Terms of Endearment: An Equilibrium Model Of Sex and Matching

Transcription

1 Ters of Endearent: An Equilibriu Model Of Sex and Matching Peter Arcidiacono Andre Beauchap Marjorie McElroy Duke University Boston College Duke University May 5, 2011 ABSTRACT: We develop a directed search odel of relationship foration hich can disentangle ale and feale preferences for types of partners and for different relationship ters using only a cross-section of observed atches. Individuals direct their search to a particular type of atch on the basis of (i) the ters of the relationship, (ii) the type of partner, and (iii) the endogenously deterined probability of atching. If en outnuber oen, they tend to trade a lo probability of a preferred atch for a high probability of a less-preferred atch; the analogous stateent holds for oen. Using data fro National Longitudinal Study of Adolescent Health e estiate the equilibriu atching odel ith high school relationships. Variation in gender ratios is used to uncover ale and feale preferences. Estiates fro the structural odel atch subjective data on hether sex ould occur in one s ideal relationship. The equilibriu result shos that soe oen ould ideally not have sex, but do so out of atching concerns; the reverse is true for en. We thank Aloysius Sio as ell as seinar participants at Boston College, Calgary, Georgeton, Pittsburgh and Princeton. This paper uses data fro Add Health, a progra project designed by J. Richard Udry, Peter S. Bearan, and Kathleen Mullan Harris, and funded by a grant P01-HD31921 fro the Eunice Kennedy Shriver National Institute of Child Health and Huan Developent, ith cooperative funding fro 17 other agencies. Special acknoledgent is due Ronald R. Rindfuss and Barbara Entisle for assistance in the original design. Persons interested in obtaining data files fro Add Health should contact Add Health, Carolina Population Center, 123 W. Franklin Street, Chapel Hill, NC (addhealth@unc.edu). No direct support as received fro grant P01-HD31921 for this analysis.

2 1 Introduction With respect to en cheating, That s a thing that girls let slide, because you have to.... If you don t let it slide, you don t have a boyfriend (UNC coed). 1 With respect to the success of his arriage,... If I had arried soeone ho as ore educated or taller than [y ife] Thuy, I don t think she ould have been happy here ith e (Korean farer). 2 These quotes fro individuals facing unfavorable gender ratios indicate circustances in hich individuals ay sacrifice either their preferred relationship ters or partner type for a higher chance of atching. This paper presents a to-sided odel of relationship foration hich identifies separate preferences for en and oen, enabling the analysis of such trade-offs. Using data on current high school relationships, e present strong evidence that, copared to oen, en have a uch stronger preference for relationships ith sex. Thus, hen en are relatively scarce, oen agree to sexual relations out of atching concerns. Disentangling ale and feale preferences regarding sexual behaviors requires that, ceteris paribus, the extra utility fro a given change in the ters of the relationship ust differ beteen en and oen. Moreover, as the search behaviors of en and oen are rarely observed, 3 e need to be able to identify the separate 1 Quoted by Willias (2010) in an article discussing social life and relationships at the University of North Carolina, Chapel Hill hich has a 40% ale and 60% feale student body. 2 Quoted by Onishi (2010) in Wed to Strangers, Vietnaese Wives Build Korean Lives, the second of to articles describing brokered arriages for rural South Korean en facing a shortage of potential brides. 3 One exception is data fro an on-line atching site collected by Hitsch, Hortacsu and Ariely (2010) and used, in part, to investigate the characteristics that en and 2

3 preferences of en and oen fro data on existing atches. In order to disentangle ale and feale preferences fro observed atches, e need to specify to features of the arket. First is ho partners are assigned. In contrast to Choo and Sio (2006), Sio (2009) and Fox (2010), the assignent of partners in our odel is not efficient: not everyone ho desires to be atched ill be. Instead, e allo for search frictions hereby the ex ante yield fro searching for a partner can only be knon probabilistically. Further, individuals are able to target their search toards individuals ith particular characteristics (directed search), potentially trading off the probability of finding a partner against partner quality. Second is ho behaviors in relationships are deterined. Here e again take a targeted search approach, soehat analogous to age posting odels such as Burdett and Mortensen (1998). In age posting odels, orkers kno ahead of tie hat age they ill receive at a particular eployer should they be hired. Our odel incorporates directed search on both partner characteristics and relationship ters. The supply of en and oen of different characteristics, coupled ith the directed search odel allos to uncover differences in ale and feale preferences. In particular, e rely on the copetitive behavior of en and oen hen searching for a partner. The ain idea is that hen en outnuber oen, e tend to observe relationships characterized by hat oen ant and conversely if oen outnuber oen value in a potential partner. They used individual eail contact decisions to estiate separate preferences for en and oen. 3

