as:

eB∗∗

Proof.

=

⎧ ⎨ 1 ⎩

˛ kω(1 + (k − 1))

See Section A.5.1.

1/(k−1)

if − ˛ + kω(1 + (k − 1)) ≤ 0 if − ˛ + kω(1 + (k − 1)) > 0



The proof of this proposition is based on the properties of the two implicit functions eB (eG ), defined by the equilibrium conditions (6) and (7). Fig. 1 depicts these two functions. The conditions on the parameters guarantee that the two implicit functions are strictly monotone, decreasing in the (eG , eB ) plane, and that they cross only once.

13

Bentolila et al. (2010) implement a similar analysis.

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

449

eB

1 eq. (7)

eq. (6) eG

1 Fig. 1. Equilibrium.

3. Results 3.1. Impact of social networks on mismatch The primary objective of the analysis is to determine the impact of social contacts as a job information channel on mismatch. The level of mismatch can be measured by the fraction of workers employed in bad matches over the fraction of workers employed in good matches. Mismatch created by a particular search method is also defined as the probability that the search method provides a bad offer over the probability that it provides a good one. Definition.

The mismatch level is defined as the fraction eB /eG . The mismatch created by the formal market is defined as

M Pr M B /Pr G = 1/. The mismatch created by social networks is expressed as

Pr N B Pr N G

=

eB +(1−)eG eG +(1−)eB

=

(eB /eG )+1− 14 . +(1−)(eB /eG )

The following lemma summarizes some properties of the mismatch level. Lemma 1. 1. The mismatch level in the society can be expressed as follows: eB qB (u, eB , eG ) 1 = = eG ˛ ˛qG (u, eB , eG )



Pr M B

Pr N B

Pr G

Pr G

+ M



(1 − ) N

(10)

where  ∈ (0, 1). 2. The mismatch level decreases with (i) the market efficiency () if ω > , (ii) the separation rate difference (˛), (iii) the homophily level () if  > 1 or ˛ > 1. Proof.

See Section A.5.2.



The first point specifies that the mismatch level depends on the ratio of the arrival rate of bad offers and the arrival rate of good offers (qB (u, eB , eG )/qG (u, eB , eG )), because workers accept the first arriving offer. Moreover, the aggregate mismatch M N N level is the linear combination of the mismatch created by the market (Pr M B /Pr G ) and social networks (Pr B /Pr G ). The second point states the impact of different parameters on the mismatch level: (1) as the market arrival becomes more efficient, good matches are formed more often; (2) when ˛ increases, good matches last longer than bad matches, which leads to a lower mismatch level in the equilibrium; and (3) if either  > 1 or ˛ > 1, a worker has a higher chance of being employed in a good match than in a bad one (eG > eB ). In this case, when the homophily level increases, the mismatch created

14

j

Pr i is the probability that a search method j ∈ {N(etwork), M(arket)} provides an offer of type i ∈ {B, G}.

450

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

by social networks becomes lower because a neighbor with the same skills as the individual is more likely to provide a good offer than a contact with different skills. Namely, a neighbor with the same skills can provide a good offer when working in a good match, i.e. with probability eG , while a worker with different skills can provide a good offer when mismatched, i.e. with probability eB . Note that if the majority of contacts have the same skills as the individual ( > 0.5), the mismatch created by social networks is lower when the mismatch level eB /eG is lower. This is because contacts with the same skills as the individual transmit good job offers only when they are employed in a good match. As the market efficiency  or the separation rate difference ˛ rises, the mismatch level decreases and the social networks create less mismatch. The efficiencies of the two search methods are interrelated: a more efficient market will imply a more efficient information transmission through social contacts. Next, the study investigates whether the presence of social networks in the matching process increases or decreases the mismatch level of the society compared to an economy where social networks do not exist. In an economy without social networks, workers can find a job only in the formal market, and the mismatch level is determined by the market efficiency parameter. Definition.

In an economy without social networks, workers can only use the market channel to find a job. The mismatch

is expressed as follows:

eB eG

=

M 1 Pr B ˛ Pr M G

=

1 1 15 ˛ .

There exists a particular case when social networks do not affect the mismatch level. This result holds true when the market arrival is uniform random and the good and bad matches are separated at the same rate. Proposition 2. If the market arrival is uniform random ( = 1) and the job separation rate is equal between bad and good matches (˛ = 1), the mismatch level is equal to 1 (eB /eG = 1) in economies both with and without social networks, independent of the network structure. Proof.

See Section A.5.3.



M When  = 1, the direct arrival provides good and bad jobs with equal probability (Pr M G = Pr B ). A worker i who finds a job in the formal market will be equally likely to work in either of the two sectors and, therefore, equally likely to transmit the offers of the two sectors to his/her contacts. Any neighbor j who finds a job through him/her will also be equally likely to work in a good match or to be mismatched. Any neighbor of worker j will also be equally likely to work in the two sectors if s/he receives an offer from worker j or in the formal market. Continuing this argument yields that both search methods M N N provide good and bad offers with equal probability (Pr M G = Pr B , Pr G = Pr B ). This leads to the result that the mismatch level eB /eG is equal to 1 when ˛ = 1 (see Eq. (10)). In this particular case, the homophily level has no impact on the mismatch level, since any social contact will be equally likely to transmit a bad or a good offer to an unemployed worker, independent of their types. The following proposition describes the impact of social networks on the mismatch level in the more general case, when parameters  and ˛ are not restricted to be equal to 1. The impact of social networks here depends on whether workers are connected to individuals with the same or different skill background. When the homophily level is high enough, the presence of social contacts in the matching process decreases the mismatch level compared to an economy without social networks. Despite representing favoring relationships, social networks create less mismatch than the formal market, independent of the value of market efficiency.

Proposition 3. For any , there exists a homophily value  > 0.5 such that the two search methods create the same amount of M N N mismatch: Pr M B /Pr G = Pr B /Pr G . For ∀ > , the mismatch level is lower in the economy with social networks than without them. Furthermore: (i) If ˛ = 1,  = 1, (ii) if ˛ > 1,  < 1, (iii)  decreases when ˛ increases. Proof.

See Section A.5.4.



First, consider the case of point (i) when bad and good matches are separated at equal rates (˛ = 1) and all links connect workers with the same skills ( = 1). In this case, the two search methods create an equal amount of mismatch. By assumption, M a worker i who finds a job in the formal market will experience a mismatch rate Pr M B /Pr G = 1/. S/he can transmit a good (bad) job offer to his/her contacts with the same skill background whenever s/he is employed in a good (bad) match. Therefore, the mismatch rate of a neighbor j who finds a job through worker i will also be equal to 1/; the same is true for any worker who finds a job via worker j, etc. The mismatch created by the formal market and social networks will both be equal to

15 Note that as the aggregate mismatch level is the linear combination of the mismatch created by the formal market and the social network, the mismatch N level is lower in an economy with social contacts than without them, as long as the network creates less mismatch than the formal market (Pr N B /Pr G < 1/).

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

451

M N N 1/: Pr M B /Pr G = Pr B /Pr G = 1/. The two search methods are equally efficient in terms of matching workers to appropriate jobs when  = 1. The aggregate mismatch level is eB /eG = 1/, which follows from Eq. (10) when ˛ = 1. If the homophily level decreases below 1, the efficiency of the network becomes lower (see Lemma 1) and the presence of social networks increases the mismatch level in the society. Second, consider point (ii). When the separation rate for bad matches increases (˛ > 1), the mismatch level becomes lower in the equilibrium because bad matches last for a shorter time. For every value of the homophily level, the arrival of offers through social contacts becomes more efficient as the mismatch level decreases: unemployed workers will hear about good job offers more frequently through their neighbors possessing the same skills who transmit good job information whenever employed in a good match. For high values of the degree of homophily, the network creates less mismatch N than the formal market (Pr N B /Pr G < 1/), and the presence of the network channel reduces the mismatch level compared to an economy without social networks. When the homophily level decreases, the network becomes more inefficient in terms of matching. There is a threshold value in homophily such that the two search channels create mismatch with equal M N N probability: Pr M B /Pr G = Pr B /Pr G . This threshold is larger than 0.5 because when  = 0.5, good and bad job offers arrive with

equal probability through the social contacts:

Pr N B

Pr N G

=

0.5eB +0.5eG 0.5eG +0.5eB

= 1. The formal market is more efficient than the network

since it provides good offers more often than bad ones ( > 1). Such a homophily threshold can be found even if the value of market efficiency () is arbitrarily large. This happens because when  rises, the mismatch level decreases, which makes the network channel more efficient as well. As mentioned earlier, market efficiency has a positive impact on the efficiency of the network. The impact of network connectivity on the mismatch level also depends on the homophily level; this is stated in the following proposition: Proposition 4.

As the number of neighbors k rises,

(i) the mismatch level falls if  > , (ii) the mismatch level increases, if  < . (iii)  remains unchanged.

Proof.

See Section A.5.5.



