Social Interactions and Endogenous Association. October 2006

Transcription

1 Socal Interactons and Endogenous ssocaton October 26 Bruce. Wenberg Oho State Unversty, IZ, and NBER Department of Economcs 945 North gh Street Columbus, Oho BSTRCT Ths paper develops a model of socal nteractons wth endogenous assocaton. People are assumed to nvest n relatonshps to maxmze ther utlty. Even n a lnear-n-means model, when assocatons are endogenous, the effect of macro-group composton on behavor s non-lnear and vares across ndvduals. We also show that larger groups facltate sortng. Usng data on assocatons among hgh school students, we provde a range of evdence consstent wth our model. Indvduals assocate wth people whose behavors and characterstcs are smlar to ther own. Ths tendency s stronger n large groups. We also show that behavors vary wthn and between macro-groups n the way predcted by endogenous assocaton. I apprecate helpful comments from Larry Blume, Ed Glaeser, Bryan Graham, Larry Katz, Kevn Lang, Lung-fe Lee, Erzo Luttmer, Pat Reagan, and Qngyan Shang and from partcpants at the 25 E Meetngs, the Unversty of labama, IZ, arvard Unversty, and the London School of Economcs. Qngyan Shang provded excellent research assstance. I am also grateful for support from the Mershon Center at Oho State Unversty and the NBER, where much of the work on ths paper was done. I am, of course, solely responsble for all errors.

2 Socal Interactons and Endogenous ssocaton I. Introducton Socal scentsts have ncreasngly turned to socal nteractons models, n whch peoples behavors are affected by ther socal groups, to understand large dspartes n behavors and outcomes, and especally the low outcomes among underrepresented and economcally dsadvantaged groups. Economsts nterested n socal nteractons have focused on controllng for the effect of selecton nto macro-groups, such as schools or neghborhoods, on estmates of socal nteractons. 2 Whle analogous forces presumably operate wthn these macro-groups, to the best of our knowledge ths paper s the frst to systematcally study the effects of endogenous assocaton wthn groups. The absence of work on ths queston s troublesome for, whle selecton nto groups almost surely bases estmates of the effects of groups up, endogenous assocaton wthn groups lmts the nterpretaton of conventonal estmates and means that they are almost surely based down. If assocatons are endogenous, socal groups wll have non-lnear effects and ndvdual characterstcs and group composton wll nteract n determnng behavor, markedly changng the polcy mplcatons of socal nteractons models. Ths paper develops a formal model of wthn-group assocatons. We assume that ndvduals choose both ther behavor and ther assocatons to maxmze ther utlty, whch depends on both ther own and ther assocates behavors and characterstcs. We specfy a cost n terms of tme and effort to an ndvdual of assocatng wth the other See, for nstance, Wlson [987, 996]; Massey and Denton [993]; and Jargowsky [997]. Early studes often contan weak controls for macro-group selecton (Datcher [982]; Corcoran, Gordon, Laren, and Solon [992]). 2 Recent attenton has focused on controllng for macro-group selecton. See surveys by Jencks and Mayer [99]; Detz [22]; and aurn, Detz, and Wenberg [22] and theoretcal work by Bayer and Ross [26]. More recent studes seek to dentfy random varaton n socal groups. Such studes nclude Bayer, Ross, and Topa [24]; Bertrand, Luttmer, and Mullanathan [2]; Borjas [995]; Case and Katz [99]; Cutler and Glaeser [997]; Evans, Oates, and Schwab [992]; Glaeser, Schenkman, and Sacerdote [996, 23]; oxby [2]; Ioanndes and Zabel [Forthcomng]; Solon, Page, and Duncan [2]; Topa [2]; Wenberg [2] and studes n footnote 4.

3 members of hs macro-group. We also assume and provde evdence that people obtan the most utlty from assocatng wth others whose behavors and characterstcs are smlar to ther own, known as homophly n socology. We specfy a model that yelds the standard mplcaton that behavors depend lnearly on the (weghted) average of the characterstcs and actons of the other members of an ndvdual s macro-group, wth the weghts n our model determned by the amount of ther nteracton. Even wth ths lnear behavoral model, when assocatons are endogenous, macro-group composton has a non-lnear effect on behavor and the effect of macro-groups vares wth ndvdual characterstcs and behavors. 3 These nonlneartes arse when assocatons are endogenous because assocates behavors and characterstcs wll be non-lnear functons of own characterstcs. To the best of our knowledge, we are the frst to generate these non-lneartes endogenously. We apply our model to understand assocaton patterns n schools. In that context, students who are nclned to substance use n a school-grade where few other students are nclned toward substance use wll be forced to assocate wth people who do not use substances, dscouragng ther own tendency to substance use. s the number of students nclned to substance use ncreases, these ndvduals wll fnd more lke-mnded assocates accentuatng ther tendences. t a polcy level, these non-lneartes and nteractons mply that even n a lnearn-means model, socal nteractons are not zero-sum. To reduce the amount of a partcular behavor, one would want to avod placng people who are prone to that behavor n groups where there are many smlar people n these groups, there wll be enough other people nclned toward the behavor wth whom to assocate to generate a 3 Krvo and Peterson [996]; Galster, Querca, and Cortes [2]; Galster [22]; and Wenberg, Reagan, and Yankow [24]; and Burke and Sass [26] provde evdence for non-lnear effects and nteractons. Non-lnear effects have been argued for at least snce Crane [99], although there are few formal mcromodels of these effects (Querca and Galster [Forthcomng] dscuss theores). 2