4 en. 4,5 Men and oen target their searches not only based upon the characteristics of the partner but also on the ters of the relationship. For exaple, a an ay choose to search for a oan of a specific race here the relationship ould include sex. With the ters of the relationship specified up front, utility is nontransferable. The probability of successfully finding a atch then depends upon the nuber of searchers on each side of the arket looking for each cobination of race and relationship ters. Searchers face a trade-off beteen having a lo probability of atching under their preferred relationship ters and a higher probability of atching under less-preferred ters. For a large class of constant elasticity of substitution atching functions, e sho that, as the gender ratio becoes ore unfavorable, the individual becoes ore likely to sacrifice relationship ters for a higher atch probability. The advantages of our odeling approach are three-fold. First, by linking choices over partners ith outcoes, in equilibriu e are able to eight the different sex ratios appropriately. Standard practice is to use only one sex-ratio hen looking at the relationships beteen sex ratios and outcoes. But the relevant sex ratios are 4 This fundaental idea has a long pedigree in the literature on intra-household allocations. McElroy and Horney (1981) and McElroy (1990) pointed to the gender ratio in the rearriage arket as one eber of a class of shifters (EEPs) for the bargaining poers of spouses and thereby intra-household allocations. Chiappori (1992) (and elsehere) suggested these sae shifters (rechristened as distribution factors ) to study intra-household elfare. 5 Many others have exained the influence of gender ratios on outcoes. See Angrist (2002) for a detailed revie of the influence of gender ratios on arriage, labor supply, and child elfare aong any others. 4

5 deterined by equilibriu forces. Second, by alloing preferences over both partner characteristics and hat happens in the relationship (in this case age and race of the partner plus hether sex occurs), e are able to capture the tradeoffs ade across the to. For exaple, increasing the nuber of senior boys favors oen. This ay, hoever, still result in relationships that are ore sexual because of feale preferences for senior boys despite feales preferring not to have sex. Finally, by orking in a nontransferable fraeork ith partner selection, e are able to identify preferences fro data on outcoes. In the transferable case, if a particular outcoe is affected by the sex ratio, it is unclear ho preferences are affected by the sex ratio because individuals ay be aking transfers that are not observed by the researcher, particularly in the case here the outcoes of relevance are discrete. We estiate the odel using data fro the National Longitudinal Study of Adolescent Health (Add Health). These data contain inforation on the universe of students at particular U.S. high schools in 1995 as ell as ansers to detailed questions about relationships for a subset of the students. The odel is estiated assuing that individuals are able to target their search toards opposite-sex partners of a particular grade and race as ell as to specify hether or not sex ill occur in the relationship. Not surprisingly, estiates of this structural odel sho that en value sexual relationships relatively ore than oen. By siulating choices in the absence of atching concerns, e find that 36% of oen and 63% of en ould prefer to be in a sexual, as opposed to a nonsexual, relationship. These counterfactual choices bear a striking reseblance to subjective reports by students found in Add Health. There, 36% of oen and 59% of en responded that sex ould be a part of their ideal relationship. Hence, our structural odel, hile estiated on observed atches, is able to back out preferences for sex that are rearkably close to the self reports, 5

6 providing soe credence to both the self-reported data and our structural estiates. Taken together, they provide strong evidence that, relative to oen, en prefer relationships that include sex. More iportantly, these estiates iply that atching concerns lead soe oen to have sex, not because they prefer this, but because they ere illing to trade off relationship ters for a higher probability of atching. With differing preferences across en and oen, observed changes in sexual behavior ay then indicate transfers in elfare fro one gender to the other. The rest of this paper proceeds as follos. The next section presents the Add Health data on high school relationships. Section 3 lays out a to-sided odel of targeted search and atching, relates the atching function to special cases found in the literature, establishes the existence of equilibriu, and ho the gender ratio affects the probability of atching. Section 4 describes the axiu likelihood estiator. Section 5 presents the resulting estiates and shos ho the structural odel can back out preferences in the absence of copetitive effects, deonstrating ho the odel atches self-reported preferences on a nuber of diensions. Section 6 shos our results are robust to different assuptions regarding searching outside the school as ell as choosing not to search. Section 7 offers an exploration of hat our results iply about feale elfare beyond the teen sex setting. 6

7 2 Data and Descriptive Characteristics We use data fro Wave I of the National Longitudinal Survey of Adolescent Health. 6 The data include an in-school survey of alost 90,000 seventh to telfth grade students at a randoly sapled set of 80 counities across the United States. 7 Attepts ere ade to have as any students as possible fro each school fill out the survey during a school day. Questions consist ainly of individual data like age, race, and grade, ith liited inforation on acadeics, extra-curricular activities and risky behavior. We use this saple to construct school level aggregates by observable characteristics, grade and race, hich serve as inputs in calculating gender ratios. The Add Health data also includes a saple of students ho ere adinistered a ore detailed survey, the in-hoe saple. The in-hoe saple includes a detailed survey about relationship histories and sexual behaviors. The relationship histories include both hat happens ithin the relationships as ell as characteristics of the partner such as race and grade. A natural proble in this survey design is the issue of hat constitutes a relationship to respondents, particularly hen en and oen ay define relationships differently. Here e follo the Add Health definition that a relationship referred to fro here on, consists of all the folloing (i) as holding hands, (ii) kissing, and (iii) saying I love you. This definition results in the ost 6 The survey of adolescents in the United States as organized through the Carolina Population Center and data ere collected in four aves, in , , and A school pair, consisting of a high school and a randoly selected feeder school (iddle school or junior high school fro the sae district) ere taken fro each counity. 7

