Distributional effects of hiring through networks

Review of Economic Dynamics 20 (2016) 90–110

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Distributional effects of hiring through networks Yoske Igarashi 1 University of Exeter, United Kingdom

a r t i c l e

i n f o

Article history: Received 15 February 2014 Received in revised form 29 January 2016 Available online 3 February 2016 JEL classification: E24 E60 Keywords: Random search Network Referral Policy analysis Welfare Dynamics

a b s t r a c t How would a policy that bans the use of networks in hiring (e.g. anti-old boy network laws) affect welfare? We answer this question in a random search model in which there are two hiring methods, formal costly channels and referral channels, and there are two types of workers, networked workers, who can be hired through both channels, and nonnetworked workers, who cannot be hired through referrals. We show that the effect of a referral-restricting policy on non-networked workers can be either positive or negative, depending on model parameters. In our calibration such a policy would make nonnetworked workers slightly worse off and networked workers substantially worse off. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Workers’ social networks are widely used in the labor market (Ioannides and Loury, 2004; Topa, 2011).2 Albert Rees (1966) points out that workers’ social networks can alleviate informational frictions that are present in the ‘unstandardized’ market of labor. Both workers and jobs are highly heterogeneous due to workers’ skills, specializations and personality, as well as jobs’ characteristics, locations, workplace culture, and so on. Therefore, firms normally have to incur a large cost to find a good match, including the cost of assessing applications, of interviews and internships. In such circumstances, workers’ social networks (e.g. relatives, friends, neighbors, school alumni, etc.) provide firms with an alternative means to resolve uncertainty about the quality of a potential match. That is, an employee occasionally acts as a ‘matchmaker’ and produces a referral.3 Because the referrer personally knows the qualities of a candidate, and because he himself is an insider of the company, the referrer can tell if the candidate is suitable for the job. This way, referrers act as a better alternative to public and private placement agencies while reducing firms’ costly screening activities.

E-mail address: [email protected]. I am most grateful to my PhD advisor Neil Wallace. I also thank Byung Soo Lee, Dario Pozzoli, David Jinkins, Edward Green, James Jordan, John Maloney, Manolis Galenianos, Rish Singhania, Rulin Zhou, Saroj Bhattarai, Vaidyanathan Venkateswaran, Yu Awaya, the Associate Editor, the two anonymous referees, and seminar participants at Cornell University, Bank of Canada, Indiana University, Penn State University, University of Exeter, and Copenhagen Business School for helpful comments and advice. 2 A large proportion of workers attempt to use their personal networks in their job search (e.g. Holzer, 1987a, 1987b; Elliott, 1999) and many of them actually find their jobs through referrals (e.g. Corcoran et al., 1980; Lin et al., 1981; Granovetter, 1995; Elliott, 1999). Also, some surveys report positive effects of the use of referrals on wages (e.g. Korenman and Turner, 1996). 3 Rees (1966) points out that referral can provide good screening as referrers care about their own reputation and that asking for referrals is usually costless for firms. He also points out that candidates can get better information about the workplace from referrers prior to hiring. 1

http://dx.doi.org/10.1016/j.red.2016.01.002 1094-2025/© 2016 Elsevier Inc. All rights reserved.

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This paper examines the welfare implications of the use of referrals. Our main interest is how policies that restrict referrals for the sake of fairness in hiring practice can affect worker welfare. Today, we observe many public institutions avoiding referrals and going through more formal channels by publicly posting job openings. In addition, employers may be required by law to keep their job advertisement up for a certain period of time before interviewing, and to have a certain number of interviews per post. Also, anti-nepotism laws and affirmative action regarding race, sex, nationality, caste, etc. restrict one from hiring a person who belongs to the same social group. These policies make hiring through referrals more difficult or less attractive for firms and often involve high costs to implement.4 Do these policies benefit people who do not have personal connections or access to referrals? To answer this question, we study a version of Galenianos’s (2014a) random search model with formal hiring and referral hiring. In our model, there are two permanent types of workers: workers with a social network and workers without one. Networked workers can be hired both through formal channels and through referrals, while non-networked workers cannot be hired through referrals. We show that a stationary equilibrium exists and it is a unique steady state under some moderate conditions. In this steady state, networked workers have lower unemployment and higher wages than non-networked workers. In this framework, we consider a hypothetical policy that bans hiring through referrals. We compare welfare before and after the policy starts, taking into account the transition to the new steady state. We show that this policy can lower the welfare of non-networked workers depending on model parameter values, typically in the economy with small bargaining power of workers. The calibrated version of the economy shows that a ban on referrals reduces the welfare of all workers; non-networked workers are made slightly worse off and networked workers substantially worse off. The former result may appear surprising if one imagines a fixed number of unemployed workers competing over a fixed number of posts so that networked workers ‘crowd out’ non-networked workers in the job queue. In a general equilibrium framework, however, the presence of network hiring not only reduces the number of posted vacancies but also the number of unemployed workers in the job queue by mitigating search frictions. When the latter effect dominates, the job-finding rate of non-networked workers is higher than in the absence of network hiring. In this way, the crowding-out effect of network hiring on nonnetworked workers is more delicate when unemployment and vacancy are endogenous. Our simple model suggests that the effectiveness of a policy that has the side effect of increasing search frictions is ambiguous even for policy-makers who care about people without any connections or access to referrals. The effect of worker networks in the labor market has been widely studied. Montgomery (1991, 1992, 1994); CalvoArmengol and Jackson (2004, 2007), Mayer (2011), Zenou (2015), and Sato and Zenou (2015) employ a partial equilibrium framework. These models feature exogenous job finding rates for workers so they cannot be used to study firms’ reaction to a policy that limits network hiring. Another set of models, Calvo-Armengol and Zenou (2005), Fontaine (2008), Kuzubas (2010), Galeotti and Merlino (2014), and Galenianos (2013), study network hiring in a general equilibrium search framework in which the firm free-entry condition determines the number of jobs posted. Many of these models add a network component to the matching function a la Mortensen–Pissarides (1994). That is, when a firm is matched with a worker after costly screening, this worker, if already employed, passes on that job offer to one of his unemployed friends. This assumption does not fit the idea of unmodeled heterogeneity behind the matching function.5 Moreover, in such models firms bear the same cost whether referral is used or not. In this paper, we take Rees’s (1966) view of referrals as a matching technology that is distinct from and cheaper than more formal hiring methods. For our purpose, we use Galenianos’s (2014a) framework because his model incorporates Ree’s view of referrals into a simple general equilibrium search framework.6 We extend his model to a model with heterogeneous types of workers (networked and non-networked) to study the effect of a referral-restricting policy on non-networked workers.7 Finally, the use of online social network services (SNS) in hiring has become common in modern society. Taking Rees’s notion of network hiring, however, our view is that the development of SNS does not necessarily change our framework. There are two potential impacts of SNS. First, SNS enables people to track more easily their friends’ current situations such as employment status and location (perhaps raising the efficiency of referral channels). Second, SNS enables workers to post ‘ads’ of themselves just like firms post job ads. Therefore, firms can direct their search to a group of workers with characteristics desirable to them. This feature can partly replace the role played by public agencies and private placement companies, that is, the role as a loose matchmaker that has some limited information about both sides (perhaps raising the efficiency of formal channels). At this point, firms’ use of SNS seems close to a formal costly method in that firms still have to engage in further costly screening. The rest of the paper proceeds as follows. Section 2 states the model and theoretical results. Section 3 demonstrates calibration and a policy experiment. Section 4 proves that the model’s assumption that one firm employs one worker is innocuous. Section 5 concludes.

4

See for instance Fair Employment in Northern Ireland Code of Practice (1989) issued by the Equality Commission for Northern Ireland. Such models emphasize the function of worker networks to pass on job opening information. However, searching for job openings is not a hard matter nowadays and does not seem to be a major source of search friction, nor does coordination failure among searchers. 6 Montgomery (1994) and Galenianos (2013) also distinguish formal channels and referral channels. Montgomery (1994) is a partial equilibrium model. Galenianos (2013) focuses on the relative use of referrals so that the use of referrals has no effect on the aggregate matching efficiency. 7 His other paper, Galenianos (2014b), allows for two types of workers that can differ in terms of productivity and network size. However, this model cannot be used to deal with our question because it does not allow a group of workers to have no network at all and the two types are assumed to have equal population. 5