4 hgh level of the behavor. In terms of estmaton, these non-lneartes also provde a potental soluton to Mansk s [993] reflecton problem that s qute dfferent from exstng solutons (Brock and Durlauf [2a,b]; Lee [26]; Blume and Durlauf [25]; Ln [25]; and Shang [26]). Many studes of socal nteractons explot relocatons of ndvduals across macro-groups. 4 Most such studes nvolve relocatng ndvduals who are at-rsk along some dmensons from socal groups where many ndvduals are at rsk to socal groups wth a mx of people who are and are not at rsk. These studes fnd surprsngly small effects of socal groups. Our model provdes an explanaton for these small effects. It ndcates that ndvduals relocated n ths way wll tend to re-segregate wthn ther new groups, attenuatng any benefts, even f fnely defned groups do have effects. Consstent wth ths hypothess, ngrst and Lang [24] and Klng, Ludwg, and Katz [25] fnd larger effects for grls than boys, who may ntegrate more nto ther new socal groups, and Clampet-Lundqust, Edn, Klng, and Duncan [25] provde evdence that the dfference s due to dfferences n ntegraton nto new neghborhoods. Emergng studes show that the assocatons that arse wthn macro-groups are not random, but emprcal work on socal nteractons has not taken account of these results. 5 Most studes relate an ndvdual s behavor to the mean n hs macro-group, mplctly assumng that all members of the macro-group nteract equally. 6 Some 4 Studes nclude Plotnck and offman [995]; Ladd and Ludwg [997]; aronson [998]; Rosenbaum, DeLuca, and Mller [999]; Katz, Klng, and Lebman [2]; Ludwg, Duncan, and rschfeld [2]; Sacerdote [2]; Oreopoulos [23]; Zmmerman [23]; Gould, Lavy, and Passerman [24a, 24b]; Jacob [24]; Wenberg, Reagan, and Yankow [24]; Klng; Ludwg; and Katz [25]; and ngrst and Lang [24]. 5 number of papers show that people prefer to assocate wth others of the same racal, ethnc, or relgous group (Moody [2]; Bayer, McMllan, and Rueben [22]; Marmaros and Sacerdote [23]; Bsn, Topa, and Verder [24]; Mayer and Puller [26] see, however, Ross [23]). People who are dfferent from ther macro-groups are less lkely to be mpacted by them (Duncan, Connell, and Klebanov [997]; DPasquale and Kahn [999], Cummngs, DPasquale, and Kahn [2]; Conley and Topa [2]; and oxby [2]). 6 Exceptons are Conley and Udry [2] and Bandera and Rasul [22] and Foster [26]. Bertrand, Luttmer, and Mullanathan [22], Munsch [22], Grodner and Knesner [26]; and oxby and 3

5 theoretcal analyses allow for varatons n assocatons wthn macro-groups. nseln [988]; Lee [2]; Brock and Durlauf [2a, 2b] specfy a fxed weghtng matrx that gves assocaton patterns. Ths approach gnores that nteractons are determned by behavors as much as behavors are determned by nteractons (see Coleman [96]; Wseman [22]; and Eder [23] provdes vvd examples). person who does not smoke, for nstance, s unlkely to choose to assocate wth someone who smokes heavly, and so the frst person s smokng s unlkely to be much affected by the other person s smokng. Weghtng-matrx studes provde no gudance as to who a newly added person wll assocate wth, nor do they ndcate how addng or removng people affects the assocatons of people who are already n the macro-group and reman n t (Sobel [2] makes these ponts n a partcularly strkng form). Ths gap s troublesome gven that most polces to address the effects of socal nteractons nvolve movng people across macro-groups. Ths study employs data from the Natonal Longtudnal Study of dolescent ealth (dd ealth), whch surveyed all students n 72 schools, askng about the respondents closest frends as well as ther background and behavors. We provde a range of evdence consstent wth our model of endogenous assocaton. We fnd that the majorty of varaton n frend s behavors and characterstcs arses wthn grades n schools. Indvduals assocate wth others whose behavors and characterstcs are smlar to ther own. Consstent wth our model, we fnd that so long as there are a few people wth a partcular characterstc n a group, people wth that characterstc assocate wth each other, and that people avod assocatng wth people who are dfferent from themselves untl most of the group s comprsed of people who are dfferent. We show Wengarth [26] use nformaton to specfy the group across nteractons operate. Conley and Topa [2] estmate propenstes for racal groups to nteract usng a structural model. These papers do not explctly study the assocaton process, nor do they study how assocaton-patterns are affected by changes n the populaton or n the behavors of group members. 4

6 that group sze facltates segregaton and show that there s a stronger relatonshp between own characterstcs and assocates characterstcs n large macro-groups. Whle estmatng the effect of endogenous sortng on behavors s complcated, we provde evdence that endogenous assocaton affects behavor. s ndcated, our model mples that the avalablty of some lke-mnded ndvduals permts people to assocate wth people wth the same behavoral tendences, accentuatng ther own behavoral tendences. Consstent wth ths mplcaton, we show the mean behavor among the assocates of people who are nclned to a partcular behavor (based on ther exogenous characterstcs) are hgher (relatve to people whose characterstcs mply that they are unlkely to have hgh levels of the behavor) n groups where there some people nclned to the same behavor. We also show that the avalablty of people who are nclned to a partcular behavor ncreases the behavors of people who are prone to that behavor relatve to people whose characterstcs mply that they are unlkely to have hgh levels of the behavor. Lastly, we provde some two-stage least squares estmates of the effect of assocates behavors on own behavor. II. Model II.. The Framework Ths secton develops a model of socal nteractons wth endogenous assocaton. People are characterzed by observable characterstcs, x, and an shock, ε. They choose an acton, y, and a set of assocatons. n ndvdual wth characterstc x and shock ε who takes acton y has a utlty of, U 2 = β xy + εy y Prvate Utlty ct { Cost of ssocaton ψ 2 α + θy y y + ω xy + x Φx + γ x y T Socal Utlty Ths utlty functon has three components: a prvate utlty; a cost of nteractng; and a socal utlty. Varables wth overbars and -superscrpts denote means among a person s assocates ( 2 y denotes the mean of the squared values of assocates actons). To 5