8 syetric distribution of responses ithin schools and allos for the ost data in the survey to be accessed. 8 The panel-structure of relationships also allos us to deterine hether they had sex prior to the current partnership. We restrict attention to schools hich enroll both en and oen. A saple of ongoing relationships shoed 54.66% of partners et in the sae school, the next closest avenue of atching as utual friends, accounting for 24.08% of atches. 9 Since the focus here ill be on a cross section of the atching distribution, e count only current relationships aong partners ho attend the sae school. The Add Health data is nationally representative at the level of the school and is dran fro all types of schools. We focus on respondents ho are in the 9th through 12th grades. 10 Schools for ho e observe feer than 10 students in the detailed intervies are dropped. We drop one all boys school, one vocational education school for high school dropouts, and e drop six schools ithout eaningful nubers of 9th-graders. 11 After these adjustents, our saple contains 74 schools, ith 11,273 8 Applying this definition 48.6% of ongoing in-school relationships cae fro en and 51.4% fro oen. With perfect reporting and agreeent over the definition e ould see parity. 9 The data on here individuals et is only available for a subsaple of relationships. The other avenues ere 9.38% prior friends, and less than 5% in their neighborhood, place of orship, and casual acquaintances each. 10 The in-hoe saple is dran fro schools ith different grades: 73% of schools have grades 9-12, 11% have grades 7-12, and 13% had other cobinations of grades(e.g. K-12). Finally 1.4% are dran fro a junior and senior high school hich are distinct schools. 11 These schools on average had around 300 students in each of the grades 10-12, 8

9 individuals. 12 Our focus is on atches ithin a school so those atched ith soeone outside of the school are dropped. We exaine the issue of outside the school atches in ore detail in section 6. The saple size reoving these individuals falls fro 11,273 to 7,915. Since e only observe atches for the in-hoe saple, e ust take into account the fraction of the in-school saple ho ould also likely be in a relationship outside of the school. We do this by assuing the fraction of the in-hoe saple in atches outside of the school conditional on gender, race, grade, and sexual history atches the fraction of the in-school saple that are in atches outside of the school. In theory, en and oen should report roughly the sae nuber of relationships but in practice this is not the case. Given that e observe double reporting in these data, i.e. en are asked to report their atches ithin a school and so are oen, e can see these differences. Men reported 584 atches here sex occurred and 487 atches here sex did not occur, hile oen reported 478 atches ith sex and 447 atches ithout sex. To deal ith this isreporting e use inforation about atches reported by oen. We drop all en and create atched-ale observations fro data reported by oen, 13 giving us a saple of atched en and oen but on average 9 students in the 9th grade. The Add Health sapling design only probabilistically included the ost relevant junior high or iddle school for a high school,the relevant 9th grade observations these six schools ere not sapled, but rather a sall feeder school. 12 Fro students beteen grades 9-12, e drop: individuals issing saple eights(1124), schools discussed above(2337), schools ither feer than 10 reported students(106). 13 Partner s past sexual experiences ere not reported, so e estiate this proba- 9

10 and unatched oen. To get the nuber of unatched en e take the original ale observations and subtract the nuber of atched en. All these operations are done at the gender-school-grade-race level. This procedure aounts to treating ale reporting of on grade and race as truthful, and feale reporting on atches as truthful. The result is 934 current in-school atches, aong 7,961 individuals fro 74 schools Descriptive Statistics Table 1 reports descriptive statistics for the saple, prior to the procedure outlined above. Once those atched outside the school are reoved, a little under 25% of the saple are in relationship ith half of those relationships involving sex. Included in the descriptive statistics is hether the individual has had sex prior to their current relationship. This variable as created fro reports of the full relationship history and takes on a value of one if the person has had sex in the past ith soeone besides their current partner. Men report significantly higher rates of prior sex. There are a disproportionate fraction of the saple in loer grades. This is due to individuals in higher grades being ore likely to atch outside of the school. Soe direct inforation on gender differences in preferences for sex can be found fro questions that ere asked of the in-hoe saple. Individuals ere asked about hether they ould ant a roantic relationship over the next year and hat sorts of things ould happen in the relationship. Included in the questions ere hether bility conditional in grade and race fro the ale reports. 14 The nubers 7,961 and 7,915 do not atch precisely because for soe ale school-grade-race subgroups the in-hoe saple intervieed feer individuals than feales atched ith, expanding the nuber of individuals e observe. 10

11 the ideal relationship ould include having sex. 15 Table 2 shos elicited preferences over sex and relationships overall and by grade. Coparing Table 1 to Table 2, ore individuals prefer having relationships than do, suggesting significant search frictions. While preferences for relationships are the sae for both en and oen, preferences for sex are not. While 59% of en ould prefer to have sex, the fraction of oen ho prefer to have sex is only 36%. Preferences for sex rise ith age. Even ith this rise, coparing the sex preferences for oen of a particular grade ith the sex preferences for en of another grade shos stronger ale preferences for sex ith one exception: 12th grade oen have stronger preference for sex than 9th grade en. Note fro Table 1 that half of current relationships entail sex, hich is higher than the self-reported preferences for oen averaged over any grade, even conditional on anting a relationship. This suggests the possibility that oen ay be sacrificing hat they ant in order to for relationships. Whether sacrifices over the ters of the relationship are ade ay in part be dictated by the characteristics of the partner. Individuals ay be illing to take ore undesirable relationship ters hen the partner is ore desirable. We no turn to characteristics of the partner, focusing in particular on grade and race. Table 3 shos the share of relationships for each possible ale/feale grade cobination. The ost coon atches are aong individuals in the sae grade. Sae grade atches ake up over 42% of all atches. The six cobinations of an older an ith a younger oan also ake up a large fraction of observations at 40%, leaving only 18% of atches for oen ith younger en. While atched oen are evenly 15 Add Health responses to questions regarding ever having sex are very siilar to the NLSY97 (cf. Arcidiacono, Khaja and Ouyang (2009)): beginning at a telfth grade sex participation rate in the lo 60% range, and falling roughly 10% per grade. 11