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2. Model The model is Galenianos (2014a) with two types of workers. Time is continuous and the horizon is infinite. 2.1. Workers and firms There is a continuum of workers and their total measure is one. Each worker has an exogenously given type j ∈ {0, 1}. Type-0 workers have no networks (‘non-networked workers’) and type-1 workers have networks (‘networked workers’). These two types are identical in all other respects. In particular, they have the same productivity. Let n0 and n1 be the proportions of non-networked and networked workers, respectively (n0 + n1 = 1). There is also a continuum of firms and their total measure is infinite. Both workers and firms are risk-neutral, maximizing expected discounted income with discount rate r. At any point in time, each worker is either employed or unemployed. If he is unemployed, he gets flow output b > 0 from home production. Each firm can employ at most one worker, so we will use the terms firm and job interchangeably.8 Each firm/job is either active (i.e., filled with a type- j worker) or unmatched.9 Active firms produce constant flow output y > b. Each unmatched firm is either posting or non-posting. Posting firms pay flow cost k to keep posting their vacancy.10 Entry is costless so firms currently not posting a vacancy can get posting status simply by starting to pay the flow cost. 2.2. Matching technologies New matches between workers and firms are created in two ways and firms can use both. Existing matches are destroyed at some exogenous rate δ , in which case the worker becomes unemployed and the firm becomes unmatched. One matching technology, representing formal channels, is described by the following version of the standard matching function. Let u j be the unemployment rate of type- j workers and u ≡ n0 u 0 + n1 u 1 be the total unemployment rate. Furthermore, let v be the measure of posting firms. Then the matching function creates a flow M (u , v ) ≡ μu 1−η v η of matches between u unemployed workers and v posting firms, where μ > 0 and η ∈ (0, 1). For each unemployed worker, the Poisson arrival rate of a job offer through the matching function is M (u , v )/u = μθ η , where θ ≡ v /u is the labor market tightness. For each posting firm, the arrival rate of a type j-worker is M (u , v )/ v × (n j u j /u ) = μθ η−1 × (n j u j /u ). The other matching technology uses in-network referrals as introduced by Galenianos (2014a). All firms know the same large number of other firms and all type-1 workers know the same large number of other type-1 workers. An existing match between an active firm (firm A) and a type-1 employee (worker A) generates another potential match with Poisson rate ρ ≥ 0. The timeline is as follows. Firm A knows another unmatched firm (firm B) and worker A knows a friend from his network (worker B), and their finding is that firm B and worker B could potentially be a good match. (This kind of finding arrives at rate ρ for each existing match.) In such an event, worker A contacts worker B. If worker B is currently unemployed, which will be the case with probability u 1 , worker A refers her to his employer (firm A). Then firm A refers worker B to the unmatched firm B and a new match is created. Therefore, from the active firm A’s point of view, the arrival rate of network matches is ρ u 1 . From the point of view of the unemployed type-1 worker B, the arrival rate of referrals is (1 − u 1 )ρ . That is, the arrival of referrals is assumed to depend upon the employment rate of their network friends because only existing matches generate referrals.11 An important feature of the network matching is that firms do not have to pay hiring cost k to get a worker referred to them. Any unmatched firm, whether it is currently posting a vacancy or not, could potentially get a referral. Hence each unmatched firm’s decision is whether they should just wait for referrals or whether they should also post a vacancy while waiting for referrals.12 2.3. Wages and values A firm that employs a type- j worker pays flow wage w j to its employee. A type- j worker has lifetime value W j if he is employed and U j if he is unemployed. Also, let the lifetime value of a firm be J j if it is filled by a type- j worker and V if it is posting. The value of a non-posting firm is normalized to zero.

8 One job per firm is a common assumption and it is innocuous under the constant returns to scale production and cost of hiring. This is shown in Section 4. 9 We refer to a match between a firm and a type- j worker as “type- j match”. 10 The real-life example of this cost includes cost of posting ads, reviewing applications, interviews, training interns, etc. It can also include some training cost, for instance, the cost incurred in training a probationary worker who quits before becoming profitable to the firm. 11 It is not necessary that all type-1 workers are friends with one another, but each type-1 worker having a large number of friends is important to keep the distribution of employment status among friends from becoming an individual state. The idea somewhat resembles ‘a large number of household members’ (Merz, 1995; Andolfatto, 1996, etc.). The network on the firm side, on the other hand, is not essential (see section 4). 12 The assumption that search through referral is costless for firms implicitly assumes that the matchmaker (‘worker A’) resolves uncertainty about the match quality and hence the firm does not have to go through costly activities such as interviews and internships. Introducing this cost would make the model significantly different from Galenianos (2014a). In the appendix, we present such a model. An easier way to add a cost for referral is to make the active firm (“firm A”) pay a one-time bonus to its employee (“worker A”) when he brings his friend. We discuss this in section 3.

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When an active firm refers its employee’s friend to an unmatched firm and a new match is created, it receives the whole value J 1 − V created by the new match. This is an implicit one-time payment from the unmatched firm to the active firm and the sharing rule is the active firm’s take-it-or-leave-it offer. Such a sharing rule would be unnecessary if firms could hire more than one worker so that firm A and firm B in the above example were the same one firm. In Section 4, we show that the model in which firms can employ multiple workers is equivalent to the one-firm-one-worker model with the active firm’s take-it-or-leave-it offer. There is a set of Bellman equations associated with the above values. The wage w j is determined by the standard Nash bargaining assumption, where the worker’s bargaining power is β ∈ (0, 1). We assume the free-entry condition V = 0 so a non-posting firm does not gain positive net profit from getting a posting status by starting to pay hiring cost. That is, posting firms and non-posting firms both have value V . Definition 1. A steady state equilibrium is (U i , W i , J i , w i , u i )i =0,1 and ( V , v ) that satisfy

rU 0 = b + μθ η ( W 0 − U 0 )

(1)

rU 1 = b + μθ η ( W 1 − U 1 ) + ρ (1 − u 1 )( W 1 − U 1 )

(2)

rW 0 = w 0 + δ(U 0 − W 0 )

(3)

rW 1 = w 1 + δ(U 1 − W 1 )

(4)

r J 0 = y − w 0 + δ( V − J 0 )

(5)

r J 1 = y − w 1 + δ( V − J 1 ) + ρ u 1 ( J 1 − V )  n ju j μθ η−1 (Jj − V) r V = −k + u

(6) (7)

j =0,1

( W j − U j )/β = ( J j − V )/(1 − β),

j = 0, 1

(1 − u 0 )δ = u 0 μθ η (1 − u 1 )δ = u 1 (μθ η + ρ (1 − u 1 )) V = 0,

(8) (9) (10) (11)

where u = n0 u 0 + n1 u 1 and θ = v /u. In the above, (1)–(7) are Bellman equations, (8) is wage determination, (9)–(10) are inflow-equal-outflow conditions, and (11) is the free-entry condition. That is, the equilibrium is defined as a solution to the system of 12 equations in 12 unknowns. What makes this model different from the standard Mortensen–Pissarides model is the three terms involving ρ ; the flow of workers is affected (i.e., the last term of (10)), some workers have a chance to be hired through networks (i.e., the last term of (2)), and some active firms have a chance to get profits from the use of their employee’s networks (i.e., the last term of (6)). Proposition 1. There exists a steady state equilibrium. The proof of the existence of a steady state is much harder than that of the Mortensen–Pissarides model due to the fact that the right hand sides of (2) and (6) include u 1 , not just θ . The proof is one-by-one elimination of variables and is found in Appendix. In Appendix, we also check the local stability of this steady state numerically. The following features are true in any steady state equilibrium: Proposition 2 (Dominance of networked workers). u 0 > u 1 , W 1 > W 0 , U 1 > U 0 , and w 1 > w 0 . The higher wage for type-1 workers arises due to two effects of networks. First, the type-1 worker has higher unemployment value because of the prospect of getting hired through his network (the last term of (2), or “Effect 1”). Therefore, he has a higher reservation value in Nash bargaining over the wage. Second, the firm can potentially benefit from having an employee who sometimes refers a friend (the last term of (6), or “Effect 2”). So the firm compensates for that benefit by paying a higher wage to type-1 workers. While type-1 workers are always better off than type-0 workers, J 1 > J 0 is not necessarily the case in equilibrium. Indeed, some parameter values lead to J 0 > J 1 . In such a case, the equilibrium wage for networked workers is so high that the firm prefers being matched with a non-networked worker. To understand when that happens, define match surplus as S j ≡ W j − U j + J j − V , j = 0, 1 so that (8) and (11) imply J j = (1 − β) S j . It can be shown that (r + δ + β μθ η )( S 1 − S 0 ) =

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S 1 ρ (u 1 − β). That is, in a ‘worker-friendly’ economy (i.e. high β and low u 1 ), the match surplus is higher for type-0 match than for type-1 match. This is the case where the above “Effect 1” dominates “Effect 2”. While both J 1 > J 0 and J 0 > J 1 are possible, both J 0 and J 1 are higher than V in any equilibrium, so posting firms hire whoever arrives first rather than remaining vacant and waiting for the next arrival of a different type of worker. Next we give some results on comparative statics and uniqueness of equilibrium. For that purpose, consider the standard Mortensen–Pissarides economy, which coincides with our model when ρ = 0. That is, networks do not play a role, thereby rendering the differences between types virtually meaningless. We denote all the endogenous variables of the MP economy with the subscript “m” (e.g. um is the total unemployment rate and θm ≡ v m /um is the labor market tightness of the MP steady state). The unique existence of the MP steady state is easy to show. Lemma 1. The MP steady state (U m , W m , J m , V m , w m , um , v m , θm ) exists and is unique. Due to Lemma 1, um is an implicit function of parameters (r , δ, μ, η, y , b, k, β) that is determined outside our model with networks. In this sense, um can be treated as exogenous to our model, although it is endogenous to the MP model. The next comparative statics result is relevant to the welfare implications of network hiring. Proposition 3 (Dependence of steady states on ρ ). In any equilibrium, u 1 is decreasing in ρ near ρ = 0. Moreover, (i) if um < β , then u 0 is increasing, and v, θ , and w 0 are decreasing in ρ near ρ = 0; and (ii) if um > β , then u 0 and u are decreasing, and θ , w 0 , and w 1 are increasing in ρ near ρ = 0. This proposition tells about the effects of networks and predicts the long-run impact of a policy that removes network hiring. Both in case (i) and in case (ii), the presence of network hiring decreases the unemployment rate of networked workers, u 1 . Also, vacancy v decreases in case (i) and the total unemployment u decreases in case (ii). Although their behaviors are not established analytically in the other case, our numerical exercise finds that when ρ increases, both u and v tend to fall in both cases. That is, the presence of networks lowers the total unemployment and scarcity of job-searchers makes firms refrain from costly job posting. This resembles the ‘crowding out’ effect of networked workers on non-networked workers. Without network hiring, the non-networked would observe more job posts. For non-networked workers, what matters is which variable, u or v, changes more, because non-networked workers are affected by networks only through θ . If θ is high, non-networked workers’ job-finding rate is high, which in turn leads to their low unemployment and high wage as the following logic shows. First, S 0 , the surplus of a match with a type-0 worker, moves in the direction opposite to that of θ (see (57) in appendix). Because J 0 = (1 − β) S 0 and the wage for the non-networked moves in the direction opposite to that of J 0 (see (5)), w 0 and θ move in the same direction. Second, (9) implies that the unemployment among the non-networked, u 0 , moves in the direction opposite to that of θ . Thus, there are only two scenarios: either u 0 increasing and θ & w 0 decreasing, or u 0 decreasing and θ & w 0 increasing, corresponding to the above two cases. The formal proof uses the implicit function theorem to examine the behavior of θ . Intuitively, the two offsetting effects of networks on firms determine the behavior of θ . On the one hand, positive ρ gives type-1 workers high outside value during wage negotiations (i.e. the last term of (2) or “Effect 1”), which leads to a high wage for them and lowers profits for firms, partially discouraging firms’ entry. On the other hand positive ρ allows filled firms potentially to gain profits from network matchings (i.e. the last term of (6) or “Effect 2”), partially encouraging firms’ entry. After Proposition 2, we saw that β > u 1 ⇔ [Effect 1 dominates Effect 2] in equilibrium. In Proposition 3, if (i) β > um is the case, the economy reaches β > u 1 once ρ (locally) departs from zero. This is the situation where for firms the negative effect of networks dominates so they cut vacancy posting a lot. The decrease in v dominates the decrease in u, leading to lower θ and hence to a lower job-finding rate through the matching function. As a result, type-0 workers are disadvantaged by the existence of networked people. If (ii) β < um is the case, the economy reaches β < u 1 once ρ departs from zero. This is the situation where for firms the positive effect of networks dominates, so they do not cut vacancy posting too much. The decrease in v is dominated by the decrease in u, leading to higher θ . As a result, type-0 workers actually benefit from the existence of networked people. The proposition suggests that the effect of networks on non-networked people depends on whether or not the economy is worker-friendly (high β and low unemployment). Once we calibrate the model, we can conjecture which region our economy is likely to be in. As is shown after the proof of Lemma 1, um is increasing in β , k and b, and satisfies limk→∞ um = 1, limβ→0 um > 0, etc. So both um < β and um > β are possible. Finally, because w 0 increases in case (ii), Proposition 2 implies that w 1 should also be increasing at least locally. The direction of w 1 in the other case is ambiguous. Another way to consider the externality from networks is to see the dependence of non-networked workers’ unemployment rate and wage on the proportion of networked people. The following proposition shows the implication of n1 departing from zero. Proposition 4 (Dependence of steady states on n1 ). (i) If u ∗1 < β , then u 0 is increasing and w 0 is decreasing in n1 near n1 = 0; and

