aa r X i v : . [ ec on . E M ] J un Matching Multidimensional Types: Theory and Application
Veli Safak ∗† Abstract
Becker (1973) presents a bilateral matching model in which scalar types describe agents. For this frame-work, he establishes the conditions under which positive sorting between agents’ attributes is the uniquemarket outcome. Becker’s celebrated sorting result has been applied to address many economic ques-tions. However, recent empirical studies in the fields of health, household, and labor economics suggestthat agents have multiple outcome-relevant attributes. In this paper, I study a matching model withmultidimensional types. I offer multidimensional generalizations of concordance and supermodularity toconstruct three multidimensional sorting patterns and two classes of multidimensional complementarities.For each of these sorting patterns, I identify the sufficient conditions which guarantee its optimality. Inpractice, we observe sorting patterns between observed attributes that are aggregated over unobservedcharacteristics. To reconcile theory with practice, I establish the link between production complemen-tarities and the aggregated sorting patterns. Finally, I examine the relationship between agents’ healthstatus and their spouses’ education levels among U.S. households within the framework for multidimen-sional matching markets. Preliminary analysis reveals a weak positive association between agents’ healthstatus and their spouses’ education levels. This weak positive association is estimated to be a product ofthree factors: (a) an attraction between better-educated individuals, (b) an attraction between healthierindividuals, and (c) a weak positive association between agents’ health status and their education levels. ∗ Carnegie Mellon University Qatar, e-mail: [email protected] † I would like to express my gratitude to my advisor Axel Z. Anderson for his guidance, strong and continuous support, andalways having faith in my capability. I would also like to thank James W. Albrecht and Luca Anderlini for their constructivecomments. I would further like to thank Laurent Bouton, Dan Cao, Chris Chambers, Roger Lagunoff, Arik M. Levinson,Yusufcan Masatlioglu, Franco Peracchi, and John Rust for helpful discussions. he attraction channel suggests that the insurance risk associated with a two-person family plan is higherthan the aggregate risk associated with two individual policies.
Introduction
Becker (1973) proposes a general framework for two-sided frictionless matching models in which scalar typesrepresent the agents on each side of the market, i.e. each agent has only one outcome-relevant attribute. Amatch between two agents (one from each side) generates a type-dependent matching output. A social plan-ner maximizes the aggregate output by matching the agents in pairs. There are two essential componentsof Becker’s theory: complementarity and sorting. The matching output exhibits strictly positive comple-mentarity when the marginal product of an agent strictly increases in his/her partner’s type. Similarly, thematching output exhibits strictly negative complementarity when the marginal product of an agent strictlydecreases in his/her partner’s type.Becker (1973) shows that if the matching output exhibits strictly positive complementarity, then theunique solution to the planner’s problem is positive sorting , i.e. the highest types are matched together,then the next highest types, etc. Likewise, negative sorting is the unique solution when the matching outputexhibits strictly negative complementarity. Economists have applied Becker’s assortative matching results to address several questions. For example, Kremer (1993) sheds light on the positive correlation betweenwages of the workers within a firm. Gabaix and Landier (2008) explain the rise in the CEOs’ salaries andits connection to the increase in firms’ sizes over time.In many applications, the agents may have multiple outcome-relevant attributes. For instance, educationand race in the dating/marriage market, workers’ social and cognitive skills in the labor market, and doctors’listening skills for diagnosis and fostering the doctor-patient relationship in the healthcare market are somewell-documented outcome-relevant attributes in literature. In the next section, I survey additional recentstudies that support the presence of multiple outcome-relevant attributes. If a single index can captureall outcome-relevant information, then a unidimensional model may be suitable. However, the single indexassumption is implausible in many applications. For example, Chiappori et al. (2012) analyze the U.S.marriage market by using a multidimensional matching model with an index restriction, i.e. two agents withdifferent attributes are identical if they have the same index value calculated by an exogenous index function.Fletcher and Padron (2015) provide empirical evidence against the implications of Chiappori et al.’s (2012) A decentralized version of this model with perfectly transferable utilities can easily be constructed by using the dual versionof the planner’s problem. For a recent literature review on the matching markets; I refer the readers to Chade et al. (2017). global positive sorting if and only if it exhibits positive sorting (a) between firms’ and their workers’cognitive skills, and (b) between firms’ and their workers’ social skills. Similarly, a matching distributionsatisfies global negative sorting if and only if it exhibits negative sorting (a) between firms’ and their workers’cognitive skills, and (b) between firms’ and their workers’ social skills.A naive application of Becker’s sorting result implies positive sorting between firms’ and their workers’cognitive skills when the marginal product of each firm’s cognitive skill strictly increases in its worker’scognitive skill. Similarly, positive sorting between firms’ and their workers’ social skills is obtained whenthe marginal product of each firm’s social skill strictly increases in its worker’s social skill according toBecker’s sorting result. In a multidimensional matching market, one may observe simultaneous positivecomplementarities between cognitive skills and social skills. However, simultaneous positive sorting betweencognitive skills and social skills may not be feasible.Consider two firms, x = ( x c , x s ) and x ′ = ( x ′ c , x ′ s ) , such that x = (10 , and x ′ = (20 , . Furthermore,suppose that there are two workers, y = ( y c , y s ) and y ′ = ( y ′ c , y ′ s ) , such that y = (10 , and y ′ = (20 , .Notice that matching x with y and x ′ with y ′ satisfies positive sorting between cognitive skills and violatespositive sorting between social skills. Similarly, matching x with y ′ and x ′ with y satisfies positive sortingbetween social skills and violates positive sorting between cognitive skills. In this paper, I show that,conditioning on the existence, Becker’s sorting results apply: when the output function exhibits strictlypositive complementarities between cognitive skills and between manual skills, if there exists a matchingscheme which satisfies global positive sorting, then (a) every optimal matching scheme satisfies global positivesorting, and (b) every matching scheme satisfying global positive sorting solves the planner’s problem. Iestablish the optimality of global sorting for a general global sorting class in Proposition 1.Since global positive sorting may not be feasible, I examine an alternative sorting pattern inspired byChiappori et al. (2017). Chiappori et al. (2017) study a marriage model in which one continuous variable2socioeconomic status) and one binary variable (smoking habit) represent the agents on each side of themarket. They categorize couples into two main groups. In the first group, both men and women are non-smokers. In the second group, at least one of the spouses smokes. They assume that the matching output ofa couple is the multiplication of spouses’ socioeconomic status. If there is a smoker in the household, thenthe output is scaled down by a constant. Under this complementarity structure, Chiappori et al. (2017)predict positive sorting between agents’ and their spouses’ socioeconomic status within each group. Noticethat one can easily apply the idea of splitting the sample into different groups and studying the sortingpatterns for each group in a more general setting.For the previous labor market example, a matching satisfies within-group positive sorting between cognitiveskills if it exhibits positive sorting between firms’ and their workers’ cognitive skills for all social skill pairsof firms and workers ( x s , y s ) . Consider four firms and four workers: { (10 , , (10 , , (20 , , (20 , } .Here, matching the (10 , ) firm with the (10 , ) worker and the (20 , ) firm with the (20 , ) worker isconsistent with within-group positive sorting between cognitive skills for the ( , ) social skill combination.Within-group sorting solves the feasibility problem: for arbitrary distributions of agents, there exists amatching scheme which satisfies (a) within-group positive sorting between cognitive skills and (b) within-group positive sorting between social skills. Furthermore, I show that, when the matching output exhibitsstrictly positive complementarities ( ♣ ) between cognitive skills and ( ♠ ) between social skills, every optimalmatching distribution satisfies (a) within-group positive sorting between cognitive skills and (b) within-group positive sorting between social skills. I establish the optimality of within-group sorting for a generalwithin-group sorting class in Proposition 1.Within-group sorting has two major drawbacks as a sorting concept. First of all, there may be multipleways to match agents without violating within-group sorting. Firms (10 ,
10) (10 ,
20) (20 ,
10) (20 , Firms (10 ,
10) (10 ,
20) (20 ,
10) (20 , Matched Worker (20 ,
20) (10 ,
20) (20 ,
10) (10 , Matched Worker (10 ,
10) (10 ,
20) (20 ,
