Social Networks and Unraveling in Labor Markets

Transcription

1 Social Networks an Unraveling in Labor Markets Itay P. Fainmesser y May 20, 2008 Abstract This paper evelops a moel of local unraveling (or early hiring) in entry-level labor markets. Information about workers prouctivity is reveale over time an transmitte creibly via a two-sie network connecting rms an workers. While employment starts only after workers nish their formal training, rms an workers can sign an enforceable future employment contract at any time. We n that increasing e ciency in the postgrauation market may increase unraveling. A thorough analysis shows that unraveling increases in the span of network, in network concentration aroun a subset of workers, an in early information accuracy. Unraveling ecreases in network concentration aroun a subset of rms. Network ensity increases unraveling when the network is sparse, but ecreases unraveling in ense networks. The moel also preicts more unraveling by workers that are connecte to rms of i erent qualities an by rms that are connecte to less workers. Finally, we analyze the e ects of unraveling on market outcomes an welfare an evaluate i erent policies with respect to their e ect on unraveling. moel provies preictions are consistent with evience from various markets, an suggest future theoretical an empirical work. (JEL: A14, D40, D85, C78, J44, L14) Keywors: Market esign, networks, matching, congestion, entry-level labor markets. Preliminary an incomplete. I am grateful to Susan Athey, Markus Mobius an Al Roth for their avice, guiance, an support. I am also extremely fortunate to have receive etaile comments from Drew Fuenberg. I woul like to thank Philippe Aghion, Attila Ambrus, Eric Buish, James Burns, Ulrich Doraszelski, Fuhito Kojima, Robin Lee, Senhil Mullainathan, Muriel Nieerle, Laura Serban, Aam Szeil, Alexaner Westkamp, an seminar participants at Harvar University for helpful comments an iscussion uring various stages of this project. y Harvar University, Department of Economics, an Harvar Business School. fainmess@fas.harvar.eu; Our 1

2 1 Introuction The timing of transactions is an important part of a market s behavior. In various contexts, timing (an early contracting in particular) has been shown to a ect the market outcomes with respect to both istribution an welfare. 1 The impact of early contracting is expecte to be crucial in markets that involve large stakes an uncertainty that is resolve over time, as in many entry-level professional job markets such as the unergrauate college amissions, the market for internships in the many resiency programs for young meical octors, an the market for juicial clerks (see Avery et.al. 2001, 2007, Nieerle an Roth 2005, an Roth 1984, 1990, 1996, an 2003). In these markets the ecision involves a choice of resience, income, an specialty that will accompany the workers years into the future, not to mention the impact these professionals have on their surrounings. It is also the case that the quality of the workers in these markets is reveale graually, where the most complete information is available only towars their grauation an prior to employment. Nevertheless, many of these markets have su ere from extremely early hiring of workers, long before stuents grauate an were reay to work (i.e. in the meical internships market, resients were hire as early as the fall of their junior year). The phenomena of early contracting in the presence of uncertainty that will be resolve in the future is commonly calle unraveling. Traitionally, unraveling is thought of as a ynamic phenomena, were early contracting by some agents may lea other agents to contract early as well, an sometimes to contract even earlier. This ynamic escalation was exhibite in many matching markets. 2 In this paper, we aim to she light on the inherently local nature of unraveling - rms hire workers who they know. This creates an averse selection problem for rms that i not plan on using their connections an increases rms tenency to hire through connections. We evaluate the impact that the network of connections has on the market: what network structure will better facilitate / prevent unraveling; who are the winners an the losers when unraveling occurs an what are the aggregate welfare implications when markets unravel. The analysis also suggests policy intervention. In aition to introucing the network structure, we n the timing of unraveling to be an 1 Nieerle an Roth (2003 an 2005) n that unraveling (early contracting) reuces mobility in the Gastroenterology fellowships market. Fréchette, Roth, an Ünver (2007), show that unraveling in the market for NCAA bowl games reuces the number of viewer. 2 See also Roth an Xing

3 important factor. Unlike previous papers in which unraveling happens in one perio, we allow unraveling to happen over time as more information is being reveale. In our moel, two types of unraveling can coexist: At an early stage in their training, workers face a large uncertainty regaring their future prouctivity an employment prospects. Consequently, all workers (even the best ones) will accept any early job o er from (even less preferre) rms, but some of the rms will prefer to wait with their o er to learn more about the workers. This results in worker riven unraveling as it is riven by workers eman for insurance. Later on, workers have more information about their esirability in the post-grauation marketplace an goo workers will refuse some less preferre o ers. At this time rms are seeking for insurance against not lling their positions an all rms make early o ers to workers of high enough potential. This late rm riven unraveling is riven by rms eman for insurance. 3 The ivision into two types of unraveling conitional on their timing allows us to focus on a three perio moel that grasps the ynamic nature of unraveling, resulting in more realistic results an new policy implications. The results of both the two perio moel an of the more general three perio moel are summarize in table 1. Notably, the impact or increasing connectivity or concentration in the network epens on the etails of this increase. An increase in the number of connecte workers an rms (network span) increases unraveling, while increasing the connectivity by aing more links between alreay connecte agents (network ensity) has a non monotonic e ect on unraveling an epens on the initial ensity. Similarly. increasing the concentration of connections aroun workers an aroun rms a ects unraveling in qualitatively i erent ways. There is a common line of reasoning connecting our results. On one han, an increase in unraveling opportunities reuces the attractiveness of the post-grauation perio an increases the incentives for unraveling. On the other han, increasing late unraveling opportunities increases the option value of a connection at earlier perios an reuces rms incentives for early unraveling. Our analysis consists of a careful evaluation of these countervailing e ects. Our results are a rst step in unerstaning the relationship between network structure an the incentives for unraveling an suggest a possible explanation to the interesting unraveling patterns observe in labor markets. In particular, the market for Gastroenterology fellowship seems to have a sparse network of connections as each epartment is connecte to a small 3 The terms worker an rm riven unraveling have been suggeste by Li an Suen (2000). They also suggest that unraveling shoul be investigate in a ynamic framework an provie a ynamic example. 3

4 number of internal meicine epartments, yet the span of the network is quite large (many epartments have at least one connection). In the market for juicial clerks the network looks quite i erent; it appears that there is high concentration aroun a small number of law schools that are the focus of unraveling an each juge seems to be connecte only to one or two schools. 4 Nevertheless, both of these markets exhibite extensive unraveling at times. In aition to exploring the e ects of the various changes in network structure, the paper allows us to evaluate claims often use by market participants, such as (1) in the presence of extensive unraveling, some highly connecte rms wait an o not hire until more information is reveale; an (2) competition between rms of i erent quality intensify unraveling. 5 claims are supporte by our moel. Both Finally, the introuction of ynamics an local competition consierations moi es some of the results that were iscusse in the literature in a way that has signi cant policy implications. Most notably, improving the e ciency in the post-grauation market has qualitatively i erent e ect on unraveling epening on its timing an early, worker riven, unraveling increases in e ciency. Intuitively, there are con icting e ects from improving e ciency. On one han, the post-grauation market becomes more appealing, so when close enough to the market, rms might not be able to hire on a local basis as high quality workers refuse many job o ers. On the other han, reucing this ability to contract late through the network creates an incentive for the rms to use the network early when the value of its connections is still high. For the earlier part of the unraveling process it turns out that the secon inirect e ect overcomes the irect one, inicating that an increase in the e ciency of a global market results in more worker riven unraveling. While our measure of the e ciency is relate to e ciency as traitionally e ne in the market esign literature, it has a broaer scope an captures exogenous changes in IT an search costs. To that extent, this paper takes a rst step towars an investigation of the impact 4 The e nition of a connection between juges an law schools can take many forms; a juge can be connecte to a school she grauate from, or might have been a classmate of one of the school s professors. For our nees, it is su cient to note that it seems that juges are quite consistent with their hiring from a subset of schools. 5 In an ongoing ebate aroun the esign of the market for juicial clerks, the California circuit has often claime that the New York circuit can a or waiting for later perios before hiring as it is well connecte to more high quality law schools. The New York circuit on the other han hols the opinion that the reason that the California circuit hires earlier on average is ue to i erence in the eman for the positions, an that this creates pressure for unraveling on more prestigious circuits. 4

