Like Attract Like? A Structural Comparison of Homogamy across Same-Sex and Different-Sex Households
aa r X i v : . [ ec on . GN ] F e b LIKE ATTRACT LIKE?A STRUCTURAL COMPARISON OF HOMOGAMY ACROSS SAME-SEXAND DIFFERENT-SEX HOUSEHOLDS
EDOARDO CISCATO ♭ , ALFRED GALICHON † , AND MARION GOUSS´E § Abstract.
In this paper, we extend Gary Becker’s empirical analysis of the marriage market tosame-sex couples. Becker’s theory rationalizes the well-known phenomenon of homogamy amongdifferent-sex couples: individuals mate with their likes because many characteristics, such as edu-cation, consumption behaviour, desire to nurture children, religion, etc., exhibit strong complemen-tarities in the household production function. However, because of asymmetries in the distributionsof male and female characteristics, men and women may need to marry “up” or “down” accordingto the relative shortage of their characteristics among the populations of men and women. Yet,among same-sex couples, this limitation does not exist as partners are drawn from the same popu-lation, and thus the theory of assortative mating would boldly predict that individuals will choosea partner with nearly identical characteristics. Empirical evidence suggests a very different picture:a robust stylized fact is that the correlation of the characteristics is in fact weaker among same-sexcouples. In this paper, we build an equilibrium model of same-sex marriage market which allowsfor straightforward identification of the gains to marriage. We estimate the model with 2008-2012ACS data on California and show that positive assortative mating is weaker for homosexuals thanfor heterosexuals with respect to age and race. Our results suggest that positive assortative matingwith respect to education is stronger among lesbians, and not significantly different when compar-ing gay men and married different-sex couples. As regards labor market outcomes, such as hourlywages and working hours, we find some indications that the process of specialization within thehousehold mainly applies to different-sex couples.
Keywords : sorting, matching, marriage market, homogamy, same-sex households, roommate problem.
JEL Classification : D1, C51, J12, J15.
Date : First circulated version: November 21, 2014. This version: August 2018. Galichon’s research has beensupported by NSF grant
Journal ofPolitical Economy
Volume 128, Number 2, February 2020, URL: https://doi.org/10.1086/704611. Introduction
How individuals sort themselves into marriage has important implications for income distribu-tion, labor supply, and inequality (Becker, 1973). Strong evidence shows that assortative matingin marriages accounts for a non-negligible part of income inequality across households (Eika et al.,2014).Individuals tend to mate with their likes, a pattern called homogamy . However, because of asym-metries between the distributions of the characteristics in male and female populations, homogamycannot be perfect among different-sex couples. In other words, heterosexuals cannot always finda “clone” of the opposite sex to match with. A large body of the literature has noticed that, upuntil recently, “men married down, women married up” due to the sex asymmetry in educationalachievement that has only recently started to fade (Goldin et al., 2006). Gender asymmetriesexist in other dimensions such as biological characteristics (windows of fertility , life expectancy,bio-metric characteristics), psychological traits, economic attributes (due to the gender wage gap),ethnic and racial characteristics (immigration is not symmetric across sexes, see Weiss et al., 2013)or demographic characteristics (some countries, such as China, have comparatively more imbalancedgender ratios).Homogamy has been famously studied by Becker’s seminal analysis of the family. Becker, 1973expects most non-labor market traits, such as “intelligence, height, skin color, age, education,family background or religion”, to be complements. However, he also suggests that some attributescould be substitutes; in particular, Becker suggests that we should observe a negative correlationbetween some labor market traits such as wage rates because of household specialization . In orderto provide a structural explanation of homogamy, Becker proposed a model of positive assortativemating (PAM) in which men and women are characterized by some socio-economic “ability” index. Women’s fertility rapidly declines with age, whereas men’s fertility does not. Biologists and anthropologists arguethat this dissymmetry could explain the well-documented preference of men for younger women (Hayes, 1995; Kenrick& Keefe, 1992). Low, 2013 evaluates this young age premium for women and names it “reproductive capital,” as itgives them an advantage on the marriage market over older women. P.-A. Chiappori et al., 2012 model a Becker-like marriage market with sorting on a unidimensional index. Theestimation of such index reveals that high values in some attributes can compensate for poor values in others, thusshowing that sorting is based on trade-offs between traits.
IKE ATTRACT LIKE? 3
In this model, the marriage market clears so that men are matched with women that are as closeas possible to them in terms of this index, which subsumes all the characteristics that matter onthe marriage market. The (strong) prediction of Becker’s model is that the rank of the husband’sindex in the men’s population is the same as the wife’s in the women’s population. However, thisdoes not imply that the partners’ indices are identical: they would be so only if the distributionsof the indices were the same for both men and women’s populations.This analysis of the marriage market has attracted wide attention in the economic literature, inspite of its shortcomings. One shortcoming is that it originally refers to different-sex unions only.However, in a growing number of countries, same-sex couples have gained legal recognition, andthe institutions of civil partnership and marriage no longer require that the partners must be ofopposite sex. This official recognition is the result of several legal disputes and social activism bythe gay and lesbian communities . The issue of whether to recognize same-sex unions has long beena topical subject in many countries, since it challenges the traditional model of the family. Fromboth an economic and a legal point of view, the definition of what “family” means has relevantpolitical implications as long as this term is present - and is generally central - in many modernconstitutions and legal systems. Consequently, family units benefit from a special attention ofpolicy-makers. Therefore, a discussion of the issues related to same-sex marriage - remarkably atpolicy level - requires a good understanding of the similarities and differences in the householddynamics among same-sex and different-sex couples. Besides, it is important to remember thatthe legal recognition of same-sex couples is only one of many transformations that the institutionof the family has gone through in the last decades (Stevenson, 2008; Stevenson & Wolfers, 2007).Finally, since more and more data on same-sex unions have been made available, the extension ofthe economic analysis of family to the gay and lesbian population can now be taken to data.While it is natural to consider an extension of Becker’s model to same-sex households, it is worthnoting that the previous considerations on asymmetries between men’s and women’s distributionsonly hold as long as each partner comes from a separate set according to his/her sex. On the same-sex marriage market, the two partners are drawn from the same population and the distributions of Public actions for gay rights acknowledgment are often considered to have started in 1969, in New York City.See Eskridge Jr, 1993 and Sullivan, 2009 for a detailed history and a full overview of the arguments in favor of andagainst same-sex marriage.
CISCATO, GALICHON, AND GOUSS´E the characteristics are the same. Hence, the assortative mating theory pushed to its limits impliesthat, in this setting, partners should be exactly identical, i.e., each individual will choose to marrysomeone with identical characteristics.In spite of such theoretical predictions, facts suggest a very different picture. Recent empiricalresults on the 1990 and 2000 American Census show that same-sex couples have less correlatedattributes than different-sex ones, at least in terms of a variety of non-labor traits, including racialand ethnic background, age and education (L. K. Jepsen & Jepsen, 2002; Schwartz & Graf, 2009).Studies on Norway, Sweden (Andersson et al., 2006) and Netherlands (Verbakel & Kalmijn, 2014)led to similar findings. In order to explain these systematic differences, the literature has suggestedseveral possible reasons. A first consideration is that gay people might be forced to pick from arestricted pool because of their smaller numbers in the population, thus having a narrower choicewhen selecting their partner, resulting in a more diverse range of potential matches (Anderssonet al., 2006; Harry, 1984; Kurdek & Schmitt, 1987; Schwartz & Graf, 2009; Verbakel & Kalmijn,2014). Furthermore, gay men and lesbians have been found to be more likely to live in urbanneighborhoods than heterosexuals, and since diversities in socio-economic traits are stronger incities, this facilitates the crossing of racial and social boundaries (D. Black et al., 2002; D. A.Black et al., 2007; Rosenfeld & Kim, 2005). In light of these observations, one could argue thatthe same-sex marriage market is faced with stronger search frictions. Nevertheless, this mightnot necessarily be the case if the potential partners gather in specific locations, as it happensin cities and neighborhoods that are considered “gay-friendly” (D. A. Black et al., 2007). Otheranalysts argue that gay people may have different preferences than heterosexuals, as they tend tobe less conservative than straight individuals. Some explanations in this regard point out that,since homosexuality is still considered in some cultures as at odds with prevailing social norms, gaymen and lesbians might grow less inclined to passively accept social conventions, and consequentlythey would end up choosing their partner with fewer concerns about his/her background traits Note that household location choice and social norms are strictly related: it has been reported that gay peopleoften leave their town of origin and escape social pressure exerted by relatives and acquaintances and go living inlarger cities reputed to be gay/lesbian-friendly (Rosenfeld & Kim, 2005). Analogously, they are aware that they havemore probabilities of avoiding discrimination by achieving higher educational levels and orienting their professionalchoices toward congenial working environments (Blumstein, Schwartz, et al., 1983; Verbakel & Kalmijn, 2014).