10) (20 , Matching scheme-1 Matching scheme-2Note that for each social skill combination of firms and workers, there is only one firm-worker couple underthese two matching schemes. The same is also true for each cognitive skill combination of firms and workers.Consequently, these matching schemes satisfy within-group positive sorting between cognitive skills and3ocial skills.Secondly, a matching scheme may satisfy within-group positive sorting and cannot be optimal forany matching output that exhibits strictly positive complementarities between cognitive skills and be-tween social skills. For matching scheme-1, a swap between the first and the last firm-worker couples,i.e. ((10 , , (20 , and ((20 , , (10 , , strictly increases the aggregate output for any matching out-put which exhibits strictly positive complementarities between cognitive skills and between social skills.Therefore, matching scheme-1 can never be an optimal matching scheme when the matching output exhibitsstrictly positive complementarities between cognitive skills and between social skills.To obtain a finer characterization of optimal matching schemes, I consider another extension of Becker’ssorting concepts and propose a weak sorting notion: a matching scheme satisfies weak positive sorting if theredoes not exist a pair of matched couples that (a) is consistent with global negative sorting, and (b) violatesglobal positive sorting. Notice that the first and the last firm-worker couples in matching scheme-1 violateglobal positive sorting. Indeed, the set of matching schemes that satisfy weak positive sorting is a subset ofthe set of matching schemes that satisfy within-group positive sorting. More importantly, I show that theset of optimal matching schemes for any matching output that exhibits strictly positive complementaritiesbetween cognitive skills and between social skills is a subset of the set of matching schemes that satisfiesweak positive sorting. I present a general version of this result in Proposition 1.Similar to within-group sorting, a matching scheme may satisfy weak positive sorting and can never beoptimal when the matching output exhibits strictly positive complementarities between cognitive skills andbetween social skills. Note that matching scheme-3 below satisfies weak positive sorting. At the same time,the following swap sequence strictly increases the aggregate output for any matching output that exhibitsstrictly positive complementarities between cognitive skills and between social skills:Swap-1: Between the first and the second couplesSwap-2: Between the third and the fourth couplesSwap-3: Between the first and the last couples matched after swap-1 and swap-24 irms (10 ,
10) (10 ,
20) (20 ,
10) (20 , Firms (10 ,
10) (10 ,
20) (20 ,
10) (20 , Matched Worker (10 ,
20) (20 ,
20) (10 ,
10) (20 , Matched Worker ( , ) ( , ) (10 ,
10) (20 , Matching scheme-3 Swap-1
Firms (10 ,
10) (10 ,
20) (20 ,
10) (20 , Firms (10 ,
10) (10 ,
20) (20 ,
10) (20 , Matched Worker (20 ,
20) (10 ,
20) ( , ) ( , ) Matched Worker ( , ) (10 ,
20) (20 ,
10) ( , ) Swap-2 Swap-3These examples demonstrate that global, within-group and weak sorting concepts are not adequate to obtaina fine characterization of the set of optimal matching distributions. In Section 1, I lay out the statistical logicbehind Becker’s sorting result and restate it by using the upper-set properties of the supermodular order.By devising multidimensional generalizations of supermodularity, supermodular order, and concordance, Icharacterize Pareto improving swaps for a large set of complementarity structures that allow for negativecomplementarities between some skills along with positive complementarities between some other skills.Although these results can be applied to wide-ranging matching markets, they lack predictive power: theset of optimal matching distributions may not be a singleton. Lindenlaub (2017) offers a multidimensionalsorting theory with higher predictive power. She adopts three key assumptions: (a) the agents on each sideof the market have the same number of outcome-relevant attributes, (b) each attribute complements one andonly one attribute on the other side of the market, and (c) the matching output exhibits either strictly positivecomplementarity in all attributes or strictly negative complementarity in all attributes. More specifically,she considers matching output functions that have the following form: Q ( x, y ) = Q c ( x c , y c ) + Q s ( x s , y s ) where the cross-partial derivative of Q i is either strictly positive for all i ∈ { c, s } or strictly negative for all i ∈ { c, s } .For this framework, she shows that the optimal matching is unique. In addition, she proves that the op-timal matching is a smooth function under additional restrictions: the agents are distributed with infinitelymany times continuously differentiable probability distribution functions; Q i is four times continuously dif-ferentiable; and Q x i ,y i is supermodular and log-supermodular. Under these additional restrictions, sheestablishes that ∂y ∗ i /∂x i > for all i ∈ { c, s } , where y ∗ := m ∗ ( x ) denotes the matched worker of type- x firmunder optimal matching function m ∗ : R → R .However, these assumptions on the output function may not be suitable for many applications. As an5xample, consider the following matching output: Q ( x, y ) = αx c y c + βx s y s + γx s y c . The results presentedby Lindenlaub (2017) apply only if αβ > and γ = 0 . In this case, once one assumes strictly positivecomplementarity between cognitive skills, one must also assume strictly positive complementarity betweensocial skills, and vice versa ( αβ > ). Also, each attribute of one side can interact with one and only oneattribute of the other side of the market ( γ = 0 ). Dupuy and Galichon (2014, Table 3) provide empiricalevidence suggesting that (a) positive and negative complementarities between different attributes exist and(b) some attributes simultaneously complement multiple attributes in the Dutch marriage market.One other drawback of the sorting theorems presented in Proposition 1 is that they predict extremesorting patterns as optimal. However, observing extreme sorting in practice is improbable. A potentialreason is that we do not observe every outcome-relevant attribute. For the previous example, assume that(a) firms and workers observe each other’s cognitive and social skills, and (b) econometricians observe firms’and workers’ cognitive skills but do not observe their social skills. In this case, econometricians observe asorting pattern between firms’ and their workers’ cognitive skills aggregated over social skills. The aggregatedmatching pattern between cognitive skills may exhibit mismatch, i.e. deviations from extreme sorting. Theframework presented in Section 1 allows me to explore this source of mismatch. Consider four types of firms( x ) and four types of workers ( y ): { (10 , , (10 , , (20 , , (20 , } . Assume that for skill vectors (10 , and (20 , , there are four firms. In addition, suppose that there is one firm for each skill vector (10 , and (20 , . Similarly, let there be four workers for each skill vector (10 , and (20 , ; and one workerfor each skill vector (10 , and (20 , . The unique optimal matching scheme between these firms andworkers is given below for the following output function: Q ( x, y ) = x c y c + 2 x s y s .Workers (10 ,
10) (10 ,
20) (20 ,
10) (20 , Firms (10 , (10 , (20 , (20 , Search and informational frictions may also cause deviations from extreme sorting.
Empirical Support for Multidimensional Types
In this section, I survey the recent empirical studies that support the presence of multiple outcome-relevantattributes in the healthcare, labor, and marriage/dating markets.7n the healthcare market, the relationship between doctors and patients is known to be multidimensional.Jagosh et al. (2011) show that effective communication enhances patient recovery. The authors argue thatthree listening skills of health professionals (listening as an essential component of clinical data gathering anddiagnosis; listening as a healing and therapeutic agent; and listening as a means of fostering and strengtheningthe doctor–patient relationship) are central to successful clinical outcomes. Stavropoulou (2011) indicatesthat six aspects of the relationship between doctors and patients affect nonadherence to medication using theEuropean Social Survey. Nonadherence to medication is also proven to be a complex and multidimensionalhealthcare problem by Hugtenburg et al. (2013). Similarly, Mazzi et al. (2018) identify four attributes ofdoctors and three characteristics of patients that are essential to successful clinical outcomes by using anintegrated survey of thirty-one European countries. Belasen and Belasen (2018) provide evidence suggestingthat different aspects of the relationships between doctors and patients affect not only the clinical outcomebut also patients’ rankings of hospitals.The empirical findings in labor economics literature also suggest that workers and firms have multidi-mensional types. Deming (2017) shows that workers’ cognitive and social skills are important determinantsof wages in the U.S. labor market. Girsberger et al. (2018) add manual skill to that list for the Swiss labormarket by using data from the Social Protection and Labour Market (SESAM) panel. Guvenen et al. (2018)analyze the skill mismatch between workers and firms in the U.S. labor market. Their analysis indicatesthat verbal and math skills have statistically significant effects on workers’ wages.Hitsch et al. (2010) study mating behavior in the U.S. dating market by using a large dataset providedby an online dating website. They show that the differences in age, educational attainment, and body massindex decrease the probability of dating. Belot and Francesconi (2012) confirm these findings by studyingthe speed dating patterns of individuals based on a British dataset. They find that physical attractivenessfactors (age, height, and body mass index) play an essential role in the earlier stages of the relationship.Klofstad et al. (2013) add political views to that list. By analyzing a large dataset provided by anotheronline dating website, the authors show that couples tend to share the same political preferences.Gemici and Laufer (2010) study cohabitation, marriage and separation patterns in the U.S. matingmarket. They report that age, educational attainment, and race are key variables in explaining agents’choices. Dupuy and Galichon (2014) add other important variables to that list by studying the Dutchmarriage market. They show that personality traits, such as emotional stability and conscientiousness, arealso important determinants of the Dutch household formation. Domingue et al. (2014) analyze the geneticsimilarities between married couples by using the Health and Retirement Study and information from 1.78illion single-nucleotide polymorphisms. Their results demonstrate that similar genetic types attract eachother. They also find that educational similarities between spouses are stronger than genetic similarities.The thorough examination of the household formation allows researchers to explain changes in the householdincome inequality. For example, Greenwood et al. (2014) argue that similarities between spouses’ educationlevels contributed to the increasing household income inequality in the U.S. between 1960 and 2005.