5 of IT an communication evices such as the phone, , an the internet on labor markets outcomes. In the following section, we present a general framework to analyze unraveling. In section 3, we analyze a two perio moel that shes light on the role of the social network as e ning information frictions in the market. We characterize the equilibria an analyze the e ect of changing various parameters of the network structure, information accuracy, an post-grauation market e ciency. In section 4, we exten the moel to inclue more than one perio of unraveling an investigate the e ects of intertemporal consierations an local competition. The results from sections 3 an 4 are summarize in table 1. Section 5 suggests policy implications arising from the analysis. In particular, we analyze i erent policies banning exploing o ers in labor market, an plocies regaring the stanarization an improvements in national examination systems. In Section 6 we iscuss the relation between unraveling an market outcome. We show that unraveling reuces mobility, an early (worker riven) unraveling reuces mobility more than late ( rm riven) contracting. We also n that connecte workers gain from unraveling while the e ect on connecte rms is ambiguous (unconnecte rms strictly lose from an increase in unraveling). While aggregate welfare is har to analyze in matching markets, we are able to measure the welfare loss from unraveling in some markets. Finally, we erive the cost of lack of coorination in labor markets an suggest a reason why some markets are able to coorinate on preventing unraveling better than others. In section 7 we exten the moel an investigate the role of local competition between rms of i erent qualities in hiring in pre-grauation perios. We n that in early perios, competition over workers between rms of i erent qualities exacerbates unraveling in the market. 5

6 In sections 8 we o er some concluing remarks. Table 1: Results summary an cross moel comparison 2 The moel There is a continuum of rms an a continuum of workers in the economy, both are of measure one. Workers have T training perios before they can be employe by rms. In the training perios, workers spen time in training institutions (i.e. law school, meical school, internship programs, etc.). Workers nee to formally nish their training before they can be employe. Each worker can work for at most one rm an each rm can employ at most one worker. Prouction takes place at the post-grauation perio an is a function of workers prouctivity; worker i s prouctivity is q i 2 fh; Lg an she receives a wage w i. For the rest of the paper, we suppress wages an make them a part of the job escription (w i = 0). Hence, wages are inepenent of the time of contracting. This assumption is consistent with the markets motivating this paper. 6;7 Moreover, this assumption can be substitute (with some technical 6 Juicial clerks wages are etermine by feeral law. In another market, meical resients wages seems to be constructe in a very iscrete way an limite to a small number of wage steps that can each be analyze within our framework. 7 The iscussion of the role of wage in the analysis of unraveling is not new. In particular, moels of matching markets can be analyze using the assignment moel (Koopmans an Beckmann 1957, Shapley an Shubik 1971) where wage is a part of the clearing mechanism, or using the marriage moel (Gale an Shapley 1962) where 6

7 buren) for some mil restrictions on preferences an/or wages. 8 Firm j s pro t from employing worker i is: 9 8 < j (q i ) = : 1 if q i = H 1 if q i = L Firms are a priori ientical (In section 7 we exten our moel an iscuss rms heterogeneity an systematic workers preferences over rms). Nevertheless, workers have iiosyncratic preferences about rms. Speci cally, let worker i s utility function be: 8 < 1 + " ij if hire by rm j u i = : 0 if not hire Where " ij s are i.i.. an rawn from a istribution G that has mean 0 an positive ensity in any point in the support [ "; "]. Let perio 0 be the post-grauation perio, when hiring ecisions must be nalize an employment starts. 10 An let perios T; T +1; :::; 1 be the training perios. In the training perios (t < 0), information about own prouctivity is graually reveale to the workers (exam graes, reinforcements from teachers, etc.) in the following way: At a given perio t < 0 worker i observes a noisy signal s i (t) 2 fh (t) ; l (t)g that summarizes all the information available about the worker up to perio t (ex-ante with equal probabilities, but conitional on the signals wages are assume out. It is interesting to note that even papers that use the assignment moel such as Li an Suen (2000), that analyze unraveling in the context of college amissions, amit that the marriage moel of Gale an Shapley (1962) might seem more appropriate than the assignment moel to stuy college amission an that "our [assignment moel] analysis applies with a greater force to assignment markets in which payments transfers are explicitly negotiate. 8 One simple way of incorporating wages into the moel is by making workers preferences be base solely on wages, an assign the rms iiosyncratic preferences in a form similar to the workers preferences we have in the present moel. Uner a minimum wage assumption (speci cally, let w 1 (1) (2) "), wage competition will lea to ynamics that are very similar to the ones we get using our framework. 9 Firms preferences over workers that are use throughout the paper exhibit perfect correlation. As Halabura (2007) shows some correlation across rms preferences is necessary for unraveling 8to occur. To introuce < 1 + " ij if q i = H some heterogeneity in rms preferences, we have explore a moel in which j (q i ) =. : 1 + " ij if q i = L All of the results from our paper seem to hol for every < for some > 0. As we are intereste in the case of similar preferences that are consiere to better facilitate unraveling, we omit the more complicate analysis. 10 During the training perios both rms pro ts an workers utilities are normalize to zero, so the only payo s in the moel are receive in perio 0. 7

8 from previous perios the istribution might change). A s i (t) = h (t) worker has probability (t) ; 1 of being q i = H an probability 1 (t) of becoming q i = L. A s i (t) = l (t) worker has the reverse probabilities. (t) is assume to be an increasing function of t an at grauation q i is reveale accurately to the worker so (0) = 1. After grauation, information about workers prouctivity can be reveale creibly to all rms. However, before grauation, the training institution oes not prouce o cial transcripts or other reports. Therefore, a worker can creibly convey his early information only through personal connections, i.e. a worker s teacher might know a HR person in one (or more) of the rms. Formally, let f 2 [0; 1] be the proportion of rms that are connecte to at least one worker (so (1 f) rms are not connecte to any worker). Let each connecte rm be connecte to r f workers an each connecte worker be connecte to r w rms. Therefore, the proportion of connecte workers is w = (f r f ) =r w ; we call f + w the span of the network an r the rank of a connecte agent of type 2 fw; fg. 11 Early contracts an the post-grauation marketplace. During the training perios, any worker an rm (irrespective of being connecte or not) can sign employment contracts. A contract speci es that worker i will be employe by rm j in the post-grauation perio, inepenent of any future realization. This limite contracting can result from incomplete contracts an from lack of veri ability of workers quality. More important, it seems to be the format of contracts in the markets motivating this paper. Motivate by recent iscussions on the the esign of a new match for the Gastroenterology fellowship market, further iscussion of the nature of the early contracts an their implications for unraveling can be foun in section 5. Workers an rms that i not sign an early contract reach the post-grauation perio (perio 0) unmatche. Without restricting the analysis to either a centralize or ecentralize market we focus on the market s outcome rather then on the speci c mechanism. In particular, information technologies (IT), search costs, an market culture a ect the ability of rms an workers to n their most preferre partner. In a perfect market, workers are likely to n their most preferre rm, an if the worker is of high prouctivity, she will be hire by that rm. This guarantees a high quality worker with an expecte utility arbitrarily close to 1 + " in the 11 For tractability, we assume that all connecte rms (workers) have the same rank r f (r w ). This is a genralization of a regular network to a two sie network framework. 8