IKE ATTRACT LIKE? 5 (Blumstein, Schwartz, et al., 1983; Meier et al., 2009; Schwartz & Graf, 2009). The detachmentfrom the community of origin and the research for more tolerant surroundings have an influenceboth on values and social norms and on the heterogeneity of interpersonal ties.A part of these explanations has to do with individual preferences, whereas another part hasto do with demographics, i.e., the distribution of the characteristics in the population. It is clearthat the explanations listed in the former paragraph, while different in nature, are not mutuallyexclusive, but all contribute to a better understanding of the equilibrium patterns. For instance,a high correlation in education may arise from individual tastes (as individuals could find moredesirable to match with a partner of similar educational background), but also from demographics(indeed, if some educational category represents a large share of individuals, this will increaseodds of unions within this category, thus mechanically increasing the correlation in education).When comparing the heterosexual and homosexual population, this is particularly relevant as theircompositions present significant differences (e.g., gay people are, on average, more educated thanheterosexuals; see D. A. Black et al., 2007).In this paper, we focus on differences in marital gains: we would like to compare the structureof complementarity and substitutability across same-sex and different-sex households. In order todo so, we need a methodology that helps us interpret the observation of matching patterns anddisentangle the role played by household interactions from external demographic factors. This isachieved through a structural approach, which allows us to estimate the parameters of the maritalsurplus function in order to fit the patterns actually observed. Hence, this approach will requirean equilibrium model of matching.In the wake of Becker, 1973, 1991, the economic literature has modeled the marriage market asa bipartite matching game with transferable utility. A couple consists of two partners coming eachfrom a separate or identical subpopulation (respectively, in the case of different-sex and same-sexunions). Both partners are characterized by vectors of attributes, such as education, wealth, age,physical attractiveness, etc. It is assumed that, when two partners with respective attributes x and y form a pair, they generate a surplus equal to Φ( x, y ), which is shared endogenously betweenthem. In the case of separate subpopulations (different-sex marriage), the landmark contributionof Choo and Siow, 2006 showed that the surplus function Φ can easily be estimated based on CISCATO, GALICHON, AND GOUSS´E matching patterns modulo a distributional assumption on unobservable variations in preferences,and was followed by a rich literature (P.-A. Chiappori et al., 2017; Fox, 2010; Galichon & Salani´e,2014, to cite a few). Dupuy and Galichon, 2014 extended Choo and Siow’s model to the case ofcontinuous attributes and propose the convenient bilinear parameterization Φ ( x, y ) = x ′ Ay , where A is a matrix called “affinity matrix” whose terms reflect the strength of assortativeness betweentwo partners’ attributes. However, the bipartite assumption is restrictive and does not allow toestimate the surplus on same-sex marriage markets, and, to the best of our knowledge, no suchestimation procedure is proposed in the literature. In a theoretical paper, P. A. Chiappori et al.,forthcoming focus on stable matchings in a finite population and show that, when the populationto be matched is doubled by cloning, the same-sex marriage problem, or “unipartite matchingproblem” can be mathematically reformulated as a heterosexual matching problem, or “bipartitematching problem” . In section 2 of the present paper, we apply an analogous reasoning to theempirically tractable, large-population, two-sided matching model of Dupuy and Galichon, 2014,in order to adapt their empirical strategy to the same-sex marriage market.A few papers already deal with the issue of assortativeness among same-sex households, althoughnone of them allows to draw conclusions on the structural parameters of the surplus function thatdrives the assortativeness. The most relevant benchmarks for the empirical results of this workare the aforementioned L. K. Jepsen and Jepsen, 2002 and Schwartz and Graf, 2009. Both papersmake use of the American census data (1990 and 1990/2000 respectively) and find that members ofdifferent-sex couples are more alike than those of same-sex ones with respect to non-labor markettraits. The heterogeneity in assortativeness is measured in a logit framework containing dedicatedparameters for homogamy. In general, in a logit framework individuals choose their best optionamong all possibilities. However, this fails to take into account the fact that matching takes placeunder scarcity constraint on the various characteristics. In the present paper, we estimate a modelof matching in which agents compete for a partner; our measures of assortativeness are given bythe parameters of the surplus function in each market (gay, lesbian and heterosexual). In another recent theoretical work, Peski, 2017 extends the NTU framework of Dagsvik, 2000 and Menzel, 2015and discusses the existence of stable matching in the unipartite case. Fox, 2018 proposes an empirically tractable TUframework that generalizes both the bipartite and the unipartite case, and applies it to the car parts industry.
IKE ATTRACT LIKE? 7
The contributions of the present paper are twofold. On a methodological level, this paper isthe first to propose a structural estimator of the matching surplus which applies to same-sexhouseholds, or, more generally, to instances of the unipartite matching problem. On an empiricallevel, we provide evidence by means of a structural analysis that, as concerns age and ethnicity,different-sex couples exhibit a higher degree of assortativeness than same-sex ones. While we find,in line with previous results, that sorting on education is stronger among lesbians with respectto different-sex couples, our results suggest that assortativeness on education is not significantlydifferent when comparing gay male and married different-sex couples. Further, we also look atlabor market traits such as hourly wages and working hours. Comparing assortativeness on labormarket outcomes between same-sex and different-sex couples hints to different family dynamics anddifferences in the household specialization process. Finally, we briefly discuss the estimates of themutually exclusive affinity indices obtained through our saliency analysis.The rest of the paper is organized as follows. Section 2 will present the model and section 3 theestimation procedure. We describe our data in section 4 and our results in section 5. Section 6concludes. 2.
The model
In what follows, it is assumed that the full type of each individual, i.e., the complete set of allindividual characteristics that matter for the marriage market (physical attributes, psychologicaltraits, socio-economic variables, sex, sexual orientation, etc.), is fully observed by market partic-ipants. Each individual is characterized by a vector of observable characteristics x ∈ X = R K ,which constitutes his or her observable type. However, following Choo and Siow, 2006, we allowfor a certain degree of unobserved heterogeneity by assuming that agents experience variations intastes that are not observable to the analyst, but are observable to the agents. In this paper,types are assumed to be continuous, as in Dupuy and Galichon, 2014, hereafter DG, and Menzel,2015. Assume that the distribution of the characteristics x has a density function f with respectto the Lebesgue measure. Without loss of generality, the marginal distribution of the attributes isassumed to be centered, i.e. E [ X ] = 0. CISCATO, GALICHON, AND GOUSS´E
Populations. A pair is an ordered set of individuals, denoted [ x , x ] where x , x ∈ X , inwhich the order of the partner matters, which implies that the pair [ x , x ] will be distinguishedfrom its inverse twin [ x , x ]. In empirical datasets, x will often be denominated “head of thehousehold” and x “spouse of the head of the household” even though this denomination is usedmainly for practical reasons and cannot be fully representative of the actual roles in the household .A couple is an unordered set of individuals ( x , x ), so that the couple ( x , x ) coincides with thecouple ( x , x ). A matching is the density of probability π ( x , x ) of drawing a couple ( x , x ).Pairs [ x , x ] and [ x , x ] stand for the same couple, so that the density π ( x , x ) is the sum of thedensity of [ x , x ] and of the density of [ x , x ], hence the symmetry condition π ( x , x ) = π ( x , x )holds. This symmetry constraint means that the position of the individual must not matter andthus that there are no predetermined “roles” within the couple that would be relevant for theanalysis .We shall impose assumptions that will ensure that everyone is matched at equilibrium, hencethe density of probability of type x ∈ X in the population is given by R X π ( x, x ′ ) dx ′ , which countsthe number of individuals of type x matched either as the head of household in a couple [ x, x ′ ], oras the spouse of the head in a couple [ x ′ , x ]. Thus, we are led to assume: Assumption 1 (Populations) . The density π ( x, x ′ ) over couples satisfies π ∈ M sym ( f ) , where M sym ( f ) = π ≥ R X π ( x, x ′ ) dx ′ = f ( x ) ∀ x ∈ X π ( x , x ) = π ( x , x ) ∀ x , x ∈ X . In contrast, in the classical bipartite problem, we try to match optimally two distinct populations(men and women) which are characterized by the same space of observable variables X , and it isassumed that the distribution of the characteristics among the population of men has density f , We will come back in section 5.3 to this assumption that the roles of partners are exchangeable, which we testusing a number of proxies for asymmetric household roles. Candelon and Dupuy, 2015 extend P. A. Chiappori et al., forthcoming’s analysis to a model where agents formcouples with endogenously assigned roles according to their characteristics. The model is applied to team formationin professional road cycling. Fox, 2018 employs a very general many-to-many matching framework where agentsself-select to be buyers or sellers upon a meeting. In both cases, hierarchy (leader vs assistants) or roles (buyers vssellers) are clearly defined upon a match and observed in the data. This is unlikely to be the case when it comes tomore complex and long-lasting relationships such as marriage.
IKE ATTRACT LIKE? 9 while the density of the characteristics among the population of women is g . In this setting, theset of feasible matchings is typically given by: M ( f, g ) = π ≥ R X π ( x, y ) dy = f ( x ) ∀ x ∈ X R X π ( x, y ) dx = g ( y ) ∀ y ∈ X Hence, π ∈ M sym ( f ) if and only if π ∈ M ( f, f ) and π ( x , x ) = π ( x , x ). Thus the feasibilityset in the unipartite problem and in the bipartite problem differ only by the additional symmetryconstraint in the unipartite problem.2.2. Preferences.
We now model preferences. Following DG, it is assumed that a given individual x does not have access to the whole population, but only to a set of acquaintances { z xk : k ∈ Z + } ,randomly drawn, which is described below. Assumption 2 (Preferences) . An individual of type x matched to an individual of type x ′ enjoysa surplus which is the sum of three terms:(i) the systematic part of the pre-transfer matching surplus enjoyed by x from his/her match with x ′ , denoted α ( x, x ′ ) .(ii) an endogenous utility transfer from x ′ to x , denoted τ ( x, x ′ ) . This quantity can be eitherpositive or negative; we assume utility is fully transferable, hence feasibility imposes τ ( x, x ′ ) + τ ( x ′ , x ) = 0 .(iii) a “sympathy shock” ( σ/ ε x , which is stochastic conditional on x and x ′ , and whose valueis −∞ if x is not acquainted with an individual x ′ . The quantity σ/ is simply a scaling factor.More precisely, the set of acquaintances is an infinite countable random subset of X ; it is such that ( z xk , ε xk ) are the points of a Poisson process on X × R of intensity dz × e − ε dε . While the stochastic structure of the unobserved variation in preference described in part (iii)of Assumption 2 may appear complex, it is in fact a very natural extension of the logit frameworkto the continuous case, as we now argue. Indeed, it will imply that the individual maximizationprogram of an agent of type x with this set of acquaintances ismax k ∈ Z + α ( x, z xk ) + τ ( x, z xk ) + σ ε xk , (2.1) where the utility of matching with acquaintance k yields a total surplus which is the sum ofthree terms, the systematic pre-transfer surplus, the transfer, and the sympathy shock. Define thesystematic quantity of surplus at equilibrium U by U (cid:0) x, x ′ (cid:1) := α (cid:0) x, x ′ (cid:1) + τ (cid:0) x, x ′ (cid:1) thus an individual of type x maximizes U ( x, z xk ) + ( σ/ ε xk over the set of his/her acquaintances,which are indexed by k . This induces an aggregate demand over the type space. Indeed, it followsfrom the continuous logit theory initiated in Dagsvik, 1994 that the conditional probability densityof an individual of type x matching with a partner of type x ′ is π ( x ′ | x ) = exp U ( x,x ′ ) σ/ R X exp U ( x, x ′ ) σ/ dx ′ . (2.2)It is clear from expression (2.2) that this is a generalization of the logit framework to the continuouscase.Note that, by the property of independence of irrelevant alternatives (IIA) of the logit model,we do not need to describe the utilities of unmatched agents as long as the distributions of theirstochastic parts are assumed to remain in the logit setting. Indeed, in the dataset we use, allagents are matched. Of course, one may worry about a potential equilibrium selection issue, i.e.,that being matched affects the distributions of the agents’ unobserved heterogeneity; however, inthe logit setting, the IIA property guarantees that the distributions are preserved even after theselection, as shown in appendix D of DG. This is the reason why we consider a model whereeveryone is matched at equilibrium.2.3. Equilibrium.