When the homophily level is above (below) the threshold identified by the previous proposition, the mismatch level decreases (increases) whenever an unemployed worker finds a job through social contacts. If the network becomes more connected, a worker has a higher chance of finding a job through social networks, because s/he will have more neighbors from whom s/he can potentially receive job information. The homophily threshold (), defined as the homophily level where the N M M two search methods create equal amount of mismatch (Pr N B /Pr G = Pr B /Pr G ), is independent from the number of neighbors, because the mismatch created by social contacts does not depend on the network connectivity.16 The number of neighbors will determine how often an unemployed worker receives job information through contacts, but the type of offer depends on the type and employment status of the worker who transmits the job information. In summary, the presence of social networks leads to a lower mismatch level when the degree of homophily is high enough. Social networks can create less mismatch than the formal market despite favoritism among social contacts, i.e. even if employed workers do not direct offers toward suitable workers when transmitting job information. If favoritism is less common than assumed here, the social networks would create even less mismatch.

3.2. Implications for wages As the mismatch level affects the wages earned, based on mismatch creation, it is possible to compare the expected wages of jobs obtainable through the two search methods. The expected wages associated to a search method are defined as the conditional wage expectation of an arriving offer through that method. The focus on these statistics is justified by the empirical literature on the expected wages of different search channels (Pellizzari, 2010; Bentolila et al., 2010): these papers estimate the ex-post relation between wages earned by an employed individual and the search method used to obtain the job. The wages earned in a bad (good) match are denoted as wB (wG ). As the empirical literature on horizontal skill mismatch suggests (see, for example, Robst, 2007), mismatched workers earn lower wages than workers employed in a good match (wG > wB ). The expected wages associated to the market channel are defined as follows:

16

Formally, k does not appear in the equation

Pr N B

Pr N G

=

eB +(1−)eG eG +(1−)eB

.

452

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

Definition 1. The expected wages associated to the market search are the conditional expectation of the wage obtained through direct arrival, conditioned on the event that an offer has arrived: wEM =

PBM wB + PGM wG PBM + PGM

=

wB + wG (uω/(1 + u))wB + (uω/(1 + u))wG = (u(1 + )ω)/(1 + u) 1+

If the market arrival is uniform random ( = 1), the expected wages are the average of the wages earned in good and bad matches. On the other hand, if  > 1, the expectation places more weight on the wages earned in a good match. As for the network channel, the wage expectation is defined in a similar way: Definition 2. The expected wages associated to network search are defined as the conditional expectation of the wage obtained through contacts, conditioned on the event that an offer has arrived through a contact: wEN =

PBN wB + PGN wG PBN + PGN

=

(eB + (1 − )eG )wB + (eG + (1 − )eB )wG eB + eG

To compare these expectations, it is sufficient to compare the weight on the good wages wG , which ultimately depends on j j the mismatch created by the two search channels (Pr B /Pr G ).17 The concrete values of the wages do not affect the comparison.

M N N If the search in the formal market creates less mismatch than the search using social contacts (Pr M B /Pr G < Pr B /Pr G ), the market channel pays a wage premium over social networks. As the homophily becomes higher, the wages earned in jobs obtained using social contacts increase, since the mismatch created by networks decreases. For sufficiently high homophily levels, networks pay a wage premium over the formal market. Several findings of the empirical literature on the relationship between wages and obtaining a job through social contacts are in line with the results presented here. The literature finds mixed evidence: some papers suggest a wage discount for finding a job via social ties compared to the formal market (see, for example, Bentolila et al., 2010; Pellizzari, 2010), while others find a wage premium (see, for example, Dustmann et al., 2011; Kugler, 2003).18 Based on different values of the homophily level, the model’s predictions are consistent with both a wage discount and a wage premium for job search via social contacts. Mouw (2003) estimates the relationship between the number of friends in the same occupation as the individual and the wages obtained via contacts. In accordance with the model predictions, he finds that having more similar friends increases the wages obtained through social contacts. Further, Antoninis (2006) differentiates between family members and friends versus professional contacts. He finds that obtaining a job via professional contacts is associated with a wage premium, while relatives and friends either have no impact on the wages or imply a wage discount. This finding is consistent with the model, since professional contacts are presumably more similar to the individual regarding the type of education than relatives and friends (see Obukhova, 2012).

4. Extension: endogenous vacancy rates and wages The model presented above assumes that the arrival rate of new openings and the wages are exogenous. This section presumes that the job finding rate depends on an endogenously determined vacancy rate and that wages are set by Nash bargaining. It analyzes whether the previously obtained results change under these modifications. It again focuses on the symmetric equilibrium, where the vacancy rates are equal in the two sectors of the economy. It is assumed that there are search frictions on the labor market, and thus workers do not immediately become aware of the posted vacancies. The number of encounters between workers and firms positively depends on the unemployment and vacancy rates, according to the following Cobb–Douglas function: ω = Au(1−) v

(11)

where v is the vacancy rate of one sector, and A and  are technology parameters with the restriction 0 <  < 1. Parameter A can be interpreted as the institutional setting or efficiency of the labor market that determines the total number of matches between workers and vacancies for given values of the vacancy and unemployment rates, and for a given value of the worker’s intensity of search for good matches ().19 Parameter  determines the elasticity of the arrival rate of offers with respect to the unemployment rate and the vacancy rate. The vacancy rate is determined by a standard job creation equation under the assumption of free entry of firms. Firms pay costs c to maintain a vacancy. Vacant jobs can be filled by two types of workers: workers who possess the skills required by

17

For example, in the case of the market, the weight on good matches is /(1 + ), which reciprocally depends on the fraction 1/. Most of these papers use survey data, such as the European Community Household Panel and National Longitudinal Survey of Youth, which contain information on the search method that first informed the worker about the job that s/he finally accepted. These datasets provide no information on the characteristics of the workers’ social connections and, henceforth, are not helpful in carrying out a direct test of the model because the homophily level is not observable. 19 While parameter A determines the total number of matches between workers and vacancies,  sets how many of those matches will be good ones. Alternatively, A can also be interpreted as a scaling parameter. 18

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

453

Table 1 Parameter values. Name of the parameter

Notation

Value

Homophily level Discount rate Productivity in good match Productivity in bad match Vacancy costs Separation rate for good match Ratio of separation rates bad/good match Number of neighbors Arrival rate technology parameter Arrival rate technology elasticity Market efficiency Utility in unemployment

 ı p p c  ˛ k A   b

[0.5,1] 0.012 1 0.95 0.74764 0.1 [1,5] [10,45] 2.91775 0.4 [2,8] 0.45

the vacancy have high productivity (denoted by p) and earn high wages wG ; workers who possess different skills than those required by the vacancy have lower productivity p < p and receive lower wages wB < wG . Upon meeting, the firm observes the type of the worker. In a labor market characterized by search friction, it is rational for the firm to accept bad matches as long as the productivity of the worker is not too low. The loss that the firm incurs by employing a low-productivity worker is compensated by the costs it saves by not maintaining the vacancy while waiting for a more suitable worker and by paying lower wages to the mismatched worker. The job creation condition says that the costs of posting a vacancy (c) are equal to the expected discounted benefits obtained by posting a vacancy. The benefits come from the future possibilities of employing a low- or a high-productivity worker and take the following form: c = qFB

p − wB p − wG + qFG ı + ˛ ı+

(12)

The derivation of this equation is standard and presented in Section A.1.1. ı denotes the discount rate. qFB is the job filling rate of a vacancy by a worker with low productivity in that job and is expressed as follows: qFB =

uqB

(13)

v

where uqB is the aggregate number of bad matches defined by Eq. (4); the difference is that the arrival rate of offers is substituted by the expression ω = Au(1−) v . The job filling rate of a vacancy by a worker who has high productivity in the job is similarly defined: qFG =

uqG

(14)

v

where uqG is expressed as Eq. (5) again substituting ω = Au(1−) v . Wages are endogenously determined by Nash bargaining, which leads to the following wage equations: wB = ˇp + (1 − ˇ)b + ˇc wG = ˇp + (1 − ˇ)b + ˇc

v u

v u

(15) (16)

where ˇ is the relative bargaining power of the worker and b is the worker’s utility while unemployed.20 The derivation of wage equations is standard in the literature and is included in Section A.1.2. The equilibrium of this modified version of the model consists of 6 endogenous variables: u, eB , eG , v, wB , wG . The equilibrium values of these variables solve the steady state conditions (6) and (7), the job creation condition (12), and the wage equations (15) and (16), such that the arrival rate of offers ω is expressed as Eq. (11) and 1 − u = eG + eB . The equilibrium is computed numerically; the results presented in the previous section hold true for the case of endogenous vacancy rates and wages as well. The impact of the network structure is also analyzed (represented by the number of neighbors and the homophily level) on the equilibrium vacancy rate, wages and the aggregate welfare of the society. The aggregate welfare is calculated as the sum of the total output produced and the utility obtained by unemployed workers minus the total vacancy maintaining costs paid:  (eB , eG , v) = (1 − eB − eG )b + eB p + eG p − vc Table 1 presents the parameter values used in the calculations. The discount rate ı is expressed as the quarterly interest rate used by Shimer (2005): ı = 0.012. The quarterly separation rate  is set to 0.1 following the estimations by Shimer

20

Note that wB < wG since p < p.