7 smplfy notaton, t s convenent to defne γ = α and a z z T, whch s the mean of some varable z among a person s assocates weghted by the total amount of nteractons. In the emprcal analyss, we consder a varety of actons and outcomes. The actons nclude substance use and behavoral problems. We also study grades, whch depend on own actons, whose utlty depends on assocates actons, and on assocates grades (f there are human captal spllovers). Interactons can be costly n terms of tme and effort. To capture these costs parsmonously, we assume that people ncur a cost, c, per unt of tme spent nteractng wth others, where T ( x, y),ε denotes the total tme that the person spends nteractng wth others. 7 We assume that assocatons are bnary. Because of random encounters, people are assumed spend t N passvely assocatng wth every other member of ther macro-group. ere N gves the sze of the macro-group, so total passve assocatons do not depend on group sze. People can also actvely assocate wth partcular members of ther macro-group, spendng an addtonal unt of tme for a total of + N. t The socal utlty has both standard and novel components. In Mansk s [994] termnology, θ and γ gve the endogenous and exogenous socal effects. The parameters ψ, ω, and Φ have no counterparts n Mansk s framework, because they do not affect the acton drectly. They affect the utlty of assocatons and actons through ther effect on assocatons. gh values of ψ reduce the utlty of assocatng wth people wth extreme values of y. The parameter ω allows the utlty of assocatng wth someone to vary wth the other person s behavor, y, and for the effect to depend on a person s own characterstcs. For nstance, people wth behavoral problems, may produce dsutlty. 6

8 The parameter Φ s analogous, but reflects the effect of assocates characterstcs on the utlty of assocatng. For nstance, people lkely obtan utlty from assocatng wth people who are attractve or athletc and people of the same race or ethncty. Once people are vewed as choosng to assocate wth a porton of ther groups, t becomes mportant to consder how changes n the number of assocates affect the strength of nteractons. One vew, whch comes closest to the lterature, s that new assocatons completely crowd out exstng ones, leavng the total effect of assocates fxed, that α =. One mght thnk that people who have many assocates are affected by ther assocates (as a whole) more than people wth few assocates, that α >. II.B. Solvng the Model Dfferentaton wth respect to y gves the person s optmal acton, ( θy + γx ) α + ε y = β x + T. (*) Ths s the standard lnear-n-means behavoral equaton, where the total socal effect wll vary across people f α. We consder a model wth a bnary, scalar characterstc (Wenberg [25] dscusses a model wth contnuous characterstcs), and begn by elmnatng the acton to focus on assocatons on characterstcs. We assume that the characterstc takes on values x {, } low and hgh types. Let, and refer to people wth these characterstcs as π denote the share of the group wth x = {, } Φ = and that the other socal utlty parameters (θ, ψ, ω, and γ) are all zero. Let π [, π ] person wth characterstc j denote the measure of people wth characterstc. We assume x wth whom a x j assocates, whch must be non-negatve and s constraned by group composton. Wth Φ > and the values of the characterstc takng on opposte sgns, people only assocate wth people of the same type, so that a type- person chooses [ π ] π,. The frst order condton for actve assocatons s, 7 The cost could be negatve f the opportunty cost of tme s low and people lke nteractng. 7

9 U π x = α ( α ) x x ( t + Nπ ) c = subject to π [, π ] Φ. (**) Fgure llustrates the assocatons of a hgh (the results for lows are reversed). U Panel shows the desred value of π, whch solves = π gnorng the group composton constrant that π π π, as a functon of. It s possble to show that the desred π s non-ncreasng (and decreasng f α < ) n π. The fgure also shows a 45º lne, whch gves the maxmum feasble value of π. Up to three regons may be present. In regon, the desred actve assocatons exceed the maxmum possble gven the populaton composton. In the second regon, the composton constrant does not bnd and actve assocatons are postve. In the thrd regon, the person chooses not to actvely assocate. s the bold curve n the fgure shows, actve assocatons ncrease n regon ; decrease n regon 2; and are zero n regon 3. To fully characterze the model, the fgures show all three regons, but the thrd may be less relevant, than the others especally wth sortng on multple dmensons. Panel B shows x, where x denotes the value of x among type people, as a U functon of π. The curve labeled = π shows the desred value of x. It s possble to show that ths curve s non-decreasng (when α < ). Intutvely, when more of the group s hgh, a greater share of passve assocatons wll be wth hghs, whch decreases hghs motvaton to actvely assocate wth other hghs (so long as α < ) partally, but not fully, offsettng the ncrease n of x the person can acheve by assocatng wth all avalable hghs, [ π x + ( π ) xl ] ( t + Nπ ) α x. lso shown are the maxmum value Max t + π Nx x, whch s concave f α <, and the value that = 8

10 emerges n the absence of actve assocatons, [ x ( π ) x ] t α π L +, whch s lnear. The bold curve shows that x ncreases monotoncally, wth the greatest ncrease at low levels (n regon ). Except on the left (and extreme rght), changes n group composton have less effect on hghs assocates than mpled by the average characterstcs n the group (gven by the dagonal lne from (, ) to (, ) x L. In regon 2, ntegratng macro-groups has x less effect on people s assocates than mpled by macro-group means because people selectvely assocate wth others who are smlar to themselves. Panel C shows that the dfference n hump-shaped n π x between hghs and lows, L x x, s. Intutvely, n groups that are largely composed of one of the groups, there s lttle scope for sortng, so the assocatons of both types are relatvely smlar. Segregaton s greatest when there are many members of both groups. II.C. Group Sze novel mplcaton of our model s that sortng should be greater n large groups than n small groups. Panel of Fgure 2 shows that ncreasng group sze reduces the desred level of π. Ths can be seen from (**) where N and π only appear multplcatvely, so as N ncreases, the same socal nfluence can be obtaned wth a proportonately lower level of π. Panel B shows that x ncreases n regon and s otherwse unchanged (the ncrease n N corresponds to more assocatons n regon, but s offset by the decrease n π n regon 2). Increasng group sze shfts the Max x curve up because the amount of weght placed on passve assocatons does not change, but for any gven level of π, there are more hghs wth whom other hghs can assocate. The model mples more sortng n large groups when ether group s assocatons are constranted. 9