9 post grauation perio. In a virtually ranom market, a high quality worker will be matche with an arbitrary rm an will have an expecte utility of 1. More generally, let 2 [0; 1] be the e ciency in the post-grauation marketplace an the expecte utility of a high quality worker in the post-grauation perio be 1 + ". The following examples provie two i erent possible stories that are consistent with our parametrization of the market s outcome an highlight its intuitive meaning. Example 1 (Ranom assignment orer with learning of preferences) Consier workers who o not know their rm speci c preferences until they communicate with the rm (interview / y-out etc.). Workers arrive in the market sequentially, each worker knows which rms have openings, but oes not know her preferences over these rms. Firms o not know the worker s prouctivity. The worker can communicate with up to n rms. During this communication the rms n out the worker s prouctivity, an the worker ns out her rm speci c preferences with respect to those rms (preferences might inclue wage, location, etc.). Following this communication process, rms can make a job o er to the worker an the worker can accept at most one job before the next worker arrives to the market. Example 2 (Decentralize marketplace with ealines) Consier a market in which the high quality workers are assigne sequentially. Worker i arrives to the market an becomes observable to all rms, the worker oes not know which rms have hire in previous perios or earlier in that perio an cannot take a job without examining an explicit job o er from a rm. All unmatche rms try to o er a job to worker i simultaneously. The technology is such that only the n rst o ers get through before the worker has to make a ecision. The worker then accepts her most preferre job among the ones o ere. 12 In both of these examples consier = (n) to be an increasing function of the quality of the information technology in the market as capture by the number of job o ers a worker can consier In the market for Clinical Psychologists o ers were mae by phone an the market was open for a limite perio of time so hospitals often ha their o er hel by a worker that was still waiting for another o er until close to the market s closing time. This mae the market stagnant an a lot of o ers an acceptances were mae aroun the market s closing time an on an exploing o er basis. As a result, each worker receive a small number of o ers before having to accept one of them. For more etails see Roth an Xing (1997). 13 In both examples, let w 0 high quality workers an f 0 rms reach the post-grauation marketplace unmatche. 9

10 Our focus i ers from that of a large part of the market esign literature; the latter focuses on the e ciency generate by the mechanism, given the information available in the market an (maybe) the reporte preferences, an traitionally full information is assume. Several recent papers analyze labor markets at the pre-interview stage, when rms choose which of a pool of caniates to interview, an possibly make an o er to. 14 Consistent with the focus of this new literature we interpret in the following way. Small implies that it is har to n your preferre partner - in example 1, small (or small n) implies that it is har for a rm (worker) to learn about the qualities of the workers ( rms); in example 2, the same implies that it is har for a worker to n out who is intereste in hiring her. This coul result from high search costs (like high interviewing costs), limite search time, etc. In fact, many well esigne centralize markets are likely to have << 1 ue to inevitable information friction. 2.1 Unraveling Unraveling has static an ynamic interpretations. In a static framework, unraveling is simply early contracting, i.e. contracting before some relevant information is available. In a ynamic context, unraveling is e ne as escalation towars earlier contracting erive by the early contracting of others. As the signal of workers prouctivity is noisy in early perios we refer to any employment contract signe prior to grauation as unraveling. We focus in our analysis on sequential equilibria in unominate strategies. An immeiate result is that in the post-grauation perio, the market is global, high prouctivity workers are hire an low prouctivity ones are not. Also, in earlier perios, the local nature of information i usion leas to a local market, namely, in equilibrium, we get that before grauation hiring is one only through connections. 15 In any early perio t < 0, let j (t) 2 [0; 1] be the probability that rm j makes an o er in perio t conitional on being connecte to a worker with a positive signal (s i (t) = h (t)). Throughout most of the analysis we focus on symmetric equilibria so j (t) = (t) for all Given our i.i.. assumption on " ij, as long as f 0 > w 0 + n, workers expecte utility is inepenent of f 0 an w 0, rises in n an is boune between 1 an 1 + ". It will become clear later that f 0 > w 0 + n for every nite n in our moel. 14 See Lee an Schwarz (2007), an Coles an Nieerle (2007). 15 In early perios information is transferre only through connections. Moreover, the expecte pro t from employing a ranom worker is at most 0. 10

11 j. Clearly, no rm contracts with a s i (t) = l (t) worker. Similarly, let (t) 2 [0; 1] be the probability that a worker with a positive signal refuses a job o er from a ranomly rawn rm. Since the " ij s are ex-ante unknown, this is the ex-ante probability that a worker refuses a job o er from any single rm. It is easy to see that (t) can be characterize by a threshol strategy by the workers. In perio t, worker i will accept an o er from rm j if an only if " ij " (t). We efer the characterization of " (t) an (t) to section 3. Let F (t; ; ) be the excess expecte pro t for a rm in perio t from not o ering a job to a connecte h (t) worker. Namely, the i erence between the expecte pro t from waiting an the expecte pro t from making an o er to the worker. We will characterize F (t; ; ) an show a straightforwar way to erive for a range of interesting cases in the following section. For these cases, a symmetric sequential equilibrium can be characterize in the following way: Corollary 1 For every sequential equilibrium strategies (t) = (t) an (t) = (t), an for every perio t, one of the following hols: 1. F (t; ; ) = 0 an (t) 2 [0; 1] 2. F (t; ; ) > 0 an (t) = 0 3. F (t; ; ) < 0 an (t) = 1 In many cases, we will be intereste in changes in unraveling. As economists oriente towars market esign an policy it will be suitable to focus on changes in the incentives of both rms an workers to contract early. De nition 1 A change in market funamentals leas to an increase in unraveling in perio t if all of the following hol weakly, an at least one hols in a strict sense: 1. The change leas to an increase (ecrease) in the equilibrium probability that a rm makes an o er to a worker with a high prouctivity signal ( (t)) in every stable (unstable) equilibrium h ; i. 2. The change leas to a ecrease in the probability that a worker refuses a job o er ( (t)) in all equilibria. Where stable (unstable) equilibria are e ne base on the stanar notions of a xe point F () stability an rise in our moel when > 0 F () < 0. 11

12 Our notion of an increase in unraveling is quite general, in particular, both the e nition an the results to follow are consistent with a large variety of equilibrium selection rules (e.g. basin of attraction, selection of the most e cient equilibrium, etc.). We also suggest in the appenix a simple motivating ynamic process to emonstrate the intuitive nature of our e nition Worker an rm riven unraveling The relation between early information accuracy () an t is worth exploring a little as in can signi cantly in uence the ynamic patterns of unraveling. Intuitively, when t is large, so is (t). We argue (an show formally later on) that the incentives of rms to hire in perio t increase in (t) (i.e. that for every > s i = h workers make o 0) an therefore increase in t. As a result, there exists t such t > t, F () < 0 inepenent of an all rms that are connecte to Also, the probability that a worker refuses a job o er in perio t ( (t)) is increasing in 0 an therefore in t; a worker that is less concerne about being low prouctivity (an therefore unesirable later on) will be more selective with her early job o ers. Hence, There exist (t) such that for every < (t < t) workers o not refuse early o ers from rms. Furthermore, as long as " is not too large, so that workers strictly prefer getting a job to taking a large risk of being unemploye for getting a more preferre job, t t. As in the introuction we say that earlier perios exhibit worker riven unraveling, as workers are always eager to contract while rms might contract or not. In the later (but still pre-grauation) perios all rms want to hire high potential workers, but high potential workers refuse less preferre o ers. We refer to the unraveling in these perios as rm riven unraveling as = 1 while 2 [0; 1]. The istinction will become useful as some phenomena are inherently i erent between the i erent types of unraveling. 3 Social networks an information frictions In this section, we start with a simple two perio moel (T = 1), in which rms an workers can sign an early contracts in perio t = 1 or wait for the post-grauation market. We characterize the unraveling in equilibrium an erive how unraveling is a ecte by changes in the network structure, the e ciency of the post-grauation marketplace an the accuracy of the early information. This moel is instructive as it shows the role of the network structure 12