12 distributed across grades, older en are substantially ore likely to be atched than younger en. Even though 9th grade en outnuber 12th grade en by alost three to to, there are 2.5 ties ore atched 12th grade en than 9th grade en. These results point toards younger oen and older en being ore desirable and hence they ay have ore control over the ters of the relationship. Table 4 shos the patterns of cross-racial atching. As can be seen fro the diagonal eleents of the table, the vast ajority of atches over 86% are saerace atches. Each race is substantially ore likely to be in a relationship ith soeone of their on race than another race. In the set of inorities, Hispanic students date outside their race ost often, folloed by those in the other category (ho are predoinantly Asian), and then blacks. Hispanic and black en see uch higher probabilities of atching ith other races than their feale counterparts hile the reverse is true for hites and those in the other category. 16 Although not shon here, Hispanic and black en ere both ore likely to have sex ith hite feale partners conditional on atching, than ith partners of their on race. This finding also suggests race-specific gender ratio differences ay affect the likelihood of these atches having sex. Black en and, to a lesser extent, black oen, ake up a larger fraction of atches than they do a percentage of the population. 16 Other studies have used ultiple sources to quantify hich races and genders do and do not engage in inter-racial dating: Lee and Edonston (2005) offer any descriptives using U.S. Census data to track inter-racial arriage over the last 40 years. The census shos a clear pattern ith black en and Asian oen arrying outside their race far ore than black oen and Asian en. Qian (1997) reports that hite en arry ost frequently Asians, Hispanics and lastly blacks. 12

13 2.2 Subjective Preferences by Type Ho do these variations in partner desirability affect hether sex occurs in the relationship? Table 5 copares elicited preferences for sex versus hether sex occurs in the relationship. The first colun shos the percentage of oen and en of different grades and races ho, conditional on being in a relationship, had sex. The second colun then shos the corresponding fraction that anted to have sex conditional on either anting a relationship or anting to have sex ithout a relationship. 17 While Table 5 shos that across the board oen are having sex ore than they ould like and en are having sex less than they ould like, soe groups ust coproise their preferences ore than others. In particular, conditional on being in a relationship, the fraction of 12th grade oen ho are having sex is uch higher than the fraction ho ould like to be having sex. This is in contrast to 9th grade oen hose preferences for sex are siilar to hat actually occurs. For en, it is the 12th grade en hose preferences align ith hat actually happens in their relationships hile 9th grade en in relationships are having substantially less sex than they ould like. Hence, older en and younger oen benefit fro being ore desirable by having relationships hich align ore ith their stated preferences hile older oen and younger en ust sacrifice their preferences over hat happens in a relationship in order to successfully atch. 17 Calculated by taking the nuber of individuals ho said that they ould ideally have sex and dividing by the nuber of individuals ho said they either anted a relationship or anted to have sex ithout a relationship. 13

14 2.3 Gender Ratio Variation Given evidence that certain characteristics influence hether one s preferences ill align ith hat happens in a relationship, the supply of these characteristics ay also have an effect on the ters of the relationship. When en, and in particular older en, are in short supply, oen ay need to sacrifice their preferences ore in order to successfully atch. We exaine ho gender ratios vary across schools in Table 6, paying particular attention to the gender ratios for hites and blacks by grade. Each cell in Table 6 gives the percentage of feale students for each grade-race pairing. 18 Table 6 shos that there is a substantial aount of variation in the percentage of feale students, particularly aong blacks. 19 Breaking out the percentage feale along different diensions (race, and grade-race groupings) spreads the initially condensed distribution. 20 The botto panel of Table 6 splits the saple of oen by the ean on-grade percent feale and calculates the fraction having sex conditional on being in a relationship. The basic split shos a correlation beteen the percent feale and the chances of having sex. Splitting the saple based on hether a oan has had sex in the past shos that higher percent feale is ore positively correlated ith current sex. Arcidiacono, Khaja and Ouyang (2009) establish that having sex in the 18 A iniu of 5 observations fro the race or grade-race pair is required for a school to enter Table 6 19 This dispersion is even ore pronounced for Hispanic and Asian students. 20 The populations have been scaled don by one inus the epirical conditional probability of atching outside the school for each age-race-gender-school group. Belo e inflate this fraction reoved, reflecting that Patch out need not equal one (i.e. soe non-atched searched outside but failed to atch. 14

15 past substantially increases the likelihood of having sex in the future. Hence, hen oen outnuber en, e see hat appears to be ore substitution toard sex aong those oen ho have had sex in the past. To foralize this notion ore, Table 7 presents probit estiates of the probability of sex conditional on atching for oen fro the ithin-school saple of atches. Rather than the on-grade gender ratio, the gender ratio used for each individual is the percent-feale in the grade-race pair here the atch occurred. For a oan atching ith a 12th grade hite ale, this variable is the fraction of 12th grade hites ho are feale. 21 Given the high likelihood of individuals atching in their on grade-race pair, this variable serves as a crude easure of the relevant gender ratio a partner faces. This is hat atters for bargaining: a partners outside options. The results fro the reduced for are clear, ith increases in the outside options for ale partners increasing the chances of sex occurring. As the second and third colun indicate, these results hold regardless of hether observed school characteristics or school effects are included. Nonetheless, the estiates in 7 do not account for the fact that the grade-race pair individuals atched ith is itself a choice. The structural odel outlined belo specifically accounts for the endogeneity of associated ith the choice of partner characteristics. The odel ill iplicitly control for all the factors included in the iddle colun of Matched GR = M 1{j = }%feale j %feale j is the percent of j-type individuals ho are feale. here j is the partners type and 22 Since controlling for school fixed effects does not significantly alter the reduced for results, e do not estiate the structural odel tih school fixed effects. Doing so ould change the nuber of paraeters fro 22 to