In this section, I study a general model of two-sided matching markets. The only difference between theframework examined by Becker (1973) and the one presented in this section is that I allow the agents to havemultiple outcome-relevant attributes while Becker (1973) assumes that each agent has only one outcome-relevant attribute. I start by outlining the general framework in detail.
Agents:
There are two sides of the matching market, namely firms and workers. A generic firm isdenoted by x , and a generic worker is denoted by y . Every firm has K productive attributes, i.e. x ∈ R K ,and each worker has L productive attributes, i.e. y ∈ R L . It is assumed that the overall measures of firmsand workers coincide. The firms and workers are distributed according to cumulative distribution functions F : R K → [0 , and G : R L → [0 , , respectively. Matching Distribution:
Matching distribution M : R K × R L → [0 , is a cumulative distribution functionassociated with a particular matching scheme. More specifically, M ( x, y ) represents the fraction of matchedfirm-worker couples with attributes less than or equal to ( x, y ) .Given F and G , matching distribution M satisfies no-single property if and only if(a) lim y →∞ L M ( x, y ) = F ( x ) for all x ∈ R K , and (b) lim x →∞ K M ( x, y ) = G ( y ) for all y ∈ R L .Let M ( F, G ) denote the set of matching distributions that satisfy no-single property given F and G . Output Function:
A match between a firm and a worker with attributes x and y generates a matchingoutput. The matching output is determined by exogenously specified output function Q : R K × R L → R ++ ,and denoted by Q ( x, y ) . For any unmatched agent, the output is assumed to be . Planner’s Problem:
Given F , G , and Q , the social planner maximizes the aggregate output by choosinga matching distribution that satisfies no-single property: max M ∈M ( F,G ) R QdM.
Two key concepts of the matching theory are complementarity and sorting. In this context, posi-tive(negative) complementarity between a firm’s i th attribute and its worker’s j th attribute means that9he marginal product of the firm’s i th attribute is increasing(decreasing) in its worker’s j th attribute. Inunidimensional matching literature, complementarities are formulated by using supermodularity. Function Q : R → R is called (strictly) supermodular if for all x ′ > x and y ′ > y , it holds that { Q ( x ′ , y ′ ) − Q ( x, y ′ ) } − { Q ( x ′ , y ) − Q ( x, y ) } ≥ ( > ) 0 . Similarly, Q is (strictly) submodular if − Q is (strictly) supermodular; and Q satisfies modularity if Q isboth supermodular and submodular. In the multidimensional setting, supermodularity can also be usedwith a slight modification to formulate multidimensional complementarities. Towards this end, I introduce i,j pairwise supermodularity . Definition 1.
Function Q : R K × R L → R ++ is (a) (strictly) i,j pairwise supermodular if Q ( x i , x − i , y j , y − j ) is a (strictly) supermodular function of ( x i , y j ) for all ( x − i , y − j ) ∈ R K − × R L − ; (b) (strictly) i,j pairwise submodular if − Q is (strictly) i,j pairwise supermodular; and (c) i,j pairwise modular if Q is both i,j pairwise supermodular and i,j pairwise submodular.By construction, the pairwise modularity concepts can be used to formulate the relationships betweenthe marginal product of a firm’s i th attribute and its worker’s j th attribute. This aspect is easy to ob-serve when the output function is smooth: Q : R K × R L → R satisfies i,j pairwise supermodularity if andonly if its cross-partial derivative with respect to x i and y j is positive. Through the use of pairwise mod-ularity concepts, I define two multidimensional complementarity classes. Consider two disjoint subsets of { , . . . , K } × { , . . . , L } : P and N. Definition 2.
Function Q : R K × R L → R ++ exhibits (strict) P,N modularity if and only if Q ( x, y ) is (a) (strictly) i,j pairwise supermodular for all ( i, j ) ∈ P ; (b) (strictly) p,q pairwise submodular for all ( p, q ) ∈ N ; and (c) m,n pairwise modular for all ( m, n ) / ∈ P ∪ N .Let C ( P, N ) and C + ( P, N ) denote the sets of functions which satisfy P,N modularity and strict P,N modu-larity for set parameters P and N.Here P represents the set of firm-worker attribute pairs for which the output function exhibits positivecomplementarity. Similarly, N represents the set of firm-worker attribute pairs for which the output functionexhibits negative complementarity. The strongest element of these complementarity classes is that theydo not allow for complementarity between a firm’s i th attribute and its worker’s j th attribute to change10igns (from positive to negative). For example, an output function which exhibits (a) strictly positivecomplementarity between a firm’s and its worker’s cognitive skills for some levels of its worker’s social skill,and (b) strictly negative complementarity between the firm’s and its worker’s cognitive skills for some levelsof its worker’s social skill cannot be formulated by using a P,N modular function. Despite this shortcoming,P,N modular functions can capture various output function forms which have been frequently used in thematching literature.Table 1: Some examples of P,N modular functions frequently used in matching literature Output function Strict P,N modularity Reference Q ( x, y ) = x ′ Ay = K P i =1 L P j =1 a ij x i y j P = { ( i, j ) : a ij > } and N = { ( i, j ) : a ij < } Dupuy and Galichon (2014) Q ( x, y ) = K P i =1 Q i ( x i , y i ) P = { , ..., K } and N = ∅ if ∂ Q i ( x i , y i ) /∂x i ∂y i > Lindenlaub (2017)
Due to the simplicity of their interpretation and their frequent use, understanding the optimal sortingpatterns for P,N modular output functions is of theoretical and empirical interest. To achieve this goal,I introduce three multidimensional sorting patterns. The first sorting pattern is a direct extension of theglobal sorting pattern presented by Becker (1973).
Definition 3.
Matching distribution M : R K × R L → [0 , satisfies (a) positive sorting between firms’ i th and their workers’ j th attributes if and only if ( x ′ i − x i ) (cid:0) y ′ j − y j (cid:1) ≥ for all ( x, y ) , ( x ′ , y ′ ) ∈ supp ( M ) , and (b) negative sorting between firms’ i th and their worker’s j th attributes if and only if ( x ′ i − x i ) (cid:0) y ′ j − y j (cid:1) ≤ for all ( x, y ) , ( x ′ , y ′ ) ∈ supp ( M ) .Similar to P,N modularity, I combine pairwise sorting patterns to construct a multidimensional sortingclass. Definition 4 (Global P,N sorting) . Matching distribution M : R K × R L → [0 , satisfies global P,N sortingif and only if the following conditions hold for all ( x, y ) , ( x ′ , y ′ ) ∈ supp ( M ) , (a) ( x ′ i − x i ) (cid:0) y ′ j − y j (cid:1) ≥ for all ( i, j ) ∈ P , and (b) (cid:0) x ′ p − x p (cid:1) (cid:0) y ′ q − y q (cid:1) ≤ for all ( p, q ) ∈ N .In other words, a matching distribution exhibits global P,N sorting if and only if it exhibits (a) positivesorting between firms’ i th and their workers’ j th attributes for all ( i, j ) ∈ P and (b) negative sorting betweenfirms’ p th and their workers’ q th attributes for all ( p, q ) ∈ N .11lthough global P,N sorting is a clear sorting pattern between two multidimensional agents, it requiressimultaneous sorting between different attributes. Consequently, its existence is tied to the distributions ofthe agents. Example 1.
Consider a labor market with equal numbers of two types of firms: (10 , and (20 , ; andequal numbers of two types of workers: (10 , and (20 , . Notice that matching a (10 , worker with a (10 , firm and a (20 , worker with a (20 , firm violates positive sorting between the second attributes.Similarly, the swap between these two couples, i.e. matching a (10 , worker with a (20 , firm and a (20 , worker with a (10 , firm, violates positive sorting between the first attributes. Therefore, it isnot possible to observe simultaneous positive sorting between the first attributes and the second attributeswithout inefficiently leaving some agents unmatched.Due to the existence issue, I study two alternative sorting patterns that exist for arbitrary F, G, P, and N . The next sorting pattern is inspired by Chiappori et al. (2017). The authors analyze a matching modelin which each side of the market is represented by one continuous variable (socioeconomic status) and onebinary variable (smoking habit). They categorize couples into two main groups. In the first group, bothmen and women are non-smokers. In the second group, at least one spouse is a smoker. They assume thatthe matching output of a couple is the multiplication of spouses’ socioeconomic status. If there is a smokerin the household, then the output is scaled down by a constant. Under this complementarity structure, theypredict positive sorting between agents’ socioeconomic status within each group. Although this sorting resultimmediately follows from Becker’s (1973) unidimensional sorting theory, the idea of within-group sorting canbe deployed in a more general setting. Definition 5.
