Labor market and search through personal contacts Roman Chuhay∗ May 3, 2012

Abstract In the paper we consider the impact of personal contacts on the labor market outcome. Unlike previous studies, we do not assume any particular network structure or vacancies communication protocol. Instead, we state four general properties of matching function that allow us to characterize the impact of social ties on the labor market equilibrium. We show that unemployment decreases in the socialization level of workers, while wage and market tightness increase. The relationship between vacancy rate and socialization level is non-monotonic. In particular, the vacancy rate decreases in the socialization level when unemployment is sufficiently low and increases otherwise. JEL Classification numbers: J63; J64; D83; E24. Keywords: Labor market; unemployment; job search; social network.

1

Introduction

The importance of friends and relatives as a source of information on employment opportunities is well recognized by the economic literature. Empirical studies document that throughout years and across different sectors, between 30% to 50% of jobs are filled through personal contacts (for survey see Ioannides and Loury, 2004). Guided by the empirical evidence, a substantial body of theoretical literature studies the impact of personal contacts on the labor market outcome1 . A common way to embed network of personal contacts in the labor market framework is to derive a matching function using the assumed structure of ∗

Higher School of Economics, Department of Economics, Pokrovskiy blvd. 11, Moscow, Russia. Email: [email protected]. I would like to thank Fernando Vega-Redondo, Marco Van der Leij, Andrea Galeotti, Vadym Lepetyuk and funding from HSE Research Laboratory for Strategic Behaviour and Institutional Design. 1 See e.g. Jackson and Calvo-Armengol, 2004; Calvo-Armengol, 2004; Calvo-Armengol and Zenou 2005)

1

social interactions.2 Matching function defines the number of matches between unemployed workers and vacancies and thus summarizes frictions present on the labor market. To facilitate the analysis authors impose various simplifying assumptions on the propagation mechanism of vacancies in the network. These include assumptions that only one worker initially becomes aware of a vacancy, vacancies can be transmitted to immediate contacts only, and an employed worker, who learns about opening selects a recipient at random (e.g. Calvo-Armengol and Jackson, 2004; Calvo-Armengol and Zenou, 2005; Galeotti and Merlino, 2011). The structure of a job contact network is also an open question and varies from one study to another.3 In this paper we identify conditions under which previously obtained results regarding the impact of personal contacts on the unemployment level can be generalized to other network structures and communication mechanisms. In particular, we show that if obtained matching function satisfies four basic assumptions formulated in our paper then unemployment decreases in the socialization level. Moreover, using these assumptions we establish new results regarding the impact of socialization effort on the equilibrium vacancy rate, market tightness and wage. The analysis departs from three assumptions, commonly employed in the labor market literature, which specify the behavior of the matching function with respect to vacancy and unemployment rates (for a survey on matching functions, see Pissarides and Petrongolo, 2001). We then add the fourth assumption that relates in a general way the network of personal contacts to the number of matches in the economy. In particular, we assume that the number of matches increases with the socialization level of workers. One can think about the socialization level as representing workers’ average connectivity4 . In this case, a plausible assumption is that an increase in the number of worker’s connections increases her probability to hear about a vacancy. The fourth assumption is in accordance with the majority of theoretical studies that derive matching function explicitly (e.g. Cahuc and Fontaine, 2003; Ioannides and Soetevent, 2006; Galeotti and Merlino, 2011). We start our analysis by deriving a condition under which the labor market equilibrium exists and is unique. We then show that equilibrium unemployment monotonically decreases as the socialization level of workers increases. A similar result is obtained numerically by Ioannides and Soetevent (2006) for particular shapes of a degree distribution (Poisson and 2

See e.g. Calvo-Armengol, 2004; Calvo-Armengol and Zenou 2005; Ioannides and Soetevent, 2006; Fontaine and Cahuc, 2008; Galeotti and Merlino, 2011 3 The paper Calvo-Armengol and Zenou (2005) assumes a regular network (all workers have the same number of contacts), Galeotti and Merlino (2011) employs the network formation mechanism that leads to a Poisson degree distribution and finally Fontaine and Cahuc (2008) assumes that the network is composed of cliques (components inside which all workers are connected). 4 Actually, socialization level can be thought of as any parameter that enhances number of matches in the economy.