Next, we define equilibrium in this framework. DenoteΦ (cid:0) x, x ′ (cid:1) := α (cid:0) x, x ′ (cid:1) + α (cid:0) x ′ , x (cid:1) = U (cid:0) x, x ′ (cid:1) + U (cid:0) x ′ , x (cid:1) IKE ATTRACT LIKE? 11 the systematic part of the joint surplus between x and x ′ . It follows from (2.2) and symmetry of π that ( σ/
2) ln π (cid:0) x, x ′ (cid:1) = U ( x, x ′ ) − a ( x ) = U ( x ′ , x ) − a (cid:0) x ′ (cid:1) , (2.3)where a ( x ) : = σ Z X f ( x ) exp U ( x, x ′ ) σ/ dx ′ . (2.4)Substituting out for U in (2.3) yields the following equation, which expresses optimality inindividual decisions: log π (cid:0) x, x ′ (cid:1) = Φ ( x, x ′ ) − a ( x ) − a ( x ′ ) σ , (2.5)At equilibrium, the value of a ( . ) is determined by market-clearing condition R X π ( x, x ′ ) dx ′ = f ( x ), that is Z X exp (cid:18) Φ ( x, x ′ ) − a ( x ) − a ( x ′ ) σ (cid:19) dx ′ = f ( x ) . (2.6)We can now define our equilibrium matching concept. Definition 1.
The density π is an equilibrium matching if and only if there is a function a ( . ) suchthat both optimality equations (2.5) and market clearing equations (2.6) are satisfied. The main results on equilibrium characterization are summarized in the following statement:
Theorem A.
Under Assumptions (1) and (2) :(i) The equilibrium matching π ( x, x ′ ) is the unique solution tomax π ∈M ( f,f ) Z Z
X ×X Φ( x, x ′ ) π ( x, x ′ ) dxdx ′ − σ E ( π ) , (2.7) where E ( π ) is defined by E ( π ) = Z Z
X ×X π ( x, x ′ ) ln π ( x, x ′ ) dxdx ′ . (2.8) (ii) The expression of π ( x, x ′ ) is given by π (cid:0) x, x ′ (cid:1) = exp (cid:18) Φ ( x, x ′ ) − a ( x ) − a ( x ′ ) σ (cid:19) , (2.9) Note that Φ is symmetric by definition, but α has no reason to be symmetric. Mathematically speaking, Φ is(twice) the symmetric part of α . where a ( . ) is a fixed point of F , which is given by F [ a ] ( x ) = σ log Z X exp (cid:18) Φ ( x, x ′ ) − a ( x ′ ) σ (cid:19) dx ′ − σ log f ( x ) . (2.10) Proof.
By DG, Theorem 1, Problem (2.7) has a unique solution which can be expressed as π (cid:0) x, x ′ (cid:1) = exp (cid:18) Φ ( x, x ′ ) − a ( x ) − b ( x ′ ) σ (cid:19) for some a ( x ) and b ( x ′ ) determined by π ∈ M ( f, f ). By the symmetry of Φ and by the symmetryof the constraints implied by π ∈ M ( f, f ), then ˜ π ( x ′ , x ) := π ( x, x ′ ) is also solution to (2.7). Byuniqueness, ˜ π = π , thus π ( x, x ′ ) = π ( x ′ , x ). As a result, b ( x ) = a ( x ), where a is determined by Z exp (cid:18) Φ ( x, x ′ ) − a ( x ) − a ( x ′ ) σ (cid:19) dx ′ = f ( x )QED.This result deserves a number of comments. First, we should note that there is an interestinginterpretation of (2.7). While the first term inside the maximum tends to maximize the sum ofthe observable joint surplus, and hence draws the solution toward assortativeness, the second term E ( π ) is an entropic term which draws the solution toward randomness. The trade-off betweenassortativeness and randomness is expressed by the ratio Φ /σ . If this ratio is large, the assortativeterm predominates, and the solution will be close to the assortative solution. If this ratio is small,the entropic term predominates, and the solution will be close to the random solution. At the sametime, note that the model parameterized by (Φ , σ ) is scale-invariant: if k >
0, then the equilibriummatching distribution π when the parameter is (Φ , σ ) is unchanged when the parameter is ( k Φ , kσ ).This will have important consequences for identification, which is discussed in the next paragraph.As a consequence of this result, we can deduce the equilibrium transfers and the utilities atequilibrium. Indeed, note that combining the expression of π as a function of U and a and equation(2.5) yields the following expression of U as a function of a : U ( x, x ′ ) = (cid:0) Φ( x, x ′ ) + a ( x ) − a ( x ′ ) (cid:1) / . (2.11)which is the systematic part of utility that an individual of type x obtains at equilibrium from amatch with an individual of type x ′ . It is equal to half of the joint surplus, plus an adjustment IKE ATTRACT LIKE? 13 ( a ( x ) − a ( x ′ )) / x and x ′ . These bargaining powersdepend on the relative scarcity of their types; indeed, a ( x ) is to be interpreted as the Lagrangemultiplier of the scarcity constraint which imposes that π ( ., x ) should sum to f ( x ). Hence, theequilibrium transfer τ ( x, x ′ ) from x to x ′ is given by τ (cid:0) x, x ′ (cid:1) = (cid:0) α ( x ′ , x ) − α (cid:0) x, x ′ (cid:1) + a ( x ) − a ( x ′ ) (cid:1) / . (2.12)Next, note that an interesting feature of Theorem A is that, while it characterizes equilibrium inthe same-sex marriage problem, it highlights at the same time the equivalence with the different-sex marriage problem: indeed, as argued in DG, Theorem 1, the equilibrium matching in thedifferent-sex marriage problem is given by the same expression as (2.7), with the only differencethat M ( f, f ) is replaced by M ( f, g ), where f and g are respectively the distribution of men andwomen’s characteristics.We will use this characterization of the equilibrium matching as the solution of an optimizationproblem in order to estimate the joint surplus Φ based on the observation of the matching density π . As it is classical in the literature on the estimation of matching models with transferable utility,the primitive object of our investigations will be the joint surplus Φ rather than the individualpre-transfer surplus α ; indeed, without observations on the transfers, there is no hope to identify α : if we estimate that there is a high level of joint surplus in the ( x, x ′ ) relationship, we will notbe able to determine if this is due to the fact that “ x likes x ′ ” or “ x ′ likes x ”. We will only be ableto estimate that there is a high affinity between x and x ′ .3. Estimation
Estimation of the affinity matrix.
Following DG, we assume a quadratic parametrizationof the surplus function Φ to focus on a limited number of parameters which could characterize thematching patterns. We parametrize Φ by an affinity matrix A so thatΦ A ( x, y ) = x ′ Ay = X ij A ij x i y j where A has to be symmetric ( A ij = A ji ) in order for Φ to satisfy the symmetry requirement. Thenthe coefficients of the affinity matrix are given by A ij = ∂ Φ( x, y ) /∂x i ∂y j at any value ( x, y ). Matrix A has a straightforward interpretation: A ij is the marginal increase (or decrease, according to thesign) in the joint surplus resulting from a one-unit increase in the attribute i for the first partner,in conjunction with a one-unit increase in the attribute j for the second. Hence, this approach isarguably the most straightforward way to model pairwise positive or negative complementaritiesfor any pair of characteristics. It does, however, not preclude nonlinear functions of the x i ’s andthe y j ’s, which can always be appended to x and y .Recall equation (2.7), the optimal matching π maximizes the social gain W ( A ) = max π ∈M ( f,f ) E π (cid:2) x ′ Ay (cid:3) − σ E π [ln π ( x, y )] (3.1)which yields likelihood π A ( x, x ′ ) of observation ( x, x ′ ), where π A is the solution to (3.1). By theenvelope theorem, ∂ W ( A ) /∂A ij = E π A (cid:2) x i y j (cid:3) . Hence, our empirical strategy is to look for ˆ A satisfying ∂ W ( ˆ A ) /∂A ij = E ˆ π (cid:2) x i y j (cid:3) , (3.2)where ˆ π is empirical distribution associated with the observed matching.As noted before, the model with parameters ( A, σ ) is equivalent to the model with parameters( kA, kσ ) for k >
0. Hence, a choice of scale normalization should be imposed without loss ofgenerality; a simple choice when a single market is considered is σ = 1, in which case the estimator A is meant as the estimator of the ratio of the affinity matrix over the scale parameter. Theobservation and comparison of multiple markets lead to slightly different normalization choices,which are discussed in section 3.5.If a sample of size n , { ( x , y ) , ..., ( x n , y n ) } is observed, then ˆ π ( x, y ) is the associated empiricaldistribution, which places mass 1 /n to each observation. In DG, an estimator of A is obtained bysolving the following concave optimization problemmin A ∈ M K W ( A ) − E ˆ π [ X ij A ij x i y j ] , (3.3) IKE ATTRACT LIKE? 15 where M K is the set of real K × K matrices. Indeed, the first order conditions associated to (3.3)are exactly given by (3.2). However, in the present case, the symmetry of A is a requirement of themodel. The population cross-covariance matrix E π [ x i y j ] is symmetric, as π satisfies the symmetryrestriction π ( x, y ) = π ( y, x ) in the population. Yet, in the sample, ˆ π has no reason to be symmetric,as the first vector of variables x typically designates the surveyed individual, while the second vectorof variables y designates the partner of the surveyed individual. Hence, the empirical matrix ofcovariances E ˆ π [ x i y j ] will only be approximately symmetric. Thus, we symmetrize the sample byadding the symmetric households, that is, if household ij is included, meaning that individual i wassurveyed and reported partner j , we add a symmetric household ji , with j surveyed and reportingpartner i . In other words, we replace the empirical distribution ˆ π ( x, x ′ ) by its symmetric part(ˆ π ( x, x ′ ) + ˆ π ( x ′ , x )) /
2. In the sequel, ˆ π will denote that symmetric part. This leads us to proposethe following definition: Definition 2.
The estimator ˆ A of the affinity matrix is obtained by ˆ A = arg min A ∈ M K {W ( A ) − E ˆ π [ X ≤ i,j ≤ K A ij X i Y j ] } , (3.4) where M K is the set of real K × K matrices. The asymptotic behaviour of ˆ A is computed in DG, theorem 2. A word of caution, is, however, inorder. Although we have artificially doubled the sample size, by complementing household ( x i , y i )with its mirror image ( y i , x i ), one should beware that the sample size remains n , not 2 n . Thus, wecan use directly the bipartite estimator on the mirrored sample, with the only modification thatone will need to multiply the standard errors by a factor √
2, as the effective sample size has notdoubled.3.2.
Categorical variables.