Matching distribution M : R K × R L → [0 , satisfies (a) within-group positive sorting between firms’ i th and their workers’ j th attributes if and only if for all ( x, y ) , ( x ′ , y ′ ) ∈ supp ( M ) such that ( ♣ ) x k = x ′ k for all k = i and ( ♠ ) y l = y ′ l for all l = j , it holds that ( x ′ i − x i ) (cid:0) y ′ j − y j (cid:1) ≥ ; and (b) within-group negative sorting between firms’ i th and their workers’ j th attributes if and only if for all ( x, y ) , ( x ′ , y ′ ) ∈ supp ( M ) such that ( ♣ ) x k = x ′ k for all k = i and ( ♠ ) y l = y ′ l for all l = j , it holds that ( x ′ i − x i ) (cid:0) y ′ j − y j (cid:1) ≤ .Similar to P,N sorting, I combine within-group sorting patterns to define a within-group sorting class.12 efinition 6 (Within-group P,N Sorting) . Matching distribution M : R K × R L → [0 , satisfies within-groupP,N sorting if and only if it satisfies (a) within-group positive sorting between firms’ i th and their workers’ j th attributes for all ( i, j ) ∈ P ; and (b) within-group negative sorting between firms’ p th and their workers’ q th attributes for all ( p, q ) ∈ N .Within-group P,N sorting is not a very informative sorting pattern as some of the nonoptimal matchingdistributions may satisfy within-group P,N sorting. Example 2.
Consider a matching market with equal numbers of two types of workers: (10 , and (20 , ;and equal numbers of two types of firms: (10 , and (20 , . Notice that matching (10 , workers with (20 , firms, and (20 , workers with (10 , firms, satisfies within-group P,N sorting for P = { (1 , , (2 , } and N = {} . However, it is clear that a swap of the partners between firm-workercouples { (10 , , (20 , } and { (20 , , (10 , } strictly improves the aggregate output for any strictlyP,N modular output function. For example, matching same types with each other is associated with strictlyhigher aggregate output for output function Q ( x, y ) = x y + x y :
100 + 400 + 400 + 100 = 1000 >
800 = 200 + 200 + 200 + 200 . The right-hand side of the equation above equals the total output produced by firm-worker couples { (10 , , (20 , } and { (20 , , (10 , } ; and the left-hand side of the equation is the total output produced by firm-workercouples { (10 , , (10 , } and { (20 , , (20 , } .As it is demonstrated in Example 2, a fine characterization of the set of optimal matching distributionscannot be obtained by using within-group P,N sorting. As such, I adopt a more constructive approach toobtain a fine description of the set of optimal matching distributions. To motivate the idea behind thisapproach, I first lay out a statistical course to obtain Becker’s (1973) sorting results. Theorem 1 (Becker’s (1973) sorting results) . In a unidimensional matching market, i.e. K = L = 1 , (a) positive sorting is an optimal matching distribution when the output function is supermodular; (b) negative sorting is an optimal matching distribution when the output function is submodular; (c) positive sorting is the unique optimal matching distribution when the output function is strictly super-modular; and (d) negative sorting is the unique optimal matching distribution when the output function is strictly submod-ular.
13t is easy to establish these sorting results by using the upper-set properties in the supermodular order.Distribution M : R × R → [0 , dominates M ′ : R × R → [0 , in the supermodular order if and onlyif R QdM ≥ R QdM ′ for all supermodular Q : R × R → R . Muller and Scarsini (2010), and Meyer andStrulovici (2013) offer an alternative characterization of the supermodular order. Definition 7.(a)
A pair of couples ( x, y ) , ( x ′ , y ′ ) ∈ R is (weakly) concordant if and only if ( x − x ′ ) ( y − y ′ ) > ( ≥ ) 0 . (b) A pair of couples ( x, y ) , ( x ′ , y ′ ) ∈ R is (weakly) discordant if and only if ( x − x ′ ) ( y − y ′ ) < ( ≤ ) 0 . (c) A concordance improving transfer is a uniform probability transfer from a discordant bivariate pairto the concordant bivariate pair that is obtained via the swap between the discordant bivariate pair. Let τ ( x, y, x ′ , y ′ ; α ) denote the concordance improving transfer with α ≥ weight which increases densities by α at ( x, y ) and ( x ′ , y ′ ) such that ( x − x ′ ) ( y − y ′ ) > ; and decreases densities at ( x, y ′ ) and ( x ′ , y ) by α . Theorem 2 (Muller and Scarsini (2010), and Meyer and Strulovici (2013)) . M : R × R → [0 , dominates M ′ : R × R → [0 , in the supermodular order if and only if M can be obtained from M ′ via concordanceimproving transfers. Based on this alternative characterization, it is easy to see that positive sorting is the dominant match-ing distribution in the supermodular order. Similarly, negative sorting is strictly dominated by any othermatching distribution in the supermodular order. Consequently, Becker’s (1973) sorting result is obtained.Following a similar logic, a fine characterization of the set of optimal matching distributions for the multi-dimensional setting can be established by identifying the changes in the matching distribution that increasethe aggregate output when the matching output is P,N modular. On this note, I define P,N modular orderin Definition 8.
Definition 8.
Distribution M : R K × R L → [0 , dominates M ′ : R K × R L → [0 , in P,N modular order,denoted by M (cid:23) P,N M ′ , if and only if R QdM ≥ R QdM ′ for all Q ∈ C ( P, N ) . Distribution M strictlydominates M ′ in P,N modular order if and only if (a) M dominates M ′ in P,N modular order; and (b) M isnot dominated by M ′ in P,N modular order.Next, I formally define the sets of P,N dominant and P,N undominated distributions that are essential tocharacterizing the set of optimal matching distributions. Definition 9.
Matching distribution M ∈ M ( F, G ) is P,N dominant if it dominates every M ′ ∈ M ( F, G ) in P,N modular order. Similarly, M ∈ M ( F, G ) is P,N undominated if there does not exist M ′ ∈ M ( F, G ) which strictly dominates M in P,N modular order.14he next theorem restates Becker’s (1973) sorting result by using multidimensional concepts. By doingso, it makes the theoretical differences between unidimensional and multidimensional matching markets clear. Theorem 3 (Unidimensional Sorting - Multidimensional Concepts) . Let K = L = 1 and P ∪ N = { (1 , } .a) For any P and N, a matching distribution is P,N dominant if and only if it is P,N undominated.b) For any P and N, there is only one P,N dominant distribution.c) For any Q ∈ C ( P, N ) , the P,N dominant distribution solves the planner’s problem.d) For any Q ∈ C + ( P, N ) , the P,N dominant distribution is the unique solution to the planner’s problem.e) The P,N dominant matching distribution is positive assortative matching when P = { (1 , } and N = {} : Λ ( x, y ) = min { F ( x ) , G ( y ) } . f ) The P,N dominant matching distribution is negative assortative matching when P = {} and N = { (1 , } : Ω ( x, y ) = max { F ( x ) + G ( y ) − , } . There are two key aspects of optimality in matching markets with unidimensional agents. First of all,the set of P,N undominated distributions is singleton when P ∪ N = ∅ . Secondly, there exists a P,Ndominant distribution. These two features make it possible to fully characterize the solution to the planner’sproblem for strictly P,N modular output functions in unidimensional case. However, they do not apply tothe multidimensional setting. Example 3.