454

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

(2005). The productivity in a good match p is normalized to 1. It is assumed that the productivity in a bad match is 5 percent lower than the productivity in a good match: p = 0.95. The elasticity parameter of the matching technology (11) with respect to the vacancy rate () is set to 0.4. This parameter ranges from 0.3 to 0.5 in most of the estimations (see Petrongolo and Pissarides, 2001); this study takes the middle of this interval. The value of the other technology parameter A and the vacancy posting costs c are calculated to match the long-term unemployment rate u = 0.0567 and vacancy rate v = 0.0359 estimated in Shimer (2005).21 This yields the following values: A = 2.91775 and c = 0.74764. The utility of the unemployed worker b is set to 0.45. This parameter can also be interpreted as the unemployment benefits, in which case the value b = 0.45 implies a plausible replacement ratio of around 45-55 percent relative to the wages. The remaining parameters are changed within the intervals indicated in Table 1.22 The results are shown in Section A.4. Fig. 2 shows the homophily threshold identified by Proposition 3 as the function of the separation rate difference (˛) and the market efficiency () parameters. First, when ˛ = 1, the homophily threshold is equal to 1 for all values of . Second, as the value of ˛ increases, the threshold decreases. These results were demonstrated in Proposition 3. In addition, the figure suggests that as the market efficiency  increases, the homophily threshold also increases. When the market is more efficient, by definition, the direct arrival creates more good matches. As the mismatch level decreases with , the network also creates more good matches. However, the indirect impact on the effectiveness of social networks is smaller than the direct effect on market efficiency. A higher homophily level is required to make the effectiveness of the two search channels equal. Fig. 3 shows the impact of network connectivity on the mismatch level and the aggregate welfare.23 The left panel shows that, in the case in which the homophily level is higher than the threshold value 0.8857, the mismatch level decreases as the number of neighbors increases. In contrast, when the homophily level is below the threshold, the mismatch level increases with the number of neighbors. When the homophily level is equal to the threshold, the mismatch level is independent of network connectivity. These findings were analytically shown by Proposition 4 for fixed vacancy rates. The right panel of Fig. 3 shows the impact of the network structure on the aggregate welfare. The aggregate welfare increases when the network exhibits a higher homophily level, because in this case the fraction of workers employed in high-productivity matches is larger and the aggregate output is higher. As the number of neighbors increases, unemployed workers have a higher chance of hearing about job openings through the network. This makes the employment rate eB + eG higher and increases the aggregate welfare. Note that, based on this result, it cannot be concluded that workers should be subsidized to form more relationships with each other. To draw such conclusions, one needs to consider a model of network formation and compare the network formed by workers to the social optimum.24 Fig. 4 shows the impact of the network parameters (k, ) on the other endogenous variables. The right upper panel suggests that as the number of neighbors increases, fewer vacancies are posted. Adding more links to the network increases the number of matches between workers and vacancies, since workers have more potential information sources (contacts). This would imply that firms should post more vacancies. However, for the same reason, the unemployment rate decreases (see the left upper panel), which makes it more difficult for firms to find unemployed workers. Moreover, as the two lower panels show, wages increase as the network becomes more connected, which occurs because the decreasing unemployment rate improves the workers’ outside options and bargaining power. As the wages increase, firms have less incentive to post a vacancy. Adding these three effects on the vacancy rate results in the firms opening less vacancies as the number of neighbors increases.25,26 Fig. 4 also shows that an increase in the homophily level leads to a lower unemployment rate. This happens because a higher homophily level implies more workers employed in good matches that are separated less frequently. This decrease in the unemployment rate leads to higher wages and fewer vacancies posted, although the latter effect is rather moderate. The impact on the vacancy rate is again the sum of different effects. Firms post fewer vacancies when the degree of homophily increases, because there are fewer unemployed workers searching for jobs and wages are higher. The higher homophily level also increases the likelihood of finding a high-productivity worker for the vacancy, which offers higher incentives for vacancy posting. 4.1. Endogenous search intensity The market efficiency parameter  can be interpreted as the search intensity with which unemployed workers direct their search toward jobs where their productivity is high. Thus far, this search intensity has been treated as an exogenous 21

This calibration exercise used the values ˛ = 2,  = 4, k = 30,  = 0.5. Cingano and Rosolia (2012) and Glitz (2013) construct job contact networks based on the individuals’ working history. Two workers are connected in the network if they worked for the same employer during a given period of time. The median number of neighbors is 43 in Glitz (2013) and 32 in Cingano and Rosolia (2012). In accordance with these studies, I set the benchmark value for the number of neighbors to 30 and perform comparative statics exercises with values between 10 and 45. 23 This graph is calculated assuming  = 4 and ˛ = 2, in which case the homophily threshold is equal to 0.8857. 24 Previous studies on network formation in the context of labor markets with homogeneous workers suggest that workers overinvest in connections relative to the social optimum (see Galeotti and Merlino, 2014). 25 Note that this effect also increases the aggregate welfare, as the firms will pay less vacancy maintaining costs. 26 For the plausible values of the parameters, a monotonic relationship is always obtained between network connectivity and the vacancy rate; the first mentioned effect on the vacancy rate is always dominated by the latter two. 22

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

455

parameter of the model. This section considers the choice of search intensity and analyzes how the network structure affects the optimal search intensity chosen by unemployed workers. Workers determine their search intensity  when they are unemployed; they maximize the discounted benefits of being in the state of unemployment (denoted by ıU): ıU = b + qB (u, eB , eG , v, )(WB − U) + qG (u, eB , eG , v, )(WG − U) − C()

(17)

where WB and WG are the value functions for the states of being employed in a bad match or a good match, respectively. The benefits of being unemployed consist of the current benefits b, which can be interpreted as unemployment benefits, and the future benefits of obtaining either a bad job or a good job. C() is the cost of directing the job search more intensively. It is assumed that the cost function is quadratic: C() = cE  2 . The first-order condition is expressed as follows:

∂qB (u, eB , eG , v, ) ∂qG (u, eB , eG , v, ) (WB − U) + (WG − U) − 2cE  = 0 ∂ ∂

(18)

This condition says that the marginal benefits and marginal costs of increasing  should be equal. The marginal benefits come from the increased likelihood of obtaining a good job and the decreased likelihood of obtaining a job in the low∂q (u,e ,e ,v,)

productivity sector (note that B B G < 0). The expressions for the two partial derivatives and the value function ∂ differences WB − U, WG − U are calculated in Section A.2. The equilibrium of the model in this case is expressed as the following endogenous variables: u, eB , v, wB , wG , . The equilibrium values are calculated by solving the steady state conditions (6) and (7), the job creation condition (12), the wage equations (15) and (16) and the first-order condition (18), such that the arrival rate of offers ω is expressed as Eq. (11). The equilibrium is solved numerically using the parameter values described in Table 1 and assuming that cE = 0.01. Fig. 5 shows how the optimal search intensity changes when the network becomes more connected or homophilous. A higher homophily level in the network is associated with a lower optimal search intensity. This happens because when the homophily level is higher, unemployed workers obtain a good job with higher probability through their social contacts, which decreases the need to make the costly investment in search intensity. A higher homophily level therefore has two opposite effects on the mismatch level: it decreases the mismatch level through the increased efficiency of the network channel while it may also increase it through the lower efficiency of the market channel. The right panel of Fig. 5 shows that the first effect dominates the second: the mismatch level decreases when the homophily level becomes higher (as in the model version with exogenous search intensity). The impact of network connectivity is similar to the homophily level. When the number of neighbors increases, unemployed workers have a higher chance of hearing about a job opening through their contacts. This job is likely to be a good match when the homophily level is higher than the threshold, and costly investment in search intensity becomes less beneficial. When the homophily level is low, unemployed workers are more likely to find a bad match through social networks when the connectivity increases. The higher likelihood of being employed in a bad match makes the marginal benefit of investing in search intensity lower, because the marginal benefits will be based on the value difference between working in a good and a bad match (WG − WB ). Instead, when the number of neighbors is low, the worker is more likely to be unemployed, and the marginal benefits of investing in search intensity is based on the value difference between working in a good match and being unemployed (WG − U). In the second case, the worker has higher incentives to invest in search intensity, because s/he can gain more by searching harder. In summary, these findings suggest that social networks serve as a substitute for the job search in the formal market: when more or better offers arrive to the unemployed worker through social contacts, s/he will invest less in the search in the formal market. 5. Discussion This section presents some of the limitations of the analysis and points out possible extensions for future study. The model assumes that workers are homogeneous regarding the number of neighbors, while in reality workers differ in their connectedness. Ioannides and Soetevent (2006) present a search and matching model with social networks and arbitrary degree distribution. They numerically show that heterogeneity in degree leads to heterogeneity in unemployment duration and wages. Following their work, the model presented here could also be extended in this direction: degree heterogeneity may lead to heterogeneity in the rate of being mismatched. However, the main conclusions about the role of homophily would carry through to this model extension, as they do not depend on the number of neighbors per se, but rather on the type of neighbors. Another limitation of the model is that the social structure is assumed to be exogenous. Galeotti and Merlino (2014) and Calvo-Armengol (2004) model network formation in the context of labor markets with homogeneous workers and jobs. The choice of the number of neighbors and the degree of homophily could be incorporated into the present model. In this extension, workers would maximize the value of unemployment by choosing their network configuration. The benefit of linking to more individuals is a shorter unemployment period, but maintaining more links also involves some costs. Similar to the literature on endogenous choice of search intensity (Pissarides, 2000), such a model with ex-ante homogeneous workers

