The Effects of Social Networks on Employment and … Effects of Social Networks on Employment and Inequality ... graphic correlation in unemployment across ... EFFECTS OF SOCIAL NETWORKS

The Effects of Social Networks on Employment and Inequality

By ANTONI CALVO-ARMENGOL AND MATTHEW O JACKSON

We develop a model where agents obtain information about job opportunitiesthrough an explicitly modeled network of social contacts We show that employmentis positively correlated across time and agents Moreover unemployment exhibitsduration dependence the probability of obtaining a job decreases in the length oftime that an agent has been unemployed Finally we examine inequality betweentwo groups If staying in the labor market is costly and one group starts with a worseemployment status then that grouprsquos drop-out rate will be higher and their em-ployment prospects will be persistently below that of the other group (JEL A14J64 J31 J70)

The importance of social networks in labormarkets is pervasive and well documentedMark Granovetter (1973 1995) found in a sur-vey of residents of a Massachusetts town thatover 50 percent of jobs were obtained throughsocial contacts Earlier work by Albert Rees(1966) found numbers of over 60 percent in asimilar study Exploration in a large number ofstudies documents similar figures for a varietyof occupations skill levels and socioeconomicbackgrounds1

In this paper we take the role of social net-works as a manner of obtaining informationabout job opportunities as a given and exploreits implications for the dynamics of employ-ment In particular we examine a simple modelof the transmission of job information through anetwork of social contacts Each agent is con-nected to others through a network Information

about jobs arrives randomly to agents Agentswho are unemployed and directly hear of a jobuse the information to obtain a job Agents whoare already employed depending on whetherthe job is more attractive than their current jobmight keep the job or else might pass alonginformation to one (or more) of their directconnections in the network Also in each periodsome of the agents who are employed randomlylose their jobs After documenting some of thebasic characteristics and dynamics of thismodel we extend it to analyze the decision ofagents to drop out of the labor force based onthe status of their network This permits us tocompare the dynamics of drop-out rates andemployment status across groups

The fact that participation in the labor force isdifferent across groups such as whites andblacks is well documented For instance DavidCard and Alan B Krueger (1992) see a drop-outrate 25 to 3 times higher for blacks compared towhites Amitabh Chandra (2000) provides abreakdown of differences in participation ratesby education level and other characteristics andfinds ratios of a similar magnitude Moreoverthe analysis of James Heckman et al (2000)suggests that differences in drop-out rates are animportant part of the inequality in wages acrossraces and that accounting for dropouts actuallyincreases the black-white wage gap23

Calvo-Armengol Universitat Auto`noma de BarcelonaDepartment of Economics Edifici B 08193 Bellaterra (Bar-celona) Spain (e-mail antonicalvouabes httpseleneuabesacalvo) Jackson Division of the Humanities andSocial Sciences 228-77 California Institute of TechnologyPasadena CA 91125 (e-mail jacksonmhsscaltecheduhttpwwwhsscaltechedujacksonmJacksonhtml) Wethank Valentina Bali Kim Border Antonio Cabrales JanetCurrie Franccedilois Fontaine Isa Hafalir Eddie LazearMassimo Morelli David Peacuterez-Castrillo and Giorgio Topafor helpful conversations and discussions We also thankseminar participants and three anonymous referees for theircomments and suggestions We gratefully acknowledge thefinancial support of Spainrsquos Ministry of Science and Tech-nology under Grant Nos SEC2001-0973 and BEC2002-02130 and the Lee Center for Advanced Networking atCaltech

1 See James Montgomery (1991) for further discussionand references

2 Ignoring dropouts biases estimated wages upwardsGiven much higher drop-out rates for blacks this can biasthe wage differential

3 The persistent inequality in wages between whites andblacks is one of the most extensively studied areas in labor

426

One is then left to explain why drop-out ratesshould differ across races An analysis of socialnetworks provides a basis for observing bothhigher drop-out rates in one race versus anotherand sustained inequality in wages and employ-ment rates even among those remaining in thelabor force as follows4 The starting point ofour model is that the better the employmentstatus of a given agentrsquos connections (eg rel-atives friends acquaintances) the more likelyit is that those connections will pass informationconcerning a job opening to the agent Thismight be for any number of reasons One is thatas the employment status of a connection im-proves it is less likely that the connection willwant to keep a new job for him or herselfAnother is that as the employment status ofother agents in the network improves the morelikely that a given agentrsquos connections will beemployed and a source of information and themore likely that they will choose to pass news tothe given agent rather than to some other agentwho already has a job This sort of informationpassing leads to positive correlation betweenthe employment status of agents who are di-rectly or indirectly connected in the networkwithin a period and across time as we establishbelow

Such correlation of employment is ob-served in the data both in the indirect form ofthe inequality mentioned previously and alsoin terms of direct measured correlation pat-terns Giorgio Topa (2001) demonstrates geo-graphic correlation in unemployment acrossneighborhoods in Chicago and also finds asignificantly positive amount of social inter-actions across such neighborhoods TimothyConley and Topa (2001) find that correlationalso exists under metrics of travel time oc-cupation and ethnicity and that ethnicity and

race are dominant factors in explaining cor-relation patterns

The positive correlation that we establish be-tween the employment of agents in a networkthen provides a basis for understanding the sus-tained difference in drop-out rates and resultinginequality in employment rates Consider twoidentical networks except that one starts witheach of its agents having a better employmentstatus than their counterparts in the other net-work Now consider the decision of an agent toeither remain in the labor market or to drop outRemaining in the labor market involves somecosts which could include things like costs ofskills maintenance education and opportunitycosts Agents in the network with worse initialstarting conditions have a lower expected dis-counted stream of future income from remain-ing in the network than agents in the networkwith better initial starting conditions This mi-nor difference might cause some agents to dropout in the worse network but remain in thebetter network The important observation isthat dropping out has a contagion effect Whensome of an agentrsquos connections drop out thatagentrsquos future prospects worsen since theagentrsquos network is no longer as useful a sourceof job information Thus some agents con-nected to dropouts also drop out due to thisindirect effect This can escalate so a slightchange in initial conditions can lead to a sub-stantial difference in drop-out decisions As alarger drop-out rate in a network leads to worseemployment status for those agents who remainin the network we find that slight differences ininitial conditions can lead to large differences indrop-out rates and sustained differences in em-ployment rates5

Before proceeding to the model let us alsomention a fourth feature that is also exhibited bythe model Unemployment exhibits duration de-pendence and persistence That is when condi-tioning on a history of unemployment theexpected probability of obtaining a job de-creases in the length of time that an agent hasbeen unemployed Such duration dependence iswell documented in the empirical literature forexample in studies by Stuart O Schweitzer and

economics James P Smith and Finis R Welch (1989)using statistics from census data break the gap down acrossa variety of dimensions and time The gap is roughly on theorder of 25 percent to 40 percent and can be partly ex-plained by differences in skill levels quality of educationand other factors [eg see Card and Krueger (1992)Chandra (2000)] See Reynolds Farley (1990) for a com-parison of labor market outcomes for 50 racial-ethnicgroups in the United States

4 In this paper we focus on differences in drop-out ratesand employment and we refer the reader to Calvo-Armengol and Jackson (2003) for an analysis of wageinequality

5 Similarly small differences in network architecturesmay also lead to sharp differences in drop-out patternsacross two groups with identical starting conditions

427VOL 94 NO 3 CALVO-ARMENGOL AND JACKSON EFFECTS OF SOCIAL NETWORKS

Ralph E Smith (1974) Heckman and GeorgeBorjas (1980) Christopher Flinn and Heckman(1982) and Lisa M Lynch (1989) To get afeeling for the magnitude Lynch (1989) findsaverage probabilities of finding employment onthe order of 030 after one week of unemploy-ment 008 after eight weeks of unemploymentand 002 after a year of unemployment

The reason that we see duration dependencein a networked model of labor markets is asimple one A longer history of unemploymentis more likely to come when the direct andindirect connections of an agent are unem-ployed Thus seeing a long spell of unemploy-ment for some agent leads to a high conditionalexpectation that the agentrsquos contacts are unem-ployed This in turn leads to a lower probabilityof obtaining information about jobs through thesocial network This network explanation is or-thogonal to standard explanations such as un-observed heterogeneity

At this point let us preview a policy predic-tion that emerges from a networked model Dueto the network effects improving the status of agiven agent also improves the outlook for thatagentrsquos connections This is the contagion effectdiscussed above in reverse As a result in anetworked model there are local increasing re-turns to subsidizing education and other poli-cies like affirmative action6 One implication isthat it can be more efficient to concentrate sub-sidies or programs so that a cluster of agentswho are interconnected in a network are tar-geted rather than spreading resources morebroadly

Before presenting the model we note that weare certainly not the first researchers to recog-nize the importance of social networks in labormarkets Just a few of the studies of labor mar-kets that have taken network transmission of jobinformation seriously are Scott A Boorman(1975) Montgomery (1991 1992 1994) Ken-neth Arrow and Ron Borzekowski (2001) Topa(2001) and Calvo-Armengol (2004)mdashnot to

mention the vast literature in sociology7 Thecontribution here is that we are the first to studyan explicit network model and prove some ofthe resulting implications for the patterns anddynamics of employment as well as the in-equality across races

I A Simple Network Model

The model we consider here is one where alljobs are identical We refer the reader to acompanion paper Calvo-Armengol and Jackson(2003) for a more general model that nests thismodel and also looks at wage dynamics andallows for heterogeneity in jobs decisions as towhether to switch jobs repeated and selectivepassing of information competing offers foremployment and other extensions of the modelpresented here In short the results presentedhere extend to wage inequality as well and arequite robust to the formulation

There are n agents Time evolves in discreteperiods indexed by t The vector st describes theemployment status of the agents at time t Ifagent i is employed at the end of period t thensit 1 and if i is unemployed then sit 0

A period t begins with some agents beingemployed and others not as described by thestatus st1 from the last period Next informa-tion about job openings arrives In particularany given agent hears about a job opening witha probability a that is between 0 and 1 This jobarrival process is independent across agents Ifthe agent is unemployed then he or she willtake the job However if the agent is alreadyemployed then he or she will pass the informa-tion along to a friend relative or acquaintancewho is unemployed This is where the networkpattern of relationships is important as it de-scribes who passes information to whom whichis ultimately crucial in determining a personrsquoslong-term employment prospects We now de-scribe these network relationships and the pro-cess of information exchange

Any two people either know each other or donot and in this model information only flowsbetween agents who know each other A graphg summarizes the links of all agents wheregij 1 indicates that i and j know each other

6 In our model improving the status of one agent haspositive external effects on other agentsrsquo expected futureemployment There are of course other factors that mightcounterbalance this sort of welfare improvement for in-stance the difficulty that an agent might have adapting tonew circumstances under affirmative action as discussed byGeorge A Akerlof (1997)