13 Figure 1: The timeline of the two perio moel t= 1 t=0 Each worker receives a noisy signal s=h,l Information is transmitte along the network an hiring ecisions are mae Post grauation marketplace. Unmatche firms an high prouctivity workers are being matche Prouction an profit an utility realizations in creating information frictions of a realistic nature. 16 The two-perio framework is also consistent with many previous moels of unraveling (Li an Rosen 1998, Li an Suen 2000, 17 Suen 2000, Halabura 2007) an highlights the value ae by explicitly moeling the network. In section 4, we examine a more general moel an the comparison of the results in the two moels will allow us to iscriminate between the role of the connections as e ning (static) information frictions an more ynamic roles. See table 1 for a comparison between the two moels with respect to results. The timeline of the moel is escribe in gure 1. For notational simplicity we suppress the inex 1 when it is clear from the context. Denote E M [] = E M [j; f; w; r f ; r w ; ; ; ] as the expecte pro t for a rm that reaches the post-grauation marketplace unmatche; changes in E M [] capture the externalities of early contracting. Let = ( 1), so rm j s expecte pro t when connecte to a s i = h worker is (1 ) = 2 1 if the rm an the worker contract in the rst perio. Let F () = E M [] 2+1: Furthermore, as worker i accepts a job o er from rm j in perio 1 if an only if 1 + " ij > (1 + "), so = (; ; G) is inepenent of. 18 Therefore, = ( 1) 2 [0; 1] for which F () = 0 are interior equilibria in which rms are ini erent between hiring a s i = h worker in the perio 1, an waiting for the post-grauation perio. Possible corner solutions involve either full unraveling ( = 1 an F () < 0), or no unraveling ( = 0 an F () > 0). 16 As there is only one perio of unraveling, the moel will allow for either worker or rm riven unraveling corresponing to low or high ( 1) respectively, but not both. Therefore, we elay any iscussion on the i erent types of unraveling to the en of this section. 17 Li an Suen (2000) suggest an extension of a multi-perio moel of unraveling. 18 In particular, is continuous an is increasing in an in. 13

14 To get an explicit expression for E M [], note that the probability that a s i = h worker i receives a job o er from rm j can be expresse as = X m=0;:::;r f 1 rf 1 0:5 m 0:5 r f m 1 1 m m + 1 where is the probability that rm j makes an o er in perio 1, (3) r f 1 m 0:5 m 0:5 r f m 1 is the probability that there are m other s i = h workers that the rm is connecte to, an 1 m+1 is the conitional probability that the rm makes the o er to worker i. can be reaily simpli e to = (1 0:5 r f ) = (0:5 r f ) (4) Let P w be the probability that worker i who is of quality s i = h gets at least one job o er, an is the fraction of workers hire in perio 1. So, where = f (r f =r w ) 1 2 P w (5) as: P w = 1 [(1 ) + ] rw (6) Using the expecte pro t of a rm in the post-grauation marketplace can be expresse E M [] = 0:5 (7) 1 By continuity of E M [] an F () an convexity of the support for an equilibrium always exists. 19 Furthermore, Proposition 1 There is at most one interior equilibrium that is unstable. Otherwise, if there is no interior equilibrium, there is a unique stable equilibrium where either all rms that are connecte to a s i = h worker make early o ers ( = 1) or no rm makes an early job o er ( = 0). 19 If F () > 0 (F () < 0) for every, the unique equilibrium is no (full) unraveling. If there exist for which F () > 0 an for which F () < 0, we are gurantee an interior solution by Weierstrass xe point theorem. 14

15 Proof. The proof is immeiate as it is easy to verify that F () < 0 for every. In light of proposition 1, our measure of unraveling can be interprete as the tenency of a market to get to the full unraveling equilibrium. The claim also call attention to the externalities involve in the unraveling process, as the incentives of a rm to unravel epen on the number of other rms that hire early. The market for Gastroenterology fellowships seems to exhibit a pattern that is consistent with proposition 1. Before the establishment an after the collapse of the centralize match, unraveling seeme to be an extensive phenomena. In particular, in 1999, just before the match was formally abolishe, only 14 out of more than 300 positions were lle in the post-grauation perio. Nevertheless, uring the most years that the match was in place, nearly all positions were lle through the match in the post-grauation perio. The importance of coorination in the Gastroenterology fellowships market can be seen from the following sent by Debbie Proctor, the gastroenterologist who took the lea in reorganizing the match, to the economists assisting in reesigning the Gastroenterology fellowships market: I m answering 3-4 s per ay especially on this issue. I want to make sure MY competition is in the match an that they on t cheat. Well, this is another way of saying that if they cheat, then I will too!...have you ever seen this before? The istrust amongst program irectors? I n it har to believe that we are unique. Maybe this is [a] social science phenomenon? 20 It is also interesting to note that there are some markets that are not well characterize by proposition 1. One example is the market for juicial clerks; even in years of extensive unraveling, there has been a substantial number of juges that i not attempt to hire early (see Avery et. al. 2001, 2007). We will suggest in section 4 a resolution of this puzzle in the form of a more general moel that allows for unraveling to happen in several points in time an characterize markets in which some of the rms will prefer to wait until close to the post-grauation perio before hiring. We now turn to evaluate the e ect of changes in the market funamentals on unraveling. 20 The quote is taken from Professor Al Roth s lecture notes for a grauate course on market esign. 15

16 3.1 Network structure an unraveling In this section we show how changes in the network structure a ect unraveling in the twoperio moel. In particular, we are intereste in network ensity, the span of the network, an network concentration aroun a subset of workers an / or rms. As network structure provies an explicit form of information frictions, the i erent e ects show that i erent ways of aing information to the market in early perios a ect in i erent ways. Intuitively, there are two main questions that can be aske about the network, who is connecte, an how they are connecte. Table 1 summaries the ning in this section an all proofs are eferre to the appenix. Network ensity an congestion. The market esign literature emphasizes the importance of unerstaning thickness an congestion in the post-grauation marketplace an their e ect on markets e ciency (i.e. Nieerle an Roth 2005 an 2007). A market can fail if it is not thick enough (if not enough participants reach the market). In this case it is bene cial for agents to seek ways of transacting outsie the market - one way is unraveling. The ip sie of thickness is congestion. If the market is very thick, but there is not su cient time to explore the available options, agents will look for ways to exten the available search an bargaining time, sometimes by unraveling. In a moel of unraveling, a relate (less stuie) question rises. Can early perios su er from congestion? The answer is yes. While no formal moel has analyze the relationship between congestion in pre-marketplace perios an unraveling, the observation that early congestion can prevent unraveling is not new. In particular, Kagel an Roth (2000) note that: when the centralize mechanisms are introuce, there is only a small rollback of the unraveling that evelope when the market was ecentralize. But because of the congestion an competition in the market, some rms an workers who inten to make early matches n themselves unable to o so, an these participate in the centralize mechanism." 21 With respect to network structure, the ensity of the network grasps the thickness (an the resulting congestion when the number of o ers mae by rms in every perio is limite) of the market in the early perios. 21 Kagel an Roth (2000) paper examines the e ect of the mechanism use in the post grauation market on the ability of the market to recover from unraveling. In their experiment the congestion in the early perio is xe an the variation comes from the matching mechanism. 16