Consider a matching market with equal numbers of two types of workers: (10 , and (20 , ,equal numbers of two types of firms: (10 , and (20 , , and a class of matching output functions withparameter γ ∈ [0 , : Q ( x, y ; γ ) = γx y + (1 − γ ) x y . Note that for all values of γ , the output functionexhibits P,N modularity for P = { (1 , , (2 , } and N = {} . The set of P,N undominated distributions is not singleton:
Consider matching distribution M underwhich every (10 , worker is matched with a (10 , firm; and every (20 , worker is matched with a (20 , firm. It immediately follows from Becker’s sorting result that M is P,N undominated since it is theunique solution to the planner’s problem when γ = 1 . Alternative matching distribution M ′ under whichevery (20 , worker is matched with a (10 , firm, and every (10 , worker is matched with a (20 , firm is also P,N undominated since it is the unique solution to the planner’s problem when γ = 0 . The set of P,N dominant distributions is empty:
Consider a matching distribution under which some (10 , workers are matched with (10 , firms; and some (20 , workers are matched with (20 , firms.This matching distribution cannot be P,N dominant as M ′ is associated with strictly higher aggregate output15hen γ = 0 . Similarly, a matching distribution under which some (20 , workers are matched with (10 , firms, and some (10 , workers are matched with (20 , firms cannot be P,N dominant as M is associatedwith strictly higher aggregate output when γ = 1 .Due to the fact that the set of P,N dominant distributions is neither singleton nor non-empty for arbitrarymodel parameters, characterization of the optimal matching distributions is a non-trivial task. I obtain afine description of the optimal matching distributions, by characterizing Pareto improving swaps in Lemma1 below. Definition 10.
A pair of firm-worker couples ( x, y ) , ( x ′ , y ′ ) ∈ R K × R L is P,N weak concordant if (a) ( x i − x ′ i ) (cid:0) y j − y ′ j (cid:1) ≥ for all ( i, j ) ∈ P ; and (b) (cid:0) x p − x ′ p (cid:1) (cid:0) y q − y ′ q (cid:1) ≤ for all ( p, q ) ∈ N .A P,N concordant pair is a P,N weak concordant pair such that ( ♣ ) some of the inequalities in (a) hold withstrict inequality for some ( i, j ) ∈ P ; or ( ♠ ) some of the inequalities in (b) hold with strict inequality forsome ( p, q ) ∈ N . Definition 11.
A P,N concordance improving transfer is a uniform probability transfer from an N,P weakconcordant pair of couples to the P,N weak concordant pair of couples that is obtained via the swap betweenthe N,P weak concordant pair of couples.
Lemma 1.
Distribution M : R K × R L → [0 , dominates M ′ : R K × R L → [0 , in P,N modular order ifand only if M can be obtained from M ′ via a sequence of P,N concordance improving transfers. This alternative characterization of dominance in P,N modular order allows me to obtain a finer de-scription of the set of optimal matching distributions. Consider matching distribution M ∈ M ( F, G ) underwhich there exists an N,P concordant pair of matched couples. Due to Lemma 1, it is easy to see that aswap between the N,P concordant pair of couples (strictly) increases the aggregate output when the outputfunction is (strictly) P,N modular. Consequently, a matching distribution under which there exists an N,Pconcordant pair of matched couples cannot be obtained as optimal for strictly P,N modular output functions.This observation rules out the nonoptimal matching distribution illustrated in Example 2, and allows me todefine a new sorting pattern. Definition 12.
Matching distribution M ∈ M ( F, G ) satisfies weak P,N sorting if there does not exist anN,P concordant pair of couples with a positive mass under M .The next proposition establishes the link between the proposed sorting patterns and the set of optimalmatching distributions for the proposed complementarity structures.16 roposition 1 (Multidimensional Sorting) . For arbitrary distributions of the agents F and G , and two disjoint sets of firm-worker attribute pairs P and N , every weak P,N assortative matching distribution satisfies within-group P,N sorting; for every Q ∈ C + ( P, N ) , every solution to the planner’s problem satisfies weak P,N sorting; and for every Q ∈ C ( P, N ) , there exists a weak P,N assortative matching distribution that solves theplanner’s problem. Suppose that there exists M ∈ M ( F, G ) that satisfies global P,N sorting. Then, for every Q ∈ C + ( P, N ) , every solution to the planner’s problem satisfies global P,N sorting; and for every Q ∈ C ( P, N ) , M solves the planner’s problem. Proposition 1 summarizes the relationship between the proposed sorting patterns and multidimensionalcomplementarities. First, it states that within-group P,N sorting is the least informative sorting patternamong all. Secondly, it offers a partial identification of the set of optimal matching distributions: (i) the setof optimal matching distributions and the set of weak P,N assortative matching distributions intersect forP,N modular output functions; and (ii) the set of optimal matching distributions is a subset of the set ofweak P,N assortative matching distributions for strictly P,N modular output functions. On the other hand, aweak P,N assortative matching distribution cannot be optimal for any strictly P,N modular output functionwhen it is strictly dominated by another matching distribution. This observation immediately follows fromthe fact that the absence of N,P concordant pairs is necessary but not sufficient for undominance in theP,N modular order. Altogether, Proposition 1 suggests that P,N sorting is a suitable general sorting class tostudy the multidimensional matching markets.Although it provides new insights into optimal matching with multidimensional agents, this sorting resultis not very practical for empirical purposes. Proposition 1 states that one cannot observe N,P concordantpairs under strictly P,N modular functions when every outcome-relevant attribute is observed by econo-metricians. In practice, econometricians only observe sorting patterns between observed attributes thatare aggregated over unobserved characteristics. Therefore, linking production complementarities betweenobserved attributes to matching patterns between these attributes when outcome-relevant and unobservedcharacteristics are present is of theoretical and empirical interest. In the next section, I examine a matchingmodel in which agents have outcome-relevant and unobserved characteristics in addition to their observedattributes. 17
Sorting with Unobserved Characteristics
Choo and Siow (2006) propose a matching model with multidimensional agents in which the agents haveoutcome-relevant characteristics that are not observed by econometricians. In this section, I examine ahomoskedastic extension of their model.Consider a two-sided matching model with equal numbers of firms and workers. Here, firm f ∈ F isdescribed by full attribute vector ˜ x f , and ˜ y w describes worker w ∈ W . Matching scheme ˜ m = { ˜ m fw } isa matrix such that the cell value associated with firm f and worker w , i.e. ˜ m fw , equals one if firm f andworker w are matched, and it equals zero otherwise. Let x f ∈ R K denote the observable attributes of firm f , and y w ∈ R L denote the observable attributes of worker w . The matching output produced by firm f and worker w , denoted by ˜ Q (cid:0) ˜ x f , ˜ y w (cid:1) , is determined by firm f ’s and worker w ’s full attributes. An optimalmatching matrix maximizes the aggregate output.Notice that an optimal matching matrix is determined by agents’ full attributes. However, econometri-cians observe only observable attributes. Consequently, the empirical goal is to estimate the complemen-tarities between observable attributes by using the optimal matching density function between observableattributes implied by an optimal matching matrix. Optimal matching density function m : R K × R L → [0 , ,associated with matching matrix ˜ m = { ˜ m fw } , is defined through aggregation over unobserved characteris-tics: m ( x, y ) = P f P w ˜ m fw { x f = x and y w = y } P f P w ˜ m fw . Notice that if the effect of unobserved characteristics on the optimal matching between observables cannotbe controlled by observed attributes, then one cannot consistently estimate the preferences on observedattributes. Example 4 demonstrates two channels through which the unobserved characteristics affect op-timal matching between observables, in a way that cannot be explained by observed attributes when thedistributions of unobserved characteristics conditional on observed attributes are unknown.
Example 4.
Consider a matching market with two firms: { (10 , , (20 , } , and two workers: { (20 , , (10 , } .Suppose that econometricians do not observe the first characteristics, and the second attributes are observ-able. Complementarity between unobserved characteristics: ˜ Q (cid:0) ˜ x f , ˜ y w (cid:1) = 5˜ x f ˜ y w + 2˜ x f ˜ y w Complementarity between unobserved and observed attributes: ˜ Q (cid:0) ˜ x f , ˜ y w (cid:1) = 2˜ x f ˜ y w + ˜ x f ˜ y w For these two matching output functions, the following optimal matching matrix and density are obtained.18orker (20 ,
10) (10 , Firm (10 , (20 ,
10 20
Firm .5 0Optimal matching densityNote that the optimal matching between firms’ and workers’ second attributes exhibits negative sorting.Negative sorting is consistent with negative complementarity and cannot be obtained under strictly positivecomplementarity according to Becker’s (1973) sorting results. Based on this observation, econometriciansmay infer negative complementarity between firms’ and workers’ second attributes whereas the underlyingprocess ˜ Q exhibits positive complementarity between these attributes.In order to limit the outcome-relevance of unobserved characteristics, three key assumptions are adoptedin empirical matching literature. Assumption 1.
There is a large number of agents for each observed type.
Under the large market assumption, one can focus on the asymptotic properties of optimal matching. Inthis context, the small sample properties of optimal sorting patterns remain an open question.
Assumption 2.