The previous analysis can be slightly adapted to deal with the caseof categorical variables, such as race. Assume that the set of categories is denoted R = { , ..., r } .Assume that the individuals are characterized by x = (cid:0) x S , x R (cid:1) , where x S ∈ R K are socio-economiccharacteristics, and x R ∈ R r is a vector of dummy variables x Ri (1 ≤ i ≤ r ) equal to 1 if individual x is of category i ∈ { , ..., r } , and zero otherwise. We work with the following specification of thesurplus Φ ( x, y ) = (cid:0) x S (cid:1) ′ A S y S + λ R (cid:8) x R = y R (cid:9) (3.5) where λ R is a term that reflects assortativeness on the categorical variable, which provides a utilityincrement λ R if both partners belong to the same category. Of course, this surplus function can beexpressed multiplicatively as Φ ( x, y ) = x ′ Ay , where A can be written blockwise as A = A S λ R I r (3.6)and hence, A is obtained by running optimization problem (3.4) subject to constraint (3.6). Notethat the envelope theorem implies that λ R is identified by the moment matching conditionPr π (cid:0) x R = y R (cid:1) = Pr ˆ π (cid:0) x R = y R (cid:1) which states that the predicted frequency of interracial couples should match the observed one.3.3. Saliency analysis.
The rank of the affinity matrix is informative about the dimensionalityof the problem, that is, how many indices are needed to explain the sorting in this market. Toanswer this question, DG introduced saliency analysis , which consists of looking for successiveapproximations of the K -dimensional matching market by p -dimensional matching markets ( p ≤ K ). Assume (without loss of generality as one can always rescale) that var ( X i ) = var ( Y j ) = 1.Then saliency analysis consists of a singular value decomposition of the affinity matrix A = U ′ Λ V ,where U and V are orthogonal loading matrices, and Λ is diagonal with positive and decreasingcoefficients on the diagonal. This idea is found in Heckman, 2007, who interprets the assignmentmatrix as a sum of Cobb-Douglas technologies using a singular value decomposition in order torefine bounds on wages. This allows to introduce new indices ˜ x = U x and ˜ y = V y which areorthogonal transforms of the former, and such that the joint surplus reflects diagonal interactionsof the new indices, i.e. Φ ( x, y ) = x ′ U ′ Λ V y = ˜ x Λ˜ y .Here, we need to slightly adapt this idea to take advantage of the symmetry of A and of therequirement that the matrix of loadings U and V should be identical. The natural solution is theeigenvalue decomposition of A , which leads to the existence of an orthogonal loading matrix U and a diagonal Λ = diag ( λ i ) with non-increasing (but not necessarily positive) coefficients on thediagonal such that A = U ′ Λ U. IKE ATTRACT LIKE? 17
This allows us to introduce a new vector of indices ˜ x = U x , which are orthogonal transforms of theprevious indices. That way, the joint surplus between individuals x and y is given byΦ ( x, y ) = x ′ U ′ Λ U y = ˜ x ′ Λ˜ y = K X p =1 λ p ˜ x p ˜ y p hence this term only reflects pairwise interactions of dimension p of ˜ x and ˜ y , which are eithercomplements (if λ p >
0) or substitute (if λ p < Theorem B.
Assume that E ˆ π [ X ] = 0 and that var ˆ π (cid:0) X i (cid:1) = 1 for all i . Then there exists anorthogonal loading matrix ˆ U and a diagonal ˆΛ = diag ( λ i ) with non-increasing coefficients on thediagonal such that ˆ A = ˆ U ′ ˆΛ ˆ U and, denoting ˜ x = ˆ U x and ˜ y = ˆ U y , the estimator of the surplus function is given by ˆΦ ( x, y ) = ˜ x ′ ˆΛ˜ y = K X p =1 λ p ˜ x p ˜ y p . Proof.
Because ˆ A is symmetric, it has the following eigenvalue decompositionˆ A = ˆ U ′ ˆΛ ˆ U where ˆ U is orthogonal, and ˆΛ = diag ( λ i ) is diagonal with non-increasing coefficients. Denoting˜ x = ˆ U x and ˜ y = ˆ U y , x ′ Ay = x ′ ˆ U ′ ˆΛ ˆ U y = ˜ x ′ ˆΛ˜ y = K X p =1 λ p ˜ x p ˜ y p . In the presence of categorical variables, the presence of a block λI r in (3.6) reflecting assortative-ness on the categorical variable implies that the singular values of A will be the singular values of˜ A in addition to λ with multiplicity r . Therefore, it is recommended to perform saliency analysissimply on the upper left block ˜ A . Selection issues.
The purpose of the present paper is to compare match formation acrosssame-sex and different-sex marriage market using the tools we developed above. In order to doso, we need to clearly delineate what is the relevant market in which agents match. We make thefollowing assumption that gay men, lesbians and heterosexuals match on segmented markets, whichwe formalize into:
Assumption 3 (Exogenous selection) . The selection into either the same-sex or different-sex mar-riage market is exogenous.
To relax this assumption, one would have to assume that all agents are pooled together in thesame market, and choose their partner’s gender among other characteristics, based on their ownsexual orientation. The marital outcome, including the gender composition of households, wouldthen be an equilibrium outcome resulting from a trade-off between socio-economic complementari-ties and other terms reflecting interactions between genders, and sexual orientations of the partners.We develop and discuss this relaxed framework in appendix A, which shows that the market can beformulated as a single unipartite one where individuals are characterized by sexual orientation andgender in addition to other socioeconomic traits, and choose the gender of their partners amongother characteristics. The more restrictive framework provided by assumption 3 can be obtainedas a limiting case of the single-market framework where the interaction between sexual orientationand gender is predominant with respect to other characteristics, and thus the partner’s gender isfully determined by own sexual orientation and gender.In the absence of data on sexual orientation in our database, assumption 3 allows to infersexual orientation from market participation, and therefore it permits to perform estimation of theaffinity matrix expressing the interactions of the socioeconomic characteristics. However, if data onsocioeconomic characteristics, gender and sexual orientation of matched partners were available,then the full matrix A could be estimated in a straightforward manner using our methodology,allowing to capture interactions not only between socio-economic terms, but also between genderand sexual orientation, etc., as explained in appendix A . While this may be out of reach with current large-scale datasets, it is not unrealistic to believe that it will bepossible to perform this type of analysis in the future. The National Survey of Family Growth, for instance, already
IKE ATTRACT LIKE? 19
As a final remark, the very fact that sexual orientation is exogenous is itself a strong assumption,and subject to current scientific debate. Researchers in biology, neuroscience, sexual medicine andpsychology have provided evidence on the influence of psycho-biological mechanisms on homosexualorientation (see Hines, 2011; Jannini et al., 2010). While there is an open debate among psycholo-gists and social scientists about the stability of sexual behaviour , research on early learning duringchildhood suggests that gender-typed behaviour –including sexual attraction– is internalized sinceinfancy and stabilizes by late adolescence (see Dillon et al., 2011; Hines, 2011).3.5. Comparison across markets.
Affinity matrices are a useful tool to analyze marital surplus,and we would like to use them to compare sorting patterns across different-sex and female/malesame-sex marriage markets. However, in order to achieve this, a discussion on normalization isneeded. Indeed, recall from the above discussion that the equilibrium matching π is the solution to W ( A, σ ) = max π ∈M ( f,f ) (cid:8) E π (cid:2) X ′ AY (cid:3) − σ E π [ln π ( X, Y )] (cid:9) , and therefore, a matching market with affinity matrix A and scaling parameter σ is observationallyequivalent to another market with the same distribution of types and affinity matrix kA and scalingparameter kσ for k >
0. Therefore, A and σ are not jointly identified, but only their ratio A/σ isidentified.It is therefore useful to adopt a normalization of (
A, σ ). For cross-market comparison purposes,the normalization σ = 1 advocated in DG can be misleading, as it assumes that the standarddeviation of the heterogeneity in preferences is the same across all markets considered. In this case,it seems more appropriate to normalize A by a factor so that the total quantity of surplus W ( A, σ )is scaled to one in each market. That is:
Assumption 4.
The affinity matrix A and the amount of heterogeneity σ are normalized so thatthe equality W ( A, σ ) = 1 holds in each market. contains detailed data on this topic, but unfortunately has no information on same-sex partnerships. This is due tothe latter being recognized at federal level only recently. Diamond, 2008a have provided the first piece of quantitative of the fluctuations of sexual orientation amongadults women using long panel data. However, psychologists avoid talking about sexuality as a “choice of lifestyle”:Diamond, 2008b, Chapter 5 considers changes in sexual orientation - and in other aspects of sexuality - as theconsequence of “complex interplays among biological, environmental, psychological, and interpersonal factors”.
When considering a single market, assumption 4 is a mere normalization, which can be imposedwithout loss of generality. It implies that the ratio of the average surplus provided by the interactionbetween characteristics i and j of two partners in a given market, divided by the average totalsurplus of a couple in that market is given by A ij E (cid:2) X i Y j (cid:3) W ( A, σ ) = A ij E (cid:2) X i Y j (cid:3) , and hence A ij E (cid:2) X i Y j (cid:3) is the share of the average surplus explained by the interaction betweencharacteristics i and j , relative to the average total surplus of a couple in that market.On the contrary, when considering multiple markets, assumption 4 is no longer an innocuousnormalization. It allows for a direct comparison of affinity matrices across markets, if one iswilling to make the restrictive assumption that the average surplus of a couple is the same in everymarket. Note that, since W ( kA, k ) = k W ( A,
1) holds for any scaling parameter k ≥
0, in practice,we impose this normalization by first computing the estimator ˆ A given by (3.4), and we then reportˆ A/ W ( ˆ A, Data
Data on same-sex couples.
Empirical studies on same-sex marriage have traditionallyneeded to cope with poor data, due to the late legal recognition of these partnerships – still un-achieved in several countries – and with misreporting issues, due to social pressure on respondents.Social scientists have largely relied on the data collected by the US Census Bureau for large sampleanalysis of same-sex unions (D. A. Black et al., 2007; L. K. Jepsen & Jepsen, 2002; Schwartz & Graf,2009). Starting from the 1990 decennial census, individuals could report themselves as “unmarriedpartners” within the household, regardless of their sex, so that gay couples could be identified.In more recent databases from the US Census Bureau, same-sex couples are still identifiable asout-of-marriage cohabiting partners. Indeed, although same-sex marriages have been officiated insome American states since 2004, they were recognized at federal level only in 2013, and currentlyavailable surveys conducted until then by the Census Bureau have not allowed reporting marriagebonds other than different-sex unions.Accordingly, the present work relies on the five-year Public Use Microdata Sample (PUMS) for2008-2012 coming from the ACS, conducted by the US Census Bureau. We restricted our sample
IKE ATTRACT LIKE? 21 to the state of California, which first legalized same-sex marriage on June 16, 2008 following aSupreme Court of California decision, and then – after some judicial and political controversiesthat impeded the officialization of same-sex weddings from November 5, 2008 to June 28, 2013 –a decision of the U.S. Supreme Court finally accomplished full legalization. Restricting the sampleto one state allows focusing on a marriage market with a uniform judicial framework. Moreover,in states where same-sex marriage is recognized, estimates on the number of married same-sexhouseholds are more reliable, and the incidence of the measurement error is smaller (Gates, 2010;Virgile, 2011).4.2. Descriptive statistics.