7 Some related references can be found in Montgomery(1991) and Granovetter (1995)

428 THE AMERICAN ECONOMIC REVIEW JUNE 2004

and gij 0 indicates that they do not know eachother It is assumed that gij gji meaning thatthe acquaintance relationship is a reciprocalone

If an agent hears about a job and is alreadyemployed then this agent randomly picks anunemployed acquaintance to pass the job infor-mation to If all of an agentrsquos acquaintance arealready employed then the job information issimply lost The probability of the joint eventthat agent i learns about a job and this job endsup in agent jrsquos hands is described by pij(s)where

pij s

a if si 0 and i j

a

yenksk0

gikif si 1 sj 0 and gij 1 and

0 otherwise

and where the vector s describes the employ-ment status of all the agents at the beginning ofthe period

In what follows we will also keep track ofsome indirect relationships as friends of afriend will play a couple of roles First they arecompetitors for job information in the short runSecond they help keep an agentrsquos friends em-ployed which is a benefit in the longer run Wesay that i and j are path-connected under thenetwork g if there exists a sequence of links thatform a path between i and j

Finally the last thing that happens in a periodis that some agents lose their jobs This happensrandomly according to an exogenous breakuprate b between 0 and 1 This is the probabilitythat any given employed agent will lose his orher job at the end of a given period and this isalso independent across agents

II The Dynamics and Patterns of Employment

As time unfolds employment evolves as afunction of both past employment status andthe network of connections which togetherrandomly determine the new employmentEmployment thus follows a finite stateMarkov process where the state is the vectorof agentsrsquo employment status at the end of aperiod and transition probabilities are depen-dent on the network of relationships We wish

to characterize the behavior of this stochasticprocess

The relationship between the one-period-ahead employment status of an agent and hispattern of connections as described by thepij(s)rsquos above is clear Having links to em-ployed agents improves irsquos prospects for hear-ing about a job if i is unemployed In additiondecreasing the competition for informationfrom two-link-away connections is helpfulThat is if friends of my friends are employedrather than unemployed then I have a higherchance of being the one that my friends willpass information to Further indirect relation-ships (more than two-links away) do not enterthe calculation for the one-period-ahead em-ployment status of an agent However once wetake a longer time perspective the evolution ofemployment across time depends on the largernetwork and status of other agents This ofcourse is because the larger network and statusof other agents affect the employment status ofirsquos connections

We first present an example which makes itclear why a full analysis of the dynamics ofemployment is subtle

Example 1 (Negative Conditional Correlation)Consider Figure 1 a network with three agentsand suppose the employment from the end ofthe last period is st1 (0 1 0) In the picturea darkened node represents an employed agent(agent 2) while unemployed agents (1 and 3)are represented by empty nodes A line betweentwo nodes indicates that those two agents arelinked

Conditional on this state st1 the employ-ment states s1t and s3t are negatively correlatedThis is due to the fact that agents 1 and 3 areldquocompetitorsrdquo for any job news that is firstheard by agent 2

Despite this negative (conditional) correla-tion in the shorter run agent 1 can benefit from3rsquos presence in the longer run Indeed agent 3rsquospresence helps improve agent 2rsquos employment

FIGURE 1 NEGATIVE CORRELATION IN CONDITIONAL

EMPLOYMENT


status Also when agent 3 is employed agent 1is more likely to hear about any job that agent 2hears about These aspects of the problemcounter the local (conditional) negative correla-tion and help induce a positive correlation be-tween the employment status of agents 1 and 3

The benefits from having other agents in thenetwork outweigh the local negative correlationeffects if we take a long-run perspective Thefollowing examples illustrate the long-run be-havior of the Markov process regulating em-ployment as shaped by the underlying networkof contacts between agents

Example 2 (Correlation and Network Structure)Consider an example with n 4 agents and leta 0100 and b 0015 If we think aboutthese numbers from the perspective of a timeperiod being a week then an agent loses a jobroughly on average once in every 67 weeks andhears (directly) about a job on average once inevery ten weeks Figure 2 shows unemploymentprobabilities and correlations between agentsrsquoemployment statuses under the long-run steady-state distribution8

If there is no network relationship at all thenwe see an average unemployment rate of 132percent Even moving to just a single link(g12 g21 1) substantially decreases theprobability (for the linked agents) of being un-employed as it drops by more than a third to83 percent The resulting unemployment rate

aggregated over the four agents is 1075 per-cent As we see from Figure 2 adding morelinks further decreases the unemployment ratebut with a decreasing marginal impact Thismakes sense as the value to having an addi-tional link comes only in providing job infor-mation when all of the existing avenues ofinformation fail to provide any The probabilityof this is decreasing in the number ofconnections

The correlation between two agentsrsquo employ-ment is (weakly9) decreasing in the number oflinks that each an agent has and the correlationbetween agentsrsquo employment is higher for di-rect compared to indirect connections The de-crease as a function of the number of links isdue to the decreased importance of any singlelink if an agent has many links The differencebetween direct and indirect connections interms of correlation is due to the fact that directconnections provide information while indirectconnections only help by indirect provision ofinformation that keeps friends friends offriends etc employed

Also note that the correlation between agents1 and 3 in the third row of Figure 2 is positive(19 percent) Thus even though agents 1 and 3are in competition for information from bothagents 2 and 4 in the shorter run their employ-ment is positively correlated in the long runThis will be true more generally as stated in thepropositions below

Next Figure 3 examines some eight-personnetworks with the same information arrival andjob breakup rates a 0100 and b 001510

Here again the probability of unemploymentfalls with the number of links and the correla-tion between two employed agents decreaseswith the distance of the shortest path of links(geodesic) between them

Also we can see some comparisons to thefour-person networks an agent has a lower un-employment rate in a complete four-person net-work than in an eight-person circle In thisexample the direct connection is worth morethan a number of indirect ones More generally

8 The numbers for more than one agent are obtainedfrom simulations in Maple We simulate the economy overa large number of periods (hundreds of thousands) andcalculate observed unemployment averages and correla-tions The programs are available upon request from theauthors The correlation numbers are only moderately ac-curate even after several hundred thousand periods

9 In some cases the correlations are indistinguishable tothe accuracy of our simulations

10 We know from Propositions 1 and 2 that Corr(s1 s5)is positive for the top network in Figure 3 However it is toosmall to accurately report its numerical value based on oursimulations

FIGURE 2 CORRELATION AND NETWORK STRUCTURE I


the trade-off between direct connections andindirect ones will depend on the network archi-tecture and the arrival and breakup rates In thisexample agents rarely lose jobs and hear aboutthem relatively frequently and so direct con-nections are employed with a high probabilityregardless of the number of their neighbors andso indirect connections are less important thandirect ones In situations with higher breakuprates and lower arrival rates indirect connec-tions can become more important

The model also provides a tool for analyzingasymmetries in the network

Example 3 (Bridges and Asymmetries) Con-sider the network in Figure 4 Again we calcu-late employment from simulations using thesame arrival and breakup rates as in the previ-ous examples

In this network the steady-state unemploy-ment probabilities are 47 percent for agents 1and 6 48 percent for agents 2 5 7 and 10 and50 percent for the rest While these are fairlyclose simple differences of an agentrsquos positionin the network affects his or her unemploymentrate even though all agents all have the samenumber of connections Here agents 1 and 6have lower unemployment rates than the others

and 3 4 8 and 9 are the worst off If wecompare agent 3 to agent 1 we note the follow-ing the average geodesic (minimum path) dis-tance between any two agents who are directlyconnected to agent 3 is 43 In a sense theagents that agent 3 is connected to are not ldquowelldiversifiedrdquo In contrast the average geodesic(minimum path) distance between any twoagents who are directly connected to agent 1 is2 Moreover agents 5 and 6 are only path-connected through agent 1 In fact 1 and 6 formwhat is referred to as a ldquobridgerdquo in the socialnetworks literature11

Example 4 (Structure Matters Densely VersusClosely Knit Networks) The model can alsoshow how other details of the network structurematter Compare the long-run average unem-ployment rates on two eight-person networkswith 12 links each In both networks all agentshave exactly three links But the average lengthof the paths connecting agents is differentacross networks Again we run simulationswith a 0100 and b 0015 see Figure 5

The average path length is lower for the circlewith diameters than for the circle with local

11 The lower unemployment (higher employment) rate ofthese agents is then consistent with ideas such as Ronald SBurtrsquos (1992) structural holes although for different reasonsthan the power reasoning behind Burtrsquos theory

FIGURE 3 CORRELATION AND NETWORK STRUCTURE II

FIGURE 4 A NETWORK WITH A BRIDGE

FIGURE 5 PATH LENGTH MATTERS


four-agents clusters meaning that the latter ismore closely knit than the former Indeed theaverage path length decreases when the span ofnetwork contacts spreads that is when the re-lationships get less introverted or less closelyknit12 The average unemployment increaseswith closed-knittedness reflecting the fact thatthe wider the breadth of current social ties themore diversified are the sources of information

The fact that the long-run employment statusof path-connected agents is positively corre-lated in the above examples is something thatholds generally In particular as we divide aand b both by some larger and larger factormdashsothat we are looking at arbitrarily short timeperiodsmdashthen we can begin to sort out theshort- and longer-run effects We call this theldquosubdivisionrdquo of periods In the limit we ap-proximate a continuous time (Poisson) processwhich is the natural situation where such tem-porary competition for jobs is short lived andinconsequential while the overall status of in-direct connections tells one a great deal aboutthe possible status of direct connections and thelonger-run effects come to dominate

PROPOSITION 1 Under fine enough subdivi-sions of periods the unique steady-state long-run distribution on employment is such that theemployment statuses of any path-connectedagents are positively correlated13

The proposition shows that despite the short-run conditional negative correlation betweenthe employment of competitors for jobs andinformation in the longer run any intercon-nected agentsrsquo employment is positively corre-lated This implies that there is a clustering ofagents by employment status and employedworkers tend to be connected with employedworkers and vice versa This is consistent withthe sort of clustering observed by Topa (2001)The intuition is clear conditional on knowingthat some set of agents are employed it is morelikely that their neighbors will end up receivinginformation about jobs and so on

Moreover the positive correlation holds notonly under the steady-state distribution but alsoacross any arbitrary time periods That is agentirsquos employment status at time t is positivelycorrelated with agent jrsquos status at time t forgeneral values of t and t

PROPOSITION 2 Under fine enough subdivi-sions of periods starting under the steady-statedistribution the employment statuses of any twopath-connected agents are positively correlatedacross arbitrary periods14

III Duration Dependence and Persistence inUnemployment

As mentioned in the introduction there aresome other patterns of unemployment that havebeen observed in the data and are exhibited bya networked model To see this let us examinesome of the serial patterns of employment thatemerge