2

negative binomial). In the framework of a reduced form of the Mortensen-Pissarides model, Galeotti and Merlino (2011) get similar result for the case of Poisson degree distribution. Our paper generalizes the result by extending it to the fully fledged Mortensen-Pissarides framework and allowing for different degree distributions and transmission mechanisms that satisfy our assumptions. There is an ambiguity in the literature regarding the relationship between the vacancy rate and network density. A numerical analysis by Ioannides and Soetevent (2006) suggests that the number of vacancies is decreasing in the network density. Armengol and Zenou (2005) conclude that the relationship is ambiguous. In the paper we show that the relationship between vacancy rate and socialization level has non-monotonic shape. In particular, if the equilibrium unemployment level is sufficiently low, then workers possess high wage bargaining power, and the economic benefits of an easier match are absorbed by a wage increase. This makes posting vacancies less attractive for firms and vacancy rate decreases. On the contrary, if unemployment is high enough then the wage does not increase significantly with the socialization level and firms have incentives to post more vacancies. In our model, like in the standard Mortensen-Pissarides setup, the wage depends on the labor market conditions summarized by a market tightness (the ratio of the number of vacancies to the number of unemployed workers). We show that despite the non-monotonic behavior of the vacancy rate, both the market tightness and the wage are monotonically increasing in the socialization level. This comes as a result of the increased wage bargaining power of workers who find a job easier due to a better access to professional contacts. To illustrate the application of our model, we consider a network structure and vacancies communication mechanism described by Galeotti and Merlino (2011). Authors specify network formation mechanism and consider two-period model with given vacancy rate and wage. We show that their matching function satisfies all outlined axioms and thus all results obtained in our paper, regarding vacancy rate, market tightness and wage hold. The rest of the paper is organized as follows. In Section 2 we state the assumptions that we require a matching function to satisfy (2.1) and describe the labor market framework (2.2). Section 3 provides condition for existence and uniqueness of the labor market equilibrium and states the main findings of the paper. As an example, in Section 4 we extend the analysis to the matching function from Galeotti and Merlino (2011). Finally, Section 5 concludes.

2

Model

In this part we describe social interactions in general terms, allowing our specifications to fit different scenarios. There is a large number n of ex-ante identical workers, who are embedded into a job contact network. The network is characterized by a socialization level of workers s. For a given structure of the network one can think about s as a network density (the average 3

number of contacts per worker). Workers can be in one of two states, either employed or unemployed. With some probability (which could be different for employed and unemployed workers) a worker learns about a vacancy directly from an employer. In this case the worker may accept the job offer or pass it to her contacts. The transmission of information in the network is costless and instant. The model consists of four main elements: matching function, labor supply and demand, wage and labor market turnover. In the first part of this section we outline properties that the matching function should satisfy. The second part describes the labor market in the Mortensen-Pissarides framework.

2.1

Matching function

In order to incorporate a job contact network into a labor market framework, the majority of theoretical models impose restrictive assumptions on the network structure and propagation mechanism of vacancies. This includes assumptions that only one worker initially becomes aware of a vacancy, when an employed worker relays the offer a recipient is selected at random and information about a vacancy does not go beyond first neighbors. We assume that a number of matches between unemployed workers and vacancies m(s, v, u) is a function of socialization level s, vacancy rate v and unemployment rate u. We require the matching function to satisfy four properties outlined below. Assumptions 1. A matching function satisfies: (A1) m(s, v, u) is positive and increasing in both u and v. (A2) m(s, v, u) ≤ min(u, v), m(s, 1, u) = u and m(s, v, 1) = v. (A3)

m(s,v,u) v

is decreasing in v and

m(s,v,u) u

is decreasing in u.

(A4) m(s, v, u) is increasing in the socialization level of workers s. Assumptions (A1)-(A3) are commonly employed in the labor market literature (for survey see Petrongolo and Pissarides, 2001). Assumption (A1) states that for a given vacancy rate, an increase in the unemployment rate generates more matches. The same is true for vacancy rate holding fixed the unemployment. The assumption (A2) implies that the number of matches cannot be greater than the minimum of u and v and assigns limiting values to the matching function. The assumption (A3) reflects the effect of the competition: the more vacancies are posted, the harder it is to fill one, and the same holds for unemployed workers. The assumption (A4) relates the network of personal contacts to the number of matches in the economy. In the paper we assume that the number of matches is increasing in the 4

socialization level. The main idea is that a network facilitates information exchange and thus increases the number of matches. Although there is no empirical evidence on the issue, a monotonically increasing effect has been found in various theoretical papers that explicitly derive matching functions (e.g. see Cahuc and Fontaine, 2003; Galeotti and Merlino, 2011; Ioannides and Soetevent, 2006). The only exception is a matching function derived by Calvo-Armengol (2004) in the case of a regular network (a network where all workers have the same number of connections). In the paper, the author states that an increase of the network density has two competing effects on the number of matches. On the one hand, workers gain better access to job offers, since there are more contacts that can relay offers to them. On the other hand, there is a higher probability that an unemployed worker receives several offers simultaneously in which case the information about all but one is lost. The paper states that there is a threshold level of network density after which the second effect dominates and the matching function decreases in the connectivity. Thus in the case of the matching function derived in Calvo-Armengol (2004) all the results of our paper apply for connectivity levels lower than the threshold value.

