Learning About Match Quality and the Use of Referrals

Viewer
Transcript

Learning About Match Quality and the Use of Referrals Manolis Galenianos∗ Pennsylvania State University

May 2012

Abstract A theoretical model of the labor market is developed to study the ﬁrm’s decision to use referrals as a hiring method. The labor market is characterized by search frictions and uncertain quality of the match between a worker and a job. Using referrals increases the arrival rate of applicants and provides more accurate signals regarding a worker’s suitability for the job. The latter informational advantage leads to predictions regarding the correlation of a referral with wages, productivity and separation rates that are consistent with the data. The model is extended by introducing heterogeneity in ﬁrm productivity and allowing the endogenous determination of signal accuracy. In equilibrium, high productivity ﬁrms choose greater accuracy of signals which diminishes the informational advantage of referrals. This generates the predictions that high productivity ﬁrms use referrals to a lesser extent which biases wage regressions that ignore ﬁrm ﬁxed eﬀects.

∗

I would like to thank Boyan Jovanovic, Giuseppe Moscarini, Uta Schoenberg, Giorgio Topa and seminar and conference participants for useful comments and the National Science Foundation for ﬁnancial support (grant SES-0922215).

1

Introduction

It is well-known that approximately half of all American workers ﬁnd their jobs through referrals by friends, acquaintances, relatives etc.1 Most work on this topic focuses on the eﬀect of referrals on workers’ labor market outcomes. The focus of the present study is on the other side of the market, namely the ﬁrms’ decision to use referrals as a method of hiring workers. This paper makes the following contributions. It develops a model where searching through a referral provides the ﬁrm with better information regarding an applicant’s suitability for the job and ﬁrms choose how intensely to use referrals. The model’s predictions are shown to be consistent with the evidence regarding referred workers’ wage, productivity and separation outcomes as well as, importantly, certain regularities regarding ﬁrms’ referral usage. Finally, the model points towards moments in the data that are worth exploring to better understand the importance of referrals. Speciﬁcally, an equilibrium model of the labor market is developed where a ﬁrm and a worker meet through the market or through a referral. The productivity of an employment relationship is match-speciﬁc and it is uncertain at the time of meeting. The ﬁrm and worker observe a public signal before deciding whether to form a match and the accuracy of the signal depends on the channel through which they met. Match quality is revealed over the course of the employment relationship. The ﬁrm exerts eﬀort in searching through each of the two channels. In the baseline model, ﬁrms are homogeneous. Under the assumption that a referral leads to more accurate information regarding match quality, the model’s predictions are consistent with a wealth of empirical evidence. First, a referred applicant is more likely to be hired.2 Conditional on being hired, the probability of a good match is greater if a worker is hired through a referral rather than the market 1

For instance Granovetter (1995). See Ioannides and Loury (2006) or Topa (2010) for recent surveys This is consistent with the evidence in the ﬁrm-level studies of Fernandez and Weinberg (1997), Castilla (2005), and Brown, Setren and Topa (2011). 2

2

which leads to the second prediction: ceteris paribus, a referred worker receives a higher wage and has higher productivity.3 The third prediction is that a referred worker has lower separation rates as he is more likely to survive learning about match quality.4 The fourth prediction is that the diﬀerentials in wages, productivity and separation rates decline with tenure, because as learning occurs only the good matches survive regardless of which channel they were formed.5 The model is subsequently extended in two ways: ﬁrms are heterogeneous in productivity and choose the accuracy of their signals subject to a cost. High productivity ﬁrms face a greater opportunity cost of employing a worker who turns out to be badly matched and therefore choose a higher level of signal accuracy. This equilibrium outcome diminishes the informational advantage of referrals for high productivity ﬁrms which leads to a number of additional predictions which are supported by the data. In equilibrium, low productivity ﬁrms are predicted to use referrals to a greater extent than high productivity ﬁrms.6 The model still predicts that a referred worker receives a higher wage, so long as the ﬁrm type is controlled for. However, when the ﬁrm type is ignored the wage premium of a referral is predicted to fall because referred workers are more likely to work at low productivity ﬁrms which pay lower wages, and it is even possible to generate a negative correlation between ﬁnding being hired through a referral and the wage.7 3

For evidence regarding wages see Simon and Warner (1992), Bayer, Ross and Topa (2008), Brown, Setren and Topa (2011) and Dustmann, Glitz and Schoenberg (2011); regarding productivity see Castilla (2005) and Pinkston (2011) who have data on direct measures of worker productivity. 4 See Loury (1983), Simon and Warner (1992), Brown, Setren and Topa (2011) and Dustmann, Glitz and Schoenberg (2011). 5 See Simon and Warner (1992), Castilla (2005), Brown, Setren and Topa (2011) and Dustmann, Glitz and Schoenberg (2011). 6 Holzer (1987) and Mardsen (1994) ﬁnd that larger ﬁrms use more formal methods to hire workers and Mardsen (1994) also reports that large ﬁrms use more formal methods to screen workers. Pellizzari (2010) ﬁnds that workers who report ﬁnding their jobs through a referral work at smaller ﬁrms. To the extent that ﬁrm productivity is positively correlated with ﬁrm size, as is generally believed to be the case, this evidence is consistent with the model’s prediction. 7 Pistaferri (1999), Pellizzari (2010), Bentolila, Michellaci and Suarez (2010) and Dustmann, Glitz and Schoenberg (2011) ﬁnd evidence of a negative correlation between the wage and a referral when controlling for worker ﬁxed eﬀects and observables but without ﬁrm ﬁxed eﬀects. Dustmann, Glitz and Schoenberg (2011) repeat their previous regression adding controls for ﬁrm ﬁxed eﬀects and ﬁnd that the coeﬃcient of a referral turns positive. The ﬁrm-level studies mentioned earlier conﬁrm this positive correlation. See Section

3

Furthermore, the wage, productivity and separation diﬀerentials between referred and nonreferred workers are predicted to be lower at highly productive ﬁrms. In this model, referrals are an informal way of alleviating a friction that is present in the market, namely uncertainty about a worker’s productivity. However, there are also formal ways of achieving the same goal such as setting up a better interviewing process, which corresponds to improving signal accuracy in this paper. Highly productive ﬁrms use the latter method to a larger extent and for this reason they hire less through referrals. Therefore, ﬁrm selection in the usage of referrals is a natural by-product of alternative methods for solving the same problem. The closest paper to the current work is Dustmann, Glitz and Schoenberg (2011). In addition to providing what is arguably the most detailed empirical analysis to date on the eﬀect of referrals on labor market outcomes, they develop a theoretical model where match quality is uncertain and referrals lead to more accurate signals regarding match quality.8 The present paper departs from their theoretical analysis in the following ways: ﬁrst, ﬁrms choose the intensity with which they search through the market and referrals; second, ﬁrm productivity is heterogeneous; third, ﬁrms choose the accuracy of the signal regarding match quality. This paper complements their work in providing an interpretation for selection by ﬁrm type regarding referral use.

2

The Labor Market With Homogeneous Firms

This Section presents the baseline model, characterizes the equilibrium and derives its main testable predictions. 3.3 for a discussion. 8 Simon and Warner (1992) develop a similar model in a partial equilibrium setting.

4

2.1

The Model

Time is continuous, the horizon is inﬁnite and the labor market is in steady state. There is measure n of workers who are ex ante homogeneous, risk-neutral, maximize expected discounted utility and discount the future at rate r. A worker is either employed or unemployed. The ﬂow utility of unemployment is z > 0 and the ﬂow utility of employment is equal to the wage. There is an endogenous measure of ﬁrms which is determined through free entry. Each ﬁrm hires one worker, is risk-neutral, maximizes expected discounted proﬁts and discounts the future at rate r. A ﬁrm is either ﬁlled and producing or vacant and searching. The ﬂow proﬁt when vacant is −K and the ﬂow proﬁt when producing is equal to output minus the wage. The output of a match is given by y¯ x where y is the ﬁrm’s productivity and x¯ is the expected match quality. Match quality can be good or bad: x ∈ {xB , xG } with xG > xB . Denoting the probability that a match is good by p, the expected match quality is given by x¯ = pxG + (1 − p)xB . It is assumed that xG = 1, xB = 0 and that y > z.9 Match quality is determined when a worker and a ﬁrm ﬁrst meet and it is good with probability γ ∈ (0, 1). At the time of the meeting, the worker and ﬁrm observe an informative signal regarding the realization of match quality and decide whether to form a match or to continue searching (the signal and matching decision are described in more detail below). The worker and the ﬁrm are symmetrically informed about the probability that their match is good. During production, match quality is perfectly revealed to the worker and ﬁrm at rate λ, at which point they decide whether to terminate or to continue their match. The assumptions on match quality imply that it is always optimal to terminate a match whose quality is revealed to be bad.10 9 Modeling worker heterogeneity as being purely match-speciﬁc is very convenient because every worker’s value of unemployment is the same regardless of output at the current match. See Galenianos (2011) for a model of referrals with permanent heterogeneity across workers. 10 This model’s qualitative predictions regarding labor market outcomes are very similar to those of Jo-

5

Matches are destroyed exogenously at rate δ, in addition to the endogenous destruction that might occur after match quality is revealed. There is no on the job search. The surplus is split through Nash bargaining where the workers’ bargaining power is β. The following assumption will be maintained: γ ≥ γ¯ =

z λ y+(y−z) r+δ

.