Our sample is limited to those individuals involved in a cohabitingpartnership, both married and unmarried, thus excluding singles but also couples whose partnersdo not live in the same home. Each couple is identified as a householder with his/her partner,where both share the same ID household number.The main database is composed of 681,060 individuals in couples who have completed theirschooling. Because we restrict ourselves to prime age couples (both partners 25-50 year old), thesize of our sample is decreased to 285,546 individuals. Out of them, 3,654 individuals (1.28% of thesample) live in same-sex couples, of which 2,034 male (0.71 %) and 1,620 female (0.57 %). 87.39% ofthe individuals in the sample are married heterosexuals and 11.33 % are cohabiting heterosexuals.For estimation purposes, after randomly selecting a subsample of different-sex couples , a total of9,820 couples are considered, of which 4,959 are married and 4,799 are not.To compare different marriage markets, following (L. K. Jepsen & Jepsen, 2002), the main sampleis divided into four subsamples: same-sex male couples, same-sex female couples, different-sexunmarried couples and different-sex married couples. This repartition is based on the assumptionthat individuals enter into separate markets according to their sexuality, in line with assumption 3.However, another criterion is used to differentiate two of the subgroups: married and unmarrieddifferent-sex couples are treated as two separate subpopulations , since empirical evidence hasreported significant differences in patterns between these two kinds of partnership (L. K. Jepsen In this period, marriage licenses issued to same-sex couples held their validity. We randomly select 4% of married couples and 30% of unmarried couples. See Mourifi´e and Siow, 2014 for a very interesting discussion of the endogenous choice of the form of maritalrelationship. & Jepsen, 2002; Schwartz & Graf, 2009). Although it is impossible to know a priori if a personis interested in a marital union rather than in a less binding relationship, this repartition can beof great interest and deepen the analysis. Nevertheless, even if California represents the largerstate-level ACS sample in the US, further splitting the gay and lesbian groups into two subgroupswould imply working with potentially very small samples. Moreover, although same-sex marriage ispermitted, it has been recognized only recently and at the end of many legal struggles, which mayhave prevented a part of those same-sex couples that wished to marry from doing so. With moredata available, considering married and unmarried same-sex couples separately would be extremelyinteresting, as proved by recent research of (Verbakel & Kalmijn, 2014) based on Dutch data.Our study takes into consideration several variables, some related to the labor market and someothers to the general background. Non-labor market traits include age, education and race. Ageand education are treated as continuous variables, with the latter defined as the highest schoolinglevel attained by the individual. Thanks to the detailed data of the ACS, the variable has beenbuilt in order to reflect as many distinct educational stages as possible. We consider five largeracial/ethnic groups: Non-Hispanic White, Non-Hispanic Black, Non-Hispanic Asian, Hispanic andOthers . Finally, among labor market variables, we compute and include hourly wage and usualamount of hours worked per week. Note that yearly wage is top-coded for very high values (over $ American demographic institutions do not include a Hispanic category in variables on race, furnishing a separatevariable for Hispanic origins, which is why there is some overlapping and the other categories bear the specification“Non-Hispanic”. The issue concerns the conceptual differences of ”race” and “ethnicity”. See for instance (Rodriguez,2000) for clarifications. The variable is computed as follows: we divide yearly wage by 52 in order to have the average weekly wagefor last year and then we divide it again by the usual number of hours worked per week, which is available in thedataset. The hourly wage is partly approximated because the exact number of weeks worked in the last 12 months isnot available. Note also that when information on labor earning or number of working weeks is missing, we set thehourly wage and the number of working weeks to 0 so that we keep all individuals in our analysis.
IKE ATTRACT LIKE? 23 couples, and also have higher wages. We observe that unmarried different-sex couples are muchyounger than married couples and same-sex couples. Unmarried heterosexual men and womenare on average four year younger than others. Cohabitation is often (but not always) a “trial”period before marriage, which can explain this age difference . Table 2 presents the distribution ofethnicity among couples: White individuals and Black women are overrepresented among lesbians,while Asians and Hispanics are under-represented in this population.Table 3 presents correlations among traits. It shows that age and educational attainment aremuch more correlated among married different-sex couples than among unmarried and same-sexones. Moreover, the correlation is stronger for lesbian couples than for gay male ones. Correlationson labor market outcomes are particularly interesting: hours worked are negatively correlated onlyfor married different-sex couples, a possible clue of stronger household specialization, whereas thecorrelation is positive albeit low for same-sex couples. On the other hand, wages display a positivecorrelation in every market, with different-sex married couples and male same-sex couples exhibitingthe lowest correlation.Table 4a, 4b and 4c present homogamy rates of couples with respect to race for different types ofcouples. The homogamy rate is the ratio between the observed number of couples of a certain typeand the counterfactual number which should be observed if individuals formed couples randomly.For instance, table 4b shows that lesbian couples among Black women form 10 times much morethan if they were formed randomly among the lesbian population.5. Results
Homogamy rates and correlations presented in section 4 are interesting measures of assortativemating and provide a good starting point for our analysis. However, they are not sufficient to reachany conclusion about the degree of assortativeness in the marriage market. By estimating theparameters of the surplus function, we compare the level of complementarity and substitutabilitybetween characteristics across different marriage markets. This approach is consistent with Becker’smodel of assortative mating, and allows us to measure the degree of assortativeness for each com-bination of characteristics ceteris paribus . In particular, we can test whether assortativeness on This would require a dynamic framework, which we don’t have in our static model. See the theoretical work ofBrien et al., 2006 and Gemici and Laufer, 2011. observables - notably, age, race and education - is weaker among same-sex couples, as found bySchwartz and Graf, 2009.While we measure the direction and strength of interactions between traits, we do not attemptto estimate preference and production terms separately. Hence, we cannot tell whether maritalgains differ across markets because of differences in household production rather than pure tastefor homogamy. In particular, we cannot tell to which extent differences in the opportunity cost ofbearing and raising children affect sorting patterns . If couples wish to have genetically relatedchildren, Assisted Reproductive Technologies imply that children inherit genetic traits from onlyone out of two partners, with possibly important implications for sorting. In our empirical analysis,we limit ourselves to the estimation of the model on carefully chosen subsamples (e.g., childlesscouples) in order to provide an intuition of where the major sources of diversity between gay anddifferent-sex couples lie. While not exhaustive, these robustness checks could constitute a usefulstarting point for future research.Finally, Becker’s model suggests that we interpret differences in assortativeness as a consequenceof differences in marital gains, rather than as a consequence of search dynamics (notably, geographicfactors and search frictions), segmentation into local markets along socio-economic traits, or pref-erences of third parties and social pressure (Kalmijn, 1998). In particular, gay individuals tend tomove away from their hometowns and may not be “out” at school or in the workplace (Rosenfeld &Kim, 2005), and this could influence the composition of their interpersonal ties . Also in this case,some of our robustness checks can help understand how these concurrent forces affect our results.Nonetheless, we believe that a model explicitly accounting for such factors would be necessary toquantify their impact on sorting patterns. Allen and Lu, 2017 propose a theoretical model which explains differences in expected matching behavior,marriage rates, non-child-friendly activities, and fertility, based on different costs of procreation and complementaritiesbetween marriage and children. For instance, online dating among heterosexuals has been found to reduce assortative matching on education(Hitsch et al., 2010). However, as dating apps and websites grow in popularity, thus giving access to a larger andlarger pool of possible matches, and tend to specialize on segmented markets (e.g., by ethnicity or religion), we wonderif this conclusion still holds.
IKE ATTRACT LIKE? 25
We report in table 5 the estimates of the affinity matrix for gays, lesbians, married and cohabitingheterosexuals.5.1.
Age, education and race/ethnicity.
Our estimates of the diagonal elements of the affinitymatrices are highly positive and significant for age, education and ethnicity, which confirms previousfindings about positive assortative mating. In line with the results by L. K. Jepsen and Jepsen, 2002and Schwartz and Graf, 2009, we find that assortativeness on age and ethnicity is comparativelyweak for male same-sex couples (0.62 for age, 0.62 for ethnicity), and progressively stronger forfemale same-sex (0.79, 1.26) and unmarried different-sex couples (1.14, 1.98), whereas marrieddifferent-sex couples exhibit the strongest complementarities (2.17, 2.49). Results on education aremore nuanced: complementarity of schooling levels is the strongest for lesbian couples (1.19), whileestimates for married same-sex (0.82) and gay couples (0.84) are not significantly different. Finally,complementarity of schooling levels is the lowest for unmarried heterosexuals (0.66).Our estimates on the level of educational sorting is partly at odds with previous findings. Em-pirical research on this topic mainly concluded that assortativeness on education is weaker on bothmale and female same-sex marriage markets with respect to different-sex marriage markets. How-ever, the social science literature provides a large set of explanations about why sorting patternsshould differ across different-sex and same-sex couples, and not all of them predict that educationalsorting is weaker among the latter. On the one hand, gay men and lesbians are expected to bemore inclined to “transgress” social norms and to cross socio-economic and racial barriers whenchoosing their partner (Rosenfeld & Kim, 2005; Schwartz & Graf, 2009). Our findings suggest thatthis effect might be prevalent as concerns age and ethnicity. On the other hand, gay people arealso expected to have stronger egalitarian preferences . Verbakel and Kalmijn, 2014 suggest thatsimilar schooling levels can lead to a more equal division of labor. Spouses that aim to concentratetheir efforts on the labor market rather than to specialize each in a different set of skills might thusexhibit a stronger level of assortativeness on education. The main reference works about mating among gay people are listed in our introduction. We refer to Schwartzand Graf, 2009 and Verbakel and Kalmijn, 2014 who, drawing from literatures from different social sciences, bothprovide a complete and updated review on this topic.