17 De nition 2 For a xe set of rms an worker (f an w), the network ensity D is the total number of connections in the network. For illustration, the networks in gure 2a an 2b have an ientical set of connecte rms an workers, an ensities of 2 an 4 corresponingly. Figure 2a Figure 2b Note that an increase in ensity oes not change the aggregate information in the system, but rather increases the information that every rm is expose to via an increase in the information overlap between rms. 22 Proposition 2 For every xe set of connecte rms an workers (f an w) there exist upper an lower network ensity threshols D = D (f; w) an D = D (f; w) such that for every low enough ensity (D < D) an increase in network ensity leas to an increase in unraveling, an for every high enough ensity D > D an increase in network ensity leas to a ecrease in unraveling. Conjecture 1 D = D. 23 Proposition 2 focuses on the trae-o between two e ects of an increase in network ensity: (1) an increase in competition for hiring each s i = h worker perio 1 increases the number of s i = h workers hire early an reuces the attractiveness of the post-grauation marketplace; an (2) a weakening of the coorination of o ers between the rms (provie by the network), increase the probability that some s = h workers receive multiple o ers while others receive non, an leas to a reuction in the number of workers that are hire early an an increase 22 Later we will investigate the consequences of information infusion to the system in the form of an increase in the number of connecte workers or in the early information accuracy. 23 Conjecture 1 is supporte by extensive numerical analysis. 17

18 in the attractiveness of the post-grauation marketplace. In a sparse network, the rst e ect ominates, while in a thick network the secon e ect is more signi cant. 24 The following lemma, that is an important part of the proof, shows that even when actions are hel xe, increasing the number of connections oes not necessarily increase contracting opportunities. Lemma 1 For every f, w, an there exist a threshol D = b D (f; w; ) such that for every D b D, the number of workers hire in perio ecreases in D () increases in D, an for every D > b D, Lemma 1 grasps an inherent trae-o between coorination an competition in promoting e ciency in a static environment. While increasing overlap a little can raise competition as more rms are competing on every worker, it comes with a cost of the coorination that a sparse network provies. For practical uses, it is instructive to focus on markets that unravel for which we are able to get an estimate of b D (f; w; ) given a large. In particular, it is useful to know whether increasing ensity in a particular market can mitigate the unraveling problem. It turns out that if unraveling in the market is relatively high ( 0:89) then every increase in ensity reuces unraveling, even if we starte with a very sparse network. To see why, look at the networks in gure 2 an assume full unraveling ( = 1) : In gure 2a, every s i = h worker will be hire early with probability one. In the more ense network in gure 2b, there is a probability 1 8 that both workers are s i = h an only one of them is hire early. Congestion in the early perios can therefore eter unraveling. We o not want to overstate the practical importance of this result. There are several market elements that shoul be consiere before applying this result to a speci c market: (1) a limite number of o ers mae at every perio; an (2) a limite number of perios before the en market. In some markets these assumptions are realistic. In Harvar Business School, MBA stuents are not allowe to miss classes for interviews, an rms are not allowe to conition an interview on a stuent missing class time. This policy is enforce as rms that try to eviate are not invite to recruit 24 Clearly, in reality there might be an interesting process in which an increase in access for information increases coorination ability between the rms as well. Such a change will not be graspe by a change in network ensity, as rms coorinate by committing to not use some of their connections. 25 Calvó-Armengol an Zenou (2005) prove a similar statement for a worker-only network an exogenously given in an employee referral moel. 18

19 in the school. In this case, recruiting perios are ivie to very speci c times an the number of interviews in every such time perio is limite. To some extent all early interviews are one uring breaks. In other entry-level market the structure is less rigi. However, it is not uncommon that recruiting scheule aligns with the acaemic calener. Despite its caveats, we n the result encouraging. Information technologies as the phone, , or the internet improve communication an reuce search cost an istance barriers. While it seems unlikely that market esigners be able to reuce the number of communication channels between i erent agents, we might be able to in uence the formation of i erent channels of communication, that o not enable an increase in the number of early o ers, by improving their interface (like esigning easy platforms to make professional webpages, an give more room on the internet to publish stuents work). The span of the network is one of the most basic network escriptors as it escribes the number of rms an workers that are connecte by the network. As such, the span of the network is a basic measure of early information transmission in our moel. Proposition 3 An increase in the span of the network (an increase of both f an w in ientical proportions) leas to an increase in unraveling. As we saw, only connecte agents will unravel. However, the following result is even stronger; an increase in the span of the network will increase unraveling for rms an workers that were connecte before the increase as well. As the span of the network increases, the expecte number of high quality workers in the post-grauation marketplace ecreases for a xe. This reuces the expecte pro t of a rm that reaches unmatche to the post-grauation marketplace an increase its incentive to hire early. Network concentration aroun a subset of rms or workers is a measure of market power an of informational avantage of some rms or workers over the others. In aition, it is relate to the span of the network. Namely, increasing network concentration aroun a subset of agents (either workers or rms) ecreases the number of connecte agents. Our next result shows that in a two perio moel this e ect ominates an increasing the network s concentration ecreases unraveling through a ecrease in the span. In section 4 we introuce ynamic consierations an show that the e ect of network concentration on unraveling is 19

20 more complex an that there are inherent i erences between increasing concentration aroun workers an aroun rms. Proposition 4 An increase in the concentration of connections aroun a subset of workers (an increase in r w accompanie by a ecrease in w) or rms (an increase in r f accompanie by a ecrease in f) leas to a ecrease in unraveling. While tempting, it will not be correct to interpret proposition 4 as allowing comparison between rms with i erent levels of connectivity within the same equilibrium. The main force riving the result are the changes in the overall connectivity of workers an rms. In section 4, after we present the more general moel of unraveling, we will be able to she light on this question as well. 3.2 Information accuracy an e ciency In this section, we go beyon the analysis of the network structure an analyze the e ects of varying the information accuracy in early perios () an the e ciency of the post-grauation marketplace (). The e ciency of the post-grauation marketplace can be a ecte by many market changes. For example, a reuction of search cost through lower interview costs or an improvement in evaluation stanars can help rms an workers to improve their iiosyncratic match. In general, many changes in Information Technology (IT) can a ect signi cantly the ability of rms an workers that are a better match to n each other. Proposition 5 An increase in the e ciency of the post-grauation marketplace () leas to a ecrease in unraveling if > 1 " 1+" an oes not a ect unraveling if < 1 " 1+". Intuitively, in the simple two perio moel, increasing e ciency a ects only the workers "pickiness" in the sense that it improves their post-grauation option an they can ecline more rms. This becomes relevant when workers have a high enough (relative to the strength of their preferences as escribe by ") an ecline job o ers from their less preferre rms. When we introuce ynamic consierations in section 4 the result becomes more subtle as refusal of late o ers by workers increases rms incentives to make early o ers. 20