16 3 Model We analyze the tradeoffs aongst three fundaental sources of expected utility fro searching for a partner: the type of partner (race, grade), the ters of the relationship (sex/no sex) and the probability of success (atching) in the search for that type and those ters. A searcher knos in advance his utility fro type and ters. And, he can target a less-preferred cobination of type and ters in order to have a higher probability of atching. It is this fundaental tradeoff that distinguishes our odel fro others. At its core, our odel ebeds search and the attendant risk of not atching into a static odel. In that sense, it is analogous to the age posting odels in hich orkers choose beteen a high age fir ith a lo probability of atching, or a lo age fir ith a high probability of atching. In order to disentangle ale and feale preferences fro data on atches, e propose a to-sided search odel ith non-transferable utility. We consider only opposite-sex one-to-one atching. 23 We categorize each ale as a type here {1, 2,..., M}. Siilarly, e each oan is given a type here {1, 2,..., W } eleents. An individual s type denotes soe collection of characteristics such as age, grade, race, or attractiveness. For ales (feales) there are then W (M) types of ates. Let i indicate the i-th eber of type. We index the possible ters of the relationship by r {1,..., R}. The possible ters could include not having sex, having sex ith protection, etc. We odel search as being copletely directed: en and oen are able to target their search on both 23 Only 2% of the saple reported concurrent sexual atches and 1% reported concurrent relationships, though clearly soe reporting probles exist. We proceed in odeling one-to-one atching given the coplexity of odeling ultiple atches and the first order iportance of the ain reported atch for preferences. 16

17 the characteristics of the partner as ell as the ters of the relationship. Each an (oan) then akes a discrete choice to search in one of M R (W R) arkets, resulting in M W R types of atches. Search is then odeled as a one-shot gae: there are no dynaics in the odel. Individuals first decide in hich arket to search. Couples are atched ith the probability of atching depending on the nuber of searchers on both sides of the arket. 3.1 Individuals An individual s expected utility for searching in a particular arket depends upon three factors: 1. the probability of atching in the arket here the probability of a -type an atching ith a -type oan in an r-type relationship is P r, 2. a deterinistic portion of utility conditional on atching given by µ r for a -type an, 3. and an individual-specific preference ter ɛ r i. 24 Note that the only individual-specific part of expected utility are the ɛ r i s. Further, the ɛ r i s are knon to the individual before aking their decision: there is no atchspecific coponent beyond hat occurs through the observed characteristics of the partner and the ters of the relationship. Hence, the only uncertainty fro the individual s perspective is their probability of finding a atch. Finally, note that the probability of atching is only affected by ale and feale type and relationship 24 The corresponding ters for oen are P r, µ r, and ɛ r i. 17

18 type: all ales of type searching in the, r arket have the sae probability of atching. We noralize the utility of not atching to zero. Expected utility fro searching in a particular arket is then the probability of atching in the arket ties the utility conditional on atching. We specify the functional for of the utility such that expected utility for a -type an searching for a -type oan of relationship type r as: E (Ui r ) = P r e µr +ɛ r i (1) We assue that the ɛ r i s are i.i.d. Type I extree value errors and are unknon to the econoetrician. Taking logs yields: ln (E (Ui r )) = µ r + ln (P r ) + ɛ r i (2) Individual i of type then chooses to search for a oan of type under relationship ters r, d i = {, r}, hen: ( r {, r} = arg ax,r µ + ln P r ) + ɛ r i We treat the ɛ r i s as observed only to the individual: it is not knon to the other participants in the arket. The probability of a -type an searching for a -type oan in an r-type relationship, φ r then follos a ultinoial logit for: Pr(, r ) = φ r = exp (µ r + ln (P r )) r exp (µ r + ln (P r )) (3) here the µ s for one of the arkets for both en and oen ust be noralized to zero. 18

19 3.2 Matching We no specify the atching process. The atching process essentially a production function, taking as inputs the nuber searching en and the nuber of searching oen in each arket and giving the nuber of atches in each arket as output. We paraeterize the nuber of atches, X, in arket {,, r} as depending upon the nuber of -type en and -type oen searching in the arket. Let N and N indicate the nuber of -type en and nuber of -type oen. Recall that φ r and φ r give the probability of -type en and -type oen ho search in arket {,, r} hich are also the arket shares of searching en and oen. The nuber of atches in arket {,, r} is then given by: X r = A [(φ r N ) ρ + (φ r N ) ρ ] 1 ρ (4) here ρ deterines the elasticity of substitution (σ = 1 ), and A easures search 1 ρ frictions. When ρ 0 CES becoes Cobb-Douglas and ρ CES becoes Leontief. Note that X r = X r for all,, and r. Under the assuption that all -type en searching in the sae arket have the sae probabilities of atching, P r is given by: P r = X r φ r N = A [(φr N ) ρ + (φ r N ) ρ ] 1 ρ [ = A 1 + (5) φ r N ( ) φ r ρ ] 1 ρ N φ r N The log of this ter then enters into the ultinoial logit probabilities of searching in particular arkets and captures the influence of the gender ratio on arket search 19