2.2 2.2.1

Labor market Labor demand

Denote by IF,t and IV,t the expected inter-temporal gains generated by a filled job and a vacancy respectively at the beginning of time period t. In the case of a filled job, a worker produces product y and is paid wage wt . Following the Mortenson-Pissarides framework we assume that there is cost γ, which a firm pays for advertising a vacancy. The Bellman equations are straightforward to obtain: 1 [(1 − δ)IF,t+1 + δIV,t+1 ] 1+r    1 1 1 = −γ + 1 − m(s, vt , ut ) IV,t+1 + m(s, vt , ut )IF,t 1+r vt vt IF,t = y − wt +

IV,t

Indeed, a filled vacancy produces y − wt until a job destruction shock arrives, in which case, the vacancy produces IV,t . In the case of an unfilled vacancy a firm incurs cost γ, and with probability v1t m(s, vt , ut ) a match between a vacancy and an unemployed worker occurs. In the steady state IF,t = IF,t+1 = IF and IV,t = IV,t+1 = IV . Moreover, the free entry condition implies that the inter-temporal gains of an open vacancy IV is driven to zero. Thus, the inter-temporal gains generated by a filled job in the steady state is: IF =

(1 + r)(y − w) r+δ 5

(1)

Combining both equations we obtain the labor demand function: 1 (y − w) m(s, v, u) = γ(δ + r) v

2.2.2

(2)

Labor market turnover

At the beginning of each period t, proportion m(s, vt−1 , ut−1 ) of unemployed workers who got a job in the previous period start to work. At the end of each period with probability δ an employed worker loses a job and becomes unemployed. The evolution of the unemployment rate is given by the difference of these two flows: ut = ut−1 − m(s, vt−1 , ut−1 ) + δ(1 − ut−1 + m(s, vt−1 , ut−1 )) In the steady state, the unemployment rate as well as the vacancy rate are constant, i.e. ut−1 = ut = u and vt−1 = vt = v, and flow out of unemployment equate flow into unemployment. We thus have: m(s, v, u) =

δ (1 − u) 1−δ

(3)

The equation (3) describes labor market turnover in the steady state. 2.2.3

Wage

In general, workers heterogeneity in terms of connectivity leads to differences in individual hiring probabilities. This may give an advantage to highly connected workers during the wage bargaining process and potentially may lead to heterogeneity in wages. However, the number of personal contacts is private information of a worker and is not observed by a firm. This makes it difficult to condition the wage on the worker’s connectivity as workers always have incentives to overstate the number of their connections. To simplify the analysis, we assume that there is a labor union that negotiates the wage on behalf of all workers, which then accept the resulting wage. Lets denote by IE,t and IU,t streams of discounted utility of employed and unemployed worker correspondingly. An employed worker receives wage wt and stays employed in the next period with probability 1 − δ. With complementary probability δ the worker loses a job and becomes unemployed. An unemployed worker receives unemployment benefits, which are without loss of generality normalized to 0. In each period with probability u1t m(s, vt , ut ) an

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unemployed worker learns about a vacancy and becomes employed. We assume that workers incur cost cs to maintain social contacts, irrespective of their employment status. IE,t = wt − cs +

IU,t

1 = −cs + 1+r

1 [(1 − δ)IE,t+1 + δIU,t+1 ] 1+r

   1 1 1 − m(s, vt , ut ) IU,t+1 + m(s, vt , ut )IE,t ut ut

(4)

(5)

where r is the discount factor. The wage is determined according to the generalized Nash bargaining process, which maximizes product of capital gains of firm and representative worker. Lets denote by β ∈ [0, 1] union’s bargaining power, then the wage is the solution to the following optimization problem: w = arg max{(IE − IU )β (IF − IV )1−β } w

The first order condition is the following: β

∂ ∂w IE

IE − IU

+ (1 − β)

∂ ∂w IF

IF − IV

=0

∂ ∂ Using Bellman equations for filled and unfilled vacancies we get ∂W IE = 1+r r+δ and ∂W IF = − 1+r r+δ . Substituting these derivatives into the optimality condition we obtain the following expression:

IE − IU = β(IF + IE − IU − IV )

(6)

Expression (6) implies that share β of the total surplus generated by a match goes to a worker.