11 Glaeser, Sacerdote, and Schenkman [996] and Graham [25] estmate the strength of socal nteractons usng varatons across large groups or varatons n group sze. Insofar as large groups facltate sortng one mght expect more varaton n large groups. Endogenous assocaton lkely bases estmates usng Glaeser, Sacerdote, and Schenkman s method upward, but estmates usng Graham s method downward. 8 t a polcy level, our results on macro-group sze ndcate that attempts to ntegrate people from dfferent groups wll be more successful f macro-groups are kept small, lmtng peoples ablty to re-segregate wthn groups. II.D. Crowdng Out s ndcated, once the number of assocatons s endogenous, one needs to thnk about how the number of assocates someone has affects the strength of the socal nfluence he experences. By focusng on assocates means, exstng models mplctly assume that addtonal assocatons fully crowd out exstng ones ( α = ), but t seems lkely that people wth more assocates wll be affected by them more. U x x Wth complete crowdng out, (**) reduces to = Φx cn =. π t + Nπ ere ncreasng π shfts weght from the average assocate to own types. When a person s very dfferent from hs group and assocates (.e. when x and x are very dfferent) a person wants to actvely assocate more, although the constrant π π s lkely to bnd. On the other hand, the more assocates a person has (the larger t Nπ + ), the less each addtonal assocate does to change x. When there s no crowdng out U ( α = ), (**) reduces to = x Φx c π. In ths case, the person ether actvely 8 I am grateful to Bryan Graham for dscussng these ponts.

12 assocates wth all people of hs type (f x Φx > c ) or none. When α (,) assocatons are qualtatvely lke those when α = n that there s some crowdng out. Fgure 3, Panel shows that n Regon 2 ncreasng crowdng out (reducng α) reduces actve assocatons (shfts down the desred level of π ), although t may ncrease t on the left of regon 2. Ths happens for both of the reasons dscussed above. Wth strong crowdng out, addtonal assocatons shft weght from the average assocate to the margnal assocate, and as the group becomes more smlar to the average assocate, there s less beneft from reducng weght on them. (ctve assocatons can ncrease at the left of regon 2 because people who are dfferent from ther group and assocates can reduce the weght on them.) Wth crowdng out, once a person s assocatng wth many people, addtonal assocatons have less effect on the mean. Panel B s drawn under the assumpton that t =, so that n the absence of actve assocatons, there s no effect of α on x. Increasng crowdng out shfts down the Max U x and = π curves. The Max x curve falls because x depends on α T, and as α declnes U α T declnes (so long as T>). The = π curve declnes for ths reason and because π declnes for the reasons above. Panel C shows that ncreasng crowdng out reduces the dfference n the mean assocatons between hghs and lows. Ths effect s partcularly large when most of the group s made up of one type or the other and n the mddle. Thus, the assumpton of complete crowdng out, whch s mplct n the lterature, lmts the scope for people to segregate wthn ther macro-groups by actve assocatng. II.E. Endogenous ctons So far we have consdered models wth sortng on exogenous characterstcs. Ths

13 secton focuses on a bnary acton, y {, }, assumng that people derve utlty from assocatng wth people takng the same acton ( θ > ). (For smplcty, we assume that the other socal utlty parameters, ψ, Φ, ω, and γ, are zero.) Condtonal on the dstrbuton of actons, sortng on an endogenous behavor s dentcal to sortng on an exogenous characterstc. Characterzng the equlbrum nvolves solvng for the share of the group takng an acton as a functon of the dstrbuton of exogenous characterstcs n the group. person wth characterstc x takes acton y y f, 2 2 x β + yε y + y θy > yl x β + ylε + y Lθy L yl. 2 2 Let ε * ( y y ) L y θy y y L L y θy L * * and ε β + ε x denote the value of ε at whch a person wth x = or x = x s ndfferent between the actons. For smplcty, we agan assume that x {, } wth characterstc. The probablty that a person * * x takes the low (hgh) acton s ρ F( ε ) ( ρ F( ε ) L where F (). denotes the cumulatve dstrbuton functon of ε. Let ρ = ρ π + ρ ( π ) ), L denote the share of the group takng the hgh acton, where π denotes the share of the group wth the hgh characterstc. The effect of π on the mean acton can be calculated by dfferentatng ths expresson wth respect to π and rearrangng. Formally, ρ π = + * * F( ε L ) F( ε ) * * ( F' ( ε ) π + F' ( ε )( π )) L * ε ρ, where 2

14 * ε ρ = y + y θ y L y ρ π y y ρ π L L y L G y π G y π L cn θ y cn θ y L y L π ρ y yl π L L ρ If there s some crowdng out, f people choose to actvely assocate, and f L L <. y > and y <, t s possble to show that L ε * ρ wll be negatve and that t wll be more negatve when ρ (and π ) are ether low or hgh. 9 t these extremes, ρ π wll be large because ts denomnator wll be small. Fgure 4 shows the share of the group and people wth the hgh and low characterstc takng the hgh acton as a functon of the share of the group wth the hgh characterstc and the share of the group wth the hgh acton assumng that ε s dstrbuted unformly wth bounds such that people are always n the nteror of the support. Ths assumpton s appealng because t mples that behavors are lnear n characterstcs and assocatons. In both panels, the share of each type takng the hgh acton s non-lnear. The steeper sectons at low and hgh levels emerge because at low levels, ntroducng a few hghs makes t possble for hghs to fnd lke-mnded assocates, whle at the hgh end, elmnatng the few remanng hghs makes t mpossble for lows to fnd lke-mnded assocates. Whle the unform case s useful for demonstratng that the model generates nonlneartes endogenously, t s probably more realstc to assume that the shock, ε, s 9 Under these assumptons, the frst term s negatve and more negatve whenever ether of the groups has fewer actve assocatons, whch wll be at extreme values of ρ (and π ). The second and thrd terms are zero whenever people s actve assocatons are unconstraned or zero and negatve when the constrant π π bnds, whch wll be at extreme values of ρ (and π ). s π ncreases, these terms become 3