As anticipated above, childrearing is a major driver of household specialization, and same-sexcouples are less likely to have children . Hence, we estimate the affinity matrix using the subsampleof childless couples for each of our four marriage markets (see summary table 8 and the full tablesin the online appendix). We find that, with respect to ethnicity, both childless same-sex andchildless different-sex couples exhibit a weaker taste for homogamy compared with couples withchildren of the same respective sexual orientation. Similar results hold for sorting on age, althoughonly differences between married different-sex couples with and without children are significantlydifferent. It thus seems that individuals who plan to have children look for a more similar partneralong these two dimensions than those who do not.When it comes to education, the picture is a bit more contrasted. As for the previously discussedtraits, one observes stronger assortativeness on education for same-sex couples with children thanfor those without. In contrast, childless different-sex couples are more assortatively matched oneducation than those with children. Different-sex couples who do not plan to have children will notbenefit from large gains from specialization and may look for a partner with similar schooling. It isinteresting to note that married childless different-sex couples are found to exhibit a higher degreeof assortativeness with respect to age, ethnicity and education with respect to same-sex childlesscouples .5.2. Labor market traits.
To describe labor market traits, we must be very cautious as theseoutcomes are potentially endogenous. Since we do not observe these traits at the moment of thematch formation but possibly much later, the specialization process at work in couples may have In our sample, among the 25-50 years-old, 14.5% of gay men have children, 37.8% of lesbians, 58.04% ofcohabiting different-sex couples and 83.5% of married different sex couples. We are aware that the subsample constitutes an “ artificial” marriage market, since individuals do not rigidlyself-select into a separate market based on their preference for having children. However, our model does not have aspecification that explicitly accounts for choices related to childbearing. We also estimate the affinity matrix for married different-sex couples with one and three children (see table 8c).Our findings are in line with what stated in the main text. The higher the number of children, the stronger theassortativeness on age and ethnicity, and the weaker the assortativeness on education.
IKE ATTRACT LIKE? 27 already begun. In particular, we expect that this specialization effect is strong in different-sex cou-ples, who are more likely to have children . Raising children takes time and many mothers leavethe labor force or reduce their working hours. Consequently, because of interrupted careers andless paid part-time jobs, their hourly wage does not rise as much as that of their male counterpartsand we observe many associations between low-wage women and high-wage men. This phenomenoncould bias our estimates. To assess the importance and the sign of this bias, we also perform theestimation on four additional selected samples where the specialization effect should be limited :1) childless couples, 2) bi-earner couples, 3) young couples (25-35 year old), 4) recently marriedcouples with no children. The last selection is only available for different-sex married couples aswe observe their wedding date; we keep couples who got married in the preceding year and whohave no children. A summary of the results is available in tables 8a, 8b, 8c and 8d. The full tablesare available in the online appendix. We first describe the general results obtained from the mainsample.First, we measure significant positive assortativeness on hourly wages for all types of couples,although the coefficient is higher for same-sex couples (0.05 for gays and 0.06 for lesbians) and fordifferent-sex unmarried couples (0.05) than for married different-sex couples (0.01). Furthermore,we observe negative assortative mating on working hours for married different-sex couples (-0.04),whereas we observe much higher and significant positive estimates for same-sex couples (0.12 forgays and 0.20 for lesbians). The coefficient for unmarried couples is also positive and significant(0.09). Assortative mating on wages and working hours is likely to be related to the presenceof children. As same-sex couples are less likely to have children, they have weaker incentivesfor specialization. Unmarried couples may also have lower preferences for children than marriedcouples. To better understand this result, we estimate our model on childless couples. We find thatmarried different-sex couples without children have a positive coefficient for both wage (0.07) andworking hours (0.12), and thus are more similar to same-sex and different-sex unmarried couples.In this regard, our results are in line with those of C. Jepsen and Jepsen, 2015. Similarly, the (Antecol & Steinberger, 2013) and (C. Jepsen & Jepsen, 2015) showed that to a lesser extent some householdspecialization also exists within same-sex households. Moreover, Antecol and Steinberger, 2013 stress that childlessdifferent-sex couples are less specialized and thus more similar to same-sex couples. assortative mating coefficients for wages and hours are higher – but to a lesser extent – for same-sex couples without children compared to those with children. We also perform the estimationon married couples with only one child and married couples with three children to disentanglethe effect of the presence of children among married couples. As couples have more children, weobserve a decrease in assortative mating coefficients for wages and hours. Similarly, as expected, theestimation on bi-earner couples shows an increase in assortative mating on wages and hours . Inthe case of young same-sex couples the comparison leads to less clear-cut conclusions due to smallsample size, while positive assortative mating on labor market traits is stronger when comparingyoung different-sex couples with those in our main sample.The cross-estimate between the wage of one partner and working hours of the other partner isalso very interesting to analyze. We find negative assortative mating on wages and working hours.The estimate is highly negative for lesbians (-0.19) and different-sex married couples (-0.09 for theinteraction between wife’s wage and husband’s hours, -0.13 for the symmetric interaction). It is alsonegative and significant for gays (-0.07). Hence, for heterosexual married couples and homosexualcouples, the match gain increases when one partner increases his/her wage and the other decreaseshis/her working hours. This result is robust to the presence of children, to the age of couples andto the bi-earner sample.5.3. Other cross-interactions and symmetry.
Other significant positive cross-effects have beenfound for some off-diagonal elements of the affinity matrix. The parameter capturing the interactionbetween wage and education is persistently high and positive. This might suggest that higher wageindividuals have a preference for more educated partners, keeping constant their wage and all othercharacteristics. Affinity between these two variables is relatively weaker for gay men (0.13) andlesbians (0.20). Complementarity between the two inputs is stronger for different-sex couples, butthe relationship is asymmetric: estimates for married couples suggest that complementarity inhusband’s wage and wife’s education is stronger (0.27) than the other way around (0.19). However,the corresponding estimates for unmarried couples go in the opposite direction (0.21 and 0.37respectively). The complementarity between the two traits might be explained by the fact that In our analysis, bi-earner couples are couples whose both members declare positive wage and number of workingweeks.
IKE ATTRACT LIKE? 29 high-income individuals - independently of their educational level - may enjoy the company ofcultured partners.Another cross-interaction that arises from the estimation is the substitutability between age andhours worked on same-sex marriage markets (-0.13 for gay men and -0.10 for lesbians). This in-teraction might be due to household bargaining dynamics, as explained by Oreffice, 2011: youngerpartners enjoy higher bargaining power and thus can afford reducing their labor supply. Interest-ingly, unmarried different-sex couples exhibit similar patterns, although the effect is weaker.Finally, as a last robustness check for our main results, we estimate a bipartite matching model ofthe same-sex marriage market where the affinity matrix is not required to be symmetric. In this case,we need to define two separate subpopulations to run a bipartite estimation, and therefore we needto define household roles. On one side of the market, we group all those partners that are registeredas “householders”, whereas on the other we group their “cohabiting partners”. This repartitionis highly artificial, since it implies that two gay individuals that are householders before finding apartner can never match: in general, it seems implausible to divide the same-sex population in twoseparate subgroups with the data that we have at hand. Nonetheless, it is interesting to check -under the strict assumption of predetermined roles - whether some asymmetry in cross-interactionsoccurs. We observe that the affinity matrices for both gay men and lesbians (respectively tables 7aand 7b) are not much different than in the unipartite case. When testing for differences betweenthe off-diagonal coefficients , we find that only the cross-interaction between age and hours issignificantly different between householders and their partner in male same-sex households. Theinteraction is significant only in one direction, relatively young cohabiting partners can reduce theirlabor supply when matching with older householders. The difference is non-significant for femalesame-sex households. We already stated that we were not very confident in the label “householder”to define a particular role in the couple. As additional robustness checks, we assign a particularrole to each partner in same-sex couples according to another characteristic. We test if there is aparticular role assigned to 1) the older partner, 2) the higher earner (total income) partner. Eachpartner belongs to a certain population depending to his/her status in the couple, then we estimatea bipartite matching model of the same-sex marriage market on these two populations. Results We provide test results in the online appendix. are presented on the online appendix. Again, they are not much different from the unipartitecase. We now detail some exceptions. When separate populations are defined according to therelative age, we now find a negative interaction between age of the older partner and education andhours worked of the younger partner among gay couples. When separate populations are definedaccording to the relative total income, symmetry is respected for all coefficients for male same-sexcouples but not for female same-sex couples. Specialized roles may appear among lesbian couplesas they are more likely to have children. There are asymmetries in the interaction between hoursand education, hours and wage and education and wage. The coefficient of interaction betweeneducation and wage is always high and positive but it is much higher between the education of thehigher earner and wage of the lower earner than the other way around.5.4.
Matching on unobservables.
Thanks to assumption 4, we can evaluate the parameter σ for each market. As anticipated in section 2, this parameter has a simple interpretation: the higher σ , the more matching appears as random to the econometrician, or, in other words, the higherthe entropy. Since our observable characteristics are meant to capture the main socio-economictraits, we expect that a higher σ implies that matching is less “deterministic”: indeed, for higher σ ,the socio-economic background of an individual matters relatively less, whereas other unobservabletraits (e.g. personality or physical appearance) may matter relatively more.We find that entropy is higher on same-sex marriage markets (1.26 for gay men and 1.23 forlesbians), whereas it is lower on different-sex marriage markets (1.04 for unmarried couples and1.00 for married). Hence, if we interpret entropy as due to the relevance of unobservables, it seemsthat, among same-sex couples, socio-economic background matters less relatively to unobservabletraits.While we privilege an interpretation of σ that is consistent with Becker’s frictionless matchingmodel, it is important to recall that differences in the search process are also captured by σ .All else held constant, stronger search frictions should result in higher entropy (Shimer & Smith,2000). However, we are not able to disentangle the effects of search frictions from the relevanceof unobservables. This may be problematic when comparing the same- and different-sex marriagemarkets because gays constitute a relatively small part of the population. If the frequency of IKE ATTRACT LIKE? 31 meetings is increasing in the size of the pool of potential partners, as suggested by the search-and-matching literature , then we should expect search frictions to be stronger for the gay population.In order to address this issue, we estimate the affinity matrix for a subsample of couples living inthe metropolitan areas of Los Angeles and San Francisco. We expect search frictions to be smallin densely populated urban areas, and meeting opportunities to be comparable for homosexualsand heterosexuals. While the point-estimates of σ are almost unchanged for different-sex couples(1.03 for unmarried, 1.00 for married), we observe an increase for both gay men (1.35) and lesbians(1.32) . Since the estimates of σ for same-sex couples are persistently higher even in areas wheresearch frictions are expected to be lower, we conjecture that the difference in entropy can, at leastin part, be explained by unobservable traits.5.5. Saliency analysis.