3

Labor market steady state

In this section we focus on the comparative statics of the labor market equilibrium. We first establish the existence and uniqueness of the labor market equilibrium and then study the impact of socialization level on employment, vacancy rate, and wage. Taking the difference between expected stream of utility for employed and unemployed worker, we can find capital gains of a worker who exits unemployment: IE − IU =

(1 + r)w r + δ + u1 m(s, v, u) 7

(7)

Substituting (7) into (6) and rearranging we can express the worker wage in terms of s and equilibrium levels of u and v as the following:  v w =β y+γ u We obtain the same wage expression as in the standard matching model without social interactions (see Eq. (1.20), p. 17, in Pissarides, 2000). Network characteristics do not enter directly into the wage expression, but equilibrium wage depends on the socialization level through the equilibrium values of u and v. Substituting the wage into the labor demand expression (2) we get the following equation: 

(1 − β)y

1 u − βγ m(s, v, u) = γ(δ + r) v u

(8)

Equations (3) and (8) are equilibrium conditions. The following proposition provides a condition under which labor market equilibrium exists and is unique: Proposition 1. For any s there is a unique labor market equilibrium {u∗ (s), v ∗ (s), w∗ (s)} if γ(r+β+δ) γ(r+β+δ) ∗ ∗ ∗ (1−β) < y < δ(1−β) . Moreover, functions u (s), v (s), and w (s) are continuous. Proof See Appendix  The first part of the condition in Proposition 1 insures that y is sufficiently high and firms hire workers when everyone is unemployed. If all workers are unemployed (A2) implies that every vacancy posted results in a match and immediately generates capital gains for a firm. Thus if firms do not post vacancies when u = 1, they never do so. The second part of the condition puts an upper bound on the productivity, ensuring that v ≤ 1. In the further analysis we assume that the condition for existence and uniqueness of the equilibrium is met. We now relate equilibrium unemployment rate u∗ (s) and vacancy rate v ∗ (s) to the workers socialization level and productivity. Proposition 2. In the equilibrium the following holds: (i) u∗ (s) decreases in the socialization level of workers s and productivity y. (ii) v ∗ (s) increases in the productivity y, while v ∗ (s) decreases in s if u∗ (s) < u ¯v and √ βδ(1−δ)(δ+r)−βδ increases otherwise, where u ¯v = (1−δ)(δ+r)−βδ .

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Proof See Appendix  According to assumption (A4 ), the individual probability to find a job u1 m(s, v, u) is increasing in socialization level s. Thus an increase in s moves the Beveridge curve (3) downwards on the (u, v) plane, implying a lower unemployment level for all v. At the same time, an increase in the number of matches lowers the cost of filling a vacancy. According to the free entry condition, the number of posted vacancies increases and the labor demand curve moves upwards. Both effects lead to a decrease of equilibrium unemployment u∗ (s). An increase in worker’s productivity lowers the equilibrium unemployment and increases the vacancy rate. A growth in productivity implies a higher surplus generated by a match between an unemployed worker and a vacancy. According to the free entry condition, this leads to an increase in number of posted vacancies v. The labor demand curve shifts upward, while the Beveridge curve is not affected. As a result the equilibrium vacancy rate is increasing and equilibrium unemployment is decreasing in the worker’s productivity. There is ambiguity in the literature regarding whether the number of vacancies is increasing or decreasing in the network density. A numerical analysis by Ioannides and Soetevent (2006) suggests that number of vacancies is decreasing in the network density, while Armengol and Zenou (2005) conclude that the effect is ambiguous. We show that the effect may go in either direction, depending on the equilibrium unemployment level. An increase in the socialization level facilitates matching and leads to two competing effects that influence the number of posted vacancies. On the one hand, an increase in socialization effort increases the probability that a firm fills a vacancy and firms post more vacancies. On the other hand, the probability that an unemployed worker finds a job increases too, which leads to a higher wage and thus lowers the gains of posting a vacancy for a firm. Proposition 2 shows that the prevailing effect depends on the equilibrium level of unemployment. For low levels of unemployment, an increase in the socialization level leads to a lower posting rate. In contrast, for sufficiently high levels of equilibrium unemployment, the number of vacancies increases as the network becomes denser. To fix ideas, we consider equilibrium with an exogenously given wage. Assume that w = w0 < y and thus equilibrium conditions are given by (3) and (2). Expressing the matching function from (3) and substituting it to (2) we obtain an equivalent system of two equations. Demonstrating that the curve given by the second equation has a negative slope, it is easy to show that vacancies should increase in s for all unemployment levels. The result points out that a decrease in the number of posted vacancies in the case of low levels of equilibrium unemployment should be attributed to the wage increase. One of the most important characteristics of the labor market is a market tightness. The market tightness indicates which side of the market has an advantage in the wage bargaining process. The following proposition demonstrates that, despite potential non-monotonic behavior of equilibrium vacancies, the market tightness and wage are both monotonically 9

increasing in the socialization level. Proposition 3. The equilibrium market tightness the socialization level s.