15 normally dstrbuted. In ths case, t s possble to show that the gap n both behavors and assocatons between people wth the hgh and low characterstc are hump-shaped. Fgure 5 shows results for normally dstrbuted shocks. The dfference n the acton between people wth the hgh and low characterstc s ncreasng n the share of the group that has the hgh characterstc and the share of the group wth the hgh acton. The relatonshp between the average behavor n the populaton and the share of the group wth the hgh characterstc remans non-lnear, but n ths case the non-lnearty s dfferent (convex at low levels and concave at hgh levels). The dfference n the mean behavor of the assocates of people wth the hgh and low characterstc are also humpshaped n the share of the group wth the hgh characterstc (and the hgh behavor). These mplcatons are tested below. II.F. Implcatons number of mportant mplcatons arse mmedately from the non-lnear less negatve, so these terms are also most negatve at extreme values of ρ (and π ). Defne ρ = ρ ρ L as the dfference n behavor between people wth the hgh and low characterstc. The effect of the mean behavor n the group on ths dfference s gven by * d ρ dε * * = ( F' ( ε L ) F' ( ε ). dρ dρ s ndcated above, ε ρ * * ε <, ( F' ( ) F' ( ε ) * s negatve and more negatve at low and hgh values of ρ. Because * * * ε L ε L s negatve when ρ s low (and ε s hgh) and t s postve when ρ * s hgh (and ε s low). Thus, ρ s ncreasng when ρ <. 5 and decreasng when ρ >. 5 and the dfference n behavor between people wth the low and hgh characterstc s hump shaped n ρ. Defne y = y y L ρ as the dfference n the mean behavor of the assocates of people wth the low and hgh characterstc. From the results above on sortng on exogenous characterstcs y y s humpshaped n ρ. Gven that ρ s hump-shaped n ρ, so s * * * dstrbuton where ( F' ( ε L ) F' ( ε ) s negatve when ε s hgh postve when unform dstrbuton where people move beyond the support. L y. Note that these results hold for any * ε s low, ncludng a 4

16 relatonshp between group composton and behavor. Frst, t breaks the zero-sum mplcaton of the standard, lnear-n-means model. number of studes across a range of dscplnes have argued for a non-lnear relatonshp between group composton and behavor (see footnote 3 for references). To the best of our knowledge, our model s the frst to generate non-lneartes endogenously. In the precedng unform example, f one quarter of the populaton had the hgh characterstc and the populaton was large enough for three groups, f the hghs were dvded equally among the three groups, 3.9% of the populaton (68.4% of hghs and 8.4% of lows) would take the hgh acton. If the populaton were dvded nto 2 groups wth no hghs and one that was three quarter hghs, 23.% of the populaton (8.6% of hghs; none of the lows wth other lows, and 3.6% of the lows n the three-quarter hgh group) would take the hgh acton. Thus, groupng the hghs wth enough other hghs to allow them to heavly assocate wth each other generates hgh overall behavor. The exstng lterature places partcular emphass on the multplers generated when peoples actons depend on the actons of the other people n ther socal groups. The strength of these endogenous effects s not dentfed n the tradtonal lnear-nmeans model of socal nteractons because the expected behavor of assocates s a lnear functon of assocates observable characterstcs (see Mansk [994]; Brock and Durlauf [2a,b]; Blume and Durlauf [26]; Ln [25]; Lee [26]; and Shang [26]). Endogenzng assocatons overturns these results n two ways. Frst, the nonlnear relatonshp between the composton of a group and the mean behavor can, n prncple, dentfy the model s parameters. The dstncton between tradtonal endogenous and exogenous effects tself becomes less nterestng once assocatons are treated as endogenous because changes n the dstrbuton of exogenous characterstcs can generate multplers through assocatons. Lastly, endogenous assocaton lmts the nterpretaton of conventonal estmates 5

17 of the effects of macro-groups because the effect of macro-groups vares across people and wth group composton. Whle one can accurately estmate the effect of movng a person lke those beng studed between the groups studed, t s mpossble to estmate the effect of movng smlar people between dfferent types of groups or people who are dfferent between the groups those beng studed. Put dfferently, one can estmate the effect of the groups studed on the people studed, but there s no sngle effect of groups to estmate. s ndcated, movng people who are at rsk along some dmenson from groups where many ndvduals are at rsk to groups wth a mx of people who are and are not at rsk s lkely to have less effect on the nfluences a person experences than one would expect based on macro-group composton because people can fnd lkemnded assocates n ther new groups. II.G. ctons and Characterstcs Drectly ffect Interactons Some behavors nherently generate utlty for one s assocates, whle others generate dsutlty. For nstance, t may be enjoyable to assocate wth people who party, but unpleasant to assocate wth people wth behavoral problems. People wll assocate wth people takng actons that generate utlty for ther assocates, causng those actons to prolferate f θ >. For the present purposes t s useful to have x,. The vector ω governs the effect of assocates actons on utlty. When ω > ( ω < 6 ), the acton rases (lowers) assocates utltes. Fgure 6 compares results from a model where ω = [ ] (sold curves) to one where = [ ] that [ ] ω (dashed curves). Both models assume β = and θ =. Thus, the only dfference s that n the second case everyone derves utlty from assocatng wth people takng the hgh acton as well as from assocatng wth people takng ther own acton. s shown n the top left panel, when people derve utlty from assocatng wth people takng the hgh acton, the mean acton