Matching output function ˜ Q can be expressed as follows: ˜ Q (cid:0) ˜ x f , ˜ y w (cid:1) = Q (cid:0) x f , y w (cid:1) + ε f (cid:0) ˜ x f , y w (cid:1) + η w (cid:0) x f , ˜ y w (cid:1) . Here, the first part of the matching output is determined by observed attributes. The deterministicmatching complementarities are to be estimated by using empirical matching density. However, notice thatthe matching outcome is not determined solely by the deterministic matching complementarities. The lasttwo parts of the equation represent idiosyncratic production shocks that affect the matching outcome. Thisseparability assumption rules out complementarities between unobserved characteristics. By doing so, itallows for a tractable aggregation over unobserved characteristics.
Remark . Consider two firms f and f ′ , and two workers w and w ′ such that x f = x f ′ = x and y w = y w ′ = y .For these four agents, Assumption 2 implies that ˜ Q (cid:0) ˜ x f , ˜ y w (cid:1) + ˜ Q (cid:16) ˜ x f ′ , ˜ y w ′ (cid:17) = ˜ Q (cid:16) ˜ x f ′ , ˜ y w (cid:17) + ˜ Q (cid:16) ˜ x f , ˜ y w ′ (cid:17) . Remark 1 states that unobserved aggregate output generated by pairs of couples ( f, w ) and ( f ′ , w ′ ) thathave the same observed attributes does not change with a partner-swap between these two pairs. Assumptions19 and 2 make it is possible to link optimal matching between full attributes to the one between observedattributes when the distribution of ε f conditional on x and the distribution of η w conditional on y are known. Assumption 3.
The idiosynratic production shocks satisfy the following conditions: (a) for all f ∈ F such that x f = x , ε f = (cid:8) ε f (cid:0) ˜ x f , y (cid:1)(cid:9) y is drawn from probability distribution F x ; and (b) for all w ∈ W such that y w = y , η w = { η w ( x, ˜ y w ) } x is drawn from probability distribution G y . Under these assumptions, the full attribute vector of firm f can be represented by (cid:0) x f , ε f (cid:1) , whereprobability distribution of ε f (cid:0) ˜ x f , y (cid:1) conditional on x f = x is F x . Similarly, the full attribute vector ofworker w can be represented by ( y w , η w ) , where probability distribution of η w ( x, ˜ y w ) conditional on y w = y is G y . For example, Choo and Siow (2006) assume that F x and G y are standard Gumbel distributions, i.e. P r (cid:8) ε f (cid:0) ˜ x f , y (cid:1) ≤ ε | x f = x (cid:9) = exp {− exp {− ε }} . In this section, I consider a simple homoskedastic extensionof this distributional assumption by relaxing Galichon and Salanie’s (2010) assumption. Let the distributionof ε f conditional on x w = x be a Gumbel distribution with location parameter α ( x ) and scale parameter σ .Similarly, let the distribution of η w conditional on y w = y be a Gumbel distribution with location parameter β ( y ) and scale parameter δ . P r (cid:8) ε f (cid:0) ˜ x f , y (cid:1) ≤ ε | x f = x (cid:9) = exp (cid:26) − exp (cid:26) − (cid:18) ε − α ( x ) σ (cid:19)(cid:27)(cid:27) P r { η w ( x, ˜ y w ) ≤ η | y w = y } = exp (cid:26) − exp (cid:26) − (cid:18) η − β ( y ) δ (cid:19)(cid:27)(cid:27) Galichon and Salanie (2010) establish the uniqueness of optimal matching density. Furthermore, theypresent a relationship between optimal matching density function and double difference of the deterministicoutput function, Q : R K × R L → R , when α ( x ) = 0 and β ( y ) = 0 for all x and y . Remember that thedifference between two independent random variables following a Gumbel distribution with the same locationand scale parameters follows a logistic distribution with zero location parameter. Consequently, Galichonand Salanie’s (2010) results hold with type-dependent location parameters as well. Lemma 2.
The optimal matching density function of observable attributes m : R K × R L → [0 , satisfiesthe following condition for all x, x ′ ∈ R K and y, y ′ ∈ R L : log (cid:26) m ( x, y ) m ( x ′ , y ′ ) m ( x ′ , y ) m ( x, y ′ ) (cid:27) = ( σ + δ ) − { Q ( x, y ) + Q ( x ′ , y ′ ) − Q ( x ′ , y ) − Q ( x, y ′ ) } . (1)Lemma 2 offers a simple relationship between the complementarity structure and optimal sorting patternbetween observable attributes. Based on this relationship, Siow (2015) derives a semi-parametric identifica-tion strategy for matching markets in which each agent has only one observable attribute, i.e. K = L = 1 .20e shows that the deterministic output function is supermodular(submodular) if and only if the left-handside of Equation (1) is greater(less) than for all x < x ′ and y < y ′ . I establish a similar result by usingP,N modularity. Definition 13.
Matching density function m ′ : R K × R L → [0 , is log P,N modular if and only if log m ′ isP,N modular. Proposition 2.
Optimal matching density between observable attributes is log P,N modular if and only ifthe deterministic output function is P,N modular.
Corollary 1.
The probability of observing P,N concordant pairs relative to N,P concordant pairs is higherwhen the deterministic output function is P,N modular.
In the previous section, Proposition 1 states that for strictly P,N modular deterministic output functions,the fraction of N,P concordant pairs equals zero when the scale parameters of the idiosyncratic outputcomponents are zero, i.e. every outcome relevant attribute is observed by econometricians. Proposition 2and Corollary 1 assert that for positive values of scale parameters, positive fraction of N,P concordant pairsmay be observed when the deterministic output function is strictly P,N modular. Furthermore, the fractionof P,N concordant pairs are higher than the fraction of N,P concordant pairs when the deterministic outputfunction is P,N modular. Thus, the matching models in which the agents have unobserved characteristicsoffer milder sorting patterns between observed attributes.Lemma 2 also allows me to obtain a comparative static result that links the changes in deterministic com-plementarities to the optimal matching density function. Bojilov and Galichon (2015) present comparativestatic results regarding the changes in the deterministic complementarities when (a) the deterministic out-put function is quadratic, i.e. Q ( x, y ) = P k P l θ k,l x k y l ; and (b) the observable attributes follow Gaussiandistributions. The quadratic functional form assumption offers a simple relationship between complemen-tarities and model parameters. More specifically, complementarity between firms’ k th and their workers’ l th observable attributes is governed by parameter θ k,l alone. I offer a similar comparative static result withoutimposing restrictions on the matching output and the distributions of the observable attributes. Definition 14.
Deterministic output function Q : R K × R L → R exhibits higher P,N modularity comparedto Q ′ : R K × R L → R if and only if, for all P,N concordant ( x, y ) and ( x ′ , y ′ ) , the following condition holds: Q ( x, y ) + Q ( x ′ , y ′ ) − Q ( x ′ , y ) − Q ( x, y ′ ) ≥ Q ′ ( x, y ) + Q ′ ( x ′ , y ′ ) − Q ′ ( x ′ , y ) − Q ′ ( x, y ′ ) . Here, a P,N modular increase implies (a) an increase in complementarity between firms’ i th and theirworkers’ j th attributes for all ( i, j ) ∈ P , (b) an decrease in complementarity between firms’ p th and their21orkers’ q th attributes for all ( p, q ) ∈ N . In this context, one can use an uneven P,N modular increase toformulate a skill-biased complementarity change, non-parametrically. Proposition 3.
The fraction of P,N concordant pairs relative to the fraction of N,P concordant pairs underoptimal matching rises with P,N modular increases in the deterministic output function.
Proposition 3 strengthens the result presented in Corollary 1. For any deterministic output function(P,N modular or otherwise), a change toward P,N modularity increases the fraction of P,N concordant pairsand decreases the fraction of N,P concordant pairs. This result allows us to make inference regarding thedynamic changes in deterministic complementarities without making any functional form assumptions on thedeterministic matching output. In particular, an increase in the fraction of all P,N concordant pairs relativeto the fraction of N,P concordant pairs is consistent with a P,N modular increase in the deterministic outputfunction. For parametric purposes, one can use a quadratic function to parameterize a P,N modular increasein the deterministic output function.
Corollary 2.
Function Q ( x, y ) = P k P l θ k,l x k y l exhibits higher P,N modularity compared to Q ′ ( x, y ) = P k P l β k,l x k y l if and only if (a) θ i,j ≥ β i,j for all ( i, j ) ∈ P ; (b) θ i,j ≤ β i,j for all ( i, j ) ∈ N ; and (c) θ i,j = β i,j for all ( i, j ) / ∈ P ∪ N . In this framework, the relationship between bivariate complementarities and bivariate optimal sortingpatterns is complicated. That makes the results presented in Proposition 3 and Corollary 2 practically useful.
Example 5.