Again consider job arrival and breakup rates

12 Here close-knittedness is measured by the averagepath length which is just the average across all pairs ofagents of the length of the shortest path between them Analternative measure of the tightness of a network is forinstance the clustering coefficient which reflects the levelof intraconnectedness among agents with a common friendBoth measures are roughly equivalent though in somecases when network links are randomly rewired the aver-age path length drops sharply in contrast with a lowercorresponding decrease of the clustering coefficient a phe-nomenon often termed a ldquosmall worldrdquo effect (see egDuncan J Watts and Steven H Strogatz 1998)

13 More formally any network g on the population of nagents arrival probability a (0 1) and breakdown prob-ability b (0 1) define a finite-state irreducible and ape-riodic Markov process M (g a b) on employmentstatuses Denote by the (long-run) steady-state distribu-tion associated to this process which is uniquely definedBy dividing a and b by some common T we obtain anassociated Markov process MT (g aT bT) that wename the T period subdivision of M with steady-statedistribution T We show that there exists some T such that

for all T T the employment statuses of any path-connected agents are positively correlated under T

14 More formally we show that there exists some T suchthat Cov(Sit Sjt) 0 for any path-connected i and j andperiods t and t under the Markov process MT for anysubdivision of T T ldquoStarting under the steady-statedistributionrdquo means that the starting state is randomly drawnaccording to the steady-state distribution and then all ex-pectations account for the dependence of the process on thisinitial randomness This is necessary as Example 1 showsthat starting from some particular states one cannot escapeshort-run negative correlation Starting from the steady-state distribution is ldquoas ifrdquo our expectations are taken wherewe let the process run for a long time and average over alldates that are t t apart from each other


of a 0100 and b 001515 Ask the follow-ing question suppose that a person has beenunemployed for at least each of the last X peri-ods What is the probability that he or she willbe employed at the end of this period Weexamine the answer to this question in Figure6 as we vary the number of periods of observedpast unemployment and the network The en-tries represent the probability that agent 1 willbe employed next period conditional upon thenetwork structure and having been unemployedfor at least the given number of consecutiveprevious periods

In the case where there is no network theprobability of becoming employed is indepen-dent of the observed historymdashit is simply theprobability that an agent hears about a job (andthen does not lose it in this period) Once thereis a network in place the length of the unem-ployment history does give us information itprovides insight into the likelihood that theagentsrsquo connections are unemployed

The patterns observed here are not particularto the example but hold more generally

PROPOSITION 3 Under fine enough subdivi-sions of periods and starting under the steady-state distribution the conditional probabilitythat an individual will become employed in agiven period is decreasing with the length oftheir observed (individual) unemployment spell

Indeed longer past unemployment historieslead to worse inferences regarding the state ofonersquos connections and the overall state of thenetwork This leads to worse inferences regard-

ing the probability that an agent will hear indi-rect news about a job That is the longer i hasbeen unemployed the higher the expectationthat irsquos connections and path connections arethemselves also unemployed This makes itmore likely that irsquos connections will take anyinformation they hear of directly and less likelythat they will pass it on to i In other words alonger individual unemployment spell makes itmore likely that the state of onersquos social envi-ronment is poor which in turn leads to lowforecasts of future employment prospects

As we mentioned in the introduction thisexplanation for duration dependence is comple-mentary to many of the previous explanationsFor instance one (among a number of) expla-nations that has been offered for duration de-pendence is unobserved heterogeneity16 Asimple variant of unobserved heterogeneity isthat agents have idiosyncratic features that arerelevant to their attractiveness as an employeeand are unobservable to the econometrician butobserved by employers With such idiosyncraticfeatures some agents will be quickly reem-ployed while others will have longer spells ofunemployment and so the duration dependenceis due to the unobserved feature of the workerWhile the network model also predicts durationdependence we find that over the lifetime of asingle worker the worker may have differentlikelihoods (which are serially correlated) ofreemployment depending on the current state oftheir surrounding network So it also predictsthat controlling for the state of the networkshould help explain the duration dependence Inparticular it offers an explanation for whyworkers of a particular type in a particular lo-cation (assuming networks correlate with loca-tion) might experience different employmentcharacteristics than the same types of workers inanother location all other variables held con-stant So for example variables such as locationthat capture network effects should interact with

15 These calculations are also from simulations wherehere we can directly calculate these conditional probabili-ties by looking at conditional frequencies on observedstrings of unemployment of X periods long The limit num-bers are obtained analytically and are simply the same ashaving no network

16 Theoretical models predicting duration dependencethough are a bit scarcer In Olivier J Blanchard and PeterDiamond (1994) long unemployment spells reduce the re-employment probability through a stigma effect that inducesfirms to hire applicants with lower unemployment durations(see also Tara Vishwanath 1989 for a model with a stigmaeffect) In Christopher A Pissarides (1992) duration depen-dence arises as a consequence of a decline in worker skillsduring the unemployment spell

FIGURE 6 DURATION DEPENDENCE


other worker characteristic variables whichwould not be predicted by other models17

A Comments on Stickiness in the Dynamicsof Employment

The duration dependence for individuals isreflective of a more general persistence in em-ployment dynamics This persistence can beunderstood by first noting a simple feature ofour model When aggregate employment is rel-atively high unemployed agents have relativelymore of their connections employed and facerelatively less competition for job informationand are more likely to hear about jobs Con-versely when aggregate employment is rela-tively low unemployed agents are relativelyless likely to hear about jobs

To illustrate this point consider the bridgenetwork in Figure 4 that connects ten agentsWe calculate the (average) individual probabil-ity that an unemployed agent finds a job withinthe current period conditional on the total num-ber of employed agents in the network Weprovide calculations for two different pairs ofparameter values a 0100 b 0015 anda b 0050 The probabilities are expressedin percentage terms

When there is no network connecting agentsthe probability that an unemployed agent finds ajob is simply the arrival rate a In contrast whenagents are connected through a network (herethe bridge network of Figure 4) the probabilityof finding a job varies with the employmentstate This conditional probability is a wheneverybody is unemployed but then increaseswith the number of employed agents in thenetwork as shown in Table 1

This state dependence of the probability ofhearing about a job then implies a persistencein aggregate employment dynamics As a net-work gets closer to full employment unem-ployed agents become even more likely to

become employed Symmetrically the lowerthe employment rate the lower the probabilitythat a given unemployed agent hears about ajob18 Although the process oscillates betweenfull employment and unemployment it exhibitsa stickiness and attraction so that the closer itgets to one extreme (high employment or highunemployment) the greater the pull is from thatextreme This leads to a sort of boom and busteffect as illustrated in Figure 7

Starting from full employment Figure 7 plotsthe dynamics of a simulation of aggregate em-ployment over 100 periods for an empty net-work (the dotted line) and for the bridgenetwork of Figure 4 (the plain line) We ran thedynamics for two different parameter pairsFirst when a 0100 and b 0015 theeconomy oscillates between full and high em-ployment in the bridge network while it oscil-lates more widely between high and lowemployment in the empty network An impor-tant feature is that the spells of unemploymentare shorter in the bridge network this is reflec-tive of the fact that unemployed agents hearabout jobs more quickly when the economy isnear full employment The fact that the arrivalrate is relatively high compared to the breakuprate means that the bridge network stays veryclose to full employment most of the time Inthe second simulation a 0050 and b 0050 and so the arrival rate and breakup ratesare equal and the economy oscillates morewidely between high and low employment inboth networks Still the empty network experi-ences lower average employment as we shouldexpect but more importantly the bridge net-work snaps back to full employment more

17 We thank Eddie Lazear for pointing this out to us

18 We have not explicitly modeled equilibrium wagesand the job arrival process Incorporating these effectsmight mitigate some of the effects our model exhibitsHowever taking the arrival process as exogenous helps usshow how the network pushes the process to have certaincharacteristics See Randall Wright (1986) for a searchmodel that generates fluctuating dynamics in a proper mar-ket setting

TABLE 1mdashPROBABILITY OF FINDING EMPLOYMENT FOR AGENTS IN THE BRIDGE NETWORK

Number of employed 0 1 2 3 4 5 6 7 8 9

a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185


quickly when pushed away from it In line withthe stickiness of the process note that the onlysituation where more than two agents are un-employed for more than five periods appearsbetween periods 60 and 80 and there we hit thelowest employment overallmdashwhich is illustra-tive of the relationship between level of unem-ployment and duration

We also point out that employment need notbe evenly spread on the network especially in a

network such as the bridge network from Figure4 As a result temporal patterns may be asyn-chronous across different parts of the networkwith some parts experiencing booms and otherparts experiencing recessions at the same timeThis asynchronous behavior is illustrated in Fig-ure 8 which plots separately over 100 periods theaggregate employment of agents 1 to 5 (the dottedline) and that of agents 6 to 10 (the plain line) inthe bridge network from Figure 4 from a simula-tion with a 0050 and b 0050

IV Dropping Out and Inequality inEmployment

We now turn to showing how the networkmodel has important implications for inequalityacross agents and how that inequality canpersist

Our results so far show that an agentrsquos em-ployment status will depend in important wayson the status of those agents who are path-connected with the agent This leads to someheterogeneity across agents as their networksand the local conditions in their networks willvary Note however that in the absence ofsome structural heterogeneity across agentstheir long-run prospects will look similar Thatis if the horizon is long enough then the im-portance of the starting state will disappear

However expanding the model slightly canintroduce substantial and sustained inequalityamong agents even if their network structuresare identical The expansion in the model comes

FIGURE 7 TIME SERIES OF EMPLOYMENT FOR NETWORKED

VERSUS DISCONNECTED AGENTS

FIGURE 8 ASYNCHRONOUS PATTERNS OF EMPLOYMENT

ACROSS NETWORK SECTIONS


in the form of endogenizing the network byallowing agents to ldquodrop outrdquo of the networkThis decision can be sensitive to starting con-ditions and have lasting and far-reaching ef-fects on the network dynamics Let us take acloser look

Suppose that agents have to decide whetherto be in the labor market network or to drop outWe model this as a once-and-for-all decisionwith no reentry allowed for agents choosing todrop out Staying in the labor market results inan expected present value of costs ci 0 Theseinclude costs of education skills maintenanceopportunity costs etc We normalize the outsideoption to have a value of 0 so that an agentchooses to stay in the labor force when thediscounted expected future wages exceed thecosts

There are two simplifications made here Oneis that we model this as if all agents make theirdecisions simultaneously In reality althoughthe bulk of a given agentrsquos connections may beat similar stages of their lives the agent mayalso be connected to other agents at differentstages in their lives The second simplificationis that the drop-out decision is made just onceImplicit in this is the idea that the bulk of thecosts (education and opportunity) appear at anearly stage of an agentrsquos career and once thosecosts are sunk there is little incentive to dropout These are both clearly crude approxima-tions but reasonable starting points

There are some obvious comparative staticsDrop-out percentages will be decreasing inwages and increasing in costs Drop-out deci-sions also depend on how well an agent isconnected With better connections (for in-stance larger numbers of links holding all elseconstant) there is a larger chance of hearingabout jobs and so the future prospects of em-ployment are higher leading to a higher thresh-old of costs where the agent would choose todrop out