20 decisions. 3.3 Equilibriu The probabilities of searching in a particular arket, the φ s, give the share of a particular set individuals ho ill search in a particular arket. These φ s also affect the probabilities of atching, the P s. We rerite equation (6) to ake the dependence of P r on φ r and φ r explicit: φ r = exp (µ r + ln [P r (φ r r exp ( µ r + ln [ P r, φ r )]) ( φ r, φ r )]) (6) Since the arket shares ust su to one for both en and oen, equilibriu in our odel is characterized by stacking the (M 1) (W 1) (R 1) shares and solving for the fixed point. Since φ is a continuous apping on a copact space, Brouer s fixed point theore guarantees that an equilibriu exists. 25 It is trivial to deonstrate the ex ante efficiency of the equilibriu: oving any player fro his chosen equilibriu sub-arket to another reduces his expected utility and therefore cannot be a Pareto ove. 3.4 Iplications of Changing the Gender Ratio Given our utility specification and atching process, e no turn to ho changing the gender ratio leads to changes in the probabilities of choosing particular arkets. To begin, consider to arkets that include type oen and type en but here the relationship ters in the to arkets are given by r and r respectively. No, fix the search probabilities, the φ s, and increase the nuber of -type en. We 25 As in acro odels of the labor arket, uniqueness depends on constant returns to scale of the atching function; see Petrongolo and Pissarides (2001). 20

21 can then see hich of the to relationship arkets becoe relatively ore attractive for en and oen respectively. We then sho ho the search probabilities ust adjust in equilibriu. Denoting G as the ratio N /N, Proposition 3.1 shos the relationship beteen the gender ratio and search behavior, ith the proof in the appendix. Proposition 3.1. If ρ < 0 and µ r µ r > µ r µ r then the folloing hold: (a) φr φ r < φr φ r (b) P r > P r and P r < P r (c) Both φr G /φ r > 0 and φr G /φ r > 0 In Proposition 3.1 the average preference of a {, } pair is such that the oen in the pair have a stronger preference for ters r over r than their ale counterparts. The first to clais are intuitive. Clai (a) states that this relative preference by oen for r over r translates into search behavior: in equilibriu, the ratio of search probabilities for r versus r ust result in oen searching ore in r relative to en. These differential search probabilities then translate into atch probabilities. Since oen are relatively ore likely than en to search in r, feale atch probabilities ust be loer in r than in r, ith the reverse holding for en, clai (b). The key result for the epirical ork is clai (c). As the gender ratio oves such that en becoe relatively ore abundant, both en and oen increase their relative search probabilities in the arket here oen have a relative preference, in this case r. The result falls directly out of the elasticity of substitution. Naely, for elasticities of substitution less than one and conditional on φ, the lack of substitutability beteen en and oen in the arket iplies that, as G increases, larger changes in atch probabilities for en (oen) ill occur here en are relatively 21

22 ore (less) abundant. The elasticity of the probability of atching ith respect to G for en and oen in the {,, r} arket conditional on φ are given by: ln (P r ) ln(g ) = φ ln (P r ) ln(g ) = φ [( φ r φ r G [( φ r G φ r ) ρ + 1] 1 ) ρ + 1] 1 With ρ < 0, as the ratio of ale to feale search probabilities (φ r /φ r ) increases, the agnitude of the elasticity falls for feales and rises for ales. With fixed search probabilities, increasing the nuber of en relative to oen akes the arket here oen have a relative preference ore attractive for both sexes, resulting in shifts by both en and oen to that arket in equilibriu. To illustrate this point, consider the Leontief case here ρ oves toard negative infinity (σ 0). The atching function is given by: X r = A in {φ r N, φ r N } The nuber of atches is given by hichever side of the arket has feer searchers. Hence, the gender ratio has an extree effect on the group that is in the ajority in a particular arket, ith no effect on the group in the inority. On the other extree is the Cobb Douglas case here ρ 0, iplying the elasticity of substitution, σ, is one. The elasticity given in Equation (4) is 0.5 in this case, regardless of the values either the search probabilities or the gender ratio. Hence, if gender ratios do affect relationship ters then this is evidence that the atching function is not Cobb Douglas. 22

23 4 Estiation Having discussed the trends in the data and the odeling approach, e no turn to integrating the data and the odel for estiation. Types of en and oen are defined at the grade/race level as suggested by the clear differences in atching patterns across race and grade. Second, e classify relationships as one of to types: those that are having sex and those that are not. An individual is defined as being in a relationship ithout sex if the person eets the standards described previously (holding hands, etc.). An individual is classified as having a relationship ith sex if the individual is currently having sex, regardless of his relationship status. With to types of relationships, four grades, and four races, there are then thirty-to arkets. The next to subsections put structure on the utility function for purposes of estiation and then sho ho to for the likelihood function given the constraints posed by the data. 4.1 Utility Rather than having separate µ s (utilities) for every type of relationship, e put soe structure on the utility function. Denote the grade associated ith an -type an as G {1, 2, 3, 4}. When a an searches for an -type oan, the grade of the partner is P G. Siilarly, R {1, 2, 3, 4} gives the race of an -type an ith the corresponding race of the potential -type partner given by P G. We specify the utility of a non-sexual relationship as a function of the partner s grade and race as ell as hether the partner is in the sae grade as the searching individual, SG = I(G = P G ) here I is the indicator function, and the sae race, SR = I(R = P R ). Denoting searching in the no-sex arket by r = 1, e forulate the deterinistic part of utility for en and oen atching in the no-sex 23