A way to bring further insights on the main drivers of preferences ofindividuals over different characteristics is to decompose the affinity matrix in orthogonal dimen-sions. As detailed in section 3.3, we conduct the decomposition analysis on all variables, with aspecific treatment for race, a categorical variable. Under the parametrization (3.5), we estimatethe affinity matrix driving the interactions of the non-race characteristics, and the coefficient λ R which measures the homogamy on race. We obtainΦ ( x, y ) = K X p =1 λ p ˜ x p ˜ y p + λ R (cid:8) x R = y R (cid:9) where ˜ x p is the p -th index associated to the individual with characteristics x obtained by thedecomposition, and x R is the race characteristics. Taking expectations with respect to the sampledistribution E ˆ π [Φ ( X, Y )] = K X p =1 λ p E ˆ π h ˜ X p ˜ Y p i + λ R E ˆ π (cid:2) (cid:8) X R = Y R (cid:9)(cid:3) , this allows us to decompose the average surplus E ˆ π [Φ ( X, Y )] into the sum of the surplus createdby the interaction between characteristics p , namely λ p E ˆ π h ˜ X p ˜ Y p i , plus the surplus created by In the search-and-matching literature, the meeting rate is usually modeled as a constant return to scale functionof vacancies of both sides of the market, usually a Cobb-Douglas function. This assumption seems to be supportedempirically on the labor market (Petrongolo & Pissarides, 2001). On the marriage market, Gouss´e et al., 2017 alsouse a Cobb-Douglas matching function. Complete tables of results obtained with the subsample of couples residing in urban areas are available in theonline appendix. Urban areas are defined as the Metropolitan Statistical Areas of Los Angeles and San Francisco. homogamy on the racial characteristics λ R E ˆ π (cid:2) (cid:8) X R = Y R (cid:9)(cid:3) . We present such a decompositionin the appendix in table 6. First, we show that the share of the average surplus created byhomogamy on the racial characteristics reaches more than 47% for different-sex couples, 42% forlesbian couples and only 33% for gays. Then, the rest of the average surplus can mainly be explainedin two orthogonal dimensions which measure relative attractiveness. These indices load on differentcharacteristics of individuals. For the different-sex and the lesbian marriage market, the first indexis almost only composed of age. It explains by itself around 36% of the surplus for married different-sex couples, 33% for unmarried different-sex couples and 29% of the surplus for female same-sexcouples. Then the second index mostly relies on education for these markets, and explains around27% of the sorting for female same-sex couples, 20% for unmarried different-sex couples and 16%for different-sex married couples. For male same-sex couples, the first index of sorting relies oneducation (35%) then comes ethnicity, then age (30%). When we consider different-sex couples,the indices of mutual attractiveness could differ between genders. For married heterosexuals, theeducation/wage index (the second index) loads positively twice as much on wages for men than forwomen, whereas there is a penalty on working hours for women that does not appear for men.6. Discussion and perspectives
The contributions of the present paper are twofold. From a methodological point of view, thispaper is the first to propose a tractable empirical equilibrium framework for the analysis of same-sexmarriage. Our methodology could be applied to many other markets (e.g. roommates, teammates,co-workers). In addition, we apply the model in order to provide an empirical analysis of sortingpatterns in the same-sex marriage market in California. We conduct a cross-market comparison: weanalyze the heterogeneity in preferences between same-sex and different-sex couples. First, we findthat, as concerns age and ethnicity, the different-sex marriage market is characterized by a stronger“preference for homogamy” than the same-sex marriage market. Meanwhile, results are morenuanced when it comes to education: while lesbians show stronger assortativeness on education,there is no significant difference in that dimension between gay male and married different-sexcouples. Second, we discuss the differences in complementarity and substitutability in the marriagesurplus function as defined in Becker’s theory of the family. Our findings suggest that labor markettraits are complementary only for same-sex and unmarried cohabiting different-sex couples. The
IKE ATTRACT LIKE? 33 presence of children seems to be a central driver of these contrasted findings. These results indicatethat the traditional concept of marriage gain based on specialization within the couple is stillrelevant today, although it applies mainly to couples with children.Families and household arrangements are evolving quickly and we need to understand the un-derlying forces of these changes. The need for effective analytical frameworks to study and describenew forms of families has recently emerged in the economic literature, mainly as concerns same-sex couples (D. A. Black et al., 2007; Oreffice, 2011) and cohabiting partners (Gemici & Laufer,2011; Stevenson & Wolfers, 2007). Let us briefly discuss both. In this paper, we found structuraldifferences between three separate subpopulations divided according to sexual preferences. How-ever, can we state with certainty that these markets are mutually exclusive? In fact, individualsmay endogenously choose into which market they are willing to match. The appendix exploresa theoretical framework to move beyond the exogenous selection hypothesis. While the lack ofindividual-level data on sexual orientation does not allow at this stage to go beyond a theoreticalmodel, we trust that, with the consolidation of the same-sex marriage and the availability of moreand more accurate data, it will soon be possible to expand our understanding on these questions.Second, cohabitation is another developing phenomenon (Gemici & Laufer, 2011; Schwartz & Graf,2009; Verbakel & Kalmijn, 2014) and is associated with a lower degree of specialization and a lowerdegree of positive assortative mating. A promising area of research would be to understand thepreferences for marriage or cohabitation jointly with sorting preferences. Mourifi´e and Siow, 2014set a first model in that direction for different-sex couples, which could be adapted to same-sexcouples using the techniques put forward in the present paper; see also the empirical analysis of(Verbakel & Kalmijn, 2014), and Ald´en et al., 2015 for the marital and fertility decisions of same-sexhouseholds.Another topic of interest is the effect of new forms of families on the traditional ones over time.Opponents of same-sex marriage have voiced the fear that it will cause the marriage institutionto lose its value and favor alternative forms of families, typically more flexible/less stable, such ascohabitation. For now, researchers have found no effect of same-sex marriage on the number ofdifferent-sex marriages or on the number of divorces (Trandafir, 2014, 2015). However, we wonderwhether the legal recognition of same-sex marriage could someway impact the preferences observedon the different markets. What changes should we expect in the behavior of heterosexuals? And could it be that same-sex couples become more homogamous as same-sex marriage is institution-alized?Finally, in Becker’s theory, a rationale for marriage is the home production complementaritiesbetween men and women skills. However, the traditional gains from marriage have diminishedfor two main reasons. First, the progress in home technology has decreased the value of domesticproduction; second, as women took control over their fertility and have become more and moreeducated, their opportunity cost of staying at home has increased (Greenwood et al., 2016; Steven-son & Wolfers, 2007). Despite the decrease in the gains to traditional marriage, the institution ofmarriage has not disappeared. On the contrary, there has been a high demand for same-sex legalmarriage in many developed countries. (Stevenson & Wolfers, 2007) argue that individuals nowlook for a mate with whom they “share passions” and the new rationale for marriage is now “con-sumption complementarities” instead of “production complementarities”. It is also possible thatthe act of marriage itself is still considered as intrinsically valuable for cultural and social reasons.In any case, this evolution may lead to even higher correlation of traits. Time will tell how thesechanges will impact macroeconomic outcomes, life quality and social distance among individuals. ♭ Economics Department, Sciences Po, Paris.Address: Sciences Po, Department of Economics, 27 rue Saint-Guillaume, 75007 Paris, France.Email: [email protected]. † New York University, Department of Economics (FAS) and Department of Mathematics (CIMS),andToulouse School of Economics, Fondation Jean-Jacques Laffont.Address: NYU Economics, 19W 4th Street, New York, NY 10012, USA. Email: [email protected]. § Economics Department, Laval University.Address: Universit´e Laval, D´epartement d’Economie, Qu´ebec, (QC) G1V 0A6, Canada. Email:[email protected].
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Hypergamy, cross-boundary marriages, and family behavior (tech. rep.). IZA Discussion Paper.
Appendix A. Pooled matching market with sexual orientation
In this appendix, we consider a model with endogenous selection of the partner’s gender. Inthis model, gender and sexual orientation are observable characteristics among others. This modeltherefore assumes that all the individuals are pooled into one market, and that the partner’sgender is endogenously chosen and determined based on market characteristics, and in particular,it is subject to trade-offs with other variables. While we include this model for completeness, weregard it as a theoretical construction in the absence of matched data including a measure of sexualof orientation, and we do not present it in the main paper. The model that we use in the mainbody of the paper operates under the limiting assumption that sexual orientation fully determinesthe partner’s gender (assumption 3), which is why we assume in the paper that there are threecompletely segmented markets: lesbian, gay, and heterosexual.In reality, segmentation may not be perfect, and some people might face trade-offs betweenmatching with someone of their preferred gender rather than an attractive person of the less pre-ferred gender. In addition, some people might be equally attracted by both genders (i.e., bisexual).Assumption 3 implies that we disregard these trade-offs.Consider a model where individuals are represented by: (i) their economic characteristics x e ; (ii)their gender x g , which is a dummy variable equal to 0 if male, 1 if female (this could be extended toa continuous gender spectrum in which case x g may vary continuously between 0 and 1); and (iii) ameasure of sexual orientation x o which is set so that x o = 1 if the individual is maximally interested in women, and x o = 0 if maximally interested in men . In this case, letting x = ( x e , x g , x o ), theaffinity model will result in an affinity matrix written blockwise as A = A ee A eg A eo ∗ A gg A go ∗ ∗ A oo where the stars denote terms that are omitted due to the symmetry of A . Several components of A are especially interesting: • A ee is the classical affinity matrix between the socio-economic and demographic character-istics of the partners, and measures the pairwise assortativeness on education, age, income,race etc. Positive entries of A ee denote complementarity between these characteristics. • A go denotes the affinity between sexual orientation and gender, which is expected to bepositive, by the very definition of x o . • A gg denotes the utility penalization of same-sex couples with respect to different-sex ones,which we expect to be negative, in particular due to the relatively higher cost of bearingchildren for these households, and perhaps also due to social pressure in traditional societies.In this model, there is a trade-off in surplus between a term that reflects homogamy on socio-economic characteristics (whose strength is determined by A ee ), a term that reflects sexual orien-tation (whose strength is determined by A go ), and a term that reflects in particular the higher costof bearing children (whose strength is determined by A gg ).Note that, in the limit where the affinity term A go between gender and sexual preference is verystrong ( A go → + ∞ ) , we get a full segmentation of the markets, where the partner’s choice isfully determined by sexual orientation. In this case, we get a sequential choice model where theagents choose in a first stage which market (same-sex or different-sex) to enter, and then choosethe remaining partner’s characteristics. Because we do not observe sexual orientation, we decided Sexual orientation can be measured for instance by the means of the Kinsey scale (see Sell, 1997, for a review),which is a number between 0 (exclusively heterosexual) and 6 (exclusively homosexual). If k is the value of theKinsey scale, x o will be set to k/ − k/ x o := x g ( k/
6) + (1 − x g ) (1 − k/ EFERENCES 41 to adopt this limiting case as our framework, thus making the assumption that A go is very largewith respect to the other terms. Appendix B. Descriptive statistics
Type of couples Age Education Wage Hours Sample size
Share
Married Heterosexuals
Unmarried Heterosexuals
Homosexuals
Men 40.00 13.93 35.19 42.71 2,034 0.71%Women 39.35 13.78 28.44 40.74 1,620 0.57%
Table 1.