18 among the assocates of people takng the low acton s ncreased. Intutvely, whle they prefer to assocate wth other people takng ther own acton (because θ = ), the effect s offset by the utlty obtaned from assocatng wth people takng the hgh acton. s shown n the top rght panel, the change has no effect on the assocatons of people takng the hgh acton. s shown n the bottom left panel, for any gven composton of the populaton, the acton s hgher when people derve utlty from assocatng wth people takng the hgh acton. s shown n the bottom rght panel, ths result arses because more of both types take the hgh acton. Thus, people assocate more wth people takng hgh levels of pleasant actons, causng those actons to spread. By contrast, people tend to avod people takng hgh levels of unpleasant actons, whch wll reduce the amount of those actons. It s well known that actons for whch θ s hgh wll spread, but these results mply that for a gven value of θ, actons for whch ω s hgh wll spread too. nalogous effects arse f the characterstcs that affect behavor also affect the utlty from assocatng. For nstance, f athletc or attractve people generate utlty for ther assocates, people wll assocate wth athletes and attractve people and, consstent wth casual emprcsm, spend a lot of tme on athletcs and appearance. Ths result can be seen n Fgure 7, whch compares results from a model where Φ = (sold curves) to one where Φ = (dashed curves). s above, β =, = x. In both cases we assume that [ ] ω = θ, and [ ]. t an optmum, people only assocate wth other people wth the same acton who have the hgh characterstc. s Fgure 7 shows, s degenerate. s the top panel of Fgure 7B shows, the mean of x s hgher except when the dstrbuton of x y s hgher for people wth 7

19 the low acton, but lower for people wth the hgh acton. Intutvely, when Φ = 2, people wth the low acton wll assocate wth anyone else takng the low acton, regardless of ther characterstcs, but when Φ = 2, they stop assocatng wth people takng the low acton who have the low characterstc, whch reduces ther total assocatons and ncreases the mean behavor among ther assocates. For people takng the hgh acton, mposng Φ = 2 reduces y especally when most of the group has the low acton and low characterstc, because t stops them from assocatng wth people wth the hgh acton, but the low characterstc. s the bottom panel of the fgure shows, except n groups where almost everyone has the low characterstc, actons are hgher n the group as a whole and among people wth the hgh and low characterstc. Thus, the actons taken by people whose characterstcs generate utlty for others wll spread. II.. Peer Pressure People often descrbe socal nteractons models as models of peer pressure. Peer pressure nvolves the wthdraw of assocaton when one person takes an acton that produces dsutlty for another. To model peer pressure, assocatons must be endogenous. To capture ths dea, we assume that people derve utlty drectly from havng people assocate wth them and, n choosng ther actons, take nto consderaton how ther acton wll affect whether other people assocate wth them. Let the utlty functon be, U 2 = β xy + εy y Prvate Utlty ct { Cost of ssocatng ψ 2 + θy y y + ω xy + x Φx + γ x y T α + δp, Socal Utlty where P gves the person s popularty, the number of people assocatng wth hm. Returnng to the case of a contnuous acton, the frst order condton for the acton s, y = βx + α P ( θy + γx ) T + δ + ε y. (* ) 8

20 The condton now ncludes a term for the effect of a change n the acton on the measure of people who wll assocate wth hm. Under peer pressure, a person s acton depends on the densty of people at the margn to nteract wth hm as well as the mean of assocates behavors and characterstcs. In ths sense, the standard lnear-n-means model of nteractons may be a poor representaton of peer pressure. The strength of recepton and transmsson of peer pressure wll lkely depend on characterstcs. On the recepton-sde, nsofar as most people n the macrogroup wll want to assocate wth them regardless of ther actons, hghly attractve people wll be less subject to peer pressure than others. People whose characterstcs make them very undesrable as assocates may also be less subject to peer pressure nsofar as few people wll assocate wth them regardless of ther actons. Thus the model suggests that the extent to whch people are subject to peer pressure as recpents follows an nverse-u n how appealng they are as assocates. On the transmsson-sde, people who are attractve may get more weght n other people s P s, so people wll be partcularly subject to peer pressure from attractve people. The strength of peer pressure from the transmsson sde wll be strctly ncreasng n how appealng someone s as an assocate. II.I. Extensons Ths secton consders two further extensons to the model endogenous popularty and dslke. One could extend the analyss of popularty by makng one of the ndvdual characterstcs, x, be the measure of people n the macro-group who assocate wth a person. Presumably, people obtan more utlty from nteractng wth people who are popular. In the context of peer pressure, people wll be partcularly nterested n havng popular people assocate wth them. When popularty has effects, an nterventon that ncreases the weght placed on one person or set of people wll generate a feedback to other people n the macro group, 9

21 whch can generate multplers as one person begns to assocate wth a person that rases the utlty that others derve from assocatng wth hm. Such an effect explan the emphass placed on affectng how much weght people place on others, by makng postve (or negatve) examples of people, such as valedctorans (Mofftt [2] dscusses a varety of polces). Lastly, one mght allow people to nvest tme down-weghtng some members of ther macro-group. People would optmally want to down-weght people who are most dfferent from them. Wthout crowdng out, people who are most dfferent from ther macro-groups wll nvest the most to down-weght others. The possblty of downweghtng reduces the benefts from placng people wth bad outcomes nto better groups. III. Data We use data from the Natonal Longtudnal Study of dolescent ealth, commonly known as dd ealth. The dd ealth data provde nformaton on wde range of youth behavors, ncludng substance use, rsky behavors, grades, and behavoral problems, and data on famly background. They contan natonally representatve data on 9,8 students enrolled n grades 7-2 n 72 schools. Students were asked about ther 5 closest male and female frends, so nformaton s avalable on both the macro groups schools and grades as well as the subgroupngs that emerge wthn them. lso, all people are surveyed symmetrcally. We restrct attenton to students enrolled n 9 th through 2 th grades n classes wth at least students. The data contan a lmted panel aspect, whch we do not utlze. ppendx Table lsts the behavors and outcomes used and ther constructon. In most cases varables were gven as the frequency of a behavor. In these cases, we coded them so as to gve a monthly frequency. lternatve surveys provde nformaton about partcular ndvduals and ask the respondents about the behavors of ther frends. Under ths approach less nformaton s avalable on frends than on respondents 2