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(ii) if u ∗1 > β , then u 0 is decreasing and w 0 is increasing in n1 near n1 = 0, where u ∗1 is the unemployment rate of (zero-measured) type-1 workers at the steady state with n1 = 0.13 Thus the implication of n1 departing from zero is somewhat consistent with the previous proposition’s implication for ρ departing from zero. Given our interpretation of n1 as the proportion of people with potentially ‘useful’ friends, we assume that the policy that bans referral hiring affects ρ , rather than n1 . In models with endogenous network formation, n1 would react to such a policy, as we will discuss briefly in the concluding remarks. Finally, the following uniqueness property is useful when we compute the equilibrium numerically in the next section. Proposition 5 (Uniqueness). Suppose η ≤ 0.5. Then, (i) at most one steady state equilibrium satisfies u 1 ≤ 0.5; (ii) k ≈ 0 is sufficient for uniqueness; and (iii) ρ ≈ 0 and um ≤ 0.5 are sufficient for uniqueness.14 3. Policy implications In this section, we study implications of the policy that bans hiring through networks, imposing ρ = 0. After such a policy is imposed, there is no longer a difference between type-0 and type-1 workers so the environment becomes that of Mortensen–Pissarides. (Hereafter, we mean ρ = 0 by “the MP model”.) As the next proposition states, the MP model has very simple dynamics. Proposition 6 (Stability of the MP steady state). Consider the dynamics of the MP equilibrium:

U˙ m = rU m (t ) − b − μθm (t )η [ W m (t ) − U m (t )]

˙ m = rW m (t ) − w m (t ) − δ[U m (t ) − W m (t )] W

(12) (13)

˙J m = r J m (t ) − ( y − w m (t )) − δ[ V m (t ) − J m (t )]

(14)

V˙ m = r V m (t ) + k − μθm (t )η−1 [ J m (t ) − V m (t )]

(15)

u˙ m = (1 − um (t ))δ − um (t )μθm (t )η ,

(16)

with Nash bargaining condition ( W m (t ) − U m (t ))/β = ( J m (t ) − V m (t ))/(1 − β), ∀t, and the free-entry V m (t ) = V˙ m (t ) = 0, ∀t. Given the initial value for um , an equilibrium path is unique. Moreover, um (t ) converges to the steady state level gradually, while the wage, values, and θm (t ) are constant. In general, simple comparison across steady states corresponding to different policy parameter values only looks at ‘long-run’ effect of a policy change. To study the overall effect of a policy change in the presence of discounting, the transition process should be taken into account. In our context, instead of simply comparing welfare of the two steady states, that is, one for ρ > 0 and the other for ρ = 0 (the MP steady state), we should compare the welfare of the former steady state with the welfare of the dynamic path that converges to the latter steady state after the referral-restricting policy is imposed.15 So, the following is our scenario. Suppose that the economy starts with ρ > 0 and it is in the steady state with the total unemployment rate u ≡ n0 u 0 + n1 u 1 . One day, say at time t 0 , hiring through networks is banned by law in a permanent and unanticipated manner, so ρ is set to zero. Is such a policy good or bad? By the last proposition, while the economy’s unemployment rate, starting with the initial level u, gradually converges to the MP steady-state level um , all the values and wages immediately jump to those of the MP steady state. Therefore, for values and hence welfares, the comparison in Proposition 3 is valid even though the whole economy doesn’t immediately jump to the MP steady state. The result in Proposition 3, however, is limited to near ρ = 0, is only qualitative and still depends on parameters. Hence we perform a simple calibration below. In the calibrated model, we first analyze how much each of the four groups of workers (type-0/type-1 and employed/unemployed) becomes better off or worse off. In addition, we compute the total resource of the economy: match output plus

13 14 15

This u ∗1 depends on ρ and is ready to affect the economy once n1 departs from zero. η ≤ 0.5 is not inconsistent with um ≤ 0.5. For instance, Shimer (2005) calibrates η to be 0.28. Note that comparison across steady states (comparative statics) is implied by Proposition 3.

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Table 1 Parameter calibration. t r

δ

ρ μ η y b k n1

β

the unit of time discount rate job destruction rate importance of networks coefficient of matching function exponent of matching function output home production vacancy cost size of networked workers worker’s bargaining power

quarter 0.012 (Shimer) 0.1 (Shimer) 1.2 0.45/0.63η 0.1 or 0.4 1 (normalization) 0.4 (Shimer) 7.1 0.85 0.024

home production minus vacancy cost. In the pre-policy steady state, the total resource is [bu + y (1 − u ) − kv ]/r. In the post-policy economy, it is

∞ Y m (t 0 ) ≡

e −r (τ −t ) [bum (τ ) + y (1 − um (τ )) − kv m (τ )]dτ , with um (t 0 ) = u .

(17)

t0

The following proposition is useful to compute it. Proposition 7.

Y m (t 0 ) = uU m + (1 − u )( W m + J m ). 3.1. Calibration and numerical exercise The unit of time is chosen as a quarter of a year. We have 10 parameters, summarized in Table 1. Match output y is normalized to one. We inherit from Shimer (2005) r = 0.012, δ = 0.1, and b = 0.4. Regarding the size of the population of networked workers, Topa (2011) mentions, “Holzer (1987a, 1987b) uses data from the 1981–1982 modules of the National Longitudinal Survey of Youth and finds that 87% of currently employed and 85% of currently unemployed workers used friends and relatives in their job search, alongside other methods.” That is, approximately 85% of workers have at some time attempted to use networks in their job search along with other methods.16 We interpret it as n1 = 0.85 and vary it later. There are five parameters left to be calibrated, k, μ, η , ρ and β . The following are our calibration targets. 1. 2. 3. 4.

u = 5.67% (the U.S. average of 1951–2003, reported in Shimer, 2005) θ = 0.63 (Hagedorn and Manovskii, 2008)17 n (1−u ) Average wage 0 1−u 0 w 0 + n1 (11−−uu 1 ) w 1 = 0.666 (two thirds of output)

Several surveys report that about 50% of people find jobs through social contacts. In our model this implies

u μθ η ≈ n1 u 1 ρ (1 − u 1 ) Also Topa (2011) mentions, “Korenman and Turner (1996) also find that the use of social contacts increases wages by about 20% in a survey of Boston youth, and by 7% in a sample of young urban males from the 1982 NLSY.” Its interpretation in our model is that the ratio of the wage of those hired through referrals, w 1 , to the average wage of those hired through the matching function, (n0 u 0 /u ) w 0 + (n1 u 1 /u ) w 1 , is 1.07–1.20. This can act as a barometer of the model performance. Target 4, together with 1, pins down μθ η , resulting in u 0 = 10.7%, u 1 = 4.78%, and ρ = 1.2.18 On average, a posted vacancy is filled in 0.19 years, and a type-1 employee brings his friend once in 4.29 years. (Target 4 holds because there are more active firms with a type-1 employee than firms that post a costly vacancy.) A match will last for 2.5 years on average. Next, targets 2 and 3 pin down k and β , resulting in k = 7.1 and β = 0.024.19 The above targets also imply the relation between μ and η , or μ = 0.45/0.63η , not determining the two parameters separately. These two parameter values 16

Other methods include newspaper ads, direct contact, visits to state agencies, private agencies, school placement offices, etc. This number does not affect the result due to a well-known neutrality among v, k and μ. If k is multiplied by 2 and μ is multiplied by 2η , then the only endogenous variable that is affected is v (v is cut in half). Therefore, when k and μ are free parameters, changing v /u in calibration does not affect other endogenous variables. See Shimer (2005, p. 38) for details. 18 To see this, note that adding up (9) and (10) gives u μθ η + n1 u 1 ρ (1 − u 1 ) = (1 − u )δ . Target 4 implies that the first and second terms of the (LHS) are approximately equal, which pins down μθ η . 19 Shimer (2005) imposes ‘Hosios rule’ and sets β = 0.72. Hagedorn and Manovskii (2008) target a productivity elasticity of wage to obtain β = 0.052. As the wage level and β have a one-to-one relation, fixing β is equivalent to fixing the wage level. We set the wage level and change it later so we can get some idea about how robust our result is. 17

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Table 2 Computed welfare criteria. Pre-policy