21 Accuracy of early information as measure by has a non-monotonic e ect on unraveling. Proposition 6 There exist threshols an such that: (1) when < an increase in the accuracy of early information () leas to an increase in unraveling; 26 an, (2) when > an increase in the accuracy of early information () leas to a ecrease in unraveling. Conjecture 2 =. 27 Increasing the accuracy of information in perio 1 (by increasing ) can a ect the incentives of both rms an workers to contract early. The e ect on workers is straightforwar, as the information accuracy grows, workers are more likely to ecline early job o ers. If is high, unraveling is rm riven, an this e ect ominates. If is low, the e ect of changes in information accuracy on rms incentives becomes more important. This e ect is riven by two forces. First, an increase in information accuracy increases the expecte pro t of a rm that hires a s i = h worker in perio 1; we call this the irect e ect. The major part of the proof is focuse on the other, inirect, e ect of increasing on the expecte pro ts of rms that o not unravel. This e ect can be summarize M 0; for the same amount of unraveling, increasing implies lower expecte pro t for rms that reach the post-grauation perio unmatche. Intuitively, as grows, there is a higher probability that a worker that was hire early, turns out to be of high quality an her absence from the post-grauation perio marketplace will reuce the expecte pro t of rms that try to get matche in this perio. If we agree that in real markets information becomes more accurate over time, propositions 5 an 6 suggest that pre-grauation perios can be naturally ivie into two time segments: early on, when is small, workers will accept any job o er in orer to be insure against unemployment. This worker riven unraveling can be in uence only by changes to rms incentives. Closer to grauation, is large, an the balance of power shifts. Firms are now reay to hire any high potential worker, while workers ecline the less esirable job o ers. This rm riven unraveling can be in uence by changing workers incentives. 26 In particular, 1 " 1+". 27 Conjecture 2 is supporte by extensive numerical analysis. 21

22 The partition suggests a tractable yet informative way of introucing ynamics. In the following sections we introuce a three perio moel that accommoate both worker an rm riven unraveling an allows us to internalize intertemporal consieration an local competition as a ecting unraveling. This also allows us to re ne our results from the above simpli e moel in a very realistic way. 4 A moel with worker an rm riven unraveling In this section we exten our moel an allow for intertemporal interaction in the unraveling process. We show how unraveling is a ecte by market structure an present an equilibrium characterization of a three perio moel that captures important i erences between various labor market. The three perio moel also provies a more accurate escription of the e ects of changes in network concentration aroun workers an changes in e ciency of the postgrauation perio s marketplace. Table 1 summarizes the similarities an i erences between the two an three perio moels with respect to results. Consier workers an rms that live for three perios (perios 2, 1, an 0, with 0 being the post grauation perio). Motivate by our iscussion at the en of the previous section, let ( 1) = 1 an ( 2) = < 1 ". This guarantees that in perio 1 workers o not 1+" ecline job o ers an the unraveling is worker riven, while in perio 2 rms make o ers to high potential workers with probability 1 an the unraveling is rm riven. Therefore, in all equilibria ( 1) = 1 an ( 2) = 0. For notational simplicity we say that in perio 2 workers get a noisy signal s i 2 fh; lg of their own prouctivity an in perio 1 workers observe q i 2 fh; Lg which is their true prouctivity. Similarly, let ( 2) = an ( 1) =.. Other e nitions follow through from the previous section. The timeline of the moel is escribe in gure An example The general three perio moel with a general network structure is very complex. Therefore, before turning to the analysis of the general moel, we suggest an example with a simpler network structure that allow for a more tractable analysis an highlights some of the important ways in which enriching the ynamics a ects the results. In this example, each worker or rm 22

23 Figure 3: The timeline of the moel t= 2 t= 1 t=0 Each worker receives a noisy signal s=h,l Information is transmitte along the network an hiring ecisions are being mae Each worker receives an accurate signal q=h,l Information is transmitte along the network an hiring ecisions are being mae Post grauation marketplace. Unmatche firms an high prouctivity workers are being matche Prouction an profit an utility realizations has at most one connection (r w = r f = 1). The implie network is a set of rm-worker pairs. One moeling ecision that eserve mentioning is the ynamics of the network between perio 2 an perio 1. In some cases, in is reasonable to assume that a rm is connecte to the same actual worker in both perios. Another approach is to let rms an workers have the same position in the network with respect to the number of workers or rms that they are connecte to, but allow them to raw a new sample in every perio. As theoretically interesting as it may be, this moeling ecision turns out to be unimportant to any of the results. In the more general case we simplify the analysis by assuming the later. To avoi uplicate analysis, an to show where the assumption comes into play we suggest in this example the earlier approach. It shoul be fairly easy to see that the results are robust to the speci c formulation in this example. Recall that E M [] = E M [ j j; f; w; ; ; ] is the expecte pro t for a rm that reaches the post-grauation marketplace unmatche. If t is the number of workers hire in perio t, then E M [] = 0: M t < 0 for t = 2; 1, so the highest E M [] is when there are no workers hire early 1 = 2 = 0. This exercise highlights the potential for escalation in early contracting an captures the ynamic nature of unraveling. A rm s expecte pro t when connecte to a s i = h worker i is 2 worker contract in the perio 2 an 1 if the rm an the E 2 [j] = E 2 [j; f; w; ; G; ; ; s i = h; the rm oes not hire i in perio 2] = (8) = (1 ) + [1 (1 )] E M [] otherwise. The rst expression in (8) is the probability that the worker connecte to the rm is reveale to be a H type, an that she accepts an o er from the rm in perio 1. The 23

24 secon expression is the probability that the rm reaches the post-grauation perio unmatche multiplie by the expecte pro t in such case. Let F () = F ( 2; ) = E 2 [j] 2 + 1: Therefore, 2 [0; 1] for which F () = 0 are interior equilibria in which rms are ini erent between hiring a s i = h worker in perio 2, an waiting for perio 1. Possible corner solutions involve either full unraveling ( = 1 an F () < 0), or no unraveling ( = 0 an F () > 0). As in the previous section, an equilibrium is stable 0 an unstable otherwise. By continuity of F () an convexity of the support for an equilibrium always exists. The following results are the counterparts of propositions 1, 3 an 6 respectively, an verify that the corresponing results from the two perio moel still hol. 28 to the appenix. The proofs are eferre Proposition 7 Consistent with the results from the two-perio moel, in the three perio moel: 1. (Equilibrium characterization) There exists an information accuracy threshol = such that for every higher information accuracy ( > ) there is at most one interior equilibrium that is unstable. Otherwise, if there is no interior equilibrium, there is a unique stable equilibrium where either all rms that are connecte to a s i = h worker make early o ers ( = 1) or no rm makes an early job o er ( = 0). If ( < ) the unique equilibrium involves no unraveling. 2. (The span of the network) An increase in the span of the network () leas to an increase in unraveling. 3. An increase in the accuracy of early information (at any pre-grauation perio) leas to an increase in (worker riven) unraveling. below. The intuition for 1 an 2 is similar to the one in the two perio moel. We expan on 3 The accuracy of early information is measure in this moel by ( 1) an ( 2). While we have set constant the accuracy of information in perio 1 (recall that ( 1) = 1), 28 As in this example we hol r w an r f constant, the analysis of the ensity an concentration results in the ynamic context are eferre to the next section. 24