22 Table reports descrptve statstcs. 2 Slghtly under half the sample s male; 6% dentfy as black (respondents were allowed to report multple races); 5% dentfy as spanc. Just over three quarters of respondents lve wth ther father, and ther mothers have 3.6 years of schoolng on average. verage grades vary from a B- to a B, wth the lowest grades n math and the hghest grades n hstory and socal studes. On average the respondents smoked a bt more than once a week, they drnk a couple of tmes a month, and get drunk about half as frequently. The most common behavors or behavor problems are havng trouble payng attenton and dong homework regularly. Skppng school and fghtng are the least common, occurrng about once a month. lso shown are the means and standard devatons of the assocate averages of the behavors and characterstcs. The mean of the assocates behavors are above the mean of own behavors, ndcatng that people wth worse behavor are lsted as frends more often than people wth better behavor. On the other hand, the mean of assocates grades tend to be slghtly hgher than own grades, ndcatng that popularty s ncreasng n grades. Smlarly, the mean of mother s educaton and father present among assocates s above the own mean for these varables, ndcatng that chldren wth better famly backgrounds may be more appealng because they are wealther or more socally adept. IV. Wthn-Group Varatons n ssocatons s ndcated, most studes of socal effects focus on varatons across macrogroups. We begn by nvestgatng how much of the varaton n frend s behavors and characterstcs arses wthn macro-groups as opposed to between macro-groups. Meanngful macro-groups should be reasonably narrowly defned yet contan a large porton of peoples assocates. School-grades meet ths crteron wth 74% of the and reports of frends behavor may be based dfferently than reports of own behavor. 2 The orgnal sample comprses 9,8 ndvduals. Once the sample s restrcted to ndvduals n grades 9 through 2 wth vald school, grade, and dentfcaton nformaton, the sample s reduced to 62,23. Restrctng the sample to people wth some dentfable frends wth vald nformaton reduces the sample to 47,57. The remanng deletons are for havng mssng data for a varety of the other varables. 2

23 surveyed frends beng n the person s school-grade. Let z j denote a behavor or characterstc of the jth frend of person. The mean value of z among person s frends s z = N j z j, where N gves the number of people wth whom s frends. ere and below, by studyng 22 z we mplctly focus on the case where α = (work that estmates α s n progress). We decompose the varance of z nto wthn and between school-grade components. The total varance n [ z ] VR[ z E[ z S, G ] + VR[ E[ z S G ] z s VR =,. The former term represents the varance arsng wthn school-grades and the later the varance arsng across schoolgrades. The share of the varance arsng wthn school-grades s the 2 R from an analyss of varance wth a full set of school-grade nteractons, whch s reported the frst column of Table 2. The estmates ndcate that for most behavors over 9% of the varaton n frends behavors arses wthn school-grades. For grades (and televson watchng) about 8% of the varaton n frends behavors and outcomes arses wthn school grades. Between 75% and 9% of the varaton n famly background, measured by mother s educaton, mother s a homemaker, and father s presence arses wthn schoolgrades. Consderably more of the varaton n the racal and ethnc composton of frends can be accounted for by school-grade effects, frequently close to half, whch s to be expected gven the extent of racal and ethnc resdental sortng. Because people often form frendshps wth people who lve near them, a porton of the wthn school-grade varaton lkely arses from resdental sortng among students n the same school-grade. Trackng and selecton nto classes may also generate wthn school-grade varatons n assocatons. The dd ealth survey does not contan data on trackng or the classes taken, but t does contan characterstcs of the census block group of resdence for a porton of respondents. Because neghborhood characterstcs are only

24 avalable for one respondent n fve, column 3 reports the 2 R from the analyss of varance descrbed above when appled to the smaller sample; column 4 reports results when the neghborhood varables nteracted wth school effects are added to the model. Swtchng to the smaller sample tends to ncrease the 2 R slghtly because the sample sze declnes dramatcally relatve to the number of effects. Incluson of neghborhood varables typcally accounts for 5% of the varance n frends behavor. Even n ths smaller sample wth neghborhood varables, wthn macro-group varatons account for 8% or more of the varaton n assocates behavors; 3%-7% of the varatons n assocates race and ethncty; 7% of the varaton n assocates grades and famly background. To the extent that people choose ther frends based on ther behavors and there s an assocaton between neghborhood characterstcs and behavor, these estmates wll overstate the effect of resdental proxmty. The substantal varatons n assocates behavors can not be vewed as the effect of sortng nsofar as exogenous varatons n assocatons combned wth socal effects on behavor wll generate varatons n assocates behavors. Nevertheless, we fnd substantal varatons n assocates famly backgrounds wthn macro-groups, and these varatons are not a consequence of assocatons. Regardless of causalty, our estmates ndcate gnorng varatons n assocatons wth macro-groups leads to substantal msmeasurng the socal nfluences to whch people are exposed. V. The Choce of ssocates The precedng analyss ndcates that the vast majorty of varaton n the composton of socal groups arses wthn narrowly-defned macro-groups. We begn by studyng the determnants of assocatons on exogenous characterstcs (race, ethncty, and famly background). It s easer to study sortng on exogenous characterstcs than behavors and outcomes because, as ndcated, f socal nteractons operate, behavors and outcomes are affected by others. That s, the relatonshp between a person s 23