A ﬁrm and a worker meet through two diﬀerent channels: the market, M , or a referral, R. The two channels diﬀer in the information that is transmitted about match quality. When a worker and a ﬁrm meet through channel i ∈ {M, R}, a binary signal si ∈ {gi , bi } is generated regarding match quality. The signal si is correct with probability qGi if the match is good and qBi if it is bad. These probabilities will be referred to as the signal’s accuracy. Denote the unconditional probability that the worker generates a good signal by πi and the posterior probability that the match is good conditional on a good signal by pi . Therefore πi = γqGi + (1 − γ)(1 − qBi ), pi =

γqGi . πi

The following assumptions are made regarding the accuracy of the signals: qGi ∈ [ 21 , 1], qBi ∈ [ 12 , 1], qki ∈ ( 12 , 1) for some k ∈ {G, B} and qGM qGR > . 1 − qBR 1 − qBM

(1)

The assumptions on signal accuracy mean that pi ∈ ( 12 , 1) and pR > pM . In particular, equation (1) implies that the overall level of signal accuracy is higher through the referral than the market channel which corresponds to a higher posterior probability of a good match conditional on a good signal. vanovic (1979) and Moscarini (2005) despite the assumption that learning is lumpy: separation rates fall and wages increase with tenure, conditional on the match surviving. Of course, the dynamics leading up to learning are much simpler in the present model. See Pries (2004) for micro-foundations of the present model’s assumptions on the structure of learning and production. Nagypal (2007) provides evidence that it is better to model separation rates that decline with tenure as learning about match quality rather than, say, learning by doing.

6

Assumption (1) is motivated by the empirical evidence that referred workers receive higher wages, are more productive, have lower separation rates than non-referred workers and these diﬀerentials decline over time.11 The fact that the advantage of referred workers is relatively short-lived suggests that a referral is mostly a mechanism for speeding up the transmission of information that the employer will eventually receive anyway. The present paper takes this observation as the starting point and evaluates its implications with respect to ﬁrms’ other decisions such as search eﬀort and, in Section 3, choice of signal accuracy in an equilibrium setting. In particular, studying the strategic interaction between the recipient of the referral, the referrer and the referred, though certainly interesting, is beyond the scope of this paper.12 An Example: A physical structure is presented where the diﬀerence between the signals received through the market and referral channels is made explicit and is consistent with Assumption (1). When a worker and a ﬁrm meet, the worker is interviewed which generates an interview signal s1 . In the event the meeting occurred through a referral, an additional personal signal s2 is generated. The two signals are conditionally independent. The interview signal is correct with probability qG1 if match quality is good and qB1 if it is bad. Similarly, the personal signal is correct with probability qG2 and qB2 if match quality is good or bad, respectively. Assume that qG2 = 1. The connection between the model and this example is the following. The market signal in the model is the same as the the interview signal from a meeting through the market. 11

See the evidence cited in the introduction and, in particular, Dustmann, Glitz and Schoenberg (2011). Note that the advantage of referred workers is most obvious when the ﬁrm type is controlled for, either in the ﬁrm-level studies or in the studies that control for ﬁrm ﬁxed eﬀects. This is explored further in Section 3 where ﬁrm heterogeneity is introduced. 12 More detailed data regarding the source of the referral would be needed to inform such a study. As far as I know, Pinkston (2011) is the only paper with heterogeneity in the source of a referral and he reports that workers referred by friends of the employer look statistically diﬀerent from workers referred by other sources, such as current employees or labor unions, which suggests the presence of nepotism in the former case. However, only 5.4% of hires have been referred by the employer’s friends (out of 55% of hires who were referred in total) suggesting that, at least in that data, nepotism is a not a ﬁrst order issue. Additionally, in a ﬁrm-level study Fernandez, Castilla and Moore (2000) ﬁnd that a referred worker becomes more likely to separate from the ﬁrm when his referrer separates himself, suggesting the presence of non-pecuniary beneﬁts of referrals.

7

Turning to referrals, a bad personal signal means that match quality is bad for sure and a match never occurs after such a signal. Therefore, the referral signal in the model is the same as the interview signal conditional on a good personal signal in the example. The resulting probabilities are:

qGM = qGR = qG1 , qBM = qB1 , qBR = 1 − (1 − qB1 )(1 − qB2 ) = qB1 + qB2 (1 − qB1 ) > qBM .

Therefore, it is clear that pR > pM in this example. When the ﬁrm and worker meet they observe the signal and decide whether to match or to continue searching. Let dsi ∈ [0, 1] denote the probability with which the match is formed when they meet through channel i and observe signal si . Attention is restricted to equilibria where a bad signal leads to separation and a good signal leads to match formation with positive probability through both channels of search: dbM = dbR = 0, dgM > 0, and dgR > 0.13 To simplify notation, let dR = dgR and dM = dgM from now on. Observe that the ﬁrm and worker can use lotteries to decide whether to form a match (however, randomization will turn out not to occur in equilibrium). The aggregate ﬂow of meetings between unemployed workers and vacancies through the two channels is given by a standard Cobb-Douglas function:

m(v, u) = µv η u1−η , 13

This formulation can be thought of as arising from the following speciﬁcation. A signal s is drawn from a continuous distribution F (s|x, i) which depends on match quality x ∈ {G, B} and whose density satisﬁes the monotone likelihood ratio property. The match will be formed only if the signal is above some threshold s¯i and qGi = 1 − F (¯ si |G, i) and qBi = F (¯ si |B, i). This speciﬁcation diﬀers from the model in that the threshold, and hence the probabilities, is endogenous but this does not materially aﬀect any of the decisions that are considered here.

8

where u denotes the number of unemployed workers, v denotes the number of vacancies, µ > 0 and η ∈ (0, 1). A vacancy chooses how much eﬀort to exert in searching through the market (eM ) and through referrals (eR ). Let EM and ER denote the aggregate eﬀort exerted in searching through the market and referrals, respectively. The ﬂow of meetings through the market and referrals are given by: EM m(v, u), EM + ER ER mR (v, u, EM , ER ) = m(v, u). EM + ER

mM (v, u, EM , ER ) =

A worker meets a vacancy through channel i at rate:

αW i =

Ei m(v, u) . EM + ER u

The rate at which vacancy j meets a worker through channel i when choosing eﬀort eM and eR is: αFj i =

ei Ei m(v, u) . Ei EM + ER v

The cost of exerting eﬀort eM and eR is given by cM e2M cR e2R CM (eM ) + CR (eR ) = + . 2 2 Attention is restricted to symmetric equilibria where every ﬁrm chooses ei = Ei for i ∈ {M, R}. Allowing the marginal cost of exerting additional eﬀort to diﬀer across the two channels breaks the connection between signal accuracy and the choice of eﬀort. Exerting additional eﬀort in searching through referrals aﬀects a ﬁrm in two ways: it increases the arrival rate of workers and improves the quality of the signal. The focus of 9

this paper is on how this informational diﬀerential aﬀects the usage of referrals relative to the market as a search channel. For this reason, it is assumed that the aggregate search eﬀort only aﬀects the proportion of meetings that occur through each channel but not the total number of meetings. For the same reason, the network that leads to the meeting is not modeled explicitly. See Galenianos (2011) for a study where an explicit network is modeled and it has an eﬀect on the aggregate matching eﬃciency. When unmatched, a worker is unemployed and a ﬁrm is vacant. When matched, a workerﬁrm pair is in one of three states: a match where learning has not yet occurred (an uncertain match) which was created through the market or through a referral; or a match where learning has occurred and match quality is good (recall that bad matches are immediately terminated), in which case the channel through which they met no longer matters. The value functions for each state are now determined. An unemployed worker meets ﬁrms through channel i at rate αW i and, if a good signal is emitted, forms a match with probability di . Denoting the value of unemployment by U yields: rU = z + αW M πM dM (WM − U ) + αW R πR dR (WR − U ),

where WM and WR denote the worker’s value of being in an uncertain match which was created through the market or a referral, respectively. A vacancy chooses eﬀort levels eM and eR , meets workers through channel i at rate αF i and, if a good signal is emitted, forms a match with probability di . The value of a vacancy j which chooses eﬀort eM and eR is given by: rV˜ j (eM , eR ) = −K + αFj M πM dM (JM − V ) + αFj R πR dR (JR − V ) −

cM e2M + cR e2R , (2) 2

where JM and JR denote the ﬁrm’s value of being in an uncertain match which was created through the market or a referral, respectively.