25 we can preict unambiguously the e ect of changes in ( 1) when it is close to 1. Basically, for high ( 1) increasing ( 1) increases the refusal rate of workers in perio 1 ( rm riven unraveling) an therefore make it less pro table for rms to wait for perio 1. Hence, for a non-constant ( 1) we can state the next result. As the proof is a erivative of the proof of proposition 8, it is omitte. Lemma 2 Increasing the accuracy of information in perio 1 increases perio 2 (worker riven) unraveling an ecreases perio 1 ( rm riven) unraveling. It turns out that an increase in the accuracy of early information, no matter at which perio has an unambiguous e ect on early, worker riven (perio 2) unraveling. This is important as the accuracy of information uring a stuent s training can be a ecte by simple policies. We suggest an analysis of one such policy in section 5. Lemma 2 an Proposition 7-(3) are consistent with the lack of information provie about workers by many training institutions. In particular, many schools cluster their graes aroun a very small support with only very few stuents grae at the extremes. In some schools this happens through grae in ation, in others, such as Harvar Business School, there are three possible graes, with 80% of the stuents receiving the same mile grae as ictate by school policy. These schools provie stuents with insurance that is similar to the one provie by unraveling in the early (worker riven) contracting an prevent unraveling in the early stages when workers accept any job to insure themselves against being ranke low by the school. 29 So far, the results from the two perio moel followe through. The following result is the rst major moi cation to the limite two-perio moel an potentially carries important policy implications Market e ciency an the importance of incorporating intertemporal consierations in moels of unraveling The two perio moel prouce a strong result: increasing the e ciency of the post-grauation match outcome can only reuce unraveling. Our next result suggests that the two perio moel 29 Our results are relate to Ostrovsky an Schwarz (2006). They n that improving the information available to rms in the post grauation market might, uner some conitions, trigger unraveling. Our results a to that an show aing information in other stages of training can also increase unraveling. Later, in proposition 8 we a another layer to this line of research an show that any improvement in the matching in the post-grauation marketplace can trigger unraveling. 25

26 is not su cient for unerstaning the impact of match e ciency on unraveling, an emphasizes the inherent trae-o between static an ynamic e ciency. Proposition 8 Increasing the e ciency of the post-grauation marketplace () leas to an increase in perio 2 (worker riven) unraveling an to a ecrease in perio 1 ( rm riven) unraveling. The main premise of the result is the unerstaning that a rm s connection is an asset that can be use in i erent times (an maybe for i erent uses), in the same way that other assets can be channele to other uses. Intuitively, increasing the e ciency ecreases the probability that a worker will accept a job o ere to her in perio 1 () : Hence, increasing reuces the probability that the rm coul use a connection to a q i = H worker in perio 1 an the future option value of the connection ecreases. Therefore, the rm will prefer to capitalize the connection at an earlier stage. The intuition goes beyon the moel at han. In many everyay applications, iniviuals an organizations ten to take early action or sign long term contracts when they believe that their strategic avantage might be jeoparize in the future. The intuition is also present in search moels an in the analysis of the classic optimal stopping problem. In a nite search an agent s threshol for stopping after the rst raw will go own as we reuce the number of raws left. Similarly the intuition oes not seem to epen on the number of pre-grauation perios in the moel as any reuction of the opportunity cost from not keeping the connection will reuce the rm s incentives to wait further to realize the bene ts of the connection. In the unraveling case there is another force pulling in the other irection. Namely, the ecrease in perio 1 s hiring increases the expecte pro t of rms that reach the post-grauation perio unmatche as the pool of workers in this perio increases. Nevertheless, this secon, inirect e ect, is ominate by the irect e ect of reuction in the option value of keeping the connection. The proof consists of eriving F incentives to hire early increase with. Formally, Recall that an showing that it is strictly negative, so that rms F = (E M [] 1) + [1 (1 )] E M [] > 0 so we can focus on the expression in the parenthesis. An increase in reuces a rm s probability of hiring a worker that is connecte to it in perio 26 (9) 1 as the

27 worker is more likely to refuse an o er. This irect e ect reuces a rm s expecte pro t in perios later than 2. Clearly, this a ects the rms negatively only if the worker is reveale to be of high quality (which happens with probability ) an is capture by the (negative) rst element in the summation (E M [] 1). On the other han, reucing the hiring in perio 1 one by other rms increases the expecte pro t for a rm that reach the post-grauation perio unmatche. This change is graspe by the positive secon element of the summation, [1 (1 M []. It turns out that the rst (irect) e ect is greater than the on the post-grauation perio. Proposition 8 is i erent in spirit than many results in the market esign literature an highlights the contrast between our notion of e ciency an the notions of stability or even mechanism e ciency. The market esign literature usually assumes that information is publicly available an costless. This assumption is reasonable in this literature as it takes agents preferences, incluing the number of their acceptable matches, as given an not a ecte by market parameters. However, to unerstan unraveling in a broaer context we allow for changes in information technologies an search costs in the market by varying. 4.2 The general moel In this section we investigate the three perio moel with a general network structure. We show that although the equilibrium characterization is richer, the comparative statics from the example carry through. We claim that the extene equilibrium characterization that allows for the existence of stable equilibria, correspons to market with more complex network structures. With respect to network structure, we verify that the results regaring the span of the network an the concentration aroun rms hol, an moify the results regaring the network ensity an the concentration aroun workers. The moi e result emonstrate the consierations coming from intertemporal an local competition. Allowing r f an r w to accept any positive integer value poses several technical i culties. Therefore, in this section we characterize the equilibrium equation analytically, but use numerical methos to solve for equilibrium an comparative statics. 30 Moreover, we are require to put more structure on the hiring process in the pre-grauation perios. We assume that rms an workers raw new connections at every perio, these connections 30 The analysis was conucte using Matlab. The coe is available upon request from the author. 27

28 substitute for the connections in the previous perio. This seems like a reasonable assumption, especially if a rm / worker is well connecte an cannot communicate with all connections in every perio. In that case, the connections counte for in the moel represent a subset of the actual connections that are activate ranomly to transmit or receive information. More importantly, to the extent that we also examine the limit case where r f = r w = 1, the example above emonstrates that our results are not sensitive to this assumption. perio Secon, while in perio 2 we let rms make the o er to the workers, we reverse roles in 1 when workers can choose rms to apply to. It will become clear in the analysis that this is a mere simpli cation an shoul not change our result qualitatively. 31 With that in min, each rm can choose whether or not to make an o er in perio 2, an any rm that i not hire before, can potentially receive an application from a worker in perio 1. As before, in every perio, an agent ( rm or worker) can make at most one o er to any of the other agents. The analysis of perio 2 is almost ientical to the analysis of the early perio in the two perio moel given in equations (3) - (6), with the only change that the probability that a worker refuses an o er is ( 2) = 0. In perio 2, let 2 be the probability that worker i that is of quality s i = h is o ere a job by rm j that is connecte to her, P w 2 be the probability that a connecte worker is o ere any job, an 2 be the fraction of workers hire. The implie probability that a connecte rm hires in perio 2 is 2 f. In perio 1, the probability that rm j receives an application from a worker i of prouctivity q i = H is, 1 = P w (1 ) (10) 2 where 1 1 P w (1 ) is the probability that the worker is of high prouctivity 2 an was not hire in perio 2, an 31 The assumption can also be justi e using a simple argument of market power; in perio 2, the unraveling is worker riven. Workers will accept any o er an o not have bargaining power as some rms might not want to make o ers. In perio 1, the unraveling is rm riven. High prouctivity workers know that any rm will be willing to hire them, an it oes not longer make sense to assume that they will be restricte to rms that make them an o er. 28