The part that is less transparent but still quiteintuitive is the interaction between the deci-sions of different agents Positive correlation ofemployment for path-connected agents bothwithin and across periods implies that havingmore agents participate is better news for agiven agent as it effectively improves theagentrsquos network connections and prospects forfuture employment Therefore the decisions tostay in the labor force are strategic comple-

ments implying that the drop-out game is su-permodular That is as more of the otherplayers decide to stay in a given playerrsquos deci-sion to stay in is increasingly favored relative todropping out The theory of supermodulargames then guarantees the existence of a max-imal Nash equilibrium in pure strategies Amaximal equilibrium is such that the set ofagents staying in the market is maximal so thatthe set of agents staying in at any other equi-librium of the game is a subset of those stayingin at the maximal equilibrium We restrict at-tention to such maximal equilibria1920

This supermodular aspect of the drop-out de-cisions is where we see the emergence of thecontagion effects discussed in the introductionThe fact that an agent drops out leads to worsefuture employment prospects for that agentrsquosconnections This in turn increases the chancethat those agents drop out and so forth Thusdrop-out decisions are not independently andidentically distributed (iid) even when thecosts of staying in the labor force are iidacross agents This effect as well as how theinitial condition of the network affects drop-outrates are illustrated in the following example

Example 5 (Initial Conditions Dropouts andContagion) To measure the contagion effectwe first ask how many people would drop outwithout any equilibrium effect that is if theyeach did the calculation supposing that every-one else was going to stay in Then we cancalculate how many people drop out in equilib-rium and any extra people dropping out are dueto somebody else dropping out which is whatwe attribute to the contagion effect

For these calculations we take the cost ofstaying in the network ci to be uniformly dis-tributed on [08 1] and fix the per period wageto be w 1 We do the calculations withcomplete networks where each participatingagent is directly linked to every other agent We

19 Formally let di 0 1 denote irsquos decision ofwhether to stay in the labor market where di 1 stands forstaying in A vector of decisions d is a maximal equi-librium if it is an equilibrium and for every other equilib-rium d we have d d where is the component-wise ordering on 0 1n

20 There is a coordination game going on (players mayall wish to drop out if all others do etc) and here lookingat the maximal equilibrium eliminates coordination errorsand allows us to focus on the network effects of the model


compute the percentage of dropouts for differ-ent values of n and we also calculate the per-centage of dropouts due to contagion We do thecalculation for two initial states everybody em-ployed s0 (1 1) and everybody unem-ployed s0 (0 0)

For Tables 2 and 3 the calculations are donefor a discount rate of 09 where we simplifythings by assuming that an agent starts in theinitial state and then jumps to the steady state inthe next ldquoperiodrdquo This just gives us a roughcalculation but enough to see the effects So anagent who stays in gets a net payoff of 01si 09pi ci where pi is the agentrsquos steady-stateemployment probability in the maximal equilib-rium We again set a 0100 and b 001521

So for instance in Table 3 when n 16and everybody is initially unemployed wehave 68 percent of the people dropping out onaverage This means that we expect about 11people to drop out on average and about 5people to stay in The 87 percent due tocontagion means that about 15 (0087 16) of the people dropping out are doing sobecause others drop out and they would bewilling to stay in if all the others were willing

to Thus about 95 of the 11 people woulddrop out even if all stayed in and 15 of the11 drop out because of the fact that someothers have dropped out

Note that the contagion effect is larger forthe worse starting state and is also larger forsmaller networks (although not entirelymonotone) This is true because the impact ofsomeone dropping out is more pronounced inworse starting states and smaller networks Inthe limit the impact of having people dropout is negligible and so the contagion effectdisappears when agents have very large num-bers of connections (holding all else fixed)For n 1 there cannot be a contagion effectso the number is 0 there as well

The nonmonotonicity of the contagion ef-fect in n is a bit subtle The possibility ofcontagion is initially nonexistent It then in-creases as the number of connections in-creases since there are more possiblecombinations of neighbor dropouts that cantake place with three connections (when n 4) than one connection (when n 2) andany single dropout can then trigger anotherEventually with large numbers of connec-tions the marginal impact of an additionalconnection to a given agent is already verylow and in fact becomes second order in thenumber of agents The fact that it shrinks somuch means that eventually the contagioneffect disappears as even having some frac-tion of onersquo s connections drop out is nolonger a problem if there are still a largenumber of connections staying in

The previous example shows that differentsocial groups with identical network relation-ships but differing by their starting employment

21 In these calculations we estimate pi by an approxima-tion formula to save on calculations as we need to iterateon both the cirsquos and the number of agents The approxima-tion formula leads to some slight biases In the simulationsfor any given n we first randomly draw the cirsquos We thencalculate the pirsquos if all agents stay in From this we can findout which agents would drop out We can then recalculatethe pirsquos for the new network and see which agents drop outwith the new pirsquos We continue this process until no addi-tional agents drop out This provides the maximal equilib-rium for this draw of cirsquos We then run iterations on thisalgorithm for new draws of cirsquos and calculate sample aver-ages over the iterations

TABLE 2mdashDROPOUTS AND CONTAGIONmdashSTARTING EMPLOYED

s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3

Drop-out percentage 583 445 262 147 97 78 68Percentage due to contagion 0 88 50 14 04 02 0

TABLE 3mdashDROPOUTS AND CONTAGIONmdashSTARTING UNEMPLOYED

s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3



state have different drop-out rates Becausedropping out hurts the prospects for the groupfurther this can have strong implications forinequality patterns We now show that thisholds more generally

In analyzing the drop-out game for theproposition below we do not alter the networkstructure when an agent drops out instead wesimply set the dropoutrsquo s employment status tobe 0 forever Thus a dropoutrsquo s social contactsdo not change and his or her contacts still passon news about jobs It is as if the agentrsquo sconnections still pass the agent informationthinking that the agent is simply unemployedrather than a dropout This might introduce abias to the extent that information is no longerpassed on to someone who has dropped outAccounting for such effects would complicatethe analysis as one would need to keep trackof any modifications of the network structureas agents drop out

PROPOSITION 4 Consider two socialgroups with identical network structures Ifthe starting state person-by-person is higherfor one group than the other then the set ofagents who drop out of the first group in themaximal equilibrium is a subset of their coun-terparts in the second group These differ-ences in drop-out rates generate persistentinequality in probabilities of employment inthe steady-state distributions with the firstgroup having weakly better employment prob-abilities than their counterparts There is astrict difference in employment probabilitiesfor all agents in any component of the net-work for which the equilibrium drop-out de-cisions differ across the two groups

So we have established that a networkedmodel can generate persistent differencesamong two social groups with identical eco-nomic characteristics except that they differ intheir starting state As mentioned in the intro-duction this is consistent with documented dif-ferences in drop-out rates among blacks andwhites as well as studies that show that ac-counting for voluntary dropouts from the laborforce negatively affect the standard measures ofblack economic progress (eg Chandra 2000Heckman et al 2000) While this comparison isstylized the fact that we consider two com-pletely identical networks except for their start-

ing states emphasizes how important startingconditions can be22

Just to show that the inequality we are seeingis not due to the isolation of the groups of agentswith different starting conditions let us exam-ine drop-out decisions when there is a bridgebetween two groups

Example 6 (Connected Social Groups andDropouts) Consider the network structurefrom Example 3 see Figure 9

Agents 1 to 5 start employed and agents 6 to10 start unemployed

We do drop-out calculations as in Example 5We take the ci to be uniformly distributed on[08 1] fix w 1 use a discount rate of 09and have agents who stay in get a net payoff of01si 09pi ci where pi is the agentrsquossteady-state employment probability in themaximal equilibrium of the drop-out game andsi is their starting employment state

The drop-out probabilities for the differentagents are illustrated in Table 4

Note that the drop-out rates are significantlyhigher for the agents who start in the unem-ployed state even though the network is con-nected The agent who forms the bridge fromthe unemployed side is less likely to drop outthan the other unemployed agents while thecounterpart agent who forms the bridge from

22 Differences in network structure rather than differ-ences in starting conditions can also lead to different drop-out decisions and sustained inequality For instance as wesaw in Example 2 under exactly the same a and b and witheach agent having two links the expected long-run unem-ployment of an agent in a network of four agents is 63percent while it is 6 percent for an agent in a network ofeight agents While the difference in this example is small(on the order of a 5-percent change in unemployment) itcan easily become amplified through the contagion effect

FIGURE 9 THE BRIDGE NETWORK WITH ASYMMETRIC

STARTING STATES


the employed side is more likely to drop outthan other employed agents23 While this exam-ple is clearly highly stylized it does providesome intuitive predictions and shows that start-ing conditions can lead to sustained differencesacross groups even in a connected network

Let us discuss the relation of our analysis ofdrop-out rates to theories of discriminationClassical theories of discrimination such as thatof Gary Becker (1957) or Thomas C Schelling(1971) postulate that individuals have an intrin-sic preference for individuals in their own soci-etal group24 Because of such preferences andexternalities individuals end up segregated inthe workplace and the resulting sorting patternsby group affiliation can breed wage inequal-ity25 Our model offers an alternative and novelexplanation for inequality in wages and em-ployment2627 Two otherwise identical individ-

uals embedded in two societal groups withdifferent collective employment histories (orwith different networks as discussed below)typically will experience different employmentoutcomes In other words social networks in-fluence economic success of individuals at leastin part due to the different composition andhistory of individualsrsquo networks When coupledwith drop-out decisions sustained inequalitycan be the result of differences in history Wediscuss some policy implications of this net-work viewpoint below

V A Look at Policy Implications

Let us mention some lessons that can belearned from our model about policy in thepresence of network effects and some of whichwe will illustrate with some examples belowOne obvious lesson is that the dynamics of themodel show that policies that affect currentemployment will have both delayed and long-lasting effects

Another lesson is that there is a positive exter-nality between the status of connected individualsSo for instance if we consider improving thestatus of some number of individuals who arescattered around the network or some group thatare more tightly clustered there will be two sortsof advantages to concentrating the improvementsin tighter clusters The first is that this will im-prove the transition probabilities of those directlyinvolved but the second is that this will improvethe transition probabilities of those connected with

23 There may also be differences between the drop-outrates of agents 2 and 3 or 8 and 9 as things are notsymmetric but these differences are beyond the accuracy ofour simulations

24 There is also an important literature on ldquostatisticalrdquodiscrimination that follows Arrow (1972) John J McCall(1972) Edmond S Phelps (1972) and others Our work isquite complementary to that work as well

25 We use the word ldquocanrdquo because it may be that someemployers discriminate while the market wages do not endup unequal As Becker (1957) points out the ultimateoutcome in the market will depend on such factors as thenumber of nondiscriminating employers and elasticities oflabor supply and demand