η = 0.1

η = 0.4

5.67% (converge to 10.9%)

5.67% (converge to 11.8%)

0.502 0.693

0.501

0.494

type-0 type-1

0.490 0.678

0.489

0.482

type-0 type-1

0.491 0.679

0.490

0.483

0.714

0.540

0.534

Unemployment rate

type-0 type-1

10.7% 4.78%

Wage

type-0 type-1

Unemployed workers’ value

Employed workers’ value

Economy’s total output



Post-policy



Wage ratio = (n u /u ) w w+(1 n u /u ) w = 1.08 . 0 0 0 1 1 1

are needed to compute the post-policy equilibrium, so we try two extreme values for η , η = 0.1 and η = 0.4. In both cases, um > β results, so we are in case (ii) of Proposition 3 provided that ρ = 1.2 is small enough. Table 2 contains the computation results. All the lifetime values are expressed in terms of flow-value equivalence (i.e., lifetime value times r). The pre-policy ratio of the wage of those hired through referrals to the average wage of those hired through the matching function (the “referral-to-formal wage ratio”) is 1.08 and it matches Korenman and Turner (1996)’s observation well. Not surprisingly, the policy has large negative effects on type-1 workers. The total resource of the economy also falls with such a policy because of the increased search frictions. Moreover, the policy slightly lowers the wage and values of type-0 workers despite the fact that in this calibration, the number of posted vacancies more than doubles once the network hiring is banned (that is, a ‘crowding-out’ effect of networked workers on non-networked workers removed). This is because the increase of search frictions resulting from the ban on network hiring also causes an increase in the total number of unemployed workers that dominates the increase in posted vacancies. As a result, the job-finding rate for non-networked workers falls by such a policy and their welfare declines. We find that these results do not change very much even in the long run; simple comparison between the pre-policy steady state and the post-policy steady state also gives similar numbers as long as the discount rate is of reasonable magnitude. While the decline of total resource due to larger search frictions is intuitive, there is a remark that is noteworthy. The higher unemployment rate due to higher search frictions means less output for the economy. However, it is also true that firms counteract by increasing vacancies for formal hiring, which keeps the increase in unemployment smaller than it would be if firms did not change their behavior after the policy. In the meantime, for this same reason, the welfare loss also comes from the increase in hiring cost. In search models, the decomposition of the long-term welfare loss into the drop of output and the increase in hiring cost is easy if we compare the two steady states before and after the policy. It is the sum of the drop in output (bu + y (1 − u )) and the increase in hiring cost (kv). In the above calibration exercise, the drop of output accounts for 19.4% and the increase in hiring cost accounts for 80.6%. 3.2. Sensitivity In the economy where workers receive higher wages, Proposition 3(i) can also be the case. In such cases, the policy will still significantly reduce the total resource of the economy, but make type-0 workers slightly better-off. Fig. 1 shows what happens if we change the targeted wage level from 0.666 and hence the workers’ bargaining power from β = 0.024. The figure shows the welfare change (the post-policy lifetime value minus the pre-policy lifetime value) for each type of worker.20 As the wage level rises above about 0.75 (75% of output), the change for type-0 workers turns positive, implying that the policy makes them better off. That is, when the worker’s wage level is high, the referral-restricting policy can benefit non-networked workers at the cost of networked workers’ welfare and the total resource. For example, under a high wage level that corresponds to equal bargaining powers between workers and firms (wage level 0.978), the same policy would increase non-networked workers’ welfare slightly (+1.5%) with a relatively small drop in the total resource of the economy (−5.4%). In such an economy, because the wage is high for both types of workers, the referral-to-formal wage ratio is only 1.014, much smaller than Korenman and Turner’s (1996) reports. Fig. 2 shows welfare change for each type of worker for various n1 in our baseline case (average wage = 0.666, β = 0.024) and in the equal bargaining powers case (average wage = 0.978, β = 0.5). In the baseline case, the effect of the policy on non-networked workers is always negative and small. In the case of equal bargaining powers, the effect of the

20 The weighted average of the unemployment value and employment value is calculated. Weights are pre-policy steady-state unemployment/employment rates.

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Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

Fig. 1. Welfare changes for various wage targets. Other calibration targets are maintained.

η is set to be 0.3.

Fig. 2. The policy effect for various n1 .

policy on non-networked workers is always positive and the drop in networked workers’ welfare is less than in the baseline case. 3.3. Discussion With hindsight, the possibility that the ban on network hiring can harm non-networked workers is not entirely surprising. This is because the policy we consider bans word-of-mouth hiring of all firms, including small businesses. Imagine for example that the government establishes a law that says all the firms, even small businesses, must post vacancies publicly and have a minimal number of interviews per post. Or, alternatively, suppose that there are two social groups, one group of people more socially connected than the other group, and that the law says that all businesses must meet an affirmative action criterion, keeping a certain distribution across different social groups among their workers. Although such polices are not the same as the hypothetical policy of our model, the implications of and possible concerns about such policies are similar to ours; while such policies are likely to increase the number of posted vacancies that non-networked workers face, they will undoubtedly increase search frictions as well, leading to an increase of the number of workers who are looking for a job. If too much search friction is created, the policies will not benefit non-networked workers. We conclude this section with a brief discussion on robustness of our model specification. Would a different specification give a very different result? In our model, it is assumed that the new value that is created by referral accrues to the active firm, not to a new firm. What if the active firm only takes fraction γ ∈ [0, 1) and the rest goes to the new firm? For a lower γ , the threshold wage level in Fig. 1, at which the policy effect on the non-networked turns from negative to positive

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99

(namely 0.75 when γ = 1), will shift toward the left (e.g. 0.67 when γ = 0.5) so the effect of the policy would become more ambiguous. However, note that ‘one-firm-one-worker’ is an expositional assumption in labor search models and is an approximation of an environment in which a firm hires multiple workers. In section 4, we show that γ = 1 corresponds to the one-firm-multiple-workers environment. That is, when γ = 1, the active firm and the new firm are the same one firm. Second, the current model assumes that benefiting from a referral is free for firms. But what if it is not? The simple specification of referral in Galenianos (2014a) does not allow the addition of a large cost, say −k , to (6), unlike the cost −k in (7) for the Mortensen–Pissarides market matching. In the MP matching, if the cost of search is high, very few firms post a vacancy. This drives up the job-filling rate for a posting firm so that it can fill the position almost instantly. In contrast, simply adding cost of referral to the Galenianos model does not allow the arrival rate of referrals for an active firm to go up with the cost, which makes the model sometimes infeasible. In the appendix, we propose an alternative way to incorporate cost of referral by modeling the referral technology a la Mortensen–Pissarides matching function. We find that such a model does not necessarily have a very different implication about the welfare effect of referrals. One particular form of cost of referral is a one-time bonus to a type-1 employee who has brought his friend successfully. In the current model, the employer does not pay any such bonus to its employee, but such a bonus can be easily added to the model by assuming that the employer and its type-1 employee split the referral benefit. Such assumption does not change results quantitatively as long as the splitting rule is close to β . Because the bonus drives down the employer’s merit of having a type-1 employee, their wage goes down. 4. Model in which firms employ multiple workers In Section 2, we assumed that each firm employs at most one worker. This one-firm-one-worker assumption led to the apparently odd referral mechanism that a network friend of a worker currently employed by firm A is referred not to firm A but to some other firm, say firm B. We also assumed that in that case firm B makes payment to firm A to the extent that firm B is indifferent between accepting and not accepting the referred worker (cf. firm A’s take-it-or-leave-it offer). One may find these assumptions unnatural. In reality, a firm can employ multiple workers. Therefore, any person referred by a current employee is employed by the same firm, not by a different firm, and hence no firm-to-firm payment results. In this section, we present a model in which firms can employ multiple workers. It is known that for the standard MP framework without network hiring, the one-firm-one-worker environment and the one-firm-multiple-worker environment are equivalent if both production and cost of hiring satisfy constant returns to scale; a firm with N employees produces N times as much as a firm with a single employee, and the cost of posting N vacancies is N times as much as the cost of posting one vacancy. Whether that equivalence holds with network hiring is not so obvious. In the following, we show that such equivalence holds if the referral mechanism in the one-firm-one-worker environment is the referring firm’s take-it-or-leave-it offer. In other words, our model presented in Section 2 is only an expositional fiction that simplifies the model in which firms employ more than one worker. To see this, let J ,m be the value of a firm with non-networked workers and m networked workers that is not posting another vacancy. Let ˜J ,m be the value of a firm with non-networked workers and m networked workers that is currently posting another vacancy. The free-entry condition is hence J ,m = ˜J ,m for all and m. We also make the normalization J 0,0 = ˜J 0,0 = 0. Bellman equations are ∀ , m ≥ 1,

r J ,m = ( y − w 0 ) + m( y − w 1 ) + mρ u 1 ( J ,m+1 − J ,m )

+ δ( J −1,m − J ,m ) + mδ( J ,m−1 − J ,m )

(18)

and

r ˜J ,m = ( y − w 0 ) + m( y − w 1 ) − k + mρ u 1 ( ˜J ,m+1 − ˜J ,m )

+ δ( ˜J −1,m − ˜J ,m ) + mδ( ˜J ,m−1 − ˜J ,m ) + μθ η−1

n0 u 0 u

n1 u 1 ,m+1 ( ˜J +1,m − ˜J ,m ) + μθ η−1 ( ˜J − ˜J ,m ),

(19)

u

with initial conditions

r J 1,0 = y − w 0 + δ( J 0,0 − J 1,0 ) r J 0,1 = y − w 1 + δ( J 0,0 − J 0,1 ) + ρ u 1 ( J 0,2 − J 0,1 ) n0 u 0 2,0 n1 u 1 1,1 r ˜J 1,0 = y − w 0 − k + δ( ˜J 0,0 − ˜J 1,0 ) + μθ η−1 ( ˜J − ˜J 1,0 ) + μθ η−1 ( ˜J − ˜J 1,0 ) u u and

r ˜J 0,1 = y − w 1 − k + δ( ˜J 0,0 − ˜J 0,1 ) + ρ u 1 ( J 0,2 − J 0,1 ) + μθ η−1

n0 u 0 u

n1 u 1 0,2 ( ˜J 1,1 − ˜J 0,1 ) + μθ η−1 ( ˜J − ˜J 0,1 ). u

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Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