456

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

could be analyzed in the symmetric equilibrium, where all workers choose the same number of neighbors, resulting in a configuration analyzed here. As for the choice of homophily level, workers benefit from connecting to others who are similar to them, as such connections increase the likelihood of receiving good job information. In addition, it is plausible to think that to link to dissimilar workers is also more costly than to connect to individuals of one’s own group. Despite these benefits of connecting to similar workers, it can be expected that individuals receive some incentives to diversify their connections between groups. Diversification may occur if members of the different groups are mostly employed in different sectors that are subject to sector-specific productivity shocks. In this case, employees within one sector who are hit by a negative productivity shock are likely to lose their jobs at the same time, and workers can benefit from connecting to the members of other groups who can provide them information about openings in other sectors unaffected by the shock. These different aspects of link formation are intended to be analyzed in future work. In the model presented here, workers and firms use the two search methods simultaneously and to an exogenously given extent. Another version of the model could be built assuming that both workers and firms choose the search intensity through the two channels. The very few articles on the endogenous choice of search methods include Kugler (2003), Cahuc and Fontaine (2009), Galenianos (2013), and Bentolila et al. (2010). The choice of search methods is based on the costs and benefits their use implies. Search in the formal market is usually thought to be more costly than search via social contacts. For workers, the benefits of different search methods depend on the arrival rate of job offers and the wages of jobs provided. The use of contacts may lead to a job faster if the number of neighbors is larger, and it may provide better jobs if the homophily level is higher. For the firm, the arrival rate of employable workers depends on the connections of their employees and the quality of the match on the types of those connections. From modeling perspectives, the choice of search intensity for network search could thus be equivalent to the choice of homophily and connectivity. The present model focuses on the role of social networks in mitigating search frictions in the labor market: unemployed workers and vacancies coexist because of search frictions, and connections between workers can facilitate the encounters of trading partners in the labor market. Another role of social networks and referrals is in providing signals about the unobserved characteristics of workers for the employers. More accurate signals about productivity during hiring also lead to matches of better quality and a wage premium for workers using social contacts (see, for example, Galenianos, 2013; Dustmann et al., 2011). The study presented herein assumes a labor market characterized by search frictions and focuses on the information transmission about the existence of vacancies to workers of appropriate observable characteristics, such as the type of education.

6. Conclusions This paper investigates the impact of social networks on the matching of workers to jobs suitable for their skills. It compares the mismatch level in an economy with social networks to an economy where workers can obtain jobs exclusively in the formal market. It also compares the efficiency of the formal market to that of social networks in creating good matches. It assumes that the social network consists of favoring relationships: workers recommend each other to jobs regardless of the resulting match quality on the job. This study shows that if the network homophily index increases, the level of mismatch in the society decreases. When the homophily level is sufficiently high, the mismatch level in an economy with social networks is lower than it would be if workers did not use social contacts for their job search. In this case, social networks create less mismatch than the formal market. Social networks can thus reduce mismatch despite favoritism. The mismatch, defined as the disagreement between the type of education of the worker and his/her occupation, leads to lower wages for the worker. Workers obtaining employment via social ties can earn higher wages if a higher fraction of their neighbors are similar to themselves in terms of education or possessed skills. The network search pays a wage premium over the formal market for high homophily levels. These predictions on wages are consistent with the empirical literature. This study also considers the impact of the network structure, represented by the number of neighbors and the homophily level, on other characteristics of the labor market equilibrium. It maintains that when the number of neighbors or the homophily level increases, the aggregate welfare and the wages increase, there are fewer vacancies posted by firms, and workers have less incentive to invest in directing their search toward good matches. This latter finding is in accordance with the idea that social contacts substitute the search in the formal market.

Acknowledgements I would like to thank Mariano Bosch and Marco van der Leij for their comments and suggestions during the development of this project. I am also grateful to Oliver Baetz, Sebastian Buhai, Antonio Cabrales, Manolis Galenianos, Jonas Hedlund, Janos Hubert Kiss, Istvan Konya, Dunia Lopez Pintado, Asier Mariscal, Elena Martinez Sanchis, Paolo Pin, Miguel Sanchez Villalba, Carolina Silva, Fernando Vega-Redondo, the participants of various conferences (ASSET 2010, Alicante; SAEe 2010, Madrid; MKE 2010, Budapest; EEA-ESEM 2011, Oslo; EEA-ESEM 2012, Malaga; China Meeting of the Econometric Society 2013, Beijing; Australasian Meeting of the Econometric Society 2013, Sydney; International Symposium on Contemporary

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

457

Labor Economics 2013, Xiamen) and reserach seminars (University of Auckland, Southwestern University of Finance and Economics), and two anonymous referees for their helpful comments and remarks. All remaining errors are mine. Appendix A. A.1. Derivation of the job creation condition and wage equations A.1.1. Job creation condition The vacancy rate is endogenously determined under the assumption of free-entry of firms. The free-entry condition implies that the value of vacancy posting is driven down to zero. I write down the value function of a vacant job in the steady state: ıV = −c + qFB (JB − V ) + qFG (JG − V )

(19)

where c is the costs of vacancy posting, qFB and qFG are the job filling rates of a vacancy for bad and good matches, respectively. JB and JG are the value functions of employing a low- or high-productivity worker, respectively. Future benefits of the firm are discounted by a rate ı. When a vacancy is filled by a worker of low productivity, the firm earns the output net of wages in a bad match which is denoted by p − wB . At a rate ˛, the firm is separated from the worker and the job becomes vacant. The value of a bad match in the steady state is given by the following expression: ıJB = p − wB + ˛(V − JB ) Formally, firms accept to hire low-productivity workers as long as the value of having a bad match is higher than the value of maintaining a vacancy: ıV < ıJB . I assume that a firm employing a suitable worker (good match) receives higher net output (p − wG > p − wB ), this is consistent with the definition of mismatch as horizontal skill mismatch. The value function of the firm employing a highproductivity worker is: ıJG = p − wG + (V − JG ) I assume that firms can freely enter to the market when they find it beneficial to do so. Free-entry implies that the value of the vacancy reduces to zero: V = 0. Applying this equality, we can express JB = Eq. (19), we get the job creation equation: c = qFB

p−wB ı+˛

and JG =

p − wB p − wG + qFG ı + ˛ ı+

p−wG . Substituting these into ı+

(20)

A.1.2. Derivation of wages In this section I derive the wages earned by a worker in a bad match (wB ), the wages earned in a good match (wG ) can be computed in a similar way. In the first step, the value functions of the worker need to be defined. The discounted value of unemployment for a worker consists of the current value of unemployment benefits and the future value of possible employment which might be in the good or in the bad sector: ıU = b + qB (WB − U) + qG (WG − U)

(21)

where qB and qG represent the job finding rates of a given unemployed, for a bad and a good match, respectively. The discounted value of being employed in a bad match is the sum of the value of wages earned and the possibility of job destruction which happens at a rate ˛ in the case of bad matches. ıWB = wB + ˛(U − WB )

(22)

Similarly, the value function of being employed in a good match consists of the wages earned and the future value of being unemployed, here the job destruction rate is : ıWG = wG + (U − WG )

(23)

Wages are negotiated by the worker and the firm upon meeting on the labor market. The solution concept is Nash bargaining. For the case of the wages in a bad match, the problem to be solved is the following: wB = argmax(WB − U)ˇ (JB − V )(1−ˇ) wB

FOC: (JB − V )1−ˇ ˇ(WB − U)ˇ−1

∂(WB − U) ∂(JB − V ) + (1 − ˇ)(JB − V )−ˇ (WB − U)ˇ = 0 ∂wB ∂wB

458

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

Similarly to the literature, we assume that ∂U/∂wB = 0, by the free-entry condition V = 0. Further we have that ∂WB /∂wB = 1 and ∂JB /∂wB = −1. This leads to the following simplified version of the FOC: ˇJB = (1 − ˇ)(WB − U) Note that for the case of a good match we arrive to an analogous expression: ˇJG = (1 − ˇ)(WG − U) Now we can substitute the expressions of the value functions. In previous section, we have seen that: JB =

p − wB ı + ˛

Using the expression for the value of the worker’s employment in a bad match (22), we get: WB =

wB + ˛U ı + ˛

Substituting these two expressions into the FOC, we can express the wages as the function of ıU and the parameters: wB = ıU + ˇ(p − ıU) Similarly, we can get the following expression for the wages in a good match: wG = ıU + ˇ(p − ıU) What is left is to express ıU as the function of the parameters. We use Eq. (21) and the two FOCs to substitute out (WB − U) and (WG − U) We obtain: ıU = b +