26 While we have not included ldquofi rmsrdquo in our model notethat to the extent to which the job information comesinitially from an employeersquos own firm there would also becorrelation patterns among which firms connected agentswork for That is if an agentrsquos acquaintance is more likelyto be getting information about job openings in the acquain-tancersquos own firm then that agent has a more than uniformlyrandom likelihood of ending up employed in the acquain-tancersquos firm This would produce some segregation patternsbeyond what one would expect in a standard labor marketmodel

27 Two other important explanations for inequality can befound in Glenn C Loury (1981) and Steven Durlauf (1996) Asin our model both papers relate social background to individ-ual earning prospects In Louryrsquos paper the key aspect of

social background is captured by family income which thendetermines investment decisions in education In Durlaufrsquoswork education is modeled as a local public good and com-munity income rather than family incomes affects humancapital formation In both cases because the social backgroundimposes constraints on human capital investment income dis-parities are passed on across generations In our paper wefocus instead on the larger societal group within which one isembedded its network structure collective employment his-tory and access to information about jobs This offers a com-plementary rather than competing explanation for sustainedinequality

TABLE 4mdashDROP-OUTS RATES IN THE BRIDGE NETWORK WITH ASYMMETRIC STARTING

STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093


these individuals Moreover concentrated im-provements lead to a greater improvement of thestatus of connections than dispersed improve-ments This will then propagate through thenetwork

To get a better picture of this consider thedrop-out game Suppose that we are in a situa-tion where all agents decide to drop out Con-sider two different subsidies in the first wepick agents distributed around the network tosubsidize while in the second we subsidize agroup of agents who are clustered together Inthe first case other agents might now just haveone (if any) connection to an agent who issubsidized This might not be enough to inducethem to stay in and so nobody other than thesubsidized agents stay in the market Here themain impact on the drop-out rate is directlythrough the subsidy In contrast in the clusteredcase a number of agents now have severalconnections who are in the market This mayinduce these other agents to stay in This canthen have a contagion effect carrying over toagents connected with them and so on Beyondthe direct impact of the subsidy this has anindirect contagion effect that decreases thedrop-out rate and then improves the future sta-tus of all of the agents involved even furtherthrough the improved network effect

Exactly how one wants to distribute subsidiesto maximize their impact is a subtle matter Welook at an example to highlight the subtleties

A Concentration of Subsidies

Let us again consider a society of eight indi-viduals again where a 0100 and b 0015Suppose the costs of staying in the network ciare drawn at random from a uniform distribu-tion with support [08 1] Initially everybody isunemployed so s0 (0 0) We work withdrop-out decisions when the discount rate is 09as in the previous examples

The experiment we perform here is the fol-lowing In each case we subsidize two agents tostay in the marketmdashsimply by lowering theircost ci to 028 The question is which two agents

we subsidize In the network each agent hasfour connections The network structure is asfollows Each agent has three linksmdashtwo imme-diate neighbors and one that is slightly furtheraway This is pictured in Figure 10

Table 5 provides the percentage of agentswho stay in the network as a function of who issubsidized (two agents in each case) and whatthe range of costs (randomly drawn) are29

There are some interesting things to noteIn the highest cost range even having one

neighbor stay in is not enough to induce anagent to stay and so the only agents staying inare the subsidized ones Here it is irrelevantwhich agents are subsidized as they are the onlyones staying in

In the lowest two cost ranges having oneneighbor stay in has a big impact and sospreading the subsidies out has the maximalimpact Agents 1 and 5 are on opposite endsof the circle and have no direct contact incommon Subsidizing agents 1 and 5 thusamounts for spreading subsidies out and it isindeed the best policy in terms of maximizing

28 This might in fact overestimate the necessary subsidycosts as one might not need to lower ci by so muchMoreover we would end up with even lower subsidy costsby concentrating subsidies which increases the future pros-pects of the subsidized agents Thus lower costs of subsi-

dization would be a further reason for concentratingsubsidies

29 Note that the different cases of who are subsidizedcover all possible configurations up to a relabeling of theagents

FIGURE 10 THE STARTING NETWORK STRUCTURE


the number of agents who stay in the marketwhen the cost is at its lowest level30 Whenthe cost is in the 082 to 1 range we begin tosee a slight change where now subsidizingagent 1 and 6 is better and these agents areslightly closer together

The places where things favor a differentsort of policy is in the middle range of costsHere costs are high enough so that it is quitelikely that an agent will drop out if she hasonly one neighbor who stays in but is lesslikely if she has two neighbors who stay inand so contagion effects are relatively highSpreading the subsidies out to agents 1 and 5or 3 and 7 etc does worse than having themclose together (1 and 2 1 and 3 1 and 4) andthe best possible subsidy here is of the form 3and 8 What matters is the number of contactssubsidized agents have in common Agents 3and 8 are well-placed since both 1 and 2 areconnected to both of them Thus this concen-trates subsidies in a way that provides a highprobability that 1 and 2 will stay in Withoutsuch a concentration of subsidies we get ahigher drop-out rate

What this suggests is that in designing sub-sidy or affirmative action programs attention tonetwork effects is important Concentrating ef-forts more locally can end up having a higher orlower impact depending on the networkconfiguration

VI Possible Empirical Tests

While as we have discussed the model gen-erates patterns of employment and dropouts thatare consistent with the stylized facts from anumber of studies one might want to look atsome additional and more direct tests of themodelrsquos predictions

Note that drop-out rates and contagion effectsdepend both on the costs ranges and on thevalues for the arrival rate and breakup rateSome comparative statics are quite obvious (1)as the expected cost increases (relative towages) the drop-out rate increases (2) as thebreakup rate increases the drop-out rate in-creases and (3) as the arrival rate increases thedrop-out rate decreases However there are alsosome more subtle comparisons that can bemade

For instance let us examine what happens asjob turnover increases Here as the arrival andbreakup rates are both scaled up by the samefactor we can see the effects on the drop-outrates Note that such a change leaves the baseemployment rate (that of an isolated agent) un-changed and so the differences are attributableentirely to the network effects Table 6 pulls outvarious rescalings of the arrival and breakuprates for the two cost ranges when n 4 andagents are related through a complete networkAs before the first figure is the drop-out rateand the second is the amount attributable tocontagion effects

As we can see higher turnover rates (higherrescalings of a and b) lead to higher drop-outrates The intuition behind this is as follows

30 It is almost a tie with 1 and 6 but slightly ahead in thenext decimal

TABLE 5mdashSUBSIDIZATION STRUCTURE AND PERCENTAGE OF AGENTS WHO STAY IN

Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1

1 and 2 529 394 278 2501 and 3 536 394 271 2501 and 4 572 434 279 2501 and 5 579 438 270 2501 and 6 579 440 270 2501 and 7 571 434 278 2501 and 8 535 394 271 2503 and 4 545 393 261 2503 and 7 577 436 274 2503 and 8 562 429 291 250


With higher turnover rates when an agentbecomes unemployed it is more likely thatsome of his neighbors are unemployed Thisis actually quite subtle as the base unem-ployment rate has not changed Howeverhigher turnover makes it more likely that sev-eral agents lose their jobs at the same time andend up competing for information This effectthen lowers the employment rate which in turnfeeds back and results in less valuableconnections

This effect provides for a testable implica-tion industries with higher turnover rates allelse held equal should have higher drop-outrates Operationalizing this requires some carehowever as we do not model the career choicesfor workers or an equilibrium in wages None-theless it is clear that the prediction is that thewage to education cost ratio must be relativelyhigher in order to induce workers to enter ca-reers where the turnover rate is high comparedto those where it is low even after correctingfor any risk aversion or income-smoothingmotives

Let us briefly mention some other possible em-pirical tests of the model To the extent that directdata on network relationships is available one candirectly test the model In fact such information inthe form of survey data (the General Social Sur-vey) has been used extensively in the sociologyliterature and also in conjunction with wage data(eg Troy Tassier 2001)

There are also other tests that are possibleFor instance there is data concerning how thereliance on networks for finding jobs variesacross professions age and race groups etc(see the table in Montgomery 1991 for in-stance to see some differences across profes-sions) Our model then predicts that theintensity of clustering duration dependenceand drop-out rates should also vary across thesesocioeconomic groups Moreover even within aspecific socioeconomic group our model pre-dicts differences across separate components of

the network as the local status of the connec-tions changes

VII Concluding Discussion

As we have mentioned several times we treatthe network structure as given except that weconsider drop-out decisions Of course peoplehave more specific control over whom theysocialize with both in direct choice of theirfriendships as well as through more indirectmeans such as education and career choices thataffect whom they meet and fraternize with on aregular basis Examining the network formationand evolution process in more detail could pro-vide a fuller picture of how the labor market andthe social structure co-evolve by mutually in-fluencing each other network connectionsshape the labor market outcomes and in turnare shaped by them31

In addition to further endogenizing the net-work we can also look deeper behind the in-formation exchange procedure32 There are awide variety of explanations (especially in thesociology literature for instance see Granovetter1995) for why networks are important in jobmarkets The explanations range from assortivematching (employers can find workers with

31 There is a growing literature on the formation ofnetworks that now provides a ready set of tools for analyz-ing this problem See Jackson (2004) for a survey of modelsapplying to networks of the form analyzed here as well asSanjeev Goyal (2004) Frank Page (2004) and Anne vanden Nouweland (2004) for issues relating to learning far-sightedness and cooperation structures

32 Recall that the results stated so far extend to a frame-work more general than the simple communication protocolwhere only direct contacts can communicate with each otherand unemployed direct contacts are treated on an equalfooting In particular the general framework accommodatesa priori ranking among contacts indirect passing of jobinformation heterogeneous jobs with different wages idio-syncratic arrival and breakup rates information passing andjob turnover dependent on the overall wage distribution etcSee Calvo-Armengol and Jackson (2003) for more details

TABLE 6mdashDEPENDENCE OF DROPOUTS AND CONTAGION ON ARRIVAL AND BREAKUP RATES

Scaled by a and b 1 3 5 7 9

005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425


similar characteristics by searching throughthem) to information asymmetries (in hiringmodels with adverse selection) and simple in-surance motives (to help cope with the uncer-tainty due to the labor market turnover) In eachdifferent circumstance or setting there may be adifferent impetus behind the network This mayin turn lead to different characteristics of howthe network is structured and how it operatesDeveloping a deeper understanding along theselines might further explain differences in theimportance of networks across differentoccupations

Another aspect of changes in the network overtime is that network relationships can change asworkers are unemployed and lose contact withformer connections Long unemployment spellscan generate a desocialization process leading to aprogressive removal from labor market opportu-

nities and to the formation of unemploymenttraps This is worth further investigation

Another important avenue for extension ofthe model is to endogenize the labor marketequilibrium so that the probability of hearingabout a job depends on the current overall em-ployment and wages are equilibrium ones Thiswould begin to give insights into how networkstructure influences equilibrium structure