It is easily shown that

J ,m = ˜J ,m = J 0 + m J 1 ,

(20)

where J 0 and J 1 are from Section 2. Other equations remain the same as in the one-firm-one-worker model. So it follows that unemployment rates, wages and the vacancy rate remain the same.21 Thus our one-firm-one-worker model with the active firm’s take-it-or-leave-it offer when a referral arises is equivalent to a one-firm-multiple-workers model with constant-returns-to-scale production and cost of hiring. 5. Concluding remarks We consider a version of the Mortensen–Pissarides labor search model with two types of workers, workers who can find a job through referrals (networked workers) and workers who cannot find a job through referrals (non-networked workers). In the steady state, networked workers have the lower unemployment rate and higher wages. Banning the use of referrals in hiring reduces the economy to the canonical Mortensen–Pissarides environment and the economy starts to converge to the MP steady state. Analytically, such a policy can have both positive and negative effects on non-networked workers. In our baseline calibration, such a policy increases the number of people in the job queue more than it increases job posting and hence it is bad for non-networked workers. Moreover, the welfare of networked workers and the total surplus of the economy both drop substantially with such a policy. While this kind of study should not hinder the pursuit of fairness in employment practice, the current paper emphasizes the importance of conducting qualitative and quantitative analyses before such policies are carried out that have a side effect of increasing frictions. There are several limitations to the current model. First, there is no heterogeneity among networked workers. One example of such heterogeneity is so-called ‘weak vs. strong’ ties (e.g. Granovetter, 1973, 1995; Lin et al., 1981; Montgomery, 1992, 1994; Elliot, 1999; Zenou, 2015; Sato and Zenou, 2015). The partial equilibrium model of Montgomery (1994) predicts that if the number of members in a network is finite, the prevalence of network hiring will make people’s employment status more persistent because some networks may have higher or lower unemployment rates than others. Another example of heterogeneity is networks that are correlated with productivity (e.g. Galenianos, 2014b). Some issues regarding network hiring could only be analyzed by a model with such heterogeneity among workers. On the one hand, some firms may want to promote network hiring because friends of capable workers tend to be similarly capable (‘homophily’). On the other hand, other firms may want to avoid network hiring because a worker may refer a low-productivity friend/relative for that person’s sake (‘nepotism’). To study the latter factor properly, workers in the model should receive either direct utility or a bribe from their friends getting a job. At the same time, workers should care about their reputation in their firm so that they will refrain from referring too bad a friend. The current paper does not have endogenous network formation (e.g. Kuzubas, 2010; Galeotti and Merlino, 2014; Galenianos, 2014b). When a model has dynamic, endogenous network formation, the network is no longer a permanent characteristic of workers but a form of human capital that is built up by effort, so it may generate interesting business cycle properties for the labor market. Moreover, the efficiency implication of network-restricting policies is now more ambiguous because they may eliminate costly competition in networking. We leave investigation of these issues to future studies. Appendix A. Stability of the steady state for ρ > 0 The local stability of steady states is crucial when applying a model. In this section, we numerically check it for the Proposition 1 steady state. The dynamics of the equilibrium with positive ρ are as follows. First, the Bellman equations are22

U˙ 0 = rU 0 (t ) − b − μθ(t )η [ W 0 (t ) − U 0 (t )]

(21)

U˙ 1 = rU 1 (t ) − b − (μθ(t )η + ρ (1 − u 1 (t )))[ W 1 (t ) − U 1 (t )]

(22)

˙ 0 = rW 0 (t ) − w 0 (t ) − δ[U 0 (t ) − W 0 (t )] W

(23)

˙ 1 = rW 1 (t ) − w 1 (t ) − δ[U 1 (t ) − W 1 (t )] W

(24)

˙J 0 = r J 0 − ( y − w 0 (t )) − δ[ V (t ) − J 0 (t )]

(25)

˙J 1 = r J 1 − ( y − w 1 (t )) − δ[ V (t ) − J 1 (t )] − ρ u 1 (t )[ J 1 (t ) − V (t )]  n j u j (t ) V˙ = r V (t ) + k − μθ(t )η−1 [ J j (t ) − V (t )]. n0 u 0 (t ) + n1 u 1 (t )

(26)

j =0,1

21 22

Equation (20) also implies that firms do not gain from mergers or dissolution. That is, the size distribution of firms does not matter. The derivation is given at the end of the proof section.

(27)

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101

Fig. 3. Graphical image of the existence proof.

The laws of motion are

u˙ 0 = (1 − u 0 (t ))δ − u 0 μθ(t )η

(28)

u˙ 1 = (1 − u 1 (t ))δ − u 1 (μθ(t )η + ρ (1 − u 1 (t ))).

(29)

The Nash bargaining wage determination [ W j (t ) − U j (t )]/β = [ J j (t ) − V (t )]/(1 −β) and free-entry condition V (t ) = V˙ (t ) = 0 hold for all t. We reduce the system to that of the four variables, u 0 , u 1 , S 0 , S 1 , where S j ≡ W j − U j + J j − V is the total surplus of a match with a type- j worker. First, (21)–(26) are combined to

S˙ 0 = [r + δ + β μθ(t )η ] S 0 (t ) − ( y − b)

(30)

S˙ 1 = [r + δ + β μθ(t )η + ρ (β − u 1 (t ))] S 1 (t ) − ( y − b).

(31)

Also, (27), the Nash bargaining condition and the free-entry condition give an explicit expression for θ(t ): for all t,

⎡ θ(t )1−η =

k

μ(1 − β)

⎣

 j =0,1

⎤ −1 n j u j (t ) n0 u 0 (t ) + n1 u 1 (t )

S j (t )⎦

.

(32)

In summary, (28)–(31) together with (32) give the dynamical system in the economy with networks. Because u 0 and u 1 are state variables whose initial values are exogenously given while S 0 and S 1 are not such variables, the system is locally stable if two or more eigenvalues of the 4 × 4 Jacobian (evaluated at the steady state) are negative. If exactly two are negative and the other two are positive, then the dimension of the stable manifold is two, uniquely determining the initial values for S 0 and S 1 (i.e., the system/path is “determinate”). Although one can obtain the 4 × 4 Jacobian of the above system analytically, getting its eigenvalues analytically is extremely complex. So we compute the eigenvalues for the model calibrated in the last section. The resulting eigenvalues are −2.07, −0.907, 0.139 and 0.120. Thus the fixed point is locally stable and determinate. Appendix B. Proofs For proofs, we use the following notations: for each j ∈ {0, 1},

Sj ≡ Wj −Uj + Jj − V

α F j ≡ μθ η−1

n ju j n0 u 0 + n1 u 1

(33)

.

To prove Proposition 1, we provide two lemmas. Lemma 2. (i) For given v, there is a unique pair (u ∗0 , u ∗1 ) that satisfies (9)–(10); and (ii) Such (u ∗0 , u ∗1 ) are strictly decreasing in v.

(34)

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Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

Proof of Lemma 2. (i) First we prove it for the case δ ≥ ρ . Define the functions



T 0 (u 0 , u 1 , v ) ≡ u 0 μ

T 1 (u 0 , u 1 , v ; ρ ) ≡ u 1 μ

η

v n0 u 0 + n1 u 1 v

− (1 − u 0 )δ

(35)

+ u 1 (1 − u 1 )ρ − (1 − u 1 )δ,

(36)

η

n0 u 0 + n1 u 1

so that the steady-state unemployment rates given v, denoted u ∗0 ( v ), u ∗1 ( v ), are given as the solution to T 0 (u ∗0 , u ∗1 , v ) = 0 and T 1 (u ∗0 , u ∗1 , v ) = 0. First, we have

T 1 (u 0 , 0, v ; ρ ) = −δ < 0



T 1 ( u 0 , 1, v ; ρ ) = μ

(37)

η

v n0 u 0 + n1 1

>0

(38)

η

 ∂ T1 v ηn1 u 1 + (1 − u 1 )ρ + δ − u 1 ρ > 0 =μ 1− ∂ u1 n0 u 0 + n1 u 1 n0 u 0 + n1 u 1

η ∂ T1 v −ηn0 u 1 =μ <0 T 10 ≡ ∂ u0 n0 u 0 + n1 u 1 n0 u 0 + n1 u 1

 η −1

 ∂ T1 v u1 T 1v ≡ > 0. = ημ ∂v n0 u 0 + n1 u 1 n0 u 0 + n1 u 1

T 11 ≡

(39) (40) (41)

The first three equations imply that for any v > 0 and u 0 ∈ [0, 1], there is a unique u 1 ∈ (0, 1) that satisfies T 1 = 0, denoted T T T by u 1 1 (u 0 ; v ).23 Then (40) implies that ∂ u 1 1 (u 0 ; v )/∂ u 0 = − T 10 / T 11 > 0, so that u 1 1 (u 0 ; v ) is increasing in u 0 . Moreover, it T

can be shown that u 1 1 (u 0 ; v ) is bounded away from 0 and 1. That is, we have T

T

T

u 1 1 (u 0 ; v ) is increasing in u 0 , and u 1 1 (0; v ) > 0, u 1 1 (1; v ) < 1.