10

The value of a vacancy given the optimal choice of eﬀort is:

V = max V˜ j (eM , eR ). eM ,eR

A worker and a ﬁrm in an uncertain match that was created through channel i ∈ {M, R} produce ﬂow of output ypi , determine the wage wi through Nash bargaining and separate at exogenous rate δ. Furthermore, they learn their match quality at rate λ and transit to state G if the match turns out to be good (with probability pi ) or separate if the match is bad (with probability 1 − pi ). The worker’s and ﬁrm’s values are given by: rWi = wi + λpi (WG − Wi ) + (δ + λ(1 − pi ))(U − Wi ), rJi = ypi − wi + λpi (JG − Ji ) + (δ + λ(1 − pi ))(V − Ji ).

(3) (4)

where WG and JG denote the worker and ﬁrm’s value of being in a good match, respectively. A worker and a ﬁrm in a good match produce ﬂow of output y, determine the wage wG through Nash bargaining and separate at exogenous rate δ. The worker’s and ﬁrm’s value when in a good match are given by:

rWG = wG + δ(U − WG ), rJG = y − wG + δ(V − JG ).

(5) (6)

Wages are determined by Nash bargaining which solves:

wk = argmaxw (Wk − U )β (Jk − V )1−β ,

(7)

where k ∈ {G, M, R}. Turning to labor market ﬂows, a worker can be in one of four states: unemployed, employed at an uncertain match which was created through the market or through a referral 11

and employed at a good match. Denote the measure of workers at each state by u, nM , nR , nG and note that

n = u + nM + nR + nG .

(8)

The labor market is in steady state. The steady state is described by the following conditions which equate the ﬂows in and out of the employment states:

uαW M dM πM = (δ + λ)nM ,

(9)

uαW R dR πR = (δ + λ)nR ,

(10)

λ(nM pM + nR pR ) = δnG .

(11)

Given equations (9), (10) and (11) it is superﬂuous to equate the ﬂows in and out of unemployment. The Equilibrium can now be deﬁned. Definition 2.1 An Equilibrium is the steady state measure of unemployed u and vacancies v, the decision rules for forming a match {dM , dR } and the eﬀort levels EM and ER such that: 1. The labor market is in steady state as described in equations (8), (9), (10) and (11). 2. The surplus is split according to (7). 3. The choice of eﬀort maximizes (2). 4. A meeting through either channel leads to a match after a good signal and does not after a bad signal. 5. There is free entry of ﬁrms: V = 0.

12

2.2

Equilibrium Characterization

This section proves the following result. Proposition 2.1 An equilibrium exists if K ∈ (K, K) where K < K.

The surplus of a match between a ﬁrm and a worker who are at state k ∈ {M, R, G} is: Sk = Wk − U + Jk − V.

Note that it is possible for the surplus to be negative, for instance if the signal accuracy of one channel is very low (it is shown below that the surplus of a good match is always positive). The optimality of the match formation rules requires that a match with negative surplus is never created in equilibrium. Therefore for i ∈ {M, R}: Si > 0 ⇒ di = 1, Si < 0 ⇒ di = 0, Si = 0 ⇒ di ∈ [0, 1].

The expected surplus of a meeting between a worker and a ﬁrm through channel i is:

S¯i = πi max[Si , 0]. Within a match (i.e. for Si ≥ 0) the solution to the Nash bargaining problem implies: Wi − U = βSi , Ji − V

= (1 − β)Si .

13

It will prove useful to simplify notation by deﬁning: m(v, u) , u m(v, u) = , v

αW = αF

so that αW Ei , EM + ER αF ei = . EM + ER

αW i = αFj i

The value functions of the unemployed worker and vacancy j can be rewritten as: [ EM S¯M ER S¯R ] + , EM + ER EM + ER [ eM S¯M eR S¯R ] cM e2M + cR e2R rV˜ j (eM , eR ) = −K + αF (1 − β) + − . EM + ER EM + ER 2 rU = z + αW β

(12) (13)

Consider the ﬁrms’ eﬀort choice. To calculate the optimal choice of eﬀort, set the derivative of equation (13) with respect to ei to zero and rearrange to arrive at: ¯

ei =

αF (1 − β) Scii EM + ER

.

(14)

Notice that Si = 0 implies that ei = 0 and therefore an interior di never occurs in equilibrium. Evaluating equation (14) at ei = Ei for both i = M and i = R yields: Ei = EM + ER

S¯i ci S¯M cM

+

S¯R cR

,

(15)

if S¯M > 0 and/or S¯R > 0. If S¯M = S¯R = 0, then EM = ER = 0. Introducing the ﬁrms’ optimal eﬀort choice into equation (12) leads to the following Lemma:

14

Lemma 2.1 In equilibrium, the value of unemployment is uniquely determined as a function of the expected match surplus, the measure of unemployed and the measure of vacancies by: S¯2

rU = z +

M αW β( cM +

S¯M cM

+

2 S¯R ) cR

S¯R cR

,

(16)

if S¯M > 0 and/or S¯R > 0 and rU = z if S¯M = S¯R = 0.

The measure of unemployed is now expressed as a function of the expected surplus and the measure of vacancies. The steady state conditions (9), (10) and (11) can be rearranged as follows:

ni =

nG =

uαW Ei πi , (δ + λ)(EM + ER )

i ∈ {M, R},

uαW λ ( EM γqGM ER γqGR ) + . δ(δ + λ) EM + ER EM + ER

Therefore

nM + nR + nG = uαW

EM Γ1M + ER Γ1R , EM + ER

(17)

where

Γ1i =

γqGi (1 − γ)(1 − qBi ) + . δ δ+λ

Combining equations (8), (15) and (17) yields:

η 1−η

n = u + µv u

15

Γ1M S¯M cM S¯M cM

+ +

Γ1R S¯R cR S¯R cR

.

(18)

The right-hand side of equation (18) is equal to zero when u = 0, is strictly increasing in u and is greater than n when u = n. Therefore, equation (18) uniquely determines the measure of unemployed given the measure of vacancies and expected match surplus. Rearrange equation (16) as follows: S¯2

u =

M + ( µv η β( cM

¯

M (rU − z)( ScM

2 S¯R ) )1 cR η , ¯ SR + cR )

and introduce this expression inside equation (18) to get: S¯2

S¯2

S¯2

S¯2

¯

¯

SM SR M M + cRR ) ) 1 ( µβ( cM + cRR ) ) 1−η µ( Γ1M + Γ1R ) ( µβ( cM n cM cR η η . = + ¯ ¯ ¯ ¯ ¯ ¯ SR SM M M v (rU − z)( ScM + ScRR ) (rU − z)( ScM + ScRR ) + cM cR

(19)

The right-hand side of equation (19) approaches inﬁnity as U → z/r, is strictly decreasing in U and it approaches zero as U → +∞. Therefore, there is a unique U that satisﬁes equation (19). The following Lemma has been proven: Lemma 2.2 In equilibrium, the value of unemployment is uniquely determined as a function of the expected match surplus and the measure of vacancies by the solution to (19).

Turning to the value functions, the surplus from a good match is calculated by combing equations (5) and (6):

(r + δ)SG = y − rU − rV.

(20)

Recall that in equilibrium rU ≥ z and further observe that rU ≤ y because y is the total size of output. Since V = 0 in equilibrium and y > z by assumption, SG > 0 in equilibrium.