v ∗ (s) u∗ (s)

and the wage w∗ (s) are increasing in

Proof See Appendix  An increase in the socialization level increases the probability to find a job and lowers unemployment. For sufficiently high levels of equilibrium unemployment Proposition 2 implies that the vacancy rate should increase. Both effects lead to a higher market tightness and as a result a higher wage. However, for sufficiently low values of equilibrium unemployment, an increase in s lowers the vacancy rate. The Proposition 3 states that the equilibrium unemployment decreases faster than vacancies and both market tightness and the wage are increasing in s.

4

Example

To illustrate application of our model, we extend the analysis to the network structure and vacancies communication mechanism described by Galeotti and Merlino (2011). The authors consider endogenous network formation in a two-period model and treat the vacancy rate and wage as given. In contrast, we incorporate their matching function in the fully fledged Mortensen-Pissarides model treating the socialization effort as given. We show that the matching function satisfies all outlined axioms and thus all results obtained in our paper hold for their network and communication setup. The network formation mechanism follows Cabrales et al. (2007). Each worker i selects a socialization level, si ≥ 0. Let s = (s1 , . . . , sn ) be a profile of socialization levels. The probability that a link between i and j is present at time period t is given by: ρij (s) = g(s)si sj , where

g(s) =

   Pn

j=1 sj

−1

, if s 6= 0

 0, otherwise If all workers other than i choose the socialization level s and i chooses level si the si s probability that there is a link between i and any other worker is ρi = si +(n−1)s . We consider symmetric equilibrium and thus the probability that there is a link between any two workers is ρ = ns . We assume that the network of personal contacts is drawn afresh at the beginning of each period. Hence, each possible link between workers in time period t is formed with probability 10

ρ. This implies that a worker’s number of contacts at each time period is distributed according  to the binomial distribution, p(k) = nk ρk (1 − ρ)n−k . Each worker with probability v, independent of the employment status, learns about a vacancy directly from an employer. If a worker is unemployed she accepts the offer and becomes employed (direct employment). If an employed worker hears about the offer, she passes the information to one of her randomly selected unemployed contacts. If there are no such, then information about the vacancy is lost. In this case in order to fill the position the firm posts the vacancy in the next period. Galeotti and Merlino (2011) show that the matching function in this case is given by the following expression: h i (1−u)v −us m(s, v, u) = u[v + (1 − v)P s (s, v, u)] = u 1 − (1 − v)e− u (1−e ) The derived matching function relates the aggregate rate at which job matches occur to the unemployment rate, vacancy rate, and social network structure given by the intensity of search through personal contacts. Proposition 4. The matching function m(s, v, u) satisfies conditions (A1)-(A4) and is concave in s, v and u. Proof See Appendix  Proposition 4 implies that by Proposition 1 a labor market equilibrium exists and is unique. The properties of the equilibrium unemployment, vacancies and wage are described by Propositions 2-3.

5

Summary and conclusions

Guided by the empirical evidence, a substantial body of theoretical literature studies the impact of personal contacts on the labor market outcome. A common way to embed network of personal contacts in the labor market framework is to derive a matching function using the assumed structure of social interactions. To facilitate the analysis authors impose various simplifying assumptions on the propagation mechanism of vacancies in the network. In this paper we identify conditions under which previously obtained result regarding the impact of personal contacts on unemployment can be generalized to other network structures and communication mechanisms. In particular, we show that if obtained matching function satisfies four basic assumptions formulated in our paper then unemployment decreases in the socialization level, while wage and market tightness increase. The relationship between vacancy rate and socialization level has non-monotonic shape. The vacancy rate decreases in the socialization level when unemployment is sufficiently low and increases otherwise. We introduce the socialization level in the labor market framework in a quite general way, requiring the matching function to be increasing in it. This general assumption turned out 11

to be enough to characterize the impact of socialization level on the most important labor market variables. The analysis of the paper, however, leaves an open question regarding the class of network structures and communication mechanisms that are consistent with assumed property. We leave that for future research.