24 arket as: µ 1 = α 1 SG + α 2 P G + µ 1 = α 1 SG + α 5 P G + 4 I(P R = j) [α 3j SR + α 4j ] (7) j=1 4 I(P R = j) [α 3j SR + α 6j ] (8) here the intercept of a non-sexual relationship is noralized to zero. To econoize on paraeters, this specification sets the extra utility associated ith being in the sae grade or being of the sae race to be the sae for en and oen. The effect of partner grade and race, hoever, is alloed to vary by gender. The specification is set such that certain race/gender cobinations ay be ore desirable than other race/gender cobinations. Further, the strength of sae preferences ay differ across races hich is captured by alloing α 3 to vary by race. 26 The utility of sexual relationships takes the utility of non-sexual relationships and adds an intercept as ell as effects for partner grade and hether the individual is j=1 in 9th grade, G = We furtherore allo hether the ith an of type has engaged in sex in the past, P S i {0, 1}, to affect the current utility of sex. Note that e are not specifying that partners have preferences for individuals ho have had sex in the past but rather those ho have had sex in the past have preferences for sex no. Hence, the types and do not include past sex. Denoting searching in the sex arket by r = 2, e specify the deterinistic part of utility for en and 26 Due to liited observations, e only estiate to coefficients on sae race: one for blacks and one for everyone else. 27 By tenth grade nearly everyone in the saple has transitioned through puberty. 24

25 oen atching in the sex arket as: µ 2 (P S i ) = µ 1 + α 7 + α 8 P S i + α 9 P G + α 10 I(G = 1) (9) µ 2 (P S i ) = µ 1 + α 11 + α 8 P S i + α 12 P G + α 10 I(G = 1) (10) Although en an oen ay differ in their preferences for sex and the effect of partner grade on the utility of sex, the effect of past sex is constrained to be the sae for en and oen Foring the likelihood function We do not observe all atches but only those in the in-hoe data set. Hoever, e do observe gender, grade, and race for the population of students at each school. By inferring population oents of past sex fro the in-hoe saple, e can construct the choice probabilities for the entire school fro the in-hoe saple. We take the relationships as defined by the oen in the Add Health. Hoever, e still need to incorporate the search decisions of the en. We take these fro the oen as ell: the nuber of ales that do not atch are given by the nuber of ales in the in-hoe saple inus the nuber of en ho ere reported to atch by the oen in the in-hoe saple. Hence, if e observe 100 hite ales in the 12th grade in the in-hoe saple at a particular school and the oen in the in-hoe saple reported 25 atches ith 12th grade hite ales, then the 25 hite ales ould be assigned to the various atch categories hile the other 75 ould be assigned as not atching. This assues that the ratios of en and oen are roughly the sae in 28 Alloing coefficients for past sex to vary by gender and other characteristics did not iprove odel fit. 25

26 the in-hoe saple as in the in-school saple. The paraeters that need to be estiated include those of the utility function and the paraeters of the atching function, ρ and A. Denote θ, as the set {α, ρ, A}. Denote N as the set of students of each type broken out by the fraction of each that has had prior sex. Hence, N contains 64 eleents here each eleent refers to a gender, grade, race, and past sex cobination. Denote y i = 1 if the ith oan of type as in a current relationship (or having sex) at the tie of the survey and is zero otherise. The oan is then considered atched if y i = 1. Note that d i, the oan s search decision, is observed only if the oan as atched. Hence, e need to integrate out over the search decision for those ho are not atched. The log likelihood for the ith oan of type is then given by: [ ] L i (θ) = I(y i = 1) I(d i = {, r}) (ln [φ r (θ, N, P S i )] + ln [P r (θ, N)]) [ +I(y i = 0) ln r r φ r (θ, N, P S i ) (1 P r (θ, N)) Note that the probability of atching is not affected by past sex except through the search probabilities. Since e are pulling atched en not fro the in-hoe saple directly but rather fro questions asked of the oan about her partner, the likelihood for en is ore coplicated. In particular, e have to integrate out over hether the en have had prior sex to for the unconditional probability of the outcoe. Let π k indicate the proportion of type en ho ere in prior sex state k. 29 Integrating out over the 29 This proportion is taken as the cell ean of the fraction reporting past sex in the in-hoe saple ithin each school and -subgroup. If data ere liited, individuals probability of past sex is estiated across schools ithin each -subgroup. ] (11) 26

27 prior sex state leads to the folloing log likelihood contribution for ith an of type : L i (θ) = [ I(y i = 1) r k=0 ( [ 1 I(d i = {, r}) ln r k=0 ] [ ])] π k φ r (θ, N, k) + ln P r (θ, N) [ 1 ] + I(y i = 0) ln π k φ r (θ, N, k) (1 P r (θ, N)) here the su over k is taken over the possible prior sex states. All of the likelihoods described so far ere for a generic school. Denote the schools in the data by s {1,..., S}. Suing the log likelihoods over all the possible types and types at each school s iplies that the paraeters can be estiated using: ˆθ = arg ax θ s N s L s i(θ) + s i=1 N s L s i(θ) i=1 here a fixed point in the search probabilities is solved at each iteration. 5 Results The estiates of the structural odel are presented in Table 8. Key to disentangling ale and feale preferences given observed atches is the effect of the different gender ratios on the search decisions. These gender ratios anifest theselves through their effect on the probability of atching. The paraeters of the atching function, ρ and A, are identified through variation in atches across schools ith different gender ratios and the overall atch rate respectively. Estiates of ρ are significant and negative, ruling out the Cobb-Douglas atching odel and ensuring that gender ratios do affect the likelihood of observing particular atches. The estiates of ρ indicate the elasticity of substitution in atch production of ith a standard error of