25 behavor and those of hs assocates may reflect the effect of hs behavor on hs assocates as opposed to the effect of selectve assocaton. Our results on sortng on exogenous characterstcs are of nterest n ther own rght nsofar as sortng on racal, ethnc, and soco-economc lnes at the formatve ages studed here lkely affects atttudes toward other groups. V.. Own Characterstcs and ssocates Characterstcs We begn by regressng the mean of assocates characterstcs on own characterstcs. Let x denote a characterstc of person. We regress the mean x among s assocates, x, on x and antcpate a postve relatonshp. Our model s x = x X SG ε. φ + γ + Π + Person s own observable characterstcs, X (other than x ), are ncluded as controls. lso ncluded are fxed effects fully nteractng school and grade, SG. Wth these school-grade effects, we estmate whether people wth hgher levels of x have assocates wth hgher values of x compared to others n the same school-grade. The estmates reported n Table 3 ndcate a strong postve relatonshp between own characterstcs and those of peoples assocates. There are a number of reasons to be careful about nterpretng these estmates as the causal effect of a person s characterstcs on hs assocatons. s dscussed above, the estmated relatonshps may reflect sortng n other arenas such as neghborhoods or n classes or tracks wthn school-grades. Moreover, our model mples that people wll selectvely assocate based on behavors as well as characterstcs. If ths form of sortng s present, our estmates wll reflect the effect of sortng on behavors that are affected by the characterstcs. To address neghborhood-based selecton, we augment our earler models by ncludng neghborhood characterstcs. The remanng columns of the table report estmates from the orgnal specfcaton for the sample wth neghborhood characterstcs 24

26 and then estmates that nclude the neghborhood varables. The substantal reducton n the sample reduces the precson of the estmates, but nether the change n sample, nor the ncluson of the neghborhood varables substantally affects the pont estmates. V. B. Macro-Group Composton Under the assumpton that people prefer to assocate wth others who are smlar to themselves, the model mples that for people who have a partcular (bnary) characterstc, there should be a concave relatonshp between the share of ther assocates that have that characterstc and the share of the group that has that characterstc. For people who do not have that characterstc, the model mples that there should be a convex relatonshp between the share of ther assocates that have that characterstc and the share of group that does. The dfference n the share of assocates who have the characterstc between people wth and wthout t wll be hump-shaped. Intutvely, once a small number of people wth a characterstc are ntroduced, the people who also have that characterstc wll be able to have assocates who are smlar to themselves; people wthout the characterstc wll not be forced to assocate wth people wth t untl most of the group has the characterstc. We test ths hypothess by regressng the share of peoples assocates who have some exogenous bnary characterstc, the characterstc, G x x, on cubcs n the share of the group who have, and these cubcs nteracted wth the person s value for the characterstc, x. We focus on race and ethncty because of the strength of sortng on them (as shown n Table 3). 3 Formally, we estmate, 3 Results are not reported for Indans because less than 3% of school-grades are 5% or more Indan. 25

27 ( ) ( ) G G 2 G 3 G G 2 G 3 x = γ x + γ x + γ x + γ x + γ x + γ x { ssocate Mean Cubc n Mean Characterstc n Grade γx + πx X x Cubc n Mean Characterstc n Grade Own Characterstc ΠS x G, Race Race { Own Full Racal Composton School Interacted Fxed Character wth Effects Characterstcs Interacted wth own Race stc + The model also ncludes ndvdual characterstcs to control for the drect effect of characterstcs on assocatons; nteractons between the racal composton of the grade and the ndvdual s race (n case, for nstance, the share of the grade that s whte affects the nteractons between sans and blacks); and school fxed effects nteracted wth the characterstc. Our estmates are dentfed from dfferences n the assocatons across students n dfferent grades n the same school. The estmated polynomals, are reported n table 4. Fgure 7 plots the mpled share of people s assocates wth a gven characterstc and the share of the group wth that characterstc, wth each pont representng a sngle school-grade. (Echenque, Fryer, and Kaufman [26] report smlar results, but wthout school-grade fxed effects, obtaned at the same tme.) s expected, the dfference n the relatonshp between people wth and wthout the characterstcs are globally concave. The curves for people wth (own) and wthout (other) the characterstc are not globally concave or convex, but they generally ndcate the expected relatonshps and the non-concavtes or nonconvextes are generally n the regons wth the least densty. V. C. Macro-Group Sze The model mples that large macro-groups facltate sortng so that there should be a stronger relatonshp between ndvdual characterstcs and assocates characterstcs n large macro-groups. 4 To assess how school-grade sze affects selectve assocaton, we estmate + ε 4 The mean and standard devaton of grade sze are 236 and 26 (4.28 and.672 n logs) when weghted by students and 48 and 8 (4.427 and.427 n logs) when the data are un-weghted. 26

28 x SG G ( N ) + θx + γx + πx x + ΠSG ε = β x * log +. lso ncluded n the regresson are ndvdual characterstcs (ncludng the one beng studed), nteractons between the characterstc under nvestgaton and the share of the school-grade wth that characterstc, and a set of school-grade fxed effects. Our estmates are dentfed from dfferences n the strength of the relatonshp between own characterstcs and those of assocates across school-grades of varyng szes. Under the hypothess that grade sze facltates selectve sortng β >. The estmates are reported n Table 5. For all of the characterstcs (except father present), an ncrease n the sze of the school-grade ncreases the relatonshp between own behavor or characterstcs and those of assocates. Thus, our estmates ndcate that larger macro-groups do facltate sortng. VI. Behavors Ths secton studes how endogenous assocaton affects behavor. s ndcated, one can not drectly study the effect of sortng on endogenous varables n the same way as sortng on exogenous varables because endogenous varables may be affected by sortng. To address ths concern, we employ a mult-step procedure n whch we predct both ndvdual and group behavors usng exogenous characterstcs. We then study how changes n (predcted) behavor n a macro-group affects the behavor of people whose behavor s predcted to be hgh (or low) based on ther characterstcs. Formally, we begn by estmatng the relatonshp between behavors and exogenous ndvdual characterstcs, y = X X SG 2 2 γ + γ + Π + ε. ere we have parttoned the matrx of characterstcs, 2 X, where X = [ X X ] 2 for the effect of macro-groups on behavors. X, nto two components, X and. The model also ncludes school-grade fxed effects to control 27