29 = X m=0;:::;r u 1 rw 1 1 m 2 f m rw m 1 2 f 1 m 'm+1 (11) is the probability that the worker applies for a position in rm j taking in consieration the probability istribution over the number of rms that i not hire in perio 2 an are connecte to the worker, an the probability that the rm is not acceptable by the worker. With some algebra this is reuce to 1 = 1 1 P w ' + rw 2 = 1 f f 2 r w f (12) The probability that the rm gets at least one application in perio 1 can be written as, P f 1 = 1 (1 1 ) r f (13) an the number of workers hire in perio 1 is, 1 = f 1 2 P f 1 (14) f As in the example E M [] = 0: so that the expecte pro t of a rm that i not hire in perio 2 is, E 1 [] = P f 1 + The rm has an option to try an hire a worker in perio 1 P f 1 E M [] (15) 2 an get an expecte pro t of 2 1 if it succees an E 1 [] otherwise, or wait for perio 1 an expect a pro t of E 1 []. The implie equilibrium equation is therefore, F () = E 1 [] (16) Again, in perio 2 there will be partial (worker riven) unraveling (an interior equilibrium) if an only if F () = 0 while full or no unraveling can be achieve when F () < 0 or F () > 0 respectively. To characterize the set of equilibria an erive its properties we constructe the gri pre- 29

30 sente in table 2: 32 Table 2: Parameters for numerical analysis lower boun upper boun step size f r f r w In search of a solution for F () = 0, for every set of parameters, we conucte a line search over 2 [0:01; 0:99] in steps of 0:01 to n a change in sign. For every change of sign we have conucte a binary search to n the root with tolerance 0: The results are summarize in table 1. The results regaring the span of the network, network concentration aroun rms an the accuracy of early information remain the same as in the two perio moel an in the example above an are omitte. Equilibrium characterization Claim 1 For most parameters, there is either a unique pure strategy equilibrium where all rms take the same action in perio 2, or a unique interior (mixe strategy) equilibrium that is unstable an two pure equilibria (full an no unraveling in perio 2 are equilibria). Stable partial unraveling equilibria arise when: 1. The accuracy of early information an the network concentration aroun rms are high; an 2. The e ciency in the post-grauation marketplace, the network span, an the network concentration aroun workers is low. Claim 1 is summarize graphically in gures 4-7. The light blue regions have a unique pure strategy equilibrium, the ark blue regions have a unique interior (mixe strategy) unstable 32 As is a trivial function of G an an we assume G to be xe, we use instea of eriving it from in the analysis. 33 We have chosen this metho after plotting F (j) for a large set of speci cations an verifying that multiplicity of equilibria is very rare an in any case, equilibria are at istance (in terms of the i erence between the s) of more then 0:1 from each other. 30

31 equilibrium an two pure strategy equilibria, the yellow region exhibit stable an / or unstable interior equilibria (epening on the other parameters), an the brown regions have a unique interior stable equilibrium. Figure 4 Figure 5 Figure 6 Figure 7 The result is fairly intuitive, when r f is large, a rm has a relatively high probability of being connecte to an unemploye high prouctivity worker in perio 1 even if many rms hire in perio 2, an when r w is small there is a high probability that this worker will apply to that rm; this probability increases with the tenency of other rms to hire early. Low values for the span of the network an the e ciency parameters increase the sensitivity of the local (perio 1) market, an ecrease the sensitivity of the post-grauation market to the hiring in perio 2. Finally, The requirement for large is somewhat less intuitive, however, for the regions of parameters that allow for a stable mixe strategy equilibrium, a lower leas to a no unraveling equilibrium. This result has practical implications. In an ongoing ebate aroun the esign of the market for juicial clerks, o cials from the California circuit claime that juges in the New York circuit can wait for later perios before hiring as they are well connecte to more high quality law schools. Claim 1 suggests that a high number of connections can inee allow for stability of 31

32 an equilibrium in which some highly connecte rms wait an o not unravel early. In fact, it is a somewhat trivial exercise to introuce a small number of rms with r 0 f > r f an show that these rms will be the ones to wait. Network ensity. The result that for low enough ensities an increase in ensity leas to an increase in unraveling is trivial. We therefore investigate the e ect of ensity on unraveling for high ensity levels. For this en, we have explores networks as ense as r w = r f = 100 an base our result on these networks. 34 Claim 2 For most parameters an for r f = r w = 100, an increase in network ensity leas to a ecrease in unraveling. In particular, unraveling increases with ensity when: 1. The accuracy of early information is high; an 2. The e ciency in the post-grauation marketplace is low. Claim 2 is summarize graphically in gure 8. The light blue regions have a unique pure strategy equilibrium an therefore are not a ecte by changes in network ensity, the ark blue regions are areas where unraveling ecreases with ensity, in the yellow region unraveling either increase or ecrease with ensity (epening on the other parameters), an the brown regions exhibit equilibria in which unraveling increases with ensity. Figure 8 34 While it is possible that higher ensities will somewhat moify the results it is unlikely, as the e ect of ensity on unraveling is alreay in nitesimally small at this point an seem to ecrease rapily. 32

33 Intuitively, high e ciency reuces the activity in perio 1 an get us closer to the two perio moel analyze in section 3 as there are almost no contracts signe in perio 1. High is likely to lea to a positive e ect of ensity on unraveling as high suggests unstable equilibria with low tenency of rms to make o ers in perio 2 (low ). When is low, the following e ect ominates any other e ect of congestion: an increase in the ensity increases the probability that a rm that is intereste in making an o er is connecte to a high potential worker in perio 2. So far, the numeric results from the general moel ha relatively small moi cations to the results from the two perio moel. The following two results show that aing intertemporal an local consierations can have some signi cant implications, an she light on important policy an esign issues. Network concentration aroun workers (r w ) Claim 3 For most parameters, an increase in network concentration aroun workers leas to an increase in unraveling. In particular, unraveling ecreases with concentration aroun workers when: 1. The accuracy of early information an the concentration aroun rms are low; an 2. The e ciency in the post-grauation marketplace an the span of the network are high. Figures 9-11 present the main nings of claim 3. The light blue regions have a unique pure strategy equilibrium an therefore are not a ecte by changes in the network, the ark blue regions are areas where unraveling ecreases with concentration aroun workers, in the yellow region unraveling either increase or ecrease with network concentration aroun workers (epening on the other parameters), an the brown regions exhibit equilibria in which unraveling increases with concentration aroun workers. 33

34 Figure 9 Figure 10 Figure 11 Increasing concentration aroun workers increases the competition that rms expect in perio 1 on the remaining workers an therefore increases rms incentives to make o ers in perio 2. When the span of the network is large or when the e ciency of the post-grauation marketplace is high, the e ect on the post-grauation perio becomes strong relative to the e ects on competition in perio 1. On the other han, when an r f are large, rms ten to value their local position more an local competition becomes more signi cant. The e ciency in the post-grauation marketplace (). We ha i erent results with respect to e ciency in our two perio moel an in our three perio example. The following claim veri es that proposition 8 erive from our simple three perio example is robust to complex network structures. The intuition is the same as in the example. Claim 4 An increase in the e ciency of the post-grauation marketplace leas to an increase in worker riven unraveling (perio 2) an to a ecrease in rm riven unraveling (perio 1). 5 Labor market policies an unraveling In this section we analyze two policies that were in a ect or uner iscussion in various markets an in i erent contexts. The rst, involving stanarization of the grae system shows that more information might be ba news in the context of unraveling, an shows how ynamic moels improve our ability to evaluate suggeste policies. The secon follows a stran of the market esign literature an some practical esign issues aroun the existence of exploing o ers in a market an re nes previous observations. 34