16

To get the surplus of a match of type i ∈ {M, R}, combine equations (3) and (4) with the Nash bargaining solution, the free entry condition and equation (20):

Si =

ypi rU r + δ + λpi − r+δ r+δ r+δ+λ

⇒ πi Si = yΓ2i − rU (Γ2i + Γ3i ),

(21) (22)

where γqGi , r+δ (1 − γ)(1 − qBi ) = . r+δ+λ

Γ2i = Γ3i

If Si < 0 then a meeting through channel i does not lead to a match regardless of signal. The condition for this to occur is

Si < 0 ⇔ U > U¯i ,

where

U¯i =

yΓ2i . r(Γ2i + Γ3i )

The assumptions on signal accuracy mean that U¯M < U¯R and the assumption that γ ≥ γ¯ implies that rU¯M > z.

17

The expected surplus of a meeting can be characterized as a function of the value of unemployment: z U ∈ [ , U¯M ) ⇒ S¯i (U ) = yΓ2i − rU (Γ2i + Γ3i ) ⇒ di = 1, r U ∈ [U¯M , U¯R ) ⇒ S¯R (U ) = yΓ2R − rU (Γ2R + Γ3R ) ⇒ dR = 1,

y U ∈ [U¯R , ) r

i ∈ {M, R},

⇒ S¯M (U ) = 0

⇒ dM = 0,

⇒ S¯M (U ) = S¯R (U ) = 0

⇒ dM = dR = 0.

Notice that U < U¯M is necessary for hires to occur through both search channels. The following Lemma combines the characterization of expected match surplus derived above with the result of Lemma 2.2 to determine the value of unemployment as a function of the measure of vacancies alone. Furthermore, it derives conditions such that a meeting through either channel leads to a match after a good signal and does not after a bad signal. Lemma 2.3 In equilibrium: 1. The value of unemployment is determined as a function of the measure of vacancies, U (v). 2. limv→0 U (v) =

z r

and limv→∞ U (v) = U¯R .

3. There exists v such that v < v ⇒ U (v) < U¯M . 4. There exists v such that if v > v then a bad signal does not lead to a match. Proof. See the Appendix. Now consider the ﬁrm’s problem. Combine equations (13), (14) and (15), to determine the value of a vacancy when search eﬀort is chosen optimally:

rV

1 = −K + (1 − β)αF 2 18

2 S¯M cM S¯M cM

+ +

2 S¯R cR S¯R cR

.

Deﬁne: 1 Φ(v) = −K + (1 − β)αF 2

[S¯M (U (v))]2 cM S¯M (U (v)) cM

+ +

[S¯R (U (v))]2 cR S¯R (U (v)) cR

,

so that Φ(v) = 0 ⇔ rV = 0. When v → 0 a ﬁrm meets with a worker instantaneously and pays no vacancy costs. Furthermore, limv→0 U (v) =

z r

and in that case the surplus of a match is positive because

z < rU¯M < rU¯R . Therefore:

lim Φ(v) > 0.

v→0

As v → ∞, the surplus of a meeting through either channel goes to zero because limv→∞ U (v) = U¯R . Therefore: lim Φ(v) = −K < 0.

v→∞

By the continuity of Φ(v), there exists v such that Φ(v) = 0. Furthermore, the requirement of v ∈ (v, v) corresponds to K ∈ (K, K). This completes the proof of Proposition 2.1.

2.3

Predictions

This Section presents the ﬁrst set of the model’s predictions. Let P denote the proportion of hires that occurs through the referral channel: P =

αF R πR . αF M πM + αF R πR

P will be referred to as the prevalence of referrals. 19

Proposition 2.2 In equilibrium: 1. A worker who is hired through a referral receives a higher wage than a worker hired through the market. 2. A worker who is hired through a referral has higher productivity than a worker hired through the market. 3. A worker who is hired through a referral has a lower separation rate than a worker hired through the market. 4. The diﬀerentials in wages, productivity and separation rates across hiring channels decline with the workers’ tenure on the job. Proof. Under the assumptions on signal accuracy: pR > pM . A higher posterior probability of a good match leads to higher productivity (ypR > ypM ), higher wage (wR > wM ) and lower separation rate: λ(1 − pR ) + δ < λ(1 − pM ) + δ. These diﬀerentials disappear after learning has occurred and the probability of being in a good match among surviving workers is equal to 1 regardless of the channel through which they were hired. There is a lot of evidence to support these predictions. Regarding point 1, Bayer, Ross and Topa (2008), and Dustmann, Glitz and Schoenberg (2011) control for workers’ observable characteristics and ﬁrm ﬁxed eﬀects and ﬁnd that referred workers earn higher wages than non-referred workers. Simon and Warner (1992) ﬁnd a wage premium for being hired through a referral in the Survey of Natural and Social Scientists and Engineers and Brown, Setren and Topa (2011) reach the same conclusion in a ﬁrm-level study. Regarding point 2, Castilla (2005) and Pinkston (2011) have direct measures of worker productivity and report that productivity is higher for workers who were referred to the ﬁrm controlling for worker observables and ﬁrm ﬁxed eﬀects.

20

Regarding point 3, See Loury (1983), Simon and Warner (1992), Brown, Setren and Topa (2011) and Dustmann, Glitz and Schoenberg (2011) all ﬁnd that the separation rates are lower for referred workers, controlling for worker observables and, in the case of Dustmann, Glitz and Schoenberg (2011) for ﬁrm ﬁxed eﬀects. Regarding point 4, in their ﬁrm-level studies, Castilla (2005) reports that productivity diﬀerentials decline with tenure and Brown, Setren and Topa (2011) report that wage diﬀerentials decline with tenure, conditional on worker observables. Furthermore, Simon and Warner (1992), Dustmann, Glitz and Schoenberg (2011) report that diﬀerentials in wages and separation rates decline with tenure conditional on worker observables and ﬁrm ﬁxed eﬀects. At this point it is worth remarking that a model where referrals alleviate problems of adverse selection (bringing in higher ability workers) or moral hazard (if it is easier to monitor referred workers) will provide predictions that are consistent with points 1 and 2 but not with points 3 and, especially, 4. The fact that the performance of referred and non-referred workers converges over time is strong evidence in favor of productivity uncertainty after a hire, which implies that learning is at the heart of what makes referrals useful.

Proposition 2.3 If γ(qGR − qGM ) ≥ (1 − γ)(qBR − qBM ) then a meeting through a referral is more likely to lead to a match than a meeting through the market. Proof. Notice that

γ(qGR − qGM ) ≥ (1 − γ)(qBR − qBM ) ⇔ πR ≥ πM .

If the condition holds, the unconditional probability that a good signal is generated is higher when a ﬁrm and a worker meet through a referral than through the market. The ﬁrm-level studies of Fernandez and Weinberg (1997), Castilla (2005), and Brown, Setren and Topa (2011) have information on the universe of applicants for their respective 21

ﬁrms during the time period under study including whether the applicant was referred to the ﬁrm. All three studies ﬁnd that referred workers are more likely to be hired conditional on their observable characteristics.

Proposition 2.4 If (qGR − (1 − qBR ))(γqGR + (1 − γ)(1 − qBR )) cR > , cM (qGM − (1 − qBM ))(γqGM + (1 − γ)(1 − qBM )) ˆ for some K ˆ where K ˆ < K then more workers are hired through the market than and K ≥ K referrals (P < 12 ). Proof. See the Appendix. This Proposition makes the the informational advantage of referrals does not necessarily lead to a majority of hires occurring through referrals. In particular, if the cost of searching through referrals is suﬃciently higher than the market, then ﬁrms exert more eﬀort in searching through the market and more workers are hired through the market. This happens despite the fact that a worker is more likely to be hired conditional on meeting a ﬁrm through a referral. This observation is relevant because, though referrals are used across the board, they do not necessarily take a majority of hires.

The model provides a framework to examine the diﬀerential in the prevalence of referrals across diﬀerent sectors of the economy, such as industries or occupations. While the evidence in this regard is somewhat scattered, the model points to some moments that could be examined. This is important because these diﬀerentials have been shown to be signiﬁcant and to correlate with the eﬃciency of matching in the case of industries (Galenianos, 2011). Suppose the economy consists of two sectors, a and b, and suppose that a signal through a referral is more accurate than a signal through the market in both of them, but it is relatively 22

more accurate in sector a. To make this distinction in a sharper way, three simplifying assumptions will be made. First, assume that a signal is correct with symmetric probability regardless of the underlying match quality:

l l qGi = qBi = qil , l ∈ {a, b}, i ∈ {M, R}.

(23)

Second, assume that sector a has a more accurate signal through referrals but a less accurate signal through the market:

b > qRa , qRa > qRb > qM

(24)

Third, the value of unemployment is the same in both sectors:

Ua = Ub

(25)

This could be the outcome of worker mobility, as will be the case in Section 3, or simply a “lucky” choice of the levels of signal accuracy.