ˇ (qB JB + qG JG ) 1−ˇ

Using the job creation equation (20), we can express the sum qB JB + qG JG as: qB JB + qG JG = c

v u

Using the last two equations, we can express ıU as: ıU = b +

ˇ v c 1−ˇ u

Substituting this equation into the expressions for wages, we obtain: wB = ˇp + (1 − ˇ)b + ˇc

v u

which is the standard wage equation in the basic Pissarides model. In the same way, the expression for the wages in a good match is: wG = ˇp + (1 − ˇ)b + ˇc

v u

Note that the wage difference is wG − wB = ˇ(p − p). A.2. Endogenous search intensity In this section I derive the first-order condition for the choice of search intensity. The first-order condition is given by Eq. (18):

∂qB (u, eB , eG , v, ) ∂qG (u, eB , eG , v, ) (WB − U) + (WG − U) − 2cE  = 0 ∂ ∂ where the two partial derivatives are given by the following expressions:

∂qB (u, eB , eG , v, ) −Au(2−) v = 2 ∂ (u + 1)



1 + (eB + (1 − )eG )

∂qG (u, eB , eG , v, ) Au(1−) v Au(2−) v − = 2 2 ∂ (u + 1) (u + 1)

1 − (1 − u)k u

 1 + (eG + (1 − )eB )



1 − (1 − u)k u



G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

459

The value function differences WB − U and WG − U can be expressed by solving the three value function equations (17), (22) and (23). From the last two equations, we can obtain expressions for WB and WG : WB − U =

wB − ıU ı + ˛

WG − U =

wG − ıU ı+

Substituting these two expressions into (17) yields: ıU = b + qB

wB − ıU wG − ıU + qG − C() ı + ˛ ı+

where the arguments of qB and qG are suppressed for shorter exposition. This equation can be solved for ıU: ıU =

(ı + ˛)(ı + )(b − C()) + qB wB (ı + ) + qG wG (ı + ˛) (ı + )(ı + ˛) + qB (ı + ) + qG (ı + ˛)

Using this expression and the equations for WB and WG , the two value function differences, (WB − U) and (WG − U), can be expressed as: WB − U =

qG (wB − wG ) + (ı + )(wB − (b − C())) (ı + )(ı + ˛) + qB (ı + ) + qG (ı + ˛)

WG − U =

qB (wG − wB ) + (ı + ˛)(wG − (b − C())) (ı + )(ı + ˛) + qB (ı + ) + qG (ı + ˛)

A.3. Robustness: simulations on fixed networks The model presented in the main text assumes that the network is randomly re-drawn at each instant of time and, consequently, the expected state of the individuals’ neighbors in the network can be calculated using the population frequency of states. In this section I simulate the model on fixed network to check whether the findings of the main text hold in this context. On fixed network, the model defines a Markov process where every individual moves between three states: unemployment, employment in a bad match, employment in a good match. The transition rates from unemployment to employment can be derived similarly to the job finding rates (4) and (5) while the transition rate from employment to unemployment is equal to the job separation rate. This Markov process is ergodic since every state belongs to the same communication class: an employed worker can lose her job at any instant of time and an unemployed worker may find any kind of job at any instant of time. Since the process is ergodic, a unique limit distribution exists and it is independent of the initial state. Hence, if the simulation is run for sufficiently long time, the initial state does not determine the long-run simulation results. Statistics derived from the limiting distribution can be estimated taking averages over the simulation path if the simulation is run for approximately infinite time. I determine the sufficient simulation length by running the model from two different initial states and checking whether the same values27 are obtained for the statistics of interest from these two simulations. This type of convergence is guaranteed after T = 4.5 × 106 periods. The baseline simulation parameters are the following: ˛ = 2,  = 3, ω = 0.3,  = 0.1. The homophily level is changed between 0.5 and 1, the number of neighbors between 1 and 10. I simulate the model for 1000 agents on a regular random graph. The initial states of the agents are uniform randomly drawn. The results are shown in Fig. 6. The results of the analytical model are confirmed in the simulations. The left panel shows the mismatch created by M N N the market minus the mismatch created by social networks (Pr M B /Pr G − Pr B /Pr G ). For low homophily values the network creates more mismatch than the market. However, as the homophily level rises, the network becomes more efficient than the market. For ˛ = 1,  = 1, the two methods are equally effective. The right panel depicts the mismatch level for different values of network connectivity and homophily level. As the network becomes more connected, the mismatch decreases for

27

The difference between the two values is smaller than 0.01.

460

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

high homophily values and increases for low homophily levels. These results are in accordance with Proposition 3 and 4 of the theoretical model.

A.4. Figures Endogenous vacancy rate and wages See Figs. 2–5.

1

Theta=2 Theta=4 Theta=6 Theta=8

Homophily threshold

0.95 0.9 0.85 0.8 0.75 0.7

1

1.5

2

2.5

3

3.5

4

4.5

5

Alpha Fig. 2. Numerical solution for the modification with endogenous vacancy rate and wages. Threshold values of homophily () for different values of market efficiency () and separation rate difference (˛).

0.25

Gamma = 0.7 Gamma=0.8 Gamma=0.886 Gamma=0.9 Gamma=1

0.95

Gamma = 0.7 Gamma=0.8 Gamma=0.886 Gamma=0.9 Gamma=1

0.945

Aggregate welfare

Mismatch level

0.2

0.15

0.94

0.935

0.1 0.93

0.05

0 10

0.925

15

20

25

30

35

Number of neighbors

40

45

0.92 10

15

20

25

30

35

40

45

Number of neighbors

Fig. 3. Numerical solution for the modification with endogenous vacancy rate and wages. Left panel: mismatch level, right panel: aggregate welfare, as the function of the number of neighbors k for different values of the homophily level . These values are calculated for  = 4, ˛ = 2 that imply  = 0.8857.

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

461

0.075 Gamma = 0.7 Gamma=0.8 Gamma=0.886 Gamma=0.9 Gamma=1

0.065 0.06

Gamma = 0.7 Gamma=0.8 Gamma=0.886 Gamma=0.9 Gamma=1

0.042 0.04

Vacancy rate

Unemployment rate

0.07

0.055 0.05

0.038 0.036 0.034 0.032

0.045

0.03

0.04 10

15

20

25

30

35

40

0.028 10

45

15

20

Number of neighbors

25

30

35

40

35

40

45

Number of neighbors

0.955

0.945

Wage in good match

0.95

Wage in bad match

0.975

Gamma = 0.7 Gamma=0.8 Gamma=0.886 Gamma=0.9 Gamma=1

0.94

0.935

0.97

Gamma = 0.7 Gamma=0.8 Gamma=0.886 Gamma=0.9 Gamma=1

0.965

0.96

0.93 10

15

20

25

30

35

40

45

10

15

20

Number of neighbors

25

30

45

Number of neighbors

Fig. 4. Numerical solution for the modification with endogenous vacancy rate and wages. Left upper panel: unemployment rate, right upper panel: vacancy rate, left lower panel: wages in a bad match, right lower panel: wages in a good match, as the function of the number of neighbors k for different values of the homophily level . These values are calculated for  = 4, ˛ = 2.

6

0.35 Gamma = 0.6 Gamma = 0.7 Gamma = 0.8 Gamma = 0.9 Gamma = 1

5.5

0.3

0.25

4.5

Mismatch level

Optimal search intensity

5

Gamma = 0.6 Gamma = 0.7 Gamma = 0.8 Gamma = 0.9 Gamma = 1

4 3.5 3 2.5

0.2

0.15

0.1

2 0.05 1.5 1 10

15

20

25

30

35

Number of neighbors

40

45

0 10

15

20

25

30

35

40

45

Number of neighbors

Fig. 5. Numerical solution for the modification with endogenous vacancy rate, wages and search intensity. Left panel: optimal search intensity, right panel: mismatch level, as the function of the number of neighbors k for different values of the homophily level . These values are calculated for ˛ = 2.

462

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

Robustness: simulations on fixed networks See Fig. 6.

M N N Fig. 6. Simulation results on fixed networks. Left panel: Pr M B /Pr G − Pr B /Pr G for different values of homophily level, right panel: mismatch level for different number of neighbors. Baseline parameter values: ˛ = 2,  = 3, ω = 0.3,  = 0.1, N = 1000, k = 6.

A.5. Proofs A.5.1. Proposition 1 Proof.