6

APPENDIX

Proof of Proposition 1 Using the first and second conditions let us define the following functions: F1 (s, v, u) = m(s, v, u) − F2 (s, v, u) = (1 − β)y

δ (1 − u) 1−δ

m(s, v, u) m(s, v, u) − βγ − γ(δ + r) = 0 v u

Existence: First we should prove that F1 (s, v, u) = 0 and F2 (s, v, u) = 0 implicitly define continuous relationship of u as a function of v. Note that function F1 (·) is continuous in all three arguments and has following properties: ∂F1 (s, v, u) > 0, F1 (s, v, 0) < 0, F1 (s, v, 1) > 0 ∂u Thus for any fixed s = s0 there exists u = g1 (v, s0 ) and it is unique. We still need to prove that g1 (v, s0 ) is continuous. Let us prove by contradiction, assume that g1 (v, s) is not continuous and there exists point of discontinuity (ˆ v , sˆ) and g1 (ˆ v , sˆ) = u ˜. Let us choose any sequence of points in the domain ∞ of g1 (·), {(vt , st )}t=0 such that this sequence has limit {ˆ v , sˆ}. Then there exists sequence of ∞ {ut }t=0 such that for ∀t, ut = g(vt , st ). Note that by properties of F1 (·), function g1 (·) has bounded image hence {ut }∞ t=0 is bounded. By Bolzano-Weierstrass theorem, any bounded sequence has convergent subsequence. Let us choose subsequences {(vtn , stn )}∞ n=0 such that ∞ ∞ {utn }n=0 has a limit u ˆ. Note that {(vtn , stn )}n=0 has the same limit as the original sequence, since any subsequence of convergent sequence converge to the same limit. By our assumption lim g1 (vtn , stn ) = u ˆ 6= u ˜ = g1 (ˆ v , sˆ). By joint continuity of F1 (s, v, u) we know that n→∞

lim F1 (utn , vtn , stn ) = F1 ( lim utn , lim vtn , lim stn ) = F1 (ˆ u, vˆ, sˆ) = 0

n→∞

n→∞

n→∞

n→∞

thus we have F1 (ˆ u, vˆ, sˆ) = 0 and F1 (ˆ s, vˆ, g1 (ˆ v , sˆ)) = 0 since u ˆ and u ˜ = g1 (ˆ v , sˆ) satisfy condition F1 = 0 then both of them belong to g1 (ˆ v , sˆ). However this is possible only if g1 (·) is

12

not uniquely determined. We obtained a contradiction since given properties of F1 (·) we know that g1 (v, s) is uniquely determined, thus we can conclude that g1 (v, s) is jointly continuous in v and s. The second function is continuous in all three arguments, recall that lim v1 h(s, v, u) = 1. v→0

We know that ∂F2 (s,v,u) > 0 and it is easy to see that F2 (s, v, 0) = −γ(β + δ + r) and ∂u . F2 (s, v, 1) = (1−β)y−βγv−γ(δ+r), which is positive for any s and v as long as y > γ(r+δ+β) 1−β Following he same lines of proof we can argue that g2 (v, s) is unique and jointly continuous. Thus a solution obviously exist if at v = 1 Beveridge curve is situated to the left of r+δ+β labor-demand curve. We obtain u = δ and u = γ (1−β)y . So condition is y < γ (r+δ+β) δ(1−β) . ∗ ∗ Now let us prove that u (s), v (s) are continuous functions in s. Let us define following function: G(v, s) = g2 (v, s) − g1 (v, s) by joint continuity of g1 (·) and g2 (·) we know that G(·) is jointly continuous in v and s. As well we imposed restrictions that insure that G(0, s) < 0 and G(1, s) > 0 for any s. Using the same line of arguments as before we can conclude that v ∗ (s) is continuous. As we know at v ∗ (s), G(v ∗ (s), s) = 0 so g1 (v, s) = g2 (v, s). Taking one of them, e.g. g1 (v ∗ (s), s) we know that v ∗ (s) is continuous and g1 (·) as well, which implies that u∗ (s) = g1 (v ∗ (s), s) is continuous too. Uniqueness: Fix the socialization level of workers s. We first prove that Beveridge curve, u is decreasing in v. Indeed let us take (u, v) and (u0 , v 0 ), s.t. both satisfy the first condition and v 0 > v. Suppose that u0 ≥ u. Then (1 − u) ≥ (1 − u0 ) consequently m(s, v, u) ≥ m(s, v 0 , u0 ). We reach contradiction since according to properties of matching function we have m(s, v 0 , u0 ) ≥ m(s, v, u0 ) ≥ m(s, v, u). Therefore u0 < u. Using (A1) and (A2) we can show that derivative of F with respect to v is negative: ∂ ∂ m(s, v, u) ∂ m(s, v, u) F2 (s, v, u) = (1 − β)y − βγ <0 ∂v ∂v v ∂v u Using (A1) and (A3) we can show that derivative of F with respect to u is positive: ∂ ∂ m(s, v, u) ∂ m(s, v, u) F2 (s, v, u) = (1 − β)y − βγ >0 ∂u ∂u v ∂u u ∂v Thus by implicit function theorem ∂u > 0. This implies that labor demand is increasing in unemployment u and the curve is upper sloping at (u, v) plane. Thus given s solution to the problem is unique.