Proposition 2.5 Suppose that equations (23), (24) and (25) hold. Then in equilibrium: 1. A worker who is hired through a referral has higher productivity, receives a higher wage and has lower separation rates than a worker who is hired through the market in both sectors. 2. The diﬀerentials in productivity, wages and separation rates between workers who were hired through a referral or the market are larger in sector a. 3. If γ >

1 2

then sector a exhibits greater prevalence of referrals.

23

Proof. Equation (24) implies that

paR > pbR > pbM > paM , b a which proves points 1 and 2 and also means that S¯Ra > S¯Rb > S¯M > S¯M .

Note that: S¯b π b S¯a π a P a > P b ⇔ ¯aR Ra > ¯bR Rb . S M πM S M πM If γ >

1 2

b a then πRa > πRb > πM > πM which proves point 3.

The predictions that the prevalence of referrals across sectors is positively correlated with smaller diﬀerentials across referred and non-referred workers has not been tested in the data. However, the ﬁrm-level study of Brown, Setren and Topa (2012) provides some supportive evidence. That paper examines a large ﬁrm which hires workers at a variety of skill levels allowing the authors to interact whether a referral took place with the level of the job (support staﬀ, junior staﬀ, medium staﬀ, senior staﬀ or executive) and the educational attainment that is required (high school diploma, associate degree, bachelor’s degree, or graduate degree). They ﬁnd that a referral increases the probability of a job oﬀer and that this eﬀect is strongest for support stuﬀ and for positions that require lower educational attainment. Furthermore, the wage premium of a referral is largest for support staﬀ, declines with the seniority of the job and is actually negative for executives, although their number is probably too small to draw ﬁrm conclusions. Finally, the separation diﬀerential is again largest for support staﬀ (i.e. it is lowest for referred support staﬀ workers) and is smaller or insigniﬁcant for more senior positions. This is consistent with the well-known fact that referrals are more prevalent at the lower end of the labor market (see Bayer, Ross and Topa 2008 among others). This observation is consistent with the model’s prediction that there is a positive correlation between the prevalence of referrals and the diﬀerentials between referred and non-referred

24

workers. Of course, the evidence is somewhat anecdotal at this point but it seems to be a promising avenue for future work. See the Conclusions for additional discussion.

3

Firm Heterogeneity and Endogenous Signal Accuracy

This Section introduces two features to the baseline model: ﬁrm heterogeneity and endogenous choice of signal accuracy.

3.1

The Extended Model

There are two types of ﬁrm which diﬀer in productivity. A ﬁrm of type H has productivity y H and a ﬁrm of type L has productivity y L . It is assumed that y H > y L and that γ ≥ γ¯ L = z λ y L +(y L −z) r+δ

.

When a ﬁrm of type t ∈ {H, L} is searching for a worker, it is subject to a ﬂow cost K(v t ) where v t is the measure of type t vacancies and K(0) = K ′ (0) = 0, K ′ (v) > 0. This assumption will guarantee the coexistence of diﬀerent types of ﬁrms in equilibrium. There are two islands and each island is populated by ﬁrms that belong to one type. From now on the islands are identiﬁed by the type of ﬁrm that populate them. Every worker chooses one island to enter and the measure of workers in island t is denoted by nt , where nH + nL = n. Each island operates as in the baseline model with two simpliﬁcations and one major change. The simpliﬁcations are that, ﬁrst, the cost of exerting search eﬀort is assumed to be the same for both channels and normalized to unity: cM = cR = 1; second, the probability that a signal is correct is assumed to be symmetric across match qualities: qGi = qBi = qi for i ∈ {M, R}. The major change is that the probability that a signal is correct is a choice variable for the

25

ﬁrm. Speciﬁcally, each ﬁrm chooses h which determines the accuracy of the signals, qM (h) and qR (h). A ﬁrm that chooses h incurs cost s(h) = σh every time the ﬁrm meets with a worker. In other words, the cost is proportional to the frequency of generating signals. The choice of h represents the ﬁrm’s investment on better screening technology, such as a better human resources department.14 We assume that qi (h) is strictly increasing and strictly concave in h and it satisﬁes the Inada condition limh→0 qi′ (h) = +∞. As in Section 2, it is assumed that qR (h) > qM (h) >

1 2

′ (h).15 for all h. Furthermore, it is assumed that qR′ (h) ≤ qM

Firms exert eﬀort in searching through the market and referrals and learn about match quality as in Section 2. The value of a vacancy j of type t is: ) ( ( jt jt jt t t t t ˜ rV (eM , eR , h) = −K(v ) + αF M πM (h)dM (JM (h) − V ) − s(h) + αF R πR (h)dtR (JRt (h) − V t ) ) e2 + e2 R −s(h) − M , 2 where

V t = max V˜ jt (eM , eR , h). eM ,eR ,h

We focus on equilibria where all ﬁrms of a given type make the same decision regarding eM , eR and h. Note that the aggregate choice of human resources aﬀects the other value functions only t through its eﬀect on signal accuracy. Therefore, given qM and qRt all value functions of 14

Note that there is hold up problem under this speciﬁcation: the ﬁrm chooses h prior to meeting with the worker and only receives share 1 − β of the resulting surplus. As a result the joint surplus of a meeting is not maximized. This is consistent with the spirit of the model regarding the importance of frictions in labor markets. The prediction that high productivity ﬁrm choose higher levels of h does not qualitatively depend on the presence of the hold-up problem 15 In terms of the example of Section 2, one can interpret an increase in h as improving the accuracy of the interviewing signal while leaving the accuracy of the personal signal unaﬀected. This would have the desired eﬀect of diminishing the accuracy diﬀerential between the market and referral signals. The parallel is imperfect because the example is phrased in terms of signals whose accuracy is asymmetric across match qualities but the intuition is clear.

26

workers and ﬁrms on island t are identical to the ones in Section 2. The Equilibrium is deﬁned as follows: Definition 3.1 An Equilibrium is the steady state measure of workers nt , unemployed ut t and vacancies v t , the decision rules for forming a match {dtM , dtR }, the eﬀort levels EM and

ERt , the choice of human resources ht and the value of unemployment U t for t ∈ {H, L} such that in each island: 1. The labor market is in steady state. 2. The surplus is split according to the Nash bargaining solution. 3. The choice of eﬀort and human resources maximizes the value of a vacancy. 4. A meeting through either channel leads to a match after a good signal and does not after a bad signal. 5. There is free entry of ﬁrms: V t = K(v t ).

and 6. The value of unemployment is the same across islands (U H = U L ) and nH > 0 and nL > 0 where nH + nL = n.

3.2

Equilibrium

We prove that Proposition 3.1 An equilibrium exists if n ≥ n and σ ≤ σ ˆ. The analysis will be kept as close as possible to Section 2. The equilibrium will be characterized in each island for a given measure of workers nt . Then the allocation of workers across islands will be considered. 27

Denote the surplus of a match between a worker and a ﬁrm that chose h by:

Sit (h) = Jit (h) + Wit (h) − V t − U t .

The value function of a type-t vacancy can be written as follows, taking into account the Nash bargaining solution: (

eM t ((1 − β)πM (h)dtM SM (h) − σh) + ERt ) e2 + e2 eR R t t + t ((1 − β)πR (h)dR SR (h) − σh) − M t EM + ER 2 ( ) 2 2 t α (1 − β) ˆt (h) + eR Sˆt (h) − eM + eR , = −K(v t ) + Ft e S M M R EM + ERt 2

rV˜ jt (eM , eR , h) = −K(v t ) + αFt

t EM

(26) (27)

where

Sˆit (h) = max[πi (h)Sit (h) − σ ˜ h, 0],

and σ ˜=

σ . 1−β

(28)

As in Section 2, dti = 0 if Sit < 0 which is implicit in equation (28).