I write Eq. (7) in the following form:

(1 + u)eG − (1 + u)uqG (u, eB , eG ) = 0 After substituting the expression for qG (u, eB , eG ) and u = 1 − eB − eG , we obtain: g(eB , eG ) ≡ ((1 − eB − eG ) + 1)eG − ω((1 − eB − eG ) + (eG + (1 − )eB )(1 − (eB + eG )k )) = 0 where 0 ≤ eG ≤ 1 and 0 ≤ eB ≤ 1 such that eB + eG ≤ 1 Similarly, I rewrite Eq. 6: f (eB , eG ) ≡ ((1 − eB − eG ) + 1)˛eB − ω(1 − eB − eG + (eB + (1 − )eG )(1 − (eB + eG )k )) = 0 In several steps I show that a unique equilibrium of the model exists using the properties of the two implicit functions eB1 (eG ) defined by g(eB , eG ) = 0 and eB2 (eG ) defined by f(eB , eG ) = 0. I state sufficient conditions for the existence of a unique equilibrium. First, I analyze eB2 (eG ) and show that it is a decreasing function under some conditions. I define the following implicit derivative: 2

∂f (eB , eG )/∂eG deB (eG ) =− deG ∂f (eB , eG )/∂eB I compute the partial derivatives of this equation and show that under some conditions both of them have positive sign. First,

∂f (eB , eG ) = ˛(1 + (1 − 2eB − eG )) + ω(1 − ) + (eB + eG )−1+k (eB (1 + k) + eG ( + k − k))ω ∂eB

(24)

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

463

Here the second and third terms are positive and the first term is negative for some values of eB and eG , while it’s positive for others. The negative term is highest in absolute value when eB = 1, eG = 0. I give conditions on the parameters that guarantee that the partial derivative is positive for every value of eB and eG between 0 and 1 such that eB + eG ≤ 1. First, I compute the second-order derivative of f(eB , eG ) with respect to eB : 2

∂ f (eB , eG ) = −2˛ + (eB + eG )−2+k k(eB (1 + k) + eG (−1 + 3 + k − k))ω ∂eB2 This takes the lowest value when eB = eG = 0 where it is negative. If we increase either eB or eG , the second-order derivative rises. It rises more if we increase eB since (1 + k) − (−1 + 3 + k − k) = (k − 1)(2 − 1) ≥ 0. Thus, it takes the highest value when eB = 1 and eG = 0 (recall that eB + eG ≤ 1):



∂ f (eB , eG ) ∂eB2 e 2

= −2˛ + kω(1 + k)

B =1,eG =0

We have two cases according to whether this expression is positive or negative.

∂ f (eB ,eG ) ≤ 0: the highest value of the second-order derivative is negative, hence, the derivative is negative 2 ∂ eB eB =1,eG =0 ∂f (eB ,eG ) for every value of eB and eG . Thus, the first-order derivative is decreasing in eB and takes the smallest value when ∂eB 2

1.

eB = 1, eG = 0. We require the first-order derivative to be positive even at this smallest value. A sufficient condition is obtained by substituting eB = 1, eG = 0 to the equation of

∂f (eB ,eG ) : ∂eB

ω(1 + k) > ˛( − 1)



∂ f (eB ,eG ) ∂eB2 2

2.

eB =1,eG =0

> 0: for some high values of eB , the first-order derivative

∂f (eB ,eG ) is increasing in eB . Thus the ∂eB

first-order derivative has an interior minimum in eB defined by 2

∂ f (eB , eG ) = −2˛ + (eB + eG )−2+k k(eB (1 + k) + eG (−1 + 3 + k − k))ω = 0 ∂eB2 There are many eB and eG values satisfying this equation. Since we are looking for the minimum value of function is increasing in eG , we can choose the solution where eG = 0:

∂f (eB ,eG ) and that ∂eB



∂ f (eB , eG ) ∂eB2 e 2

= −2˛ + eB−2+k keB (1 + k)ω = 0

G =0

Solving this equation for eB gives:



eB∗ Thus,

=

2˛ k(1 + k)ω

1/(k−1)

∂f (eB ,eG ) is minimal when eG = 0 and eB = eB∗ . We require this partial derivative to be positive at this value: ∂eB



∂f (eB , eG ) ∂eB e

∗ B =eB ,eG =0

= ˛(1 + (1 − 2eB∗ )) + w(1 − ) + eB∗k (1 + k)w > 0

Second, I analyze the other partial derivative:

∂f (eB ,eG ) . I give a condition which implies that it is positive. I compute the ∂eG

partial derivative and the second-order derivative of f(eB , eG ) with respect to eG :

∂f (eB , eG ) = −˛eB  + ω + ω(eB + eG )−1+k (eB (1 + (−1 + k)) + eG (1 − )(1 + k)) ∂eG

(25)

2

∂ f (eB , eG ) = (eB + eG )−2+k k(eB (2 + (−3 + k)) + eG (1 − )(1 + k))ω≥0 ∂eG2 The second-order derivative is positive and the first-order derivative is thus increasing in eG . I would like to give conditions that the first-order derivative is positive for every value of eG and eB . It takes the smallest value when eG = 0:



∂f (eB , eG ) ∂eG e

G =0

= −˛eB  + ω( + eBk (1 + (−1 + k)))

464

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

Here the first term is negative while the second is positive. When eB = 0, this expression takes a positive value ω. When eB becomes higher, it starts to decrease: it’s derivative with respect to eB is

∂(∂f (eB , eG )/∂eG )|eG =0 = −˛ + eBk−1 kω(1 + (k − 1)) ∂eB

(26)

which takes a negative value when eB = 0. There are two possibilities:

∂f (eB ,eG ) decreases in eB for all the range 0 ≤ eB ≤ 1 and hence takes the minimum value at eB = 1. This happens ∂eG e =0 G ∂f (eB ,eG ) when the derivative of with respect to eB is negative for eB = 1: ∂eG e =0

1.

G

−˛ + kω(1 + (k − 1)) ≤ 0



Hence,

∂f (eB ,eG ) takes the minimum value at eB = 1 and this minimum value should be positive. The condition for this: ∂eG e =0 G

∂f (eB , eG ) |eG =0,eB =1 = −˛ + ω(1 + k) > 0 ∂eG Since in this case the partial derivative



of eB and eG .

∂f (eB ,eG ) takes a positive value at it’s minimum, it is positive for the whole range ∂eG

∂f (eB ,eG ) increases in eB and thus there is an interior minimum of it 0 < eB∗∗ < 1. This happens ∂eG e =0 G ∂f (eB ,eG ) whenever the derivative of with respect to eB is positive for eB = 1: ∂eG e =0

2. For high values of eB ,

G

−˛ + kω(1 + (k − 1)) > 0 The interior minimum is characterized by the derivative in (26) to be equal to zero: −˛ + eBk−1 kω(1 + (k − 1)) = 0 Solving for eB∗∗ :

 eB∗∗ =

˛ kω(1 + (k − 1))

1/(k−1)

∂f (eB ,eG ) is thus positive whenever it is positive at eG = 0, eB = eB∗∗ : ∂eG



∂f (eB , eG ) ∂eG e

∗∗ G =0,eB =eB

Summarizing,

de2 B (eG ) deG



= −˛eB∗∗  + ω  + eB∗∗k (1 + (−1 + k)) > 0

< 0 whenever

∂f (eB ,eG ) ∂f (eB ,eG ) > 0 and > 0. The sufficient conditions for this are, as shown above, ∂eG ∂eB

the following: 1. ˛(1 + (1 − 2eB∗ )) + ω(1 − ) + (eB∗ )k (1 + k)ω > 0 where eB∗ is given by:

eB∗ =

⎧ ⎪ ⎨ ⎪ ⎩

if − 2˛ + kω(1 + k) ≤ 0

1



2˛ k(1 + k)ω

 1 k−1

if − 2˛ + kω(1 + k) > 0

and

 2. −˛eB∗∗ + ω  + eB∗∗k (1 + (−1 + k)) > 0 where eB∗∗ is given by:

eB∗∗

=

⎧ ⎪ ⎨ ⎪ ⎩

if − ˛ + kω(1 + (k − 1)) ≤ 0

1



˛ kω(1 + (k − 1))

 1 k−1

if − ˛ + kω(1 + (k − 1)) > 0

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

465

At the second place, I analyze the properties of the implicit derivative deB1 /deG where eB1 (eG ) is defined by g(eB , eG ) = 0. I show that the conditions just detailed in the previous paragraph are sufficient to guarantee that deB1 /deG is negative. This implicit derivative can be expressed as: 1

deB (eG ) (∂g(eB , eG ))/(∂eG ) =− deG (∂g(eB , eG ))/(∂eB )

(27)

I write down the partial derivative in the denominator first:

∂g(eB , eG ) = (1 + (1 − eB − 2eG )) + ω( − ) + ω(eB + eG )k−1 (eG (1 + k) + eB ( + k(1 − ))) ∂eG Here the first term is negative for some values of eB and eG while the second and third terms are positive. Comparing this equation to the expression of

∂f (eB ,eG ) in (24), we can observe that they are almost the same with three differences: ∂eB

1. Here the possibly negative first term is smaller in absolute value than in the equation of 2. The second term, which is positive, is higher here than in the equation of