Consider a matching market in which firms and workers have two binary observable attributes.Assume that σ + δ = 1 and θ , θ , θ , θ , = for quadratic output function Q ( x, y ) = P k P l θ k,l x k y l . The density functions of firms and workers, p F and p G , are given below: p F ( x ) = ( . x = (0 , or x = (1 , . x = (0 , or x = (1 , p G ( y ) = ( . y = (0 , or y = (1 , . y = (0 , or y = (1 , . See Lindenlaub (2017) for a parametric examination of skill-biased changes in the U.S. labor market. m : { , } × { , } → (0 , as follows by using iterative proportional fitting procedure .Workers (0 ,
0) (0 ,
1) (1 ,
0) (1 , Firms (0 , .0848 .0097 .0013 .0042 (0 , .2739 .0848 .0042 .0371 (1 , .0371 .0042 .0848 .2739 (1 , .0042 .0013 .0097 .0848Optimal matching density function between agents Workers0 1Firms 0 .208 .2921 .292 .208Optimal matching density function between thesecond attributesLogarithm of the optimal matching density function exhibits P,N modularity for P = { (1 , , (2 , } and N = {} , which is consistent with Proposition 2. Based on Proposition 2, one can also predict that thedeterministic output function is P,N modular for P = { (1 , , (2 , } and N = {} given the optimal matchingdensity. On the other hand, the optimal matching density exhibits negative correlation between the secondattributes of firms and workers. Consequently, an econometrician may predict negative complementaritybetween the second observable attributes of firms and workers while the deterministic output function exhibitspositive complementarity between these attributes, i.e. θ , = 1 > .An analysis based on two univariate marginals of multidimensional objects is potentially deceptive notonly for the binary case. Let F : R K → [0 , and G : R L → [0 , be the distribution functions offirms’ and workers’ observable attributes. Assume that the deterministic output function is quadratic, i.e. Q ( x, y ; θ ) = P k P l θ k,l x k y l . Due to Galichon and Salanie (2015), the logarithm of optimal matching densityfunction is obtained as follows for some φ and ϕ : log m ( x, y ; θ , F, G ) = W + Q ( x, y ; θ ) + φ ( x ; θ , F, G ) + ϕ ( y ; θ , F, G ) σ + δ (2)where W = − log ( P x ′ ∈ X P y ′ ∈ Y exp (cid:26) Q ( x ′ , y ′ ; θ ) + φ ( x ′ ; θ , F, G ) + ϕ ( y ′ ; θ , F, G ) σ + δ (cid:27)) . Let m k,l ( a, b ) denote the fraction of couples for which ( ♣ ) firms’ k th attributes equal a and ( ♠ ) workers’ l th attributes equal b . By using Equation (2), the logarithm of bivariate k, l marginal matching density functioncan be formulated as follows: log m k,l ( a, b ; θ , F, G ) = θ k,l σ + δ ab + W + Λ k,l ( a, b ) (3) For details, see Deming and Stephan (1940). Λ k,l ( a, b ) = log P x ′ ∈ R K P y ′ ∈ R L I { x k = a, y l = b } exp φ ( x ′ ; θ , F, G ) + ϕ ( y ′ ; θ , F, G ) + P ( i,j ) =( k,l ) θ i,j x i y j σ + δ . Suppose that a firm’s k th attribute may take f k different values: { a , · · · , a f k } , and a worker’s l th attributemay take w l different values: { b , · · · b w l } such that a i +1 > a i for i = 1 , . . . , f k − and b j +1 > b j for j = 1 . . . w l − . Let s k,l denote the logarithm of the local odds ratio for the i th value of a firm’s k th attributeand the j th value of a worker’s l th attribute: s k,l ( i, j ) := log m k,l ( a i , b j ) + log m k,l ( a i +1 , b j +1 ) − log m k,l ( a i +1 , b j ) − log m k,l ( a i , b j +1 ) . Based on Equation (3), it is easy to see that s k,l ( i, j ) = θ k,l σ + δ ( a i +1 − a i ) ( b j +1 − b j ) + Ω ( i, j ) (4)where Ω ( i, j ) = Λ k,l ( a i , b j ) + Λ k,l ( a i +1 , b j +1 ) − Λ k,l ( a i , b j +1 ) − Λ k,l ( a i +1 , b j ) .Notice that the right-hand side of Equation (4) is not only a function of θ k,l but also all other modelparameters, i.e. the distributions of observable attributes and other complementarity parameters. Conse-quently, an inference based on bivariate sorting patterns is potentially inconsistent and biased. I addressthis issue in a separate paper by proposing a multidimensional dependence class and devising two cardinalmeasures of multidimensional dependence. The relationship between agents’ health status and their spouses’ education levels has been well-studied inmedical literature. Jaffe et al. (2005) find the mortality risk among men with cardiovascular disease is higherfor those who are married to less-educated women. In addition, they also document that the mortality riskamong women with breast cancer is higher for those who are married to less-educated men. Jaffe et al.(2006) note similar findings for men with cardiovascular disease, and show that one’s wife’s education levelis a stronger predictor of her husband’s mortality than his own education level. Kravdal (2008) and Skalickáand Kunst (2008) report that one’s mortality risk decreases with former and current spouses’ education levelsin Norway. Nilsen et al. (2012) also document a strong association between spousal education and one’sself-rated health in Norway. Brown et al. (2014) confirm that spousal education is positively associated withself-rated health in the U.S. 24n literature, educational attainment has also been proven to be an important predictor of mortalitydifferentials (see Kunst and Mackenbach, 1994; Elo and Preston, 1996; Borrell et al., 1999; Mackenbach etal., 1999; Manor et al., 2000; Krokstad et al., 2002; Manor et al., 2004). Since better-educated individualsare more likely to be healthy, an attraction between better-educated or healthier individuals may generate aspurious positive association between agents’ health status and their spouses’ education levels. This channelmay disqualify one’s spouse’s education level as a robust predictor of one’s health status. Identifying robustpredictors of one’s health status is essential for health insurance carriers. In this section, I examine whether ornot one’s spouse’s health status and education level are robust predictors of his/her own health status by usingthe empirical matching framework described in the previous section. This empirical exercise demonstrateshow one can apply the aforementioned sorting theory to address several policy-related questions.
In this paper, I use the IPUMS-CPS data series for 2010-2017. The census data contains individual-levelinformation regarding education and self-rated health levels for 266,569 couples. i th couplein the sample, I denote the woman’s attributes by x i = ( x i,E , x i,H ) , and the man’s attributes by y i =( y i,E , y i,H ) . Index1 2 3 4 5Education
Less than high school High school Some college College Post-college
Health
Poor Fair Good Very Good ExcellentTable 2: Variables25 .2 Association Concepts and Measures
Since the variables of interest are ordinal, I examine the association between these variables by using a rank-correlation measure (Kruskal’s gamma) for each year. Here, I define key concepts and measures to calculateassociation between spouses’ education levels and health status.
Definition 15.