Sample means (25-50 year old)
Ethnic Heterosexual Gay Lesbian AllWhite 42.7 67.3 63.5 43.0Black 2.9 2.5 5.2 2.9Others 0.6 0.8 1.2 0.6Asian 16.7 8.8 5.8 16.6Hispanic 37.1 20.7 24.3 36.9
Total
Table 2.
Distribution of race by sexual orientation (25-50 year old)
Type of couples Age Education Wage Hours
Heterosexual married couples 0.76 0.71 0.16 -0.12Heterosexual unmarried couples 0.68 0.64 0.30 0.11Gay couples 0.56 0.56 0.15 0.03Lesbian couples 0.66 0.65 0.20 0.08
Table 3.
Couples’ Pearson correlation coefficients
EFERENCES 43
White Black Others Asian HispanicWhite 1.12 0.79 0.46 0.94 0.67Black 12.31 0.00 0,44 0.57Others 40.00 2,14 1.21Asian 3.08 0.33Hispanic 2.39 (a)
Gays
White Black Others Asian HispanicWhite 1.28 0.39 0.74 0.70 0.47Black 10.67 1.00 0.82 0.53Others 20.00 0.91 0.87Asian 8.00 0.13Hispanic 2.69 (b)
Lesbians
WomenMen
White Black Others Asian HispanicWhite 1.96 0.32 0.87 0.37 0.28Black 0.49 24.02 1.38 0.34 0.36Others 0.84 0.62 60.91 0.40 0.46Asian 0.15 0.08 0.27 5.08 0.07Hispanic 0.26 0.16 0.44 0.09 2.32 (c)
Heterosexuals
Table 4.
Homogamy rates (25-50 year old). The homogamy rate is the ratio between the observednumber of couples of a certain type and the counterfactual number which should be observed ifindividuals formed couples randomly.
Preamble to appendices C, D and E
In the next three appendices, we present our estimation results. In appendix C, we present ourestimates for the main sample. Table 5 presents our estimates of the affinity matrix of each market(table 5a for the male same-sex marriage market, table 5b for the female same-sex marriage market,table 5c for the married different-sex marriage market and table 5d for the unmarried different-sexmarket). Table 6 presents the results of our saliency analysis, i.e., the decomposition of the affinitymatrices in orthogonal dimensions.In appendix D, table 7 presents our estimates of the affinity matrix when we perform a bipartiteestimation of the same-sex marriage market without requiring the affinity matrix to be symmetric.In this case, we define two separate subpopulations to run a bipartite estimation. On one side of themarket, we group all those gay individuals that are registered as “householders”, whereas on theother we group their “cohabiting partners”. Table 7a displays our estimates for the male same-sexmarriage market whereas 7b table presents our estimates for the female same-sex marriagemarket.Finally, in appendix E, table 8 presents our estimation results on additional selected samples:1) childless couples, 2) bi-earner couples, 3) couples living in the metropolitan area of Los Angelesor San Francisco, 4) young couples (25-35 year old). For different-sex married couples, table 8calso shows our results for couples with one child only, for couples with three children and more,and for recently married couples with no children. In this table, we do not show all the coefficientsof the affinity matrix but only the diagonal coefficients. Each sub-table presents the results for aparticular market and each row displays the estimates for a particular selected sub-sample of thismarket.
EFERENCES 45
Appendix C. Main estimation results
Age Educ. Wage Hours RaceAge 0.62 -0.06 -0.02 -0.13 (0.04) (0.06) (0.03) (0.04)
Education 0.84 0.13 -0.07 (0.09) (0.06) (0.06)
Wage 0.05 -0.07 (0.02) (0.03)
Hours 0.12 (0.04)
Race 0.62 (0.06) σ (a) Gays (1,017 couples)
Age Educ. Wage Hours RaceAge 0.79 -0.10 (0.05) (0.07) (0.06) (0.05)
Education 1.19 0.20 -0.01 (0.12) (0.10) (0.07)
Wage -0.19 (0.04) (0.05)
Hours 0.20 (0.05)
Race 1.26 (0.07) σ (b) Lesbians (810 couples)
WomenMen Age Educ. Wage Hours RaceAge 2.17 -0.20 -0.01 -0.03 (0.05) (0.03) (0.02) (0.03)
Education -0.04 -0.04 (0.03) (0.03) (0.03) (0.03)
Wage 0.09 0.27 0.01 -0.13 (0.03) (0.04) (0.00) (0.02)
Hours 0.06 0.09 -0.09 -0.04 (0.02) (0.02) (0.01) (0.02)
Race 2.49 (0.04) σ (c) Married heterosexuals (6,228 couples)
WomenMen Age Educ. Wage Hours RaceAge 1.14 -0.06 -0.06 (0.03) (0.02) (0.02) (0.02)
Education -0.07 0.66 0.37 0.05 (0.02) (0.02) (0.04) (0.02)
Wage (0.04) (0.05) (0.01) (0.03)
Hours -0.05 -0.02 (0.02) (0.02) (0.02) (0.02)
Race 1.98 (0.04) σ (d) Unmarried heterosexuals (5,645 couples)
Table 5. Affinity matrix : The tables display estimates of the affinity matrix A obtained witha sample of couples where both partners are aged between 25 and 50. If the entry A ij is positiveand significant, then trait i and j are found to be complements in the marital surplus function.On the contrary, if A ij is negative and significant, i and j are substitutes. Standard errors are inparentheses. Boldfaced estimates are significant at the 5 percent level. I1 I2 EthnicityAge
Education
Wage
Hours -0.19 -0.19
Share of systematic surplus
35% 30% 33% (a)
Gays (1,017 couples)
I1 I2 EthnicityAge
Education
Wage
Hours -0.13 -0.03
Share of systematic surplus
29% 27% 42% (b)
Lesbians (810 couples)
I1 I2 Ethnicity
Men Women Men Women
Age
Education -0.06 -0.11 0.86 0.97
Wage
Hours
Share of systematic surplus
36% 16% 48% (c)
Married heterosexuals (6,228 couples)
I1 I2 Ethnicity
Men Women Men Women
Age
Education -0.13 -0.12 0.97 0.93
Wage -0.02 -0.04 0.30 0.51
Hours -0.04 -0.05 -0.01 0.08
Share of systematic surplus
33% 20% 47% (d)
Unmarried heterosexuals (5,645 couples)
Table 6. Indices of attractiveness : Each column displays the estimates of factor loadings explainingthe composition of the p -th index of attractiveness ˜ x p and the corresponding share of average systematicsurplus E ˆ π [Φ( X, Y )] explained by such index (see section 5.5). For each market, we present the two indicesthat explain the largest shares of surplus, as well as the share of surplus explained by ethnicity. Estimatesare obtained with a sample of couples where both partners are aged between 25 and 50.EFERENCES 47
Appendix D. Bipartite estimation for same-sex couples: head/spouse
PartnerHead Age Educ. Wage Hours RaceAge 0.60 -0.06 -0.00 -0.17 (0.04) (0.05) (0.03) (0.04)
Education -0.02 (0.06) (0.08) (0.05) (0.06)
Wage -0.01 (0.04) (0.07) (0.02) (0.04)
Hours -0.05 -0.09 -0.04 (0.04) (0.05) (0.03) (0.04) (0.03)
Race 0.65 (0.06) σ (a) Gays (1,017 couples)
PartnerHead Age Educ. Wage Hours RaceAge 0.74 (0.05) (0.07) (0.05) (0.04)
Education (0.07) (0.12) (0.09) (0.07) (0.04)
Wage (0.05) (0.10) (0.03) (0.05)
Hours -0.11 -0.07 -0.15 0.21 (0.04) (0.07) (0.05) (0.04)
Race 1.18 (0.07) σ (b) Lesbians (810 couples)
Table 7. Affinity matrix : The tables display estimates of A in a bipartite market where oneside is represented by the population of “heads of household” and the other side by the “head’spartners”. Contrarily to the matrices A estimated in 5a and 5b, now A does not need to besymmetric: symmetry tests can be found in the online appendix. We use a sample of same-sexcouples where both partners are aged between 25 and 50. Standard errors are in parentheses.Boldfaced estimates are significant at the 5 percent level. Appendix E. Further robustness checks
Age Educ. Wage Hours RaceAll 0.62 0.84 0.05 0.12 0.62 (0.04) (0.09) (0.02) (0.04) (0.06)
Childless (0.04) (0.09) (0.02) (0.04) (0.07)
Both working (0.05) (0.12) (0.02) (0.09) (0.07)
Urban (0.05) (0.13) (0.02) (0.05) (0.08) (0.49) (0.28) (0.13) (0.13) (0.16) (a)
Gay couples
Age Educ. Wage Hours RaceAll 0.79 1.19 (0.05) (0.12) (0.04) (0.05) (0.07)
Childless (0.06) (0.15) (0.06) (0.06) (0.09)
Both working (0.06) (0.16) (0.04) (0.09) (0.08)
Urban -0.00 0.13 (0.08) (0.21) (0.04) (0.07) (0.10) -0.18 (0.34) (0.27) (0.31) (0.10) (0.15) (b)
Lesbian couples
Age Educ. Wage Hours RaceAll 2.17 0.82 0.01 -0.04 (0.05) (0.03) (0.00) (0.02) (0.04)
Childless (0.04) (0.04) (0.01) (0.02) (0.04)
One child (0.05) (0.03) (0.01) (0.02) (0.04)
Three children -0.02 (0.05) (0.02) (0.00) (0.02) (0.04)
Newlyweds, childless (0.05) (0.09) (0.01) (0.03) (0.05)
Both working (0.05) (0.04) (0.01) (0.04) (0.04)
Urban (0.05) (0.03) (0.00) (0.02) (0.04) (0.17) (0.05) (0.02) (0.03) (0.05) (c)
Married couples
Age Educ. Wage Hours RaceAll 1.14 0.66 0.05 0.09 1.98 (0.03) (0.02) (0.01) (0.02) (0.04)
Childless (0.03) (0.04) (0.01) (0.02) (0.04)
Both working (0.03) (0.03) (0.02) (0.04) (0.03)
Urban (0.03) (0.02) (0.0 ) (0.02) (0.04) (0.11) (0.04) (0.03) (0.03) (0.04) (d)
Unmarried couples