∂f (eB ,eG ) since ˛ ≥ 1. ∂eB

∂f (eB ,eG ) since  ≥ 1. ∂eB

3. Apart from the previous two differences, the two expressions are the same if we exchange the roles of eB and eG . ∂f (eB ,eG ) is positive for any 0 ≤ eB ≤ 1 and 0 ≤ eG ≤ 1 such that eB + eG ≤ 1. ∂eB ∂g(eB ,eG ) Clearly, the same steps could be carried out here to show that is positive for the same values, we only need to change ∂eG ∂f (eB ,eG ) ∂g(eB ,eG ) is positive are sufficient for the roles of the variables eB and eG . Thus, the same conditions that implied that ∂eB ∂eG

Previously I showed that under some conditions

to be positive since in here the negative term is smaller in absolute value while the positive term is higher, everything else being equal. Next, consider the other partial derivative in (27), it can be expressed as:

∂g(eB , eG ) = −˛eG + ω( +  − 1) + ω(eB + eG )k−1 (eB (k + 1)(1 − ) + eG (1 + (k − 1)) ∂eB We can compare this expression to the equation of

∂f (eB ,eG ) in (25). They are almost the same with the differences that ∂eG

1. the first negative term is smaller here in absolute value since  ≥ 1, 2. the second term, which is positive, is higher here again since  ≥ 1, 3. the roles of eB and eG are exchanged. ∂f (eB ,eG ) is positive for any 0 ≤ eB ≤ 1 and 0 ≤ eG ≤ 1 such that eB + eG ≤ 1. Here the same steps could be ∂eG ∂g(eB ,eG ) carried out to show that is positive only changing the roles of the variables eB and eG in the analysis. The conditions ∂eB ∂g(eB ,eG ) obtained there imply that is positive since here the positive term is higher and the negative term is lower in absolute ∂eB

Above I showed that

value.

Since both

de1 ∂g(eB ,eG ) ∂g(eB ,eG ) B (eG ) and are positive under the conditions detailed above, the implicit derivative de is negative. ∂eG ∂eB G

I have shown that f(eB , eG ) and g(eB , eG ) define two monotone decreasing functions eB (eG ) in the (eG , eB ) space. Now, I demonstrate that they indeed cross. First, I show that eB1 (eG ) intersects with the y-axis at a higher point than eB2 (eG ). eB1 (eG ) is defined by g(eB , eG ) = 0 and g(eB , 0) = −ω((1 − eB ) + (1 − )eB (1 − eBk )). Clearly, g(1, 0) = 0, so eB1 (0) = 1. eB2 (0) is defined by f (eB , 0) = ((1 − eB ) + 1)˛eB − ω(1 − eB + eB (1 − eBk )) = 0. We have that f(0, 0) = − ω < 0 and f(1, 0) = ˛ > 0. Since f(eB , eG ) is monotone increasing in eB , we know that 0 < eB2 (0) < 1. Thus eB2 (0) < eB1 (0). Second, I consider the intersections with the x-axis and show that there the order is reversed: [eB1 ]

−1 [eB1 ] (0)

−1

(0) < [eB2 ]

−1

(0).

k )) = 0. We have that g(0, 0) = − ω and g(0, is defined by g(0, eG ) = ((1 − eG ) + 1)eG − ω((1 − eG ) + eG (1 − eG

1) = . Since g(eB , eG ) is increasing in eG , we have that 0 < [eB1 ] k) )eG (1 − eG

−1 [eB2 ] (0)

−1

(0) < 1. [eB2 ]

−1

−1 [eB1 ] (0).

(0) is defined by f (0, eG ) = −ω(1 − eG + (1 −

= 0. This equation holds for eG = 1, thus =1> Consequently, eB1 (eG ) crosses the y-axis at a higher point than eB2 (eG ) but intersects with the x-axis at a lower point. This implies that the two lines have to cross at some intermediate value. Since the two functions are strictly monotone decreasing, there can only be one crossing point. This is illustrated in Fig. 1 in the main text. 

466

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

A.5.2. Lemma 1 Consider the first point. The mismatch level is given by the following expression: N qB 1 Pr M 1 eB B + Pr B = = = eG ˛qG ˛ Pr M + Pr N ˛ G G

1 ˛

=



Pr M B Pr M G

+

Pr N B Pr N G





Pr M B N Pr M G + Pr G

+

Pr N B



N Pr M G + Pr G

1 = ˛



Pr M B

Pr M G

M N Pr M G Pr G + Pr G

+

Pr N B

Pr N G



M N Pr N G Pr G + Pr G

(1 − )

where  ∈ (0, 1). The first equality uses the equilibrium relations (6) and (7), the second equality uses that the good and bad arrival rates are sums of good and bad arrival rates by search methods (market and networks). Consider the second point. First, assume that  rises. Consider what happens to the two equilibrium equations f(eB , eG ) and g(eB , eG ) as defined in the proof of Proposition 1. f(eB , eG ) increases above zero. To get back to equilibrium eB or eG has to decrease since both

∂f (eB ,eG ) and ∂eB

∂f (eB ,eG ) are positive under the conditions of Proposition 1. Hence, the solution of f(eB , eG ) = 0, eB2 (eG ) moves downward. On ∂eG g(e ,e )

the contrary, g(eB , eG ) becomes lower after an increase of  when  < ω. ∂ B G = (1 − eB − eG )(eG  − ω) which is negative when  < ω. Thus, eB has to increase to reach a new equilibrium where g(eB , eG ) = 0 and eB1 (eG ) moves to the right. Looking at Fig. 1, both changes imply that in the new equilibrium eG is higher and eB is lower. Consequently, the mismatch decreases. Second, suppose that ˛ becomes higher. g(eB , eG ) does not change. f(eB , eG ) becomes higher implying that eB has to decrease for the new equilibrium. The loci f(eB , eG ) moves downward. Looking at Fig. 1, this causes that in the new equilibrium eB is lower and eG is higher. The mismatch decreases. Third, assume that either  > 1 or ˛ > 1. This implies that in equilibrium eB < eG since for  = 1 and ˛ = 1, eB = eG (Proposition 2) and when any of these parameters increases, the mismatch level decreases. Now consider a rise of . Since eB < eG , f(eB , eG ) increases because it’s derivative with respect to  is −ω(eB − eG )(1 − (eB + eG )k ) > 0. To get to the new equilibrium eB has to decrease. On the contrary, g(eB , eG ) decreases and eG has to increase to get to the new equilibrium. Hence, the loci f(eB , eG ) = 0 moves downward while the loci g(eB , eG ) = 0 moves to the right in Fig. 1. These changes imply that eB is lower while eG is higher in the new equilibrium. The mismatch decreases. A.5.3. Proposition 2 Proof. (i) We assume that  = 1, ˛ = 1. We substitute these parameter values into the equations of the arrival rates (8) and (9). Then we divide the two equilibrium conditions (6) and (7) and we get the following equation: qB 1 + Qk(eB + (1 − )eG ) eB = = qG eG 1 + Qk(eG + (1 − )eB ) where Q =

1−(1−u)k . uk

Rearranging this latter equality we obtain:

eB (1 + Qk(eG + (1 − )eB )) = eG (1 + Qk(eB + (1 − )eG )) 2 − eB2 ) = 0 eG − eB + (1 − )Qk(eG

(eG − eB )(1 + (1 − )Qk(eG + eB )) = 0 since the second term is always positive, the multiplication can be zero only if eG = eB . This is true for any  and k. (ii) If we look at the good and bad arrival rates through social contacts in (1) and (3), we can see that if eG = eB , they are equal for any value of , u and k. Hence, the network provides good and bad jobs at the same rate.  A.5.4. Proposition 3 Proof.

We compare the mismatch created by the two search methods. We can write: PGM PBM PGN PBN

=

=

eG + (1 − )eB  + (1 − )(eB /eG ) = e eB + (1 − )eG  eB + 1 −  G

This latter ratio increases if the mismatch decreases (we showed in the proof of the previous proposition that PBN /PGN was increasing in the mismatch level).