Setting the derivative of equation (27) with respect to eM and eR to zero yields:

ei =

αFt (1 − β)Sˆit (h) . t EM + ERt

(29)

Combining equations (27) and (29) evaluated at ei = Eit yields: ) 1 ( (1 − β)αFt )2 ( ˆt 2 2 t jt ˆ ˜ (SM (h)) + (SR (h)) . rV (h) = 2 EM + ER

(30)

The following Lemma characterizes the optimal choice of h and the resulting value of unemployment. Lemma 3.1 In equilibrium: 28

1. The optimal choice of h of a type-t ﬁrm is determined as a function of the value of unemployment: ht (U t ). 2. The optimal choice of h is decreasing in σ. 3. The expected surplus of a meeting in island t is determined as a function of the value of unemployment: Sˆit (U t ). Proof. See the Appendix. The steps that lead to Lemma 2.2 can be replicated: impose the symmetric eﬀort condition on equation (29) and introduce it inside the steady state conditions. The outcome of repeating these steps (which are omitted because they are identical to Section 2) is to derive an expression that determines island t’s value of unemployment U t as a function of t the expected match surplus (SˆM , SˆRt ), the measure of vacancies (v t ) and the measure of island

t’s workers (nt ). It is straightforward to show that: Lemma 3.2 There exists a value of unemployment U t (nt ) for island t if nt ∈ (nt , nt ) for some nt < nt such that the market is in steady state, the surplus is split through Nash bargaining, the choices of eﬀort and human resources are optimal, a good signal leads to a match, a bad signal does not and the measure of vacancies is determined through free entry. The ﬁrm’s ﬂow cost depend on the measure of vacancies on the island and therefore the bounds on the measure of workers in Lemma 3.2 correspond to the bounds on the ﬂow cost of vacancies in Proposition 2.1. Otherwise, the proof is very close to that of Proposition 2.1 and is therefore omitted. The ﬁnal step is to show that the value of unemployment across the two islands is equalized for an interior measure of workers which satisﬁes the conditions above. 29

Lemma 3.3 If n ≥ n and σ ˜≤σ ˆ then there exist nH∗ > 0 and nL∗ > 0 where nH∗ + nL∗ = n and U H (nH∗ ) = U L (nL∗ ) = U ∗ . Proof. See the Appendix. This completes the proof of Proposition 3.1.

3.3

Predictions

This Section presents the predictions of the extended model.

Proposition 3.2 In equilibrium, high productivity ﬁrms choose a higher level of human resources: hH > hL . Proof. See the Appendix. Holzer (1987) and Mardsen (1994) report that larger ﬁrms use more formal methods to hire workers and Mardsen (1994) also reports that large ﬁrms use more formal methods to screen workers. To the extent that ﬁrm size is positively correlated with ﬁrm productivity, this ﬁnding is consistent with the prediction.

Proposition 3.3 In equilibrium, conditional on the ﬁrm’s type: 1. A worker who is hired through a referral receives a higher wage than a worker hired through the market. 2. A worker who is hired through a referral has higher productivity than a worker hired through the market. 3. A worker who is hired through a referral has a lower separation rate than a worker hired through the market.

30

4. The diﬀerences in wages and separation rates across hiring channels decline with the workers’ tenure on the job.

Proposition 3.3 follows from ptR > ptM .

Proposition 3.4 In equilibrium, if γ ≥ 12 : 1. Low productivity ﬁrms exhibit greater prevalence of referrals than high productivity ﬁrms: P L > P H . 2. The wage premium of a referred worker is lower if the ﬁrm type is not controlled for than if it is. 3. If y L < ϵy H for some ϵ ∈ (0, 1) then the average wage of a worker hired through a referral is lower than the average wage of a worker hired through the market. Proof. See the Appendix. Propositions 3.3 and 3.4 are consistent with the empirical ﬁnding that a referral is correlated with a higher wage when ﬁrm ﬁxed eﬀects are controlled, for while the correlation is negative when there are no ﬁrm eﬀect. This ﬁnding is most clearly demonstrated in Dustmann, Glitz and Schoenberg (2011) who perform wage regressions with and without ﬁrm ﬁxed eﬀects in a large matched employer-employee data set. They ﬁnd that their estimate for the eﬀect of a referral on the wage changes signs from negative to positive after the ﬁrm eﬀects are introduced. This ﬁnding is also consistent with the contrast between ﬁrm-level studies, where the ﬁrm eﬀect is controlled by design, which ﬁnd positive correlations with the wage (Brown, Setren and Topa 2012) and productivity (Castilla 2005) and studies where ﬁrm ﬁxed eﬀects are not included and a negative correlation is reported (Pisteferri 1999, Pellizzari 2010, Bentolila, Michelacci and Suarez 2010). It should be remarked, that including ﬁrm size, rather than 31

ﬁrm ﬁxed eﬀects, in the wage regression still generates a negative correlation of wage and a referral, although the magnitude is smaller than if ﬁrm size is not included (Pellizzari 2010). Therefore, ﬁrm selection in terms of their use of referrals is not simply a matter of size. What is more, Propositions 3.2, 3.3 and 3.4 provide an interpretation for why these studies ﬁnd seemingly opposing conclusions: the informational friction in the market can be alleviated in an informal way (using referrals) or a formal way (investing in better interviewing) and ﬁrms sort according to productivity in their preferred option: low productivity ﬁrms ﬁnd it easier to rely on referrals while high productivity ﬁrms improve their interviewing. In other words, there is no contradiction in the ﬁnding that the ﬁrms with greater value for “better” workers use referrals to a lesser extent, even though referrals are associated with “better” workers. Finally, the model provides an additional prediction. Proposition 3.5 In equilibrium, the diﬀerentials in productivity, wages and separation rates between workers who were hired through a referral or the market are larger in low productivity ﬁrms.

I am not aware of any publicly available work that has tested this prediction but evaluating it would be a useful test of this paper’s theory.

4

Conclusions

This paper presents a model where ﬁrms choose how intensely to search through referrals vs the market and the principal beneﬁt of using referrals is that they provide more accurate signals regarding a worker’s suitability for the job. This baseline model captures a large number of stylized facts regarding the correlation of referrals with wages, productivity, separation rates and the interaction of these variables with tenure. 32

The extension to include ﬁrm heterogeneity demonstrates that this framework can be used to examine the observed regularities in referral use. When the signal’s accuracy is an endogenous choice, high productivity ﬁrms will choose to improve the signal to larger extent thereby reducing the beneﬁt of referrals. As a result, high productivity ﬁrms use referrals to a lesser extent which is consistent with the data and provides an interpretation for the sometimes reported negative correlation between a worker’s wage and a referral which is due to ﬁrm selection. This framework can be further extended to study the large diﬀerence between referral use across industries and occupations. Referrals are used to a larger extent in jobs of lower socioeconomic background and which require lower levels of education (see Topa 2010). In the context of the present paper, greater use of referrals is consistent with relatively higher accuracy in the referral signal which also implies larger diﬀerentials in wages, productivity and separation rates between referred and non-referred workers. Some supportive evidence is present in Brown, Setren and Topa (2012) but this is clearly an issue that needs to be studies further, especially to better understand the source of such diﬀerential. One possibility is that the diﬀerence in signal accuracy might be due to the type of skills that are required for the job: perhaps the ﬁrst type of jobs require skills that are easy to observe and convey for the referrer (e.g. reliability) while the skills required for more complex jobs are harder to observe or convey (e.g. creativity).

5

Appendix

Lemma 2.3: In equilibrium: 1. The value of unemployment is determined as a function of the measure of vacancies, U (v). 2. limv→0 U (v) =

z r

and limv→∞ U (v) = U¯R .

33

3. There exists v such that v < v ⇒ U (v) < U¯M . 4. There exists v such that if v > v then a bad signal does not lead to a match. Proof. 1. Deﬁne 1

Ψ(U, v) = (µβ) η

( Q1 (U ) ) η1 1−η ( Q1 (U ) ) 1−η 1 n η + µη β η Q2 (U ) − , rU − z rU − z v

(31)

where

Q1 (U ) =

S¯M (U )2 cM S¯M (U ) cM

Q2 (U ) =

Γ1M S¯M (U ) cM S¯M (U ) cM

+ +

S¯R (U )2 cR S¯R (U ) cR

+ +

,

Γ1R S¯R (U ) cR . S¯R (U ) cR

Notice that Ψ(U, v) = 0 if and only if equation (19) holds. Note that

lim Ψ(U, v) = +∞.