13

Proof of Proposition 2 u∗ (s) is decreasing in s Let us do the following exercise, we take two values of search intensity s0 and s, s.t. s0 > s and compare position of curves on the (u, v) plane. Observe that for a given value u we have m(s0 , v, u) > m(s, v, u), thus to equate the right hand side v should decrease. Recall that ∂ ∂v m(s, v, u) > 0. This in turn implies that Beveridge curve should shift downwards on the (u, v) plane. ∂ In Proposition 1 we have shown that ∂v F2 (s, v, u) < 0. Taking derivative of F2 (s, v, u) with respect to s and rearranging we obtain:   1 ∂ ∂ 1 F2 (s, v, u) = (1 − β)y − γβ m(s, v, u) > 0 ∂s v u ∂s Using the implicit function theorem we can conclude that ∂v ∂s > 0. This implies that the labor demand curve shifts upwards. Combining both results it is easy to see that u∗ (s0 ) < u∗ (s). u∗ (s) is decreasing in y and v ∗ (s) is increasing in y An increase in worker productivity y shifts the labor demand curve leaving the Beveridge curve unaffected. Thus the effect of increase in y on u∗ (s) and v ∗ (s) depends on the sign of ∂v ∂y . Deriving F2 (·) with respect to y we find that: ∂ 1 F2 (s, v, u) = (1 − β) m(s, v, u) > 0 ∂y v ∂v 2 (·) < 0 by the implicit function theorem ∂y > 0 and the labor Using the fact that ∂F∂v demand curve shifts upwards. This in turn implies that increase in y decreases u∗ (s) and at the same time increases v ∗ (s). √ βδ(δ+r)−βδ ∗ ∗ v (s) decreases in s if u (s) < δ(1−β)+r and increases otherwise Substituting matching function into the second condition we can rewrite it in the following way:



(1 − β)y

 δ(1 − u) u − βγ = γ(δ + r) v (1 − δ)u

The second condition does not depend on s and thus is unaffected by the change in s. This implies that it should hold for all socialization levels s. Expressing the inverse of v from the second equation we get:   1 γ (1 − δ)(δ + r) β = + v (1 − β)y δ(1 − u) u 14

Deriving right-hand side of the expression with respect to u we obtain: γ (1 − δ)(δ + r)u2 − βδ(1 − u)2 × (1 − β)y δu2 (1 − u)2 √ βδ(1−δ)(δ+r)−βδ The derivative is negative if u < u ¯, where u ¯ = (1−δ)(δ+r)−βδ and positive otherwise. This in turn implies that the second curve is increasing on the interval [0, u ¯] and decreases afterwards. Despite non-monotonic shape of the second curve the Proposition 1 implies that equilibrium is unique. An increase in s shifts the Beveridge curve downwards on the plane (u, v) leaving the second curve unaffected. This implies that equilibrium unemployment u∗ (s) is decreasing in s while vacancies v ∗ (s) are decreasing in s if u < u ¯ and increasing otherwise.

Proof of Proposition 3 To identify the effect of s on the market tightness we express uv from the second equation:   u γ (1 − δ)(δ + r)u = +β v (1 − β)y δ(1 − u) This expression relates inverse of the equilibrium market tightness to the equilibrium unemployment rate. The derivative of the right-hand side with respect to s is given by: γ(1 − δ)(δ + r) u0s × (1 − β)yδ (1 − u)2 u v

We know that the equilibrium unemployment level is decreasing in s, which implies that decreases in s too and thus market tightness uv is increasing in s.  This in turn implies that the equilibrium wage W ∗ (s) = β Y + γ uv is increasing in s.

Proof of Proposition 4 (A1a) m(s, v, u) is increasing and concave in v

(1−u)v (1−u)v ∂ −us −us m(s, v, u) = ue− u (1−e ) + (1 − v)(1 − u)v(1 − e−us )e− u (1−e ) = ∂v

[u + (1 − v)(1 − u)v(1 − e−us )]e−

15

(1−u)v (1−e−us ) u

>0

−su (1−u)v+2su2

∂2 (esu − 1) (1 − u) ((esu − 1) (1 − v)(1 − u) + 2uesu ) − (1−e m(s, v, u) = − e ∂v 2 u

)

<0

u

(A1b) m(s, v, u) is increasing and concave in u

(1−e ∂ 1 m(s, v, u) = 1 − (1 − v) ((u + v)esu − v(1 + su(1 − u))) e− ∂u u

−su (1−u)v+su2

)

u

Due to the exponential term it is difficult to identify the sign of the derivative. Let us proof first that function is concave:   (1−e 1 ∂2 2 3 su su m(s, v, u) = − (1−v)v su (2 + s(1 − u))e + v (1 − e + us(1 − u)) e− ∂u2 u3