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

467

First of all, we look at the case when ˛ = 1 and  = 1. Here the two search methods provides equal expected wages if: PGM PBM

=

PGN PBN

⇔=

eG eB

Now, we see that this exactly is what the equilibrium conditions imply. The equilibrium mismatch level is given by the following expression when ˛ = 1 and  = 1: eB 1 + Q keB = eG  + QkeG Equivalently, eB  + QkeG eB = eG + QkeB eG Solving this equation for , we get that in equilibrium  = eG /eB . Hence, if ˛ = 1 and  = 1, the two methods generate mismatch to the same extent. By Lemma 1, we also know that the mismatch in this economy with social network coincides with the mismatch of an economy without the network channel. When we decrease  from 1, the performance of the network decreases because 1. PGN /PBN decreases as  becomes smaller if eB < eG which is the case when  > 1 and 2. as  decreases the mismatch increases which implies a further decrease of PGN /PBN . Hence, the market gives higher expected wage and the presence of social networks increases the mismatch in the society when  < 1. Now, if ˛ increases from value 1, by the previous proposition we know that the mismatch level decreases. This leaves the market efficiency unchanged, since it depends only on , and makes the network arrival more efficient. If the mismatch decreases, PGN /PBN increases. Hence when  = 1 the network creates less mismatch than the market. If  decreases, PGN /PBN decreases for the reason explained in the previous paragraph. When  = 0.5, the market is always more efficient than the network: PGN PBN

=

PM 0.5eG + 0.5eB = 1 <  = GM 0.5eB + 0.5eG PB

Thus if  = 1, the network efficiency is higher, for  = 0.5 the market efficiency is higher. Hence, we have that if ˛ > 1, there exists a threshold value of homophily  where the two search methods create the same amount of mismatch. By the same reasoning we have that  decreases if ˛ increases: the network becomes more efficient since ˛ decreases mismatch. So we find equal network and market efficiency for a lower . Lemma 1 implies that when the network creates less mismatch than the market, the mismatch level is lower in the economy with than without social networks.  A.5.5. Proposition 4 Proof. First, we show that  is independent of the network connectivity k. We again write the mismatch level as the linear combination of the probability ratios: eB 1 = eG ˛ where  =



PBM

PBN

PG

PG

+ M

PM G

P N +P M G G only PGN



(1 − ) N

. We know that PGN /PBN does not depend on k directly, it changes only with the mismatch level (eB /eG ). If k

increases, = kQ (eG + (1 − )eB ) reacts: it increases. First, assume that  =  and k rises. Fix eB and eG on the right side of the equation. As k rises, PGN becomes higher and  becomes lower, i.e. the weight on the social network arrival rises. Since when  = , PGN /PBN = PGM /PBM by definition, the mismatch level eB /eG does not change on the left-hand side in the equation in spite of the change in . The same should be true on the right-hand side too: eB /eG remains unchanged and this means that PGN /PBN , depending on eB /eG , does not change either. We also know that as k increases, eB and eG rise in absolute value. This makes PGN higher what follows from the proof of Proposition 1. There I showed that

∂g(eB ,eG ) ∂g(eB ,eG ) ≥0 and ≥0 and the derivative of g(eB , eG ) is equivalent to the derivative ∂eB ∂eG

of PGN if we substitute  = 0 to g(eB , eG ). Since PGN rises,  decreases again but, just as before, the mismatch level eB /eG remains unchanged. Hence, the value of PGN /PBN does not change and it is equal to PGM /PBM for the same value of homophily:  does not change. Second, suppose that  >  which implies that PGN /PBN > PGM /PBM . Assume that k rises. The direct effect of this rise is that N PG becomes higher and  becomes lower. Hence, the weight on the network arrival rises and on the left-hand side eB /eG decreases. The indirect effect of the rise of k also decreases mismatch. First, the decrease of eB /eG makes PGN /PBN higher. Second, as k rises both eB and eG becomes higher, this makes PGN to be higher and  to be lower. The weight on the network arrival increases again making the mismatch even lower. Thus, both the direct and indirect effects of an increase in k make the mismatch level lower when  > .

468

G. Horváth / Journal of Economic Behavior & Organization 106 (2014) 442–468

As for the case of  < , a similar argument holds with an opposite sign. Here PGN /PBN < PGM /PBM and an increase in k again leads to a higher weight on the network arrival which this time causes an increase in the mismatch. An increase in the mismatch makes the network arrival even more ineffective.  References Antoninis, M., 2006. The wage effects from the use of personal contacts as hiring channels. J. Econ. Behav. Organ. 59, 133–146. Beaman, L., Magruder, J., 2012. Who gets the job referral? Evidence from a social network experiment. Am. Econ. Rev. 102, 3574–3593. Bentolila, S., et al., 2010. Social contacts and occupational choice. Economica 77, 20–45. Berkhout, P., Hartog, J., Webbink, D., 2010. Compensation for earnings risk under worker heterogeneity. South. Econ. J. 76, 762–790. Boorman, S.A., 1975. A combinatorial optimization model for transmission of job information through contact networks. Bell J. Econ. 6, 216–249. Bramoulle, Y., Goyal, S., 2013. Favoritism (Unpublished manuscript, http://www.econ.cam.ac.uk/people/faculty/sg472/wp13/Favoritism62.pdf.pagespeed. ce.mtfZY14SBt.pdf). Bramoulle, Y., Saint-Paul, G., 2010. Social networks and labor market transitions. Labour Econ. 17, 188–195. Cahuc, P., Fontaine, F., 2009. On the efficiency of job search with social networks. J. Public Econ. Theory 11, 411–439. Calvo-Armengol, A., 2004. Job contact networks. J. Econ. Theory 115, 191–206. Calvo-Armengol, A., Jackson, M.O., 2004. The effects of social networks on employment and inequality. Am. Econ. Rev. 94 (3), 426–454. Calvo-Armengol, A., Jackson, M.O., 2007. Networks in labor markets: wage and employment dynamics and inequality. J. Econ. Theory 132, 27–46. Calvo-Armengol, A., Zenou, Y., 2005. Job matching social networks and word-of-mouth communication. J. Urban Econ. 57, 500–522. Cingano, F., Rosolia, A., 2012. People I know: job search and social networks. J. Labor Econ. 30 (2), 291–332. Dustmann, Ch., Glitz, A., Schoenberg, U., 2011. Referral-based Job Search Networks. Center for Research and Analysis of Migration Discussion Paper No. 14/11. Galenianos, M., 2012. Hiring Through Referrals, Pennsylvania State University Working Paper. http://papers.ssrn.com/sol3/papers.cfm?abstract id=1802812 Galenianos, M., 2013. Learning about match quality and the use of referrals. Rev. Econ. Dyn. 16, 668–690. Galenianos, M., 2014. Hiring through referrals. J. Econ Theory. 152, 304–323. Galeotti, A., Merlino, L.P., 2014. Endogenous job contact networks. Int. Econ. Rev. (forthcoming). Glitz, A., 2013. Coworker Networks in the Labour Market. IZA Discussion Papers No. 7392. Granovetter, M.S., 1995 [1974]. Getting a Job: A Study of Contacts and Careers. University of Chicago Press, Chicago. Holzer, H.J., 1987. Informal job search and black youth unemployment. Am. Econ. Rev. 77, 446–452. Ioannides, Y.M., Soetevent, A.R., 2006. Wages and employment in a random social network with arbitrary degree distribution. Am. Econ. Rev. Pap. Proc. 96 (2), 270–274. Korpi, T., Tahlin, M., 2008. Educational mismatch, wages and wage growth: overeducation in Sweden 1974–2000. Labour Econ. 16, 183–193. Kugler, A., 2003. Employee referrals and efficiency wages. Labour Econ. 10, 531–556. Leuven, E., Oosterbeek, H., 2011. Overeducation and mismatch in the labor market. Handbook of the Economics of Education, vol. 4. North Holland, Amsterdam, pp. 283–326. Montgomery, J.D., 1991. Social networks and labour-market outcomes: toward an economic analysis. Am. Econ. Rev. 81 (5), 1408–1418. Moscarini, G., 2001. Excess worker reallocation. Rev. Econ. Stud. 68, 593–612. Mouw, T., 2003. Social capital and finding a job: do contacts matter? Am. Sociol. Rev. 68, 868–898. Nordin, M., Persson, I., Rooth, D.-O., 2010. Education–occupation mismatch: is there an income penalty? Econ. Educ. Rev. 29, 1047–1059. Obukhova, E., 2012. Motivation vs. relevance: using strong ties to find a job in Urban China. Soc. Sci. Res. 41, 570–580. Pellizzari, M., 2010. Do friends and relatives really help in getting a good job? Ind. Labor Relat. Rev. 63, 494–510. Petrongolo, B., Pissarides, Ch., 2001. Looking into the black box: a survey of the matching function. J. Econ. Lit., 390–431. Pissarides, Ch.A., 2000. Equilibrium Unemployment Theory. The MIT Press, Cambrindge, Massachusetts. Ponzo, M., Scoppa, V., 2010. The use of informal networks in Italy: efficiency or favoritism? J. Soc. Econ. 39, 89–99. Robst, J., 2007. Education and job match: the relatedness of college major and work. Econ. Educ. Rev. 26, 397–407. Sahin, A., Song, J., Topa, G., Violante, G.L., 2011. Measuring Mismatch in the U.S. Labor Market, Mimeo. Shimer, R., 2005. The cyclical behavior of equilibrium unemployment and vacancies. Am. Econ. Rev. 95, 25–49. Shimer, R., 2007. Mismatch. Am. Econ. Rev. 97, 1074–1101. Thisse, J.-F., Zenou, Y., 2000. Skill mismatch and unemployment. Econ. Lett. 69, 415–420. van der Leij, M.J., Buhai, S., 2008. A Social Network Analysis of Occupational Segregation. Fondazione Eni Enrico Mattei Working Paper 31.2008.