U →z/r

(32)

When U ∈ [U¯M , U¯R ) equation (31) is simpliﬁed as follows: Q1 (U ) = S¯R (U ), Q2 (U ) = Γ1R , 1

⇒ Ψ(U, v) = (µβ) η

¯R (U ) ) 1−η ( S¯R (U ) ) η1 1−η ( S 1 n η + µη β η Γ1R − . rU − z rU − z v

(33)

Finally, if U ≥ U¯R then Q1 (U ) = 0 and: Ψ(U, v) = −

34

n < 0. v

(34)

These observations together with the continuity of Ψ(U, v) prove that there exists U (v) such that Ψ(U (v), v) = 0 which therefore deﬁnes the value of unemployment as a function of the measure of vacancies. Note that it is the ratio of workers to vacancies that matters. 2. Equation (32) together with limv→0 Ψ(U, v) = −∞ imply that limv→0 U (v) = a/r. Equation (34) together with 1

lim Ψ(U, v) = (µβ) η

v→∞

( Q1 (U ) ) η1 1−η ( Q1 (U ) ) 1−η 1 η + µη β η Q2 (U ) ≥ 0, rU − z rU − z

imply that limv→∞ U (v) = U¯R . 3. From equation (33), Ψ(U, v) is strictly decreasing in U so long as U ∈ (U¯M , U¯R ). If U (v) < U¯M then a meeting where a good signal is generated leads to a match regardless of the channel of search. A suﬃcient condition is Ψ(U¯M , v) < 0 which corresponds to v < v¯. 4. We now show that matches are not formed after a bad signal if U ≥ U M and that U¯M > U M so that an equilibrium is possible. Denote the probability the match quality is good after a bad signal through channel i by:

pbi =

γ(1 − qGi ) . γ(1 − qGi ) + (1 − γ)qBi

Assumption 1 implies that qGR − qBR > qGM − qBM which in turn means that pbM > pbR . From equation (21), the expected surplus of hiring a worker through channel i after a bad signal is:

Sbi =

ypbi rU r + δ + λpbi − , r+δ r+δ r+δ+λ

and SbM > SbR .

35

It is suboptimal to match after a bad signal if

rU ≥ rU M =

yΓb2M , Γb2M + Γb3M

where γ(1 − qGM ) , r+δ (1 − γ)qBM = , r+δ+λ

Γb2M = Γb3M

and it can be readily veriﬁed that U¯M > U M . If Ψ(U M , v) > 0 > Ψ(U¯M , v), then U (v) ∈ (U M , U¯M ) and the desired result holds. This inequality can be straightforwardly (if tediously) be rewritten in terms of v ∈ (v, v).

Proposition 2.4: If cR (qGR − (1 − qBR ))(γqGR + (1 − γ)(1 − qBR )) > , cM (qGM − (1 − qBM ))(γqGM + (1 − γ)(1 − qBM ))

(35)

ˆ for some K ˆ where K ˆ < K then more workers are hired through the market than and K ≥ K referrals (P < 12 ). Proof. Notice that U = U¯M ⇒ EM = 0 ⇒ αF M = 0. Therefore, the value of unemployment needs to be low enough. A condition is derived such that P <

1 2

upper bound on U is determined such that P < 12 . Observe that α F M πM αF R πR

=

M αF EME+E πM R R αF EME+E πR R

36

=

S¯M π cM M ¯ SR π cR R

.

when rU = z. Then the

Furthermore: S¯i (y − rU )γ qGi rU (1 − γ) 1 − qBi = − , ci r+δ ci r+δ+λ ci and therefore when rU = z the following holds:

(y − z)γ [ qGM πM r+δ cM

αF M πM > αF R πR ⇔ qGR πR ] (1 − qBR )πR ] z(1 − γ) [ (1 − qBM )πM − − > . cR r+δ+λ cM cR

The assumption that γ ≥ γ¯ leads to: z(1 − γ) (y − z)γ ≥ , r+δ r+δ+λ and the terms in the square brackets are equivalent to equation (35). Finally, the highest level of U such that P ≤

1 2

is given by the solution to:

) ) πM ( πR ( yΓ2M − rUˆ (Γ2M + Γ3M ) = yΓ2R − rUˆ (Γ2R + Γ3R ) cM cR M Γ2M − πcRR Γ2R ) y( πcM ˆ ⇒ rU = πM . (Γ2M + Γ3M ) − πcRR (Γ2R + Γ3R ) cM Therefore, if U < Uˆ then there are more hires through the market channel. For U < Uˆ the ˆ number of vacancies need to be below a threshold which corresponds to K < K. The last step is to check that U < Uˆ can occur in equilibrium. Speciﬁcally, this means checking that Uˆ > U M since U > U M is a necessary condition for equilibrium: M Γ2M − y( πcM

πM (Γ2M cM

+ Γ3M ) −

πR Γ ) cR 2R πR (Γ2R + cR

Γ3R )

>

yΓb2M . Γb2M + Γb3M

It is a matter of tedious algebra to show that if equation (35) holds, then Uˆ > U M .

37

Lemma 3.1: In equilibrium: 1. The optimal choice of h of a type-t ﬁrm is determined as a function of the value of unemployment: ht (U t ). 2. The optimal choice of h is decreasing in σ. 3. The expected surplus of a meeting in island t is determined as a function of the value of unemployment: Sˆit (U t ). Proof. 1. Equation (28) can be rewritten as follows (see Section 2 ): [ y t γq (h) ] ( γqi (h) (1 − γ)(1 − qi (h)) ) i Sˆit (h) = max − rU t + −σ ˜ h, 0 . r+δ r+δ r+δ+λ

(36)

Note that: ( ytγ γ 1−γ ) Sˆit ′(h) = qi′ (h) − rU t ( − ) −σ ˜, r+δ r+δ r+δ+λ ( ytγ γ 1−γ ) Sˆit ′′(h) = qi′′ (h) − rU t ( − ) < 0. r+δ r+δ r+δ+λ The assumptions about qit (h) mean that Sˆit ′(0) > 0 > limh→∞ Sˆit ′(h). Deﬁne hti (U t ) by Sˆit ′(hti (U t )) = 0 and note that hti (U t ) also maximizes [Sˆit (h)]2 . Deﬁne U¯it as follows:

rU¯it

=

¯ t )) yγqit (hti (U i −σ ˜ hti (U¯it ) r+δ ¯ t )) ¯ t ))) , γqit (hti (U (1−γ)(1−qit (hti (U i i + r+δ r+δ+λ

and notice that U¯it exists and it is unique. If U t ≥ U¯it then Sˆit (h) ≤ 0 for all h. If U t < U¯it t then Sˆit (h) > 0 for h ∈ (hti (U t ), hi (U t )) and Sˆit (h) = 0 otherwise. The assumptions on qM (h) t t t and qR (h) imply that U¯M < U¯Rt , htR (U t ) < htM (U t ) and, when applicable, hM (U t ) < hR (U t )

and htM (U t ) ≥ htR (U t ).

38

Deﬁne

t Ωt (U t , h) = [SˆM (h)]2 + [SˆRt (h)]2 .

(37)

t If U t ≥ U¯Rt then Ωt (U t , h) = 0 and ht (U t ) = 0. When U t ∈ [U¯M , U¯Rt ) then Ωt (U t , h) = t ¯ t (U t )) [SˆRt (h)]2 and ht (U t ) = htR (U t ). If U t < U¯M , Ωt (h) is strictly positive on (htR (U t ), h R

¯ t (U t )) and and it is strictly increasing on (htR (U t ), htR (U t )), strictly decreasing on (htM (U t ), h M Ωt ′(htR (U t )) > 0 > Ωt ′(htM (U t )). In this case, the optimal ht (U t ) is characterized by the root of ( t ) t Ωt ′(U t , h) = 2 SˆM (h)SˆM ′(h) + SˆRt (h)SˆRt ′(h) ,

(38)

which is maximized on (htR (U t ), htM (U t )). Further, notice that if htM (U t ) ≤ htR (U t ), then t t t SˆM (ht (U t )) > 0. This occurs if U < UˆM for some UˆM which will be the relevant condition

for matches to occur through both channels. 2. It is now shown that equation (38) is decreasing in σ ˜ when it is equal to zero. This suﬃces to prove that ht (U t ) is decreasing in σ ˜ for the stable solutions. ( ) ∂Ωt ′(U t , h) t t = 2 − hSˆM ′(h) − SˆM (h) − hSˆRt ′(h) − SˆRt (h) ∂σ ˜ ( Sˆt ) t t ) < 0. = 2 − SˆM − SˆRt − hSˆM ′(1 − M Sˆt R

where the second equality results from using the root of equation (38). 3. The expected surplus of a meeting through channel i is given by: [ y t γq (ht (U t )) ] t t ( (1 − γ)(1 − qi (ht (U t ))) ) i t t t γqi (h (U )) t t ˆ Si (U ) = max − rU + − σh (U ), 0 . r+δ r+δ r+δ+λ

39

Lemma 3.3: If n ≥ n and σ ≤ σ ˆ then there exist nH∗ > 0 and nL∗ > 0 where nH∗ + nL∗ = n and U H (nH∗ ) = U L (nL∗ ) = U ∗ . Proof. It is clearly suboptimal for all workers to go to the low ﬁrm-type island. To rule out that every worker goes to the high-type island ﬁrst recall that the vacancy cost at an island is an increasing function of the number of vacancies in that island. Therefore, if n is high enough (n ≥ n, for some n) then the resulting measure of ﬁrms (and labor market tightness) is low enough so that U H (n) < limnL →0 U L (nL ) and therefore nL∗ > 0. The surplus of a match through channel i in island t after a bad signal is emitted is:

Sbit =

y t γptbi rU ∗ r + δ + λptbi − , r+δ r+δ r+δ+λ

ptbi =

γ(1 − qi (ht )) . γ(1 − qi (ht )) + (1 − γ)qi (ht )

where

If qi (ht ) is high enough then Sbit ≤ 0 for all i and t. Additionally, if qM (ht ) is high enough, t then SˆM > 0 and a hire occurs through both channels. Both conditions are satisﬁed if σ is

low enough: σ ≤ σ ˆ for some σ ˆ.