(42)

Similarly, we have

T 0 (0, u 1 , v ) = −δ < 0



T 0 (1, u 1 , v ) = μ

v n0 1 + n1 u 1

η >0

(43)

η

 ∂ T0 v ηn0 u 0 +δ>0 =μ 1− ∂ u0 n0 u 0 + n1 u 1 n0 u 0 + n1 u 1

η ∂ T0 v −ηn1 u 0 =μ <0 T 01 ≡ ∂ u1 n0 u 0 + n1 u 1 n0 u 0 + n1 u 1

 η −1

 ∂ T0 v u0 T 0v ≡ > 0. = ημ ∂v n0 u 0 + n1 u 1 n0 u 0 + n1 u 1 T 00 ≡

(44) (45) (46)

Again, the first three equations imply that for any v > 0 and u 1 ∈ [0, 1], there is a unique u 0 ∈ (0, 1) that satisfies T 0 = 0, T denoted by u 0 0 (u 1 ; v ).24 Then a similar argument leads to T

T

T

u 0 0 (u 1 ; v ) is increasing in u 1 , and u 0 0 (0; v ) > 0, u 0 0 (1; v ) < 1. T

T

(47) T

So let u0 ≡ u 0 0 (0; v ) and u¯ 0 ≡ u 0 0 (1; v ), as seen in Fig. 3. Then consider the inverse function of u 0 = u 0 0 (u 1 ; v ) (the inverse in terms of u 1 ), and denote it as u 1 = should intersect at least once because

T u 1 0 (u 0 ; v ) ,

which is defined on [u0 , u¯ 0 ]. The two functions

T

T

(48)

T u 1 0 (u¯ 0 ; v )

(49)

<1=

Now we are ready to show the uniqueness of such an intersection, or

24

T

and u 1 1 (u 0 ; v )

u 1 1 (u0 ; v ) > 0 = u 1 0 (u0 ; v ) T u 1 1 (u¯ 0 ; v )

23

T u 1 0 (u 0 ; v )

1−η

When u 0 = 0, (36) becomes T 1 (0, u 1 , v ; ρ ) = μ( v /n1 )η u 1

+ (1 − u 1 )(u 1 ρ − δ). So (37) still holds.

1−η When u 1 = 0, (35) becomes T 0 (u 0 , 0, v ) = μ( v /n0 )η u 0 − (1 − u 0 )δ . So (43) still holds.

Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

∂ ∂ u0

T

T

T

T

103

u 1 0 ( u 0 ; v ) − u 1 1 ( u 0 ; v ) > 0.

(50)

Note that

∂ ∂ u0

u 1 0 (u 0 ; v ) − u 1 1 (u 0 ; v ) T

∂ u10 ∂ uT1 − 1 ∂ u0 ∂ u0 T 00 (− T 10 ) = − T 11 (− T 01 ) T 00 T 11 − T 10 T 01 = , (− T 01 ) T 11

=

(51)

and the numerator is proved to be positive because by (39)–(40), (44)–(45), and the assumption δ > ρ ,

T 11 T 00 − T 10 T 01



= μ

η

v n0 u 0 + n1 u 1



× μ

−μ

η

v n0 u 0 + n1 u 1

η

v n0 u 0 + n1 u 1



= (1 − η) μ

+μ

+μ

1−

v



ηn1 u 1 n0 u 0 + n1 u 1

1−

v n0 u 0 + n1 u 1

 +δ

n0 u 0 + n1 u 1

η −ηn0 u 1 v −ηn1 u 0 μ n0 u 0 + n1 u 1 n0 u 0 + n1 u 1 n0 u 0 + n1 u 1  η 2

η

n0 u 0 + n1 u 1



ηn0 u 0

n0 u 0 + n1 u 1

v

 + (1 − u 1 )ρ + δ − u 1 ρ

1−

η

1−

(52)



ηn1 u 1 n0 u 0 + n1 u 1

ηn0 u 0 n0 u 0 + n1 u 1

δ  ((1 − u 1 )ρ + δ − u 1 ρ )

+ δ((1 − u 1 )ρ + δ − u 1 ρ )

(53)

> 0,

(54) T

T

which concludes that (51) is positive. Therefore, the two functions u 1 0 (u 0 ; v ) and u 1 1 (u 0 ; v ) intersect only once, which proves (i). Next we prove the lemma for the case ρ > δ . In this case, we can restrict the domain of u 1 to [0, δ/ρ ] ⊂ [0, 1] for the following reason. Consider a hypothetical situation that there is no employment through the matching function. In such a case, the dynamics of u 1 are given by u˙ 1 = (1 − u 1 )δ − u 1 ρ (1 − u 1 ), so its fixed point is u 1 = δ/ρ . The steady state u 1 should be bounded from above by this level. T If we restrict attention to u 1 ∈ [0, δ/ρ ], (39) still holds even if ρ > δ . Also the last part of (42) is modified to u 1 1 (1; v ) < T δ/ρ . Additionally, the definition of u¯ 0 is modified so that u¯ 0 ≡ u 0 0 (δ/ρ ; v ) and (49) is modified to u 1T 1 (u¯ 0 ; v ) < δ/ρ = T0 u 1 (u¯ 0 ; v ). All the other arguments remain unchanged.

(ii) Note that by the implicit function theorem,

⎡ ⎣

∂ u ∗0 ∂v ∂ u ∗1 ∂v

⎤

− 1    T 0v ⎦ = − T 00 T 01 T 10

T 11

T 1v



−1 T 11 − T 01 = T 00 T 11 T 00 − T 10 T 01 − T 10 < 0,



T 0v T 1v



where the last inequality follows from (39)–(41), (44)–(46), and (54).

(55) (56)

2

Lemma 3. Given v, there exist unique S 0 and S 1 that satisfy (1)–(6) and (8)–(11), denoted S 0∗ ( v ) and S 1∗ ( v ). Both are strictly decreasing in v, satisfying lim v →∞ S ∗j ( v ) = 0 and lim v →0 S ∗j ( v ) < ∞.

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Proof of Lemma 3. By (1), (3), (5), (8) and (11), we have

S0 =

y −b r + δ + μθ η β

(57)

.

By (2), (4), (6), (8) and (11), we have

S1 =

y −b r + δ + μθ η β + ρ (β − u 1 )

(58)

.

Clearly S 0 > 0 holds, and S 1 > 0 also follows from (10). By Lemma 2, u 0 , u 1 and thus μθ η are uniquely determined, given v. So S 0 and S 1 are unique, given v. By Lemma 2, μθ η is strictly increasing in v, so the denominators of S 0 and S 1 in (57), (58) are both strictly increasing in v. Therefore, both S 0 and S 1 are strictly decreasing in v. Moreover, both the denominators go to infinity as v → ∞ and go to some positive constants as v → 0. So lim v →∞ S ∗j ( v ) = 0 and lim v →0 S ∗j ( v ) < ∞. 2 Proof of Proposition 1. By Lemmas 2 and 3, given v, (1)–(6) and (8)–(11) pin down u ∗0 ( v ), u ∗1 ( v ), S 0∗ ( v ) and S 1∗ ( v ), all of which are strictly decreasing in v. The remaining equation, which determines v, is (7):

k = α ∗F 0 ( v )(1 − β) S 0∗ ( v ) + α ∗F 1 ( v )(1 − β) S 1∗ ( v ), where for i = 0, 1,

α ∗F i ( v ) ≡ μ



n0 u ∗0 ( v ) + n1 u ∗1 ( v )

1−η

v

ni u ∗i ( v )

. n0 u ∗0 ( v ) + n1 u ∗1 ( v )

(59)

(60)

When v → 0, α ∗F ( v ) → ∞. So the (RHS) of (59) goes to infinity. i When v → ∞, η

ni u ∗i ( v ) 1 α ∗F i ( v ) = μ 1−η (ni u ∗i ( v ))1−η → 0, n u∗ (v ) + n u∗ (v ) v 0 0

1 1

because the expression in the bracket is bounded above by one. So the (RHS) of (59) goes to zero. Therefore, by the intermediate value theorem, there exists a v that satisfies (59). Once v is determined, all the other equilibrium variables are uniquely pinned down. 2 Proof of Proposition 2. First, subtracting (10) from (9) gives

(μθ η + δ)(u 0 − u 1 ) = u 1 (1 − u 1 )ρ ,

(61)

which implies that u 0 > u 1 . For the rest of the proof, we consider the two cases separately: (i) S 1 ≥ S 0 and (ii) S 0 > S 1 . Note that combining (1) and (2) gives

r (U 1 − U 0 ) = μθ η {( W 1 − U 1 ) − ( W 0 − U 0 )} + (1 − u 1 )ρ ( W 1 − U 1 );

(62)

combining (3) and (4) gives

w 1 − w 0 = r ( W 1 − W 0 ) + δ{( W 1 − U 1 ) − ( W 0 − U 0 )};

(63)

combining (5) and (6) gives

(r + δ)( J 1 − J 0 ) = w 0 − w 1 + ρ u 1 ( J 1 − V );

(64)

and combining (62) and (63) gives

w 1 − w 0 = (r + δ + μθ η ){( W 1 − U 1 ) − ( W 0 − U 0 )} + (1 − u 1 )ρ ( W 1 − U 1 ).

(65)

(i) Suppose S 1 ≥ S 0 in the equilibrium. Then by the Nash bargaining condition (8), we have W 1 − U 1 ≥ W 0 − U 0 and J 1 ≥ J 0 . Then (62) implies U 1 > U 0 . This together with W 1 − U 1 ≥ W 0 − U 0 implies W 1 > W 0 . Also, (65) implies w1 > w0. (ii) Now suppose that S 0 > S 1 , so that the Nash bargaining condition implies W 0 − U 0 > W 1 − U 1 and J 0 > J 1 . Then by (64), w 1 > w 0 . Hence by (63), W 1 > W 0 . This together with W 0 − U 0 > W 1 − U 1 implies U 1 > U 0 . The proposition is proved up to this point. Lastly, we note that S 1 > S 0 (hence J 1 > J 0 and W 1 − U 1 > W 0 − U 0 ) is not necessarily the case. Subtracting (57) from (58) gives

( S 1 − S 0 )(r + δ + μθ η β) = S 1 ρ (u 1 − β). So if u 1 < (>)β , then S 1 < (>) S 0 .

2

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105

Proof of Lemma 1. The steady state equilibrium of the standard Mortensen–Pissarides model is given as the solution to the following system of equations: η

rU m = b + μθm ( W m − U m )

(67)

rW m = w m + δ(U m − W m )

(68)

r J m = y − w m + δ( V m − J m ) η −1

r V m = −k + μθm

(69)

( Jm − V m)

(70)

W m − U m = β Sm

(71)

Vm = 0

(72)

η

um μθm = (1 − um )δ,

(73)

Sm ≡ W m − U m + J m − V m

(74)

θm ≡ v m /um .