Finally we point out that although our focusin this paper is on labor markets this model caneasily be adapted to other sorts of behaviorswhere social networks play a key role in infor-mation transmission An example is whether ornot individuals take advantage of certain avail-able welfare programs Recent studies by Mari-anne Bertrand et al (2000) and Anna Aizer andJanet Currie (2002) point to the importance ofsocial networks in such contexts

APPENDIX

We first provide some definitions that are necessary for the proofs that follow Here we specializethe definitions to random vectors S (S1 Sn) whose components take on values of 0 or 1 Wefollow the convention of representing random variables by capital letters and realizations by smallletters

AssociationWhile first-order stochastic dominance is well suited for capturing distributions over a single

agentrsquos status we need a richer tool for discussing interrelationships between a number of agents atonce and this is needed in the proofs that follow The following definition is first due to James DEsary et al (1967)

Let be a joint probability distribution describing S is associated if Cov( f g) 0 for all pairs of nondecreasing functions f 0 1n 3 and

g 0 1n 3 where Cov( f g) is the covariance E[ f(S)g(S)] E[ f(S)]E[g(S)]If S1 Sn are the random variables described by a measure that is associated then we say that

S1 Sn are associated Note that association of implies that Si and Sj are nonnegatively correlatedfor any i and j Essentially association is a way of saying that all dimensions of S are nonnegativelyinterrelated33

Strong AssociationAs we often want to establish strictly positive relationships and not just nonnegative ones we

need to define a strong version of association Since positive correlations can only hold betweenagents who are path-connected we need to define a version of strong association that respects sucha relationship

Consider a partition of 1 n that captures which random variables might be positivelyrelated which here will be determined by the components of the graph g

A probability distribution governing S is strongly associated relative to the partition if it isassociated and for any and nondecreasing functions f and g

33 This is still a weaker concept than affiliation which requires association for all conditionals It is imperative that wework with the weaker notion as affiliation will not hold in our setting


Covf g 0

whenever there exist i and j such that f is increasing in si for all si g is increasing in sj for all sj and i and j are in

One implication of strong association is that Si and Sj are positively correlated for any i and j in

DominationConsider two probability distributions and dominates if

E f13 E f13

for every nondecreasing function f 0 1n 3 The domination is strict if strict inequality holdsfor some nondecreasing f

Domination captures the idea that ldquohigherrdquo realizations of the state are more likely under thanunder In the case where n 1 domination reduces to first-order stochastic dominance

LEMMA 5 Consider two probability distributions and on 0 1n dominates if and onlyif there exists a Markov transition function 0 1n3 P(0 1n) [where P(0 1n) is set of allthe probability distributions on 0 1n] such that

s s

sss

where is a dilation (that is ss 0 implies that s s) Strict domination holds if ss 0 forsome s s

Thus is derived from by a shifting of mass ldquoupwardsrdquo (under the partial order ) over statesin S This lemma follows from Theorem 1840 in Charalambos Aliprantis and Kim C Border (2000)

Let

E E 0 1ns E s sf s E

E is the set of subsets of states such that if one state is in the event then all states with at least ashigh employment status (person by person) are also in Variations of the following useful lemmaappear in the statistics literature (eg see Section 33 in Esary et al 1967) A proof of this versioncan be found in Calvo-Armengol and Jackson (2003)

LEMMA 6 Consider two probability distributions and on 0 1n

E E

for every E E if and only if dominates Strict domination holds if and only if the firstinequality is strict for at least one E E The probability measure is associated if and only if

EE EE

for every E and E E The association is strong (relative to ) if the inequality is strictwhenever E and E are both sensitive to some 34

34 E is sensitive to if its indicator function is A nondecreasing function f 0 1n3 is sensitive to (relativeto ) if there exist s and s such that f(s) f(s s) and s and s s are in the support of


The proof of the following lemma is straightforward and omitted

LEMMA 7 Let be associated and have full support on values of S If f is nondecreasing and isincreasing in Si for some i and g is a nondecreasing function which is increasing in Sj for some jand Cov(Si Sj) 0 then Cov( f g) 0

Fix M (g a b) Let PT denote the matrix of transitions between different srsquos under the T-periodsubdivision MT (g aT bT) So Pss

T is the probability that St s conditional on St1 s Let

PsET

sE

PssT

LEMMA 8 Consider an economy M (g a b) Consider s S and s S such that s s andany t 1 Then for all T and E E

PsET PsE

T

Moreover if s s then the inequality is strict for at least one E

PROOF OF LEMMA 8Let us say that two states s and s are adjacent if there exists such that s s and s s

(that is s 1 and s 0) We show that PsET PsE

T for adjacent s and s as the statement thenfollows from a chain of comparisons across such s and s

Let be such that s s By adjacency si si for all i Since sk sk for all k it follows by our definition of pij(s) that pij(s) pij(s) for all j

and for all i These inequalities imply that ProbsT (St) dominates Probs

T(St) where ProbsT is the

probability distribution conditional on St1 s Given that 1 s s 0 we then have thatProbs

T (St) dominates ProbsT(St) The conclusion that PsE

T PsET follows from Lemma 6

To see the strict domination consider E ss s Since there is a positive probability that hears 0 offers under s the inequality is strict

Given a measure on 0 1n let PT denote the measure induced by multiplying the (1 n)vector by the (n n) transition matrix PT This is the distribution over states induced by a startingdistribution multiplied by the transition probabilities PT

LEMMA 9 Consider an economy M (g a b) and two measures and on S For all T if dominates then PT dominates sT Moreover if strictly dominates then PT strictly dominatesPT

PROOF OF LEMMA 9

PT13E PT13E s

PsET s s

By Lemma 5 we rewrite this as

PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET

As the second term depends only on s we rewrite that sum on s so we obtain

PT13E PT13E s

s

sssPsET sPsE

T

Since is a dilation ss 0 only if s s So we can sum over s s

PT13E PT13E s

sss

ssPsET PsE

T

Lemma 8 implies that PsET PsE

T whenever s s Thus since ss 0 and yenss ss 1 the resultfollows

Suppose that strictly dominates It follows from Lemma 5 that there exists some s s suchthat ss 0 By Lemma 8 there exists some E E such that PsE

T PsET Then [PT](E) [PT](E)

for such E implying (by Lemma 6) that PT strictly dominates PT

We prove Proposition 1 and Proposition 2 as follows

PROOF OF PROPOSITION 1Since PT represents an irreducible and aperiodic Markov chain it has a unique steady-state

distribution that we denote by T The steady-state distributions T converge to a unique limitdistribution (see H Peyton Young 1993) which we denote

Let P T be the transition matrix where the process is modified as follows Let pi(s) yenjN pji(s)Starting in state s in the hiring phase each agent i hears about a new job (and at most one) withprobability pi(s)T and this is independent of what happens to other agents while the breakup phaseis as before with independent probabilities bT of losing jobs Let T be the associated (again unique)steady-state distribution and limT T (which is well defined as shown in the proof of Claim 1below)

The following claims establish the proposition

Claim 1

Claim 2 is strongly associated

The following lemma is useful in the proof of Claim 1Let P be a transition matrix for an aperiodic irreducible Markov chain on a finite-state space Z

For any z Z let a z-tree be a directed graph on the set of vertices Z with a unique directed pathleading from each state z z to z Denote the set of all z-trees by Tz Let

(A1) pz Tz

zz Pzz13

LEMMA 10 Mark Freidlin and Alexander D Wentzell (1984)35 If P is a transition matrix foran aperiodic irreducible Markov chain on a finite-state space Z then its unique steady-statedistribution is described by

35 See Chapter 6 Lemma 31 also see Young (1993) for the adaptation to discrete processes


(z)pz

yenzZ

pz

where pz is as in (A1) above

PROOF OF CLAIM 1Given s 0 1n we consider a special subset of the set of Ts which we denote T s This is the

set of s-trees such that if s is directed to s under the tree then s and s are adjacent As PssT goes

to 0 at the rate 1T when s and s are adjacent36 and other transition probabilities go to 0 at a rateof at least 1T2 it follows from Lemma 10 that for large enough TT(s) may be approximated by

yenT s

ss PssT 13

yens

yenT s

ssPssT 13

Moreover note that for large T and adjacent s and s PssT is either bT o(1T2) (when si si for

some i) or pi(s)T o(1T2) (when si si for some i) where o(1T2) indicates a term that goes tozero at the rate of 1T2 For adjacent s and s let Pss

T bT when si si for some i and pi(s)Twhen si si for some i37 It then follows that

(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13

By a parallel argument this is the same as (s)

PROOF OF CLAIM 2Equation (A2) and Claim 1 imply that

s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13

Multiplying top and bottom of the fraction on the right-hand side by T we find that

(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13

where P is set as follows For adjacent s and s (letting i be the agent for whom si si) Pss b whensi si and Pss pi(s) when si si

38 and Pss 0 for nonadjacent s and sThe proof of the claim is then established via the following steps

36 Note that since s and s are adjacent then PssT 0

37 We take T high enough such that all coefficients of the transition matrix P are between 0 and 138 If pi(s) 1 for some i and s we can divide top and bottom through by some fixed constant to adjust without changing

the steady-state distribution


Step 1 is associated

Step 2 is strongly associated

PROOF OF STEP 1We show that for any T and any associated P T is associated From this it follows that if we

start from an associated 0 at time 0 (say an independent distribution) then 0(P T)k is associatedfor any k Since T limk0(P T)k for any 0 (as T is the steady-state distribution) and associa-tion is preserved under (weak) convergence39 this implies that T is associated for all T Then againsince association is preserved under (weak) convergence this implies that limT T isassociated

So let us now show that for any T and any associated P T is associated By Lemma 6we need to show that

(A4) EE EE 0

for any E and E in E Write

EE EE s

sP sEET P sE

T E

Since St is independent conditional on St1 s it is associated40 Hence

P sEET P sE

T P sET

Substituting into the previous expression we find that

EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E

Both P sET and (P sE

T (E)) are nondecreasing functions of s Thus since is associated it followsfrom (A5) that

EE EE s

sP sET

s

sP sET E

Then since yens (s)(P sET (E)) 0 (by the definition of ) the above inequality implies (A4)

PROOF OF STEP 2We have already established association Thus we need to establish that for any f and g that are

increasing in some si and sj respectively where i and j are path-connected

Cov f g 0

39 See for instance P5 in Section 31 of Ryszard Szekli (1995)40 See for instance P2 in Section 31 of Szekli (1995)


By Lemma 7 it suffices to verify that

Cov Si Sj 0

For any transition matrix P let Psij yens Psssisj and similarly Psi yens Psssi Thus these arethe expected values of the product SiSj and Si conditional on starting at s in the previous periodrespectively Let

CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT

It suffices to show that for each i j for all large enough T

CovijT 0

The matrix P T has diagonal entries P ssT which tend to 1 as T 3 while other entries tend to 0

Thus we use a closely associated matrix which has the same steady-state distribution but for whichsome other entries do not tend to 0

Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s

One can directly check that the unique steady-state distribution of P T is the same as that of P T andthus also that

CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT

Note also that transitions are still independent under P T This implies that starting from any s thedistribution P s

T is associated and so

P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT

Note that P siT converges to Psi where Psi is the rescaled version of P (defined above)

Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj

Thus to complete the proof it suffices to show that


(A6) s

sPsiPsj s

sPsi s

sPsj

Viewing Psi as a function of s this is equivalent to showing that Cov(Psi Psj) 0 From Step 1 weknow that is associated We also know that Psi and Psj are both nondecreasing functions of s

First let us consider the case where gij 141 We know that Psi is increasing in si and also thatPsi is increasing in sj for gij 1 Similarly Psi is increasing in sj Equation (A6) then follows fromLemma 7 (where we apply it to the case where Si Sj) as both Psi and Psj are increasing in sj

Next consider any k such that gjk 1 Repeating the argument above since Psj is increasing sjwe apply Lemma 7 again to find that Si and Sk are positively correlated Repeating this argumentinductively leads to the conclusion that Si and Sk are positively correlated for any i and k that arepath-connected

Proposition 1 now follows from Claim 2 since T 3

PROOF OF PROPOSITION 2We know from Claim 2 that is strongly associated The result then follows by induction using

Lemma 9 and then taking a large enough T so that T is close enough to for the desired strictinequalities to hold

PROOF OF PROPOSITION 3For any t t 0 let hi0

tt be the event that Sit Sit1 Sit1 Sit 0 Let hi1

tt be the eventthat Sit 1 and Sit1

Sit1 Sit 0 So hi0tt and hi1

tt differ only in irsquos status at date t Wewant to show that

(A7) PSit 1 1hi00t PSit 1 1hi0

1t

Since (paying close attention to the subscripts and superscripts in the definition of h t t ) P (Sit1

1hi01t) is a weighted average of P (Sit1 1hi0

0t) and P (Sit1 1hi10t) (A7) is equivalent to showing

that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t

PSit 1 1 hi00t 13 PSit 1 0 hi0

0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t

From the two above equations we rewrite (A8) as

(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t

Rearranging terms (A9) is equivalent to

41 If i is such that gij 0 for all j i then strong association is trivial So we treat the case where at least two agents arepath-connected


PSit 1 1 hi00tPSit 1 0 hi1

0t PSit 1 1 hi10tPSit 1 0 hi0

0t

For any let Ei0 be the set of s such that si 0 and Ei1

be the set of s such that si 1Letting be the limiting steady-state distribution we divide each side of the above inequality by

(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00

Thus to establish (A7) it is enough to show that

(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00

Let us show (A10) as the argument for (A11) is analogousThen

PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1

s0Ei00 Ps0s1Ps1s2 Pstst 1

Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1

s0Ei10 Ps0s1Ps1s2 Ps ts t 1

Note that by Proposition 1 (s0Ei10 ) strictly dominates (s0Ei0

0 ) (with some strict inequalitiessince i is connected to at least one other agent) Then by the above equations and Lemma 9 appliediteratively42 we derive the desired conclusion that (A10) is satisfied

PROOF OF PROPOSITION 4Let di 0 1 A vector of decisions d is an equilibrium if for each i 1 n di 1 implies

42 To be careful at each stage we are applying the lemma to P where Pss only has positive probability on s where si 0 except at time t 1 when si 1 It is easy to see that Lemma 9 extends to this variation Also as seen in its proof thelemma preserves some strict inequalities that correspond to the employment status of agents who are path-connected to i Forinstance for j connected to i (Ej1

0 Ei10 ) (Ej1

0 Ei00 ) Through Lemma 9 this translates to a higher probability on Ej1

t

(conditional on starting at Ei10 rather than Ei0

0 ) at each subsequent time through time t which then leads to a strictly higherprobability of i receiving a job offer at time t 1


E t

itSitS0 s di ci

and di 0 implies the reverse inequality The maximal equilibrium corresponding to a starting stateS0 s is denoted d(s) Let s s and d 0 1n We first show that for any T

ETfStS0 s d13 ETf StS0 s d13

Lemma 8 implies that any T-period subdivision and for every nondecreasing f

ETfS1S0 s d13 ETf S1S0 s d13

Lemma 9 and a simple induction argument then establish the inequality for all t 1 The inequalityis strict whenever f is increasing and s s Next let d d For a fine enough T-period subdivisionand for every nondecreasing f given that dropouts have value zero it follows that

ETfS1S0 s d13 ET f S1S0 s d13

As before the inequality extends to all t 1 by induction Again f increasing and d d imply astrict inequality Combining these observations we find that for any T when s s and d d

(A12) ETfStS0 s d13 ET f StS0 s d13

Consider the maximal equilibrium d(s) By (A12) for any T and all t

ETSitS0 s ds13 ET SitS0 s ds13

Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t

itET SitS0 s ds13 ci

and so also for all d d(s) if i is such that d(s)i 1 then

(A13) t

itET SitS0 s d13 ci

Set di d(s)i for any i such that d(s)i 1 Fixing d for such irsquos find a maximal equilibrium ats for the remaining irsquos and set d accordingly By (A13) it follows that d is an equilibrium whenconsidering all agents It follows that d d(s) Given the definition of maximal equilibrium it thenfollows that d(s) d d(s)


REFERENCES

Aizer Anna and Currie Janet ldquoNetworks Neigh-borhoods and the Utilization of Publicly-Funded Prenatal Care in Californiardquo MimeoUCLA 2002

Akerlof George A ldquoSocial Distance and SocialDecisionsrdquo Econometrica September 199765(5) pp 1005ndash28

Aliprantis Charalambos D and Border Kim CInfinite dimensional analysis HeidelbergSpringer-Verlag 2000

Arrow Kenneth ldquoModels of Job Discrimina-tionrdquo in A Pascal ed Racial discriminationin economic life Lexington MA LexingtonBooks 1972

Arrow Kenneth and Borzekowski Ron ldquoLim-ited Network Connections and the Distri-bution of Wagesrdquo Mimeo StanfordUniversity 2001

Becker Gary The economics of discriminationChicago University of Chicago Press 1957

Bertrand Marianne Luttmer Erzo F P andMullainathan Sendil ldquoNetwork Effects andWelfare Culturesrdquo Quarterly Journal of Eco-nomics August 2000 115(3) pp 1019ndash56

Blanchard Olivier J and Diamond PeterldquoRanking Unemployment Duration andWagesrdquo Review of Economic Studies July1994 61(3) pp 417ndash34

Boorman Scott A ldquoA Combinatorial Optimiza-tion Model for Transmission of Job Informa-tion Through Contact Networksrdquo Bell Journalof Economics 1975 6(1) pp 216ndash49

Burt Ronald S Structural holes The socialstructure of competition Cambridge MAHarvard University Press 1992

Calvo-Armengol Antoni ldquoJob Contact NetworksrdquoJournal of Economic Theory March 2004115(1) pp 191ndash206

Calvo-Armengol Antoni and Jackson MatthewO ldquoNetworks in Labor Markets Wage andEmployment Dynamics and InequalityrdquoMimeo Caltech and Universitat Autonomade Barcelona httpwwwhsscaltechedujacksonmdyngenpdf 2003

Card David and Krueger Alan B ldquoSchool Qual-ity and Black-White Relative Earnings ADirect Assessmentrdquo Quarterly Journal ofEconomics February 1992 7(1) pp 151ndash200

Chandra Amitabh ldquoLabor-Market Dropoutsand the Racial Wage Gap 1940ndash1990rdquo

American Economic Review May 2000 (Pa-pers and Proceedings) 90(2) pp 333ndash38

Conley Timothy G and Topa Giorgio ldquoSocio-Economic Distance and Spatial Patterns inUnemploymentrdquo Mimeo New York Univer-sity 2001

Durlauf Steven ldquoA Theory of Persistent IncomeInequalityrdquo Journal of Economic Growth1996 (1) pp 75ndash93

Esary James D Proschan Frank and WalkupDavid W ldquoAssociation of Random Vari-ables with Applicationsrdquo Annals of Math-ematical Statistics October 1967 38(5)pp 1466 ndash74

Farley Reynolds ldquoBlacks Hispanics and WhiteEthnic Groups Are Blacks Uniquely Disad-vantagedrdquo American Economic ReviewMay 1990 (Papers and Proceedings) 80(2)pp 237ndash41

Flinn Christopher and Heckman James ldquoNewMethods for Analyzing Structural Models ofLabor Force Dynamicsrdquo Journal of Econo-metrics 1982 18(1) pp 115ndash68

Freidlin Mark and Wentzell Alexander D Ran-dom perturbations of dynamical systemsNew York Springer-Verlag 1984

Goyal Sanjeev ldquoLearning in Networksrdquo in GDemange and M Wooders eds Group for-mation in economics Networks clubs andcoalitions Cambridge Cambridge Univer-sity Press 2004

Granovetter Mark ldquoThe Strength of WeakTiesrdquo American Journal of Sociology May1973 78(6) pp 1360ndash80

Getting a job A study of contacts andcareers 2nd Ed Chicago University of Chi-cago Press 1995

Heckman James J and Borjas George ldquoDoesUnemployment Cause Future Unemploy-ment Definitions Questions and Answersfrom a Continuous Time Model of Heteroge-neity and State Dependencerdquo EconomicaAugust 1980 47(187) pp 247ndash83

Heckman James Lyons Thomas M and ToddPetra E ldquoUnderstanding Black-White WageDifferentials 1960ndash1990rdquo American Eco-nomic Review May 2000 (Papers and Pro-ceedings) 90(2) pp 344ndash49

Jackson Matthew O ldquoA Survey of Models ofNetwork Formation Stability and Efficiencyrdquoin G Demange and M Wooders eds Groupformation in economics Networks clubsand coalitions Cambridge Cambridge Uni-


versity Press 2004 httpwwwhsscaltechedujacksonmnetsurvpdf

Loury Glenn C ldquo Intergenerational Transfersand the Distribution of Earningsrdquo Econo-metrica July 1981 49(4) pp 843ndash67

Lynch Lisa M ldquoThe Youth Labor Market in theEighties Determinants of ReemploymentProbabilities for Young Men and WomenrdquoReview of Economics and Statistics February1989 71(1) pp 37ndash54

McCall John J ldquoThe Simple Mathematics ofInformation Job Search and Prejudicerdquo inA Pascal ed Racial discrimination in eco-nomic life Lexington MA LexingtonBooks 1972

Montgomery James ldquoSocial Networks and La-bor Market Outcomes Toward an EconomicAnalysisrdquo American Economic Review De-cember 1991 81(5) pp 1408ndash18

ldquoJob Search and Network Composi-tion Implications of the Strength-of-Weak-Ties Hypothesisrdquo American SociologicalReview October 1992 57(5) pp 586ndash96

ldquoWeak Ties Employment and In-equality An Equilibrium Analysisrdquo Ameri-can Journal of Sociology March 1994 99(5)pp 1212ndash36