Proposition 3.2: In equilibrium, high productivity ﬁrms choose a higher level of human resources: hH > hL . Proof. The choice of h is characterized by the root of the following equation: 1 Ω′ = SˆM SˆM ′ + SˆR SˆM ′, 2

(39)

where the type superscripts and arguments are omitted for notational convenience. It will be shown that equation (39) is increasing in y at its root. This shows that the optimal choice of h is increasing in y. 40

The derivative with respect to y is given by ′ 1 ∂Ω′ ∂ SˆM ˆ′ ∂ SˆM ∂ SˆR ˆ′ ∂ SˆR′ ˆ ˆ = S + SM + S + SR 2 ∂y ∂y M ∂y ∂y R ∂y ( ) γ ′ ′ ˆ ′ ′ ˆ ˆ ˆ = qM S M + qM S M + qR S R + qR S R . r+δ

At the root of equation (39): ′ SˆM SˆM , SˆR′ = − SˆR ′ > 0 > S¯R′ because qR > qM . Therefore: where SˆM

Ω′ = 0 ⇒

) ′ 1 ∂Ω′ γ ( SˆM ′ ˆ = (qM SˆR − qR SˆM ) + qM SM + qR′ SˆR . 2 ∂y r + δ SˆR

Finally: ( yqR ) ( yqM qR γ (1 − qR )(1 − γ) qM γ − rU ( + ) − h − qR − rU ( r+δ r+δ r+δ+λ r+δ r+δ ) (1 − qM )(1 − γ) + )−h r+δ+λ ) rU (1 − γ) ( = qR (1 − qM ) − qM (1 − qR ) + h(qR − qM ) > 0, r+δ+λ

qM SˆR − qR SˆM = qM

which completes the proof.

Proposition 3.4: In equilibrium, if γ ≥ 12 : 1. Low productivity ﬁrms exhibit greater prevalence of referrals than high productivity ﬁrms: P L > P H . 2. The wage premium of a referred worker is lower if the ﬁrm type is not controlled for than if it is.

41

3. If y L < ϵy H for some ϵ ∈ (0, 1) then the average wage of a worker hired through a referral is lower than the average wage of a worker hired through the market. Proof. 1. Note that

PL > PH ⇔

L L H H SˆM πM SˆM πM < . SˆRL πRL SˆRH πRH

The following will be proven: d ( SˆM πM ) > 0, dy SˆR πR where the type superscripts and arguments are omitted to simplify notation. First, note that: d(SˆM /SˆR ) ∂(SˆM /SˆR ) ∂(SˆM /SˆR ) dh = + . dy ∂y ∂h dy As shown in Proposition 3.2: ) ∂(SˆM /SˆR ) 1 γ ( ˆ = qM SR − qR SˆR > 0. ∂y (SˆR )2 r + δ Proposition 3.2 also proved that

dh dy

> 0. Furthermore:

) ( yγ ∂(SˆM /SˆR ) 1 [( ′ γ 1−γ ) = qM (h) − rU ( − ) − 1 SˆR (h) ∂h r+δ r+δ r+δ+λ (SˆR )2 ( ) ] ( yγ γ 1−γ ) − qR′ (h) − rU ( − ) − 1 SˆM (h) , r+δ r+δ r+δ+λ which is strictly positive because

SˆR > SˆM , ′ (h) > qR′ (h). qM

42

Finally: d(πM /πR ) ∂(πM /πR ) dh = dy ∂h dy ) dh 1 ( = πM ′πR − πR ′πM 2 dy (πR ) ) dh 2γ − 1 ( ′ ′ = (qM qR − qR′ qM )(2γ − 1) + (qM − qR′ )(1 − γ) > 0. t 2 dy (πR ) 2,3. When the ﬁrm type is not controlled for, the wage diﬀerential conﬂates the diﬀerence in ﬁrm productivity levels with the wage premium of referred workers, thereby reducing the wage premium of a referral. This is not suﬃcient to generate a negative correlation between the wage and a referral and requires that the wage of worker in high productivity ﬁrms be suﬃciently larger than that at low productivity ﬁrms. This is satisﬁed if the productivity diﬀerential is high enough.

References [1] Bayer, Patrick, Stephen L. Ross and Giorgio Topa (2008). “Place of work and place of residence: informal hiring networks and labor market outcomes.” Journal of Political Economy 116 (6), 1150-1196. [2] Bentolila, Samuel, Claudio Michelacci and Javier Suarez (2010). “Social contacts and occupational choice.” Economica 77, 20-45. [3] Bewley, Truman (1999). Why wages don’t fall during a recession. Cambridge, MA. Harvard University Press. [4] Brown, Meta, Elizabeth Setren and Giorgio Topa (2011). “Do Referrals Lead to Better Matches? Evidence from a Firms Employee Referral System.” Mimeo.

43

[5] Castilla, Emilio J. (2005). “Social networks and employee performance in a call center.” American Journal of Sociology 110 (5), 1243-1283. [6] Datcher, Linda (1983). “The impact of informal networks on quit behavior.” Review of Economics and Statistics 65 (3), 491-495. [7] Dustmann, Christian, Albrecht Glitz and Uta Schoenberg (2011). “Referral-based job search networks.” Mimeo. [8] Fernandez, Roberto M., Emilio J. Castilla and Paul Moore (2000). “Social capital at work: networks and employment at a phone center.” American Journal of Sociology 105 (5), 1288-1356. [9] Fernandez, Roberto M. and Nancy Weineberg (1997). “Sifting and sorting: personal contacts and hiring in a retail bank.” American Sociological Review 62, 883-902. [10] Galenianos, Manolis (2011). “Hiring Through referrals.” Mimeo. [11] Granovetter, Mark (1995). Getting a job: a study of contacts and careers. 2nd edition Cambridge, MA. Harvard University Press. [12] Holzer, Harry (1987). “Hiring procedures in the ﬁrm: their economic determinants and outcomes.” NBER WP 2185. [13] Ioannides, Yannis and Linda Datcher Loury (2006). “Job information networks, neighborhood eﬀects, and inequality.” Journal of Economic Literature XLII, 1056-1093. [14] Jovanovic, Boyan (1979). “Job matching and the theory of turnover.” Journal of Political Economy 87 (5), 972-990. [15] Moscarini, Giuseppe (2005). “Job Matching and the Wage Distribution.” Econometrica, 73(2), 481-516.

44

[16] Nagypal, Eva (2007). “Learning by doing vs. learning about match quality: can we tell them apart?” Review of Economic Studies 74, 537-566. [17] Pellizzari, Michele (2010). “Do friends and relatives really help in getting a good job?” Industrial and Labor Relations Review 63 (3), 494-510. [18] Pinkston, Joshua (2011). “How Much Do Employers Learn from Referrals?” Industrial Relations, forthcoming. [19] Pistaferri, Luigi (1999). “Informal networks in the Italian labor market.” Giornale degli Economisti 58 (3-4), 355-375. [20] Pries, Michael (2004). “Persistence of Employment Fluctuations: A Model of Recurring Job Loss.” Review of Economic Studies, 71(1), January 2004, 193-215. [21] Simon, Curtis and John Warner (1992). “Matchmaker, matchmaker: The eﬀect of old boy networks on job match quality, earning and tenure.” Journal of Labor Economics, 10 (3), 306-330. [22] Topa, Giorgio (2010). “Labor Markets and Referrals.” The Handbook of Social Economics, forthcoming.

45

Employment Duration and Match Quality over the Business Cycle