(75)

where

Again, (67)–(69) and (71), (72) are reduced to

Sm =

y−b

(76)

η,

r + δ + β μθm

while (70) and (71) imply 1 −η

Sm =

kθm

(1 − β)μ

(77)

.

The last two equations pin down ( S m , θm ). In fact, S m is given as a unique positive solution to f S ( S m ) = 0, where

f S ( Sm) ≡ β μ

μ(1 − β)



η

1 1−η

1−η

Sm

k

+ (r + δ) S m − ( y − b).

(78)

Once S m is determined, so is the v m –um ratio. Meanwhile the steady state condition (73) is transformed to

 vm =

δ

δ η

1

− um 1 −η μ μu m

η

,

which means v m is decreasing in um . Combining it with (75) uniquely pins down v m and um . (Comparative Statics of the MP model.) Eliminating S m from (76) and (77) gives

y −b k

(1 − β) =

r+δ

μθ η−1

+ βθ.

(79)

This implies that increasing β , b, or k leads to lower θm , resulting in higher um . Eliminating θm by using (73), one can see that limβ→0 um > 0. So for sufficiently low β , um > β is the case. Also, limk→∞ um = 1 is easy to check. 2 Proof of Proposition 3. The equilibrium triple, (u 0 , u 1 , v ), is pinned down by three equation. The first two are T 0 = 0 and T 1 = 0, where function T 0 and T 1 are defined by (35) and (36). The third equation, say T v = 0, is defined by (7) combined with (57), (58):

T v (u 0 , u 1 , v ; ρ ) ≡ −k + α F 0 (1 − β)

+ α F 1 (1 − β) where

y −b r + δ + μθ η β y−b

r + δ + μθ η β + ρ (β − u 1 )

,

α F j , defined by (34), and μθ η are functions of u 0 , u 1 , v and ρ . By the implicit function theorem, ⎡ ⎤⎡ ⎤ ⎤ ⎡ T 00 T 01 T 0v u 0 (ρ ) T 0ρ ⎣ T 10 T 11 T 1v ⎦ ⎣ u (ρ ) ⎦ = − ⎣ T 1ρ ⎦ . 1 T v0 T v1 T v v v (ρ ) T vρ

(80)

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Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

Invoking that u 0 = u 1 = um when

u 1 (ρ )|ρ =0 = −

ρ = 0, one can show that  η um (κ1 + μ uv m (n1 η(1 − β)um + β(1 − n1 η)(1 − um ))) m

κ2 κ3

,

where

κ1 ≡ (r + δ)(1 − η)(1 − um ) > 0

κ2 ≡ δ + μ

vm

η

>0

um



κ3 ≡ (r + δ)(1 − η) + β μ

κ4 ≡ r + δ + β μ

vm

vm

η >0

um

η

> 0,

um

so u 1 is decreasing in ρ near ρ = 0. Calculations similar to the above lead to

u 0 (ρ )|ρ =0 = v (ρ )|ρ =0 =

n1 ημ



vm um

η

um (β − um )

κ2 κ3 n1 v {−κ1 − κ2 (β − um ) − μ



vm um

η

(η(1 − β)um + β(1 − η)(1 − um ))}

κ2 κ3

Using (57) and (58),

S 0 (ρ )|ρ =0 = S 1 (ρ )|ρ =0 =

n1 η β S m μ



vm um

η

(β − um )

κ3 κ4 (um − β) S m ((r + δ)(1 − η) + β μ



vm um

η

κ3 κ4

(1 − n1 η)) .

(i) If um < β , we have u 0 (ρ )|ρ =0 > 0, v (ρ )|ρ =0 < 0, S 0 (ρ )|ρ =0 > 0, and S 1 (ρ )|ρ =0 < 0. Then (8) implies that W 0 − U 0 and J 0 are increasing and that W 1 − U 1 and J 1 are decreasing in ρ . By (5), w 0 = y − (r + δ) J 0 , so w 0 is decreasing. Then by (3), W 0 is decreasing as well. W 0 − U 0 increasing and W 0 decreasing imply U 0 decreasing. Also, u 0 increasing and (9) imply that μθ η is decreasing. Hence θ is decreasing. (ii) If β < um , we have u 0 (ρ )|ρ =0 < 0, S 0 (ρ )|ρ =0 < 0, and S 1 (ρ )|ρ =0 > 0, while the sign of v (ρ )|ρ =0 is indeterminate. The results are then opposite of those of (ii) except for v. Moreover, w 0 , W 0 and U 0 being increasing and Proposition 2 imply that w 1 , W 1 and U 1 are also increasing near ρ = 0. Also, because both u 0 and u 1 are decreasing, u is decreasing. 2 Proof of Proposition 4. As in Proposition 3, the implicit function theorem is applied in terms of parameter n1 . We evaluate at n1 = 0 to find u 0 (n1 )|n1 =0 and S 0 (n1 )|n1 =0 . At n1 = 0, the externality of networks on non-networked people disappears, and the equations for the non-networked (their population is now one) and the equations for the networked (zeromeasured) are somewhat separated. That is, (1), (3), (5), (7), (8), (9) and (11) solve the variables for the non-networked (U 0 , W 0 , J 0 , w 0 , u 0 , v and θ ) as the solution in the standard MP environment. Once θ is known, (10) gives u 1 , and the other variables for the networked follow. Call this u 1 as u ∗1 and the corresponding S 1 as S 1∗ . Although the measure of the networked is zero at n1 = 0, this u ∗1 is affected by ρ and it affects u 0 (n1 )|n1 =0 and S 0 (n1 )|n1 =0 :

u 0 (n1 )|n1 =0 =

S 0 (n1 )|n1 =0 =

ρ u 1 ημ ρ u 1 ημ

 

vm um

vm um

η η

κ4 (β − u ∗1 ) S 1∗ /( y − b) κ2 κ3 β(β − u ∗1 ) S 1∗

u m κ3 κ4

.

Therefore, if β > u ∗1 (resp. u ∗1 > β ) at n1 = 0, then u 0 and S 0 are increasing (resp. decreasing) in n1 in the vicinity of n1 = 0. Thus J 0 = β S 0 is also increasing (resp. decreasing), and (5) implies w 0 is decreasing (resp. increasing). 2

Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

107

Fig. 4. The (RHS) of equation (59).

Proof of Proposition 5. (i) (ii) We want to show that (59) in the proof of Lemma 1 has unique solution v. For that, it suffices to show that (60) is strictly decreasing. Differentiating (60) with respect to v and using (55), (39)–(41) and (44)–(46), we have



∂ α∗

F0 (v )

∂v

n0 u 0 μθ η

=−

 ∂v



v 2 {n0 u 0 (δ + μθ η (1 − η))(δ + μθ η + q) + n1 u 1 (δ + μθ η )(δ + μθ η (1 − η) + q)}

and

∂ α ∗F 1 ( v )

n0 u 0 (1 − η )(δ + μθ η )(δ + μθ η + q)  + n1 u 1 δ 2 (1 − η) + δ(1 − η)(2μθ η + q) + μθ η (μθ η (1 − η) + q)

n1 u 1 μθ η

=−

,

n1 u 1 (1 − η )(δ + μθ η )(δ + μθ η + q)  + n0 u 0 δ 2 (1 − η) + δ(1 − η)(2μθ η + q) + μθ η (μθ η (1 − η) + q(1 − 2η))

v 2 {n0 u 0 (δ + μθ η (1 − η))(δ + μθ η + q) + n1 u 1 (δ + μθ η )(δ + μθ η (1 − η) + q)}

 ,

where q ≡ (1 − 2u ∗1 ( v ))ρ . In Lemma 2, we showed that u ∗1 ( v ) is strictly decreasing in v, so the above two equations imply that the α ∗F ( v )’s are strictly decreasing in the region such that q ≥ 0 or u ∗1 ( v ) ≤ 0.5. So, the (RHS) of (59) is also strictly i decreasing in v in the region such that u ∗1 ( v ) ≤ 0.5, as is shown in Fig. 4. Therefore, there is at most one equilibrium such that u 1 ≤ 0.5. The (RHS) of (59) does not necessarily have humps as is depicted in Fig. 4. All we know is that it goes to infinity as v → 0, goes to zero as v → ∞, and it is strictly decreasing for sufficiently large v. Parameter k appears only in this equation, so if k is large enough, the equilibrium is unique and u 1 > 0.5. If k is small enough, the equilibrium is unique and u 1 can be any small number. (iii) Proposition 3(i) implies that if ρ ≈ 0, then u 1 < um is the case in any equilibrium. Then part (i) implies the uniqueness of the steady state equilibrium. 2 Proof of Proposition 6. Defining the surplus of a match S m (t ) = W m (t ) − U m (t ) + J m (t ) − V m (t ), the Nash bargaining condition and the free-entry condition together with (12)–(14) imply

S˙ m = (r + δ + β μθm (t )η ) S m (t ) − ( y − b).

(81)

In the meantime, (15), the Nash bargaining condition and the free-entry condition imply

θm (t ) =

μ(1 − β) k



1 1−η

1

S m (t ) 1−η , ∀t .