Page Frank ldquoFarsighted Stability in NetworkFormationrdquo in G Demange and M Wooderseds Group formation in economics Net-works clubs and coalitions CambridgeCambridge University Press 2004

Phelps Edmond S ldquoThe Statistical Theory ofRacism and Sexismrdquo American EconomicReview September 1972 62(4) pp 659ndash61

Pissarides Christopher A ldquoLoss of Skill DuringUnemployment and the Persistence of Em-ployment Shocksrdquo Quarterly Journal of Eco-nomics November 1992 107(4) pp 1371ndash97

Rees Albert ldquo Information Networks in LaborMarketsrdquo American Economic Review May

1966 (Papers and Proceedings) 56(2) pp559ndash66

Schelling Thomas C ldquoDynamics Models of Seg-regationrdquo Journal of Mathematical Sociol-ogy 1971 1(1) pp 143ndash86

Schweitzer Stuart O and Smith Ralph E ldquoThePersistence of the Discouraged Worker Ef-fectrdquo Industrial and Labor Relations Review1974 27(2) pp 249ndash60

Smith James P and Welch Finis R ldquoBlack Eco-nomic Progress after Myrdalrdquo Journal ofEconomic Literature June 1989 27(2) pp519ndash64

Szekli Ryszard Stochastic ordering and depen-dence in applied probability New YorkSpringer-Verlag (Lecture Notes in Statistics 97) 1995

Tassier Troy ldquoLabor Market Implications of theBreadth of Weak Tiesrdquo Mimeo Universityof Iowa 2001

Topa Giorgio ldquoSocial Interactions Local Spill-overs and Unemploymentrdquo Review of Eco-nomic Studies April 2001 68(2) pp 261ndash95

van den Nouweland Anne ldquoStatic Networks andCoalition Formationrdquo in G Demange and MWooders eds Group formation in econom-ics Networks clubs and coalitions Cam-bridge Cambridge University Press 2004

Vishwanath Tara ldquoJob Search Stigma Effectand Escape Rate from UnemploymentrdquoJournal of Labor Economics October 19897(4) pp 487ndash502

Watts Duncan J and Strogatz Steven H ldquoCol-lective Dynamics of lsquoSmall-Worldrsquo NetworksrdquoNature June 1998 393(4) pp 440ndash42

Wright Randall ldquoJob Search and Cyclical Un-employmentrdquo Journal of Political EconomyFebruary 1986 94(1) pp 38ndash55

Young H Peyton ldquoThe Evolution of Conven-tionsrdquo Econometrica January 1993 61(1)pp 57ndash84


One is then left to explain why drop-out ratesshould differ across races An analysis of socialnetworks provides a basis for observing bothhigher drop-out rates in one race versus anotherand sustained inequality in wages and employ-ment rates even among those remaining in thelabor force as follows4 The starting point ofour model is that the better the employmentstatus of a given agentrsquos connections (eg rel-atives friends acquaintances) the more likelyit is that those connections will pass informationconcerning a job opening to the agent Thismight be for any number of reasons One is thatas the employment status of a connection im-proves it is less likely that the connection willwant to keep a new job for him or herselfAnother is that as the employment status ofother agents in the network improves the morelikely that a given agentrsquos connections will beemployed and a source of information and themore likely that they will choose to pass news tothe given agent rather than to some other agentwho already has a job This sort of informationpassing leads to positive correlation betweenthe employment status of agents who are di-rectly or indirectly connected in the networkwithin a period and across time as we establishbelow

Such correlation of employment is ob-served in the data both in the indirect form ofthe inequality mentioned previously and alsoin terms of direct measured correlation pat-terns Giorgio Topa (2001) demonstrates geo-graphic correlation in unemployment acrossneighborhoods in Chicago and also finds asignificantly positive amount of social inter-actions across such neighborhoods TimothyConley and Topa (2001) find that correlationalso exists under metrics of travel time oc-cupation and ethnicity and that ethnicity and

race are dominant factors in explaining cor-relation patterns

The positive correlation that we establish be-tween the employment of agents in a networkthen provides a basis for understanding the sus-tained difference in drop-out rates and resultinginequality in employment rates Consider twoidentical networks except that one starts witheach of its agents having a better employmentstatus than their counterparts in the other net-work Now consider the decision of an agent toeither remain in the labor market or to drop outRemaining in the labor market involves somecosts which could include things like costs ofskills maintenance education and opportunitycosts Agents in the network with worse initialstarting conditions have a lower expected dis-counted stream of future income from remain-ing in the network than agents in the networkwith better initial starting conditions This mi-nor difference might cause some agents to dropout in the worse network but remain in thebetter network The important observation isthat dropping out has a contagion effect Whensome of an agentrsquos connections drop out thatagentrsquos future prospects worsen since theagentrsquos network is no longer as useful a sourceof job information Thus some agents con-nected to dropouts also drop out due to thisindirect effect This can escalate so a slightchange in initial conditions can lead to a sub-stantial difference in drop-out decisions As alarger drop-out rate in a network leads to worseemployment status for those agents who remainin the network we find that slight differences ininitial conditions can lead to large differences indrop-out rates and sustained differences in em-ployment rates5

Before proceeding to the model let us alsomention a fourth feature that is also exhibited bythe model Unemployment exhibits duration de-pendence and persistence That is when condi-tioning on a history of unemployment theexpected probability of obtaining a job de-creases in the length of time that an agent hasbeen unemployed Such duration dependence iswell documented in the empirical literature forexample in studies by Stuart O Schweitzer and

economics James P Smith and Finis R Welch (1989)using statistics from census data break the gap down acrossa variety of dimensions and time The gap is roughly on theorder of 25 percent to 40 percent and can be partly ex-plained by differences in skill levels quality of educationand other factors [eg see Card and Krueger (1992)Chandra (2000)] See Reynolds Farley (1990) for a com-parison of labor market outcomes for 50 racial-ethnicgroups in the United States

4 In this paper we focus on differences in drop-out ratesand employment and we refer the reader to Calvo-Armengol and Jackson (2003) for an analysis of wageinequality

5 Similarly small differences in network architecturesmay also lead to sharp differences in drop-out patternsacross two groups with identical starting conditions

















pij s

a if si 0 and i j

a

yenksk0


0 otherwise













EMPLOYMENT


































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES




































































pij s

a if si 0 and i j

a

yenksk0


0 otherwise













EMPLOYMENT


































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES























































pij s

a if si 0 and i j

a

yenksk0


0 otherwise













EMPLOYMENT


































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES






































































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES


























































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES












































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES

































































a 0100 b 0015 100 104 120 145 179 207 254 257 287 344a 0050 b 0050 50 59 62 69 86 93 113 122 150 185



































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES






















































































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES










































































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES






























































s0 (1 1) n 1 n 2 n 4 n 8 n 16 n 32 n 3



s0 (0 0) n 1 n 2 n 4 n 8 n 16 n 32 n 3
















STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES


































































STARTING STATES















STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES


































































STATES

Agent 1 2 3 4 5 6 7 8 9 10

Drop-out rate 047 042 042 042 042 091 093 093 093 093




























Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES















































































Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES































































Agents subsidized

Cost range

080 to 1 082 to 1 084 to 1 086 to 1















005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES

































































005 0015 015 0045 025 0075 035 0105 045 0135ci [08 1] 6927 7627 8326 8824 9620ci [06 1] 243 283 345 375 425







APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES


























































APPENDIX













Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































Covf g 0




E f13 E f13




s s

sss



Let




E E


EE EE








PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES

























































PsET

sE

PssT


PsET PsE

T














PROOF OF LEMMA 9

PT13E PT13E s

PsET s s


PT13E PT13E s

PsET

s

sss s

We rewrite this as


PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































PT13E PT13E s

s

sssPsET

s

sPsET


PT13E PT13E s

s

sssPsET sPsE

T


PT13E PT13E s

sss

ssPsET PsE

T











Claim 1




(A1) pz Tz

zz Pzz13




(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































(z)pz

yenzZ

pz





yenT s

ss PssT 13

yens

yenT s

ssPssT 13




(A2) s limT3

yenT s

ssPssT 13

yens

yenT s

ssPssT 13



s limT3

yenT s

ssPssT 13

yens

yenT s

ss PssT 13


(A3) s

yenT s

ssPss13

yens

yenT s

ssPss13












(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES


























































(A4) EE EE 0


EE EE s

sP sEET P sE

T E


P sEET P sE

T P sET


EE EE s

sP sET P sE

T P sET E

or

(A5) EE EE s

sP sET P sE

T E



EE EE s

sP sET

s

sP sET E




Cov f g 0




Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES






















































Cov Si Sj 0


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT


CovijT 0



Let

P ssT TP ss

T if s s1

ss

TP ssT

if s s


CovijT

s

TsP sijT

s

TsP siT

s

TsP sjT



P sijT P si

TP sjT

Therefore

CovijT

s

TsP siTP sj

T s

TsP siT

s

TsP sjT


Pss TPss if s s1

ss

T Pss if s s

It follows that

limT3

CovijT

s

sPsiPsj s

sPsi s

sPsj



(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































(A6) s

sPsiPsj s

sPsi s

sPsj













1t




that


0t

By Bayesrsquo rule

PSit 1 1hi00t

PSit 1 1 hi00t


0t

and

PSit 1 1hi10t

PSit 1 1 hi10t


0t


(A9)PSit 1 1 hi0

0t


0t

PSit 1 1 hi10t


0t






0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES























































0t



(Ei00 )(Ei1

0 ) to obtain

PSit 1 1 hi00t

Ei00

PSit 1 0 hi10t

Ei10

PSit 1 1 hi1

0t

Ei10

PSit 1 0 hi00t

Ei00


(A10)PSit 1 1 hi0

0t

Ei00

PSit 1 1 hi1

0t

Ei10

and

(A11)PSit 1 0 hi1

0t

Ei10

PSit 1 0 hi0

0t

Ei00


PSit 1 1 hi00t

Ei00

s0Ei0

0

s1Ei0

1

st 1E i1

t 1

s0

Ei00

Ps0s1Ps1s2 Pstst 1

which we rewrite as

PSit 1 1 hi00t

Ei00

s0

s1Ei0

1

st 1Ei1

t 1


Similarly

PSit 1 1 hi10t

Ei10

s0

s1Ei0

1

st 1E i1

t 1






0 Ei10 ) (Ej1


t




E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































E t

itSitS0 s di ci











Thus

t

itETSitS0 s ds13

t

itET SitS0 s ds13

If d(s)i 1 then

t

itETSitS0 s ds13

t



(A13) t

itET SitS0 s d13 ci



REFERENCES





















































REFERENCES













































































The Effects of Social Networks on Employment and … Effects of Social Networks on Employment and Inequality ... graphic correlation in unemployment across ... EFFECTS OF SOCIAL NETWORKS

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