−su (1−u)v+2su

)

u

This implies that minimum of the first derivative of matching function with respect to u lays on boards. One can easily verify that the derivative is positive on both ends: m0u (s, v, 0) = (esv − 1 + v) e−sv > 0, since esv > 1 and m0u (s, v, 1) = v (1 + (es − 1) v) e−s > 0. Thus we can conclude that m0u (·) > 0 for u ∈ [0, 1]. (A2) m(s, v, u) ≤ min(u, v), m(s, 1, u) = u and m(s, v, 1) = v By construction the matching function is unemployment rate multiplied by the individual hiring probability and thus m(s, v, u) ≤ u. By the model specification, the share u of vacancies goes to unemployed workers and results in immediate match. On the contrary share 1 − u of vacancies is received by employed workers and not all of them result in match. This happens due to the miscoordination among workers when one worker can receive multiple offers or in the case when employed worker does not have unemployed neighbors. This implies that by construction m(s, v, u) ≤ v. Last two properties can be easily verified by taking limits. (A3)

m(s,v,u) v

is decreasing in vacancies v and

m(s,v,u) u

is decreasing in unemployment u

v ∂ m(s, v, u) − m(s, v, u) ∂ m(s, v, u) = ∂v <0 ∂v v v2 Trivially follows from the fact that the matching function is concave in v. −su (1−u)v+su2

(esu − 1 − su(1 − u)) (1 − v)v − (1−e ∂ m(s, v, u) =− e ∂u u u2 Note that esu = 1 + su +

(su)2 2!

+

(su)3 3!

+ . . . > 1 + su(1 − u). 16

)

u

<0

<0

(A4) m(s, v, u) is increasing and concave in socialization level of workers s (1−e ∂ m(s, v, u) = uv(1 − u)(1 − v)e− ∂s

−su (1−u)v+su2

)

u

(1−e ∂2 su − m(s, v, u) = −uv(1 − u)(1 − v) (e u + v(1 − u)) e ∂s2

−su (1−u)v+2su2

)

u

<0

References [1] Boorman S.A., 1975, “A Combinatiorial Optimization Model for Transmission of Job Information Through Contact Networks,” The Bell Journal of Economics, Vol. 6, pp. 216-249. [2] Calvo-Armengol A., Zenou Y., 2005, “Job matching, social network and word-ofmouth communication,” Journal of Urban Economics, 57, pp. 500522. [3] Calvo-Armengol, A., Jackson, M.O., 2004, “The Effects of Social Networks on Employment and Inequality,” American Economic Review, 94(3), pp. 426454. [4] Fontaine F., 2008, “Why are similar workers paid differently? The role of social networks,” Journal of Economic Dynamics and Control, Vol. 32, Issue 12, pp. 39603977. [5] Fontaine F., and Cahuc P., 2009, “On the efficiency of job search with social networks,” Journal of Public Economic Theory, vol. 11, Issue 3, pp. 411439. [6] Galeotti A., Merlino L.P., 2011, “Endogenous Job Contact Networks,” manuscript, University of Essex. [7] Glaeser E.L., Laibson D., and Sacerdote B., 2002, “An Economic Approach to Social Capital,” The Economic Journal, vol. 112, No. 483, pp. F437-F458. [8] Granovetter, M., 1973, “The Strength of Weak Ties,” American Journal of Sociology, Vol. 78, Issue 6, pp. 1360-1380. [9] Ioannides Y.M., Loury L.D., 2004, “Job Information Networks, Neighborhood Effects, and Inequality,” Journal of Economic Literature, vol. 42, No. 4, pp. 1056-1093.

17

[10] Ioannides Y.M., Soetevent A.R., 2006, “Wages and Employment in a Random Social Network with Arbitrary Degree Distribution,” The American Economic Review, Vol. 96, pp. 270-274. [11] Montgomery J.D., 1991, “Social Networks and Labor-Market Outcomes: Toward an Economic Analysis,” The American Economic Review, Vol. 81, pp. 1408-1418. [12] Pissarides C.A., 2000, “Equilibrium Unemployment Theory,” second ed., MIT Press, Cambridge. [13] Rees A., 1966, “Information networks in labor markets,” The American Economic Review Vol. 56, pp. 559-566.

18

Labor market and search through personal contacts

May 3, 2012 - Keywords: Labor market; unemployment; job search; social network. ..... link between workers in time period t is formed with probability. 10 ...

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