28 The first panel of Table 8 shos ho sex is valued above and beyond the relationship itself. Consistent ith the elicited preferences in Table 2, ales on average have stronger preferences for sex than feales. Partner grade is easured here as 0 through 3, ith zero corresponding to freshen and three to seniors. For ales, but not for feales, grade of the partner influences the utility of sex ith en preferring to have sex ith older partners. Those ho have had sex in the past also have a uch stronger preference for sex in the present. The second panel shos ho partner characteristics affect the value of a relationship. Here e see that oen prefer to be atched ith older en and that this preference is stronger than the preference by en to have sex ith older oen rather than younger oen. In contrast, en only prefer older oen hen the relationship is accopanied by sex. Individuals also prefer to be atched ith those in the sae grade and inorities prefer to atch ith one another. Consistent ith Table 4, sae race preferences are particularly strong for blacks. The estiates sho that cross-race atches are ore attractive ith hites than ith other races. Hoever, the relative preferences for ales and feales of particular races atch those in the prior literature. Naely, oen prefer black en ore than en prefer black oen hile en prefer oen of other races, a category doinated by Asians, ore than oen prefer en of other races. In Table 9 e report the equilibriu atch probabilities for hites across grades and depending on hether the individual searchers are ale and feale. 30 The first colun P captures the probability of a an in grade nine atching ith oen of various grades. As expected, freshen ales see the loest atch probabilities of any group, and those probabilities decline ith feale grade. The coparable nubers for oen are the intersections of the ros ith the P coluns (the second colun ithin each cell). Ninth grade oen see significantly higher probabilities of atching relative to ninth grade ales. Note that 30 Focusing on hites reoves probles fro seeing too fe individuals in the school. 28

29 in alost every cell in the table, the axiu nuber for oen (under the P coluns) is hen a oan looks for a sex partner, and the axiu nuber for en (under the P coluns) is hen a an looks for a relationship ithout sex. This pattern of atching is due to the preference for sex aong en, and captures the trade-off beteen preferred relationship ters and higher atch probabilities. Beteen the general equilibriu effects and the non-linear nature of the specification, the agnitudes of the utility coefficients are difficult to interpret. Hoever, e can use the coefficient estiates to back out the fraction of en and oen ho prefer sex to no sex absent concerns about atching. Naely, e can turn off the effects of the probability of atching and see hat choices ould have been ade in the absence of having to copete for partners. Backing out ale and feale preferences for sex in this ay yields estiates of the fraction of each group ho prefer to have sex. We then copare the odel estiated preferences to the stated preferences discussed in Table 5. Table 10 shos that the odel does a rearkably good job of atching the elicited preferences for sex given that the elicited easures ere not used in estiation. The fit is particularly good for oen here the elicited preferences sho 36.3% of oen prefer sex to 35.9% of oen predicted by the odel, hile for en the odel predicts 62.5% copared to the elicited 59.1%. The odel-predicted preferences for en and oen are very close across races and for individuals ith no prior sex. The odel over-predicts interest in sex for 10th grade en, and both en and oen ho have had sex in the past. We next sho ho ale and feale probabilities differ due to atching concerns. Table 11 shos differences beteen ale and feale search probabilities both ith and ithout alloing the probabilities of atching to affect the search decision. Taking into account the probability of atching, ale probabilities of searching in the set of arkets that include sex are thirteen percentage points higher than feales. Without atching concerns, there is a tenty-six percentage point difference. This eans both sides of the arket are sacrificing their preferred relationship ters to increase their chances of atching. 29

30 6 Robustness Checks In this section e relax assuptions on ho searches outside the school and hether all individuals search, shoing that the general pattern of our results regarding differences in preferences beteen ales and feales are robust to alternative assuptions. Previously e reoved individuals atched ith soeone outside of the school. This assuption is consistent ith either a) individuals searched and atched in the outside arket in a first stage or b) the probability of atching being one in the outside arket. To see b), note that only those ho atched in the outside arket ill have searched in the outside arket. Denote 0 as the outside arket. The probability of an -type an searching in the {,, r} arket is: Pr(, r ) = exp (µ r + ln (P r )) r exp (µ r + ln (P r )) + exp(µ 0 ) (12) Using the independence of irrelevant alternatives (IIA) property, note this probability conditional on searching in the school is: Pr(, r, d 0) = φ r = exp (µ r + ln (P r )) r exp (µ r + ln (P r )), (13) hich is the probability in (6) that e used to for our likelihoods, assuing everyone ho did not atch in the outside arket searched ithin the school. When the probability of atching in the outside arket is not one, the IIA property still holds, but e no longer have a good easure of the nuber of searchers in the outside arket. We allo for the probability of atching in the outside arket to be less than one by assuing that atch rates are the sae for all those ho search outside the school. Naely, e kno the characteristics of those ho atched in the outside arket. If 10% of freshen girls and 16% of senior girls atch in the outside arket, for a candidate atch rate of 80%, the iplied freshen girl search rate ould 12.5% in the outside arket ith the corresponding search rates for senior girls being 20%. We then scale don the nuber 30