(82)

Substituting the last equation into (81), we have

 η 1−η 1 ˙S m = β μ μ(1 − β) S m (t ) 1−η + (r + δ) S m (t ) − ( y − b) ≡ f S ( S m (t )). k

(83)

So the match surplus is autonomous, not depending on other variables such as the unemployment rate. Since f s ( S m ) > 0, the match surplus should be constant at the steady state level for any convergent path. So, θm (t ) is also constant. But then, (16) is an autonomous system with the (RHS) having negative slope. Therefore, um (t ) is convergent. By the Nash bargaining condition, S m (t ) being time-invariant implies that W m (t ) − U m (t ), J m (t ) − V m (t ) and J m (t ) are also time-invariant. Then (14) implies w m (t ) is time-invariant. So (12) and (13) become an autonomous system of U m (t ) and W m (t ), respectively. Since the slopes of these systems are positive, U m (t ) and W m (t ) should also be at the steady-state levels from the beginning. 2

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Y. Igarashi / Review of Economic Dynamics 20 (2016) 90–110

Proof of Proposition 7. Define Y m (t ) ≡ um (t )U m (t ) +(1 − um (t ))( W m (t ) + J m (t )). This is equal to W m (t ) + J m (t ) − um (t ) S m (t ). Therefore,

˙ m + ˙J m − u˙ m S m − um S˙ m Y˙ m = W η

= r ( W m + J m ) − y − δ(U m − W m + V m − J m ) − u˙ m S m − um [(r + δ + β μθm ) S m − ( y − b)] η

= −bum − y (1 − um ) + r ( W m + J m − um S m ) − u˙ m S m + [δ(1 − um ) − um μθm ] S m η

+ um (1 − β) S m μθm η

= −bum − y (1 − um ) + r ( W m + J m − um S m ) + um (1 − β) S m μθm = −bum − y (1 − um ) + kv m + r ( W m + J m − um S m ) = −bum − y (1 − um ) + kv m + rY m , where the second equality uses (13), (14) and (81), the fourth equality uses (16), and the fifth uses (82). The last expression implies

∞ Y m (t ) =

e −r (τ −t ) [bum (τ ) + y (1 − um (τ )) − kv m (τ )]dτ ,

t

that is, the discounted sum of the economy’s total resources.

2

Proof. (Derivation of differential equations in Appendix A.) Suppose τ ≥ 0 is a random variable that is the time until an event occurs. Let (τ ) and φ(τ ) be its distribution function and density function, respectively. That is,

(t 2 |t 1 ) ≡ P(τ ≤ t 2 |τ > t 1 ) = P(t 1 < τ ≤ t 2 |τ ≥ t 1 )

(t 2 ) − (t 1 ) . = 1 − (t 1 ) Differentiating with respect to t 2 , we have

φ(t 2 |t 1 ) ≡ φ(t 2 |τ > t 1 ) =

φ(t 2 ) 1 − (t 1 )

.

Let λ(τ ) ≡ φ(τ )/(1 − (τ )), the hazard rate. Then we have

∂φ(t 2 |t 1 ) = λ(t 1 )φ(t 2 |t 1 ), ∂ t1

(84)

which we use below. Another formula we use is as follows. Let

T D (t , T ) ≡

f (s)e −r (s−t ) ds,

t

where f is a given function. Then

∂ D (t , T ) = − f (t ) + r D (t , T ). ∂t

(85)

The type-0 worker’s unemployment value satisfies

∞ U 0 (t ) =

∞ φ( T |t ) D (t , T )dT +

t

φ( T |t )e −r (T −t ) W 0 ( T )dT ,

t

where f (s) = b is a constant function and φ( T |t ) is a density for arrival of employment. (So, λ(t ) = μθ(t )η .) Differentiating it with respect to t and using (84)–(85), we have

U˙ 0 (t ) = rU 0 (t ) − b − λ(t )[ W 0 (t ) − U 0 (t )]. The derivation is similar for the other values.

2

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109

Appendix C. Alternative model with cost of referrals In this section we depart from Galenianos’s (2014a) model of referral and present an example of alternative specifications. In Galenianos (2014a), it is implicitly assumed that all unmatched, potential jobs are costlessly waiting for the chance to be filled through a referral. Now suppose alternatively that firms have to pay a cost to get the status of “vacant for a referral”. Denote this cost by k R , the measure of vacancies of this type (per unit population of networked workers) by v R , and the value of such a vacancy by V R (as opposed to k, v and V for formal market matching). We assume that v R is determined by another free-entry condition V R = 0 in equilibrium. The number of referrals generated by existing matches η 1−η

is assumed to be M R ( v R , u 1 ) ≡ (1 − u 1 )ρ v R u 1 with η ∈ (0, 1).25 It is necessary to assume that the employer of the referring worker does not take the entirety of the newly created value but only γ ∈ [0, 1) proportion of it. The steady state equilibrium, (U i , W i , J i , w i , u i )i =0,1 and ( V , v , V R , v R ), is characterized by a set of equations similar to (1)–(11), where (2), (6) and (10) are replaced by

rU 1 = b + μθ η ( W 1 − U 1 ) + ( M R ( v R , u 1 )/u 1 )( W 1 − U 1 )

(2’)

r J 1 = y − w 1 + δ( V − J 1 ) + ( M R ( v R , u 1 )/(1 − u 1 ))γ ( J 1 − V )

(6’)

(1 − u 1 )δ = u 1 (μθ η + M R ( v R , u 1 )/u 1 )

(10’)

with an extra Bellman equation for the second type of vacancy added:

r V R = −k R + ( M R ( v R , u 1 )/ v R )(1 − γ )( J 1 − V R ).

(87)

While we keep the calibration targets in section 3, we would need more targets due to more parameters. Here we set k R = k, η = η and γ ≈ 1 for the comparison purposes.26 With this model, the qualitative features of the policy in section 3 are still true and a figure that is very similar to Fig. 1 results. That is, eliminating network hiring slightly decreases the welfare of type-0 workers in the economy if they have low wages but benefits them otherwise. While lowering the cost of referral does not change such results, lowering γ does, as is the case with our main model. We do not have analytical results for this model. More detailed investigation is left for future studies. References Andolfatto, D., 1996. Business cycles and labor-market search. The American Economic Review 86 (1), 112–132. Calvo-Armengol, A., Jackson, M.O., 2004. The effects of social networks on employment and inequality. The American Economic Review 94 (3), 426–454. Calvo-Armengol, A., Jackson, M.O., 2007. Networks in labor markets: wage and employment dynamics and inequality. Journal of Economic Theory 132, 27–46. Calvo-Armengol, A., Zenou, Y., 2005. Job matching, social network and word-of-mouth communication. Journal of Urban Economics 57, 500–522. Corcoran, M., Datcher, L., Duncan, G.J., 1980. Most workers find jobs through word of mouth. Monthly Labor Review 103 (8), 33–35. Elliott, J.R., 1999. Social isolation and labor market insulation: network and neighborhood effects on less-educated urban workers. The Sociological Quarterly 40 (2), 199–216. Equality Commission for Northern Ireland, Fair Employment in Northern Ireland Code of Practice 1989. (Available from their website under ‘publication – Fair employment and treatment’). Fontaine, F., 2008. Why are similar workers paid differently? The role of social networks. Journal of Economic Dynamics and Control 32, 3960–3977. Galenianos, M., 2013. Learning about match quality and the use of referrals. Review of Economic Dynamics 16, 668–690. Galenianos, M., 2014a. Hiring through referrals. Journal of Economic Theory 152, 304–323. Galenianos, M., 2014b. Endogenous Referral Networks and Worker Heterogeneity. University of London, Royal Holloway College. Galeotti, A., Merlino, L.P., 2014. Endogenous job contact networks. International Economic Review 55 (4), 1201–1226. Granovetter, M., 1973. The strength of weak ties. American Journal of Sociology 78 (6), 1360–1380. Granovetter, M., 1995. Getting a Job: a Study of Contacts and Careers. The University of Chicago Press. Hagedorn, M., Manovskii, I., 2008. The cyclical behavior of equilibrium unemployment and vacancies revisited. The American Economic Review 98 (4), 1692–1706. Holzer, H., 1987a. Job search by employed and unemployed youth. Industrial & Labor Relations Review 40 (4), 601–610. Holzer, H., 1987b. Informal job search and black youth unemployment. The American Economic Review 77 (3), 446–452. Ioannides, Y., Loury, L.D., 2004. Job information networks, neighborhood effects, and inequality. Journal of Economic Literature 42 (4), 1056–1093. Korenman, S., Turner, S., 1996. Employment contacts and minority-white wage differences. Industrial Relations 35 (1), 106–122. Kuzubas, T.U., 2010. Endogenous social networks in the labor market. Unpublished manuscript. Bogazici University. Lin, N., Ensel, W.M., Vaughn, J.C., 1981. Social resources and strength of ties: structural factors in occupational status attainment. American Sociological Review 46, 393–405. Mayer, A., 2011. Quantifying the effects of job matching through social networks. Journal of Applied Economics 14 (1), 35–59. Merz, M., 1995. Search in the labor market and the real business cycle. Journal of Monetary Economics 36 (2), 269–300. Montgomery, J.D., 1991. Social networks and labor-market outcomes: toward an economic analysis. The American Economic Review 81 (5), 1408–1418.

25 Here we maintain the assumption that the proportion of type-1 workers that are currently employed, 1 − u 1 , affects the referral matching. The Poisson arrival rate of a referral matching for a type-1 unemployed worker, that for an active firm with a type-1 employee and that for a firm posting for referral matching, are obtained by dividing M R by u 1 , (1 − u 1 ), and v R , respectively. Re-scaling M R by a function of n1 does not affect the following exercise because ρ is a free parameter in our calibration that is determined so that half of the hirings will occur through referrals. 26 Setting γ = 1 is no longer possible because then no companies would seek costly referral. Also, setting η = 0 does not quite reduce the model to the Galenianos (2014a) model because v R will still appear in the denominator of the job-filling rate of referrals.

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Montgomery, J.D., 1992. Job search and network composition: implications of the strength-of-weak-ties hypothesis. American Sociological Review 57, 586–596. Montgomery, J.D., 1994. Weak ties, employment, and inequality: an equilibrium analysis. American Journal of Sociology 99 (5), 1212–1236. Mortensen, D., Pissarides, C., 1994. Job creation and job destruction in the theory of unemployment. The Review of Economic Studies 61, 397–415. Rees, A., 1966. Labor economics: effects of more knowledge – information networks in labor markets. The American Economic Review 56 (1/2), 559–566. Sato, Y., Zenou, Y., 2015. How urbanization affect employment and social interactions. European Economic Review 75, 131–155. Shimer, R., 2005. The cyclical behavior of equilibrium unemployment and vacancies. The American Economic Review 95 (1), 25–49. Topa, G., 2011. Labor Markets and Referrals. The Handbook of Social Economics, vol. 1. North Holland, pp. 1193–1221. Zenou, Y., 2015. A Dynamic model of weak and strong ties in the labor market. Journal of Labor Economics 33 (4), 891–932.