The i th and the j th couples exhibit concordance(discordance) between(a) women’s health status and education levels if and only if ( x i,H − x j,H ) ( x i,E − x j,E ) > ( < ) 0 ;(b) men’s health status and education levels if and only if ( y i,H − y j,H ) ( y i,E − y j,E ) > ( < ) 0 ;(c) men’s education levels and their wives’ health status if and only if ( x i,H − x j,H ) ( y i,E − y j,E ) > ( < ) 0 ;(d) men’s health status and their wives’ education levels if and only if ( y i,H − y j,H ) ( x i,E − x j,E ) > ( < ) 0 ;and(e) men’s and their wives’ health status if and only if ( x i,H − x j,H ) ( y i,H − y j,H ) > ( < ) 0 .To measure the association between agents’ health status and education levels, I calculate the followingKruskal’s gamma statistics for each survey year. Γ W,WH,E = C W,WH,E − D W,WH,E C W,WH,E + D W,WH,E Γ M,MH,E = C M,MH,E − D M,MH,E C M,MH,E + D M,MH,E Γ W,MH,E = C W,MH,E − D W,MH,E C W,MH,E + D W,MH,E Γ M,WH,E = C M,WH,E − D M,WH,E C M,WH,E + D M,WH,E Γ W,MH,H = C W,MH,H − D W,MH,H C W,MH,H + D W,MH,H
Here, C a,bk,l denotes the fraction of pairs of couples that exhibits concordance between a ’s attribute- k and b ’s attribute- l for a, b ∈ { W ( W omen ) , M ( M en ) } and k, l ∈ { E ( Education ) , H ( Health ) } . Similarly, D a,bk,l denotes the fraction of pairs of couples that exhibits discordance between a ’s attribute- k and b ’s attribute- l (see Definition 15). Kruskal’s gamma takes values between -1 and 1, inclusively. Values close to -1 indicatestrong negative association, and those close to 1 indicate strong positive association. Agents’ Own Health Status and Education Levels
As it is illustrated in Table 3, there is a weak positive association between agents’ own health status andeducation levels. The association is slightly more pronounced among women. Tables 10 and 11 indicate thatthe lower tail distribution of self-rated health does not vary substantially by education level. On average,a level increase in education is associated with .2398 level increase in self-rated health among women, and.2104 level increase among men. 26able 3: Agents’ Own Health Status and Education Levels
Survey YearStatistic 2010 2011 2012 2013 2014 2015 2016 2017 Γ W,WH,E .3121 .2934 .3176 .301 .3044 .2807 .2614 .2707 Γ M,MH,E .2708 .263 .2748 .2741 .2563 .2475 .2297 .2486
Agents’ Own Health Status and Their Spouses’ Education Levels
Table 4 shows that there is a weak positive association between agents’ own health status and their spouses’education levels. The association is slightly more pronounced for men. Tables 12 and 13 indicate thatlower tail distribution of agents’ own health status does not vary substantially by their spouses’ educationlevels. On average, a level increase in one’s spouse’s education is associated with .2189 level increase in one’sown health among men, and .2015 level increase among women. These results suggest that the associationbetween men’s health status and their wives’ education levels is higher than the association between men’shealth status and their own education levels. On the other hand, the association between women’s healthstatus and their husbands’ education levels is lower than the association between women’s health status andtheir own education levels.Table 4: Agents’ Own Health Status and Their Spouses’ Education Levels
Survey YearStatistic 2010 2011 2012 2013 2014 2015 2016 2017 Γ W,MH,E .2756 .2565 .2753 .2602 .255 .238 .2331 .244 Γ M,WH,E .2771 .2669 .2791 .273 .2731 .2597 .2239 .2522
Agents’ Own and Their Spouses’ Health Status
Table 5 reports a strong positive association between agents’ own and their spouses’ health status. Individualsare most likely to be married to spouses with the same self-rated health (see Table 14). This strong positiveassociation has an important actuarial implication: the risk associated with a two-person family plan ishigher than the aggregate risk associated with two individual plans.Table 5: One’s Health and One’s Spouse’s Health
Survey YearStatistic 2010 2011 2012 2013 2014 2015 2016 2017 Γ W,MH,H .7586 .7469 .7328 .7423 .7384 .7486 .7396 .749827he nation’s first and largest private online marketplace for health insurance, eHealth, Inc. , reportsaverage insurance premiums and deductibles for individual and family plans every year. According to thelatest report, a two-person family insurance costs $717, whereas an individual plan costs $310 to a singleman, and $332 to a single woman. Although per capita insurance premium is higher for married individuals,insurance policies with lower insurance premiums also have higher deductibles. According to the same report,a two-person family plan has $8,113 deductible. In addition, an individual plan has $4,457 deductible formen and $4,259 deductible for women. Based on these figures, it is not clear whether or not the insurersassess the extra risk associated with family plans adequately. The evaluation of the extra risk implied by thepositive association between spouses’ health status requires an in-depth analysis which is beyond the scopeof this paper. Here, I estimate a parametric matching model to explain the association patterns presented above. Themain goal is to decompose the mechanism that generates the association patterns into two channels: attrac-tion and distribution. The empirical framework presented in Section 2 allows me to dissociate these twoeffects. Therefore, this decomposition reveals spurious association patterns. For the purposes of parametricestimation, I adopt two additional assumptions.
Assumption 4.
Deterministic output function Q : { , , , , } × { , , , , } → R is quadratic: Q ( x r , y c ) = X k ∈{ H,E } X l ∈{ H,E } θ k,l x r,k y c,l where x r denotes the attribute vector of a type-r woman, and y c denotes the attribute vector of a type-c manfor r, c ∈ { , ..., } . https://news.ehealthinsurance.com/_ir/68/20169/eHealth%20Health%20Insurance%20Price%20Index%20Report%20for%20the%202016%20Open%20Enrollment%20Period%20-%20October%202016.pdf ealthEducation 1 2 3 4 51 Assumption 5.
The scale parameters of idiosyncratic output components add up to 1, i.e. σ + δ = 1 . Under Assumptions 1-5, the probability of observing a marriage between a type-r woman and a type-cman among all marriages of type-r women, denoted by p ( x r , y c ) , can be formulated as follows: p ( x r , y c ) := m ( x r , y c ) P j =1 m ( x r , y j ) = exp { Q ( x r , y c ) + ϕ ( y c ) } P j =1 exp { Q ( x r , y j ) + ϕ ( y c ) } . (5)An equivalent model can be obtained by choosing the first type of man as base category: p ( x r , y c | θ ) =
11 + P j =1 exp ( P k ∈{ H,E } P l = { H,E } θ k,l x r,k ( y j,l − y j, ) + ( ϕ ( y j ) − ϕ ( y )) ) c = 1exp ( P k ∈{ H,E } P l = { H,E } θ k,l x r,k ( y j,l − y j, ) + ( ϕ ( y c ) − ϕ ( y )) ) P j =1 exp ( P k ∈{ H,E } P l = { H,E } θ k,l x r,k ( y j,l − y j, ) + ( ϕ ( y j ) − ϕ ( y )) ) c = 1 . (6)Let f m ( x r ) denote the fraction of type-r married women in a sample of N married couples. The likelihoodfunction is formulated as follows: L N ( θ ) = N Y i =1 m (cid:16) x ( i ) , y ( i ) | θ (cid:17) = N Y i =1 25 Y r =1 25 Y c =1 { m ( x r , y c | θ ) } d ( i ) r,c = Y r =1 25 Y c =1 { f m ( x r ) p ( x r , y c | θ ) } n r,c where m (cid:0) x ( i ) , y ( i ) | θ (cid:1) : probability of observing the i th couple for parameter vector θ ; m ( x r , y c | θ ) : probability of observing a marriage between a type-r woman and a type-c man for parametervector θ ; d ( i ) r,c : binary identifier for the i th couple that equals 1 if the i th marriage is between a type-r woman and atype-c man; and n r,c : number of marriages between type-r women and type-c men.I estimate the complementarities by maximizing the following log-likelihood function:29 N ( θ ) = log L N ( θ ) = X r =1 n r log { f m ( x r ) } + X r =1 25 X c =1 n r,c log { p ( x r , y c | θ ) } (7)where n r is the total number of type-r women in the sample.The log-likelihood function has 28 parameters: 4 attraction parameters, { θ k,l } , and 24 deterministicsympathy parameters, { ϕ ( y j ) − ϕ ( y ) } . It is a well-known fact that the numerical methods which use agradient ascent algorithm are less accurate for high-dimensional parameter spaces. In order to overcome thisproblem, I estimate the parameters by using simulated annealing. As it is illustrated in Tables 10 and11, the distributions of the agents vary only slightly over time. For this reason, I use the entire sample andestimate only one set of parameters. The estimated parameters are reported in Table 7.Table 7: Estimated ComplementaritiesMenHealth EducationWomen Health .7625 -.0375Education -.0226 .5572 These results indicate an attraction between individuals with the same education levels and health status.The interpretation of these parameters is a little bit complicated. To interpret θ H,H , consider four agents x, x ′ , y , and y ′ such that (a) ( x E − x ′ E ) = ( y E − y ′ E ) = 0 , and (b) ( x H − x ′ H ) ( y H − y ′ H ) > . For these agents,one is e { . ( x H − x ′ H )( y H − y ′ H ) } times more likely to observe concordance than discordance between spouses’health status (see Equation (1)). In other words, everything held constant, one is at least . e . times more likely to observe concordance than discordance between spouses’ health status. Furthermore, thelikelihood of concordance rises with ( x H − x ′ H ) ( y H − y ′ H ) . The educational attraction parameter can alsobe interpreted in the same way: everything held constant, one is at least . e . times more likelyto observe concordance relative to discordance between spouses’ education levels.Contrary to weak positive association between agents’ own health status and their spouses’ educationlevels, the estimation results suggest a disaffection between healthier individuals and more educated indi-viduals: everything held constant, one is slightly more likely to observe fewer educated individuals marryinghealthier individuals. Altogether, the attraction analysis suggests that the weak positive association betweenagents’ own health status and their spouses’ education levels is a product of three factors: (a) an attractionbetween better-educated individuals, (b) an attraction between healthier individuals, and (c) a weak positive https://en.wikipedia.org/wiki/Simulated_annealing . If we use a code to generate the empiricalhousehold distribution, it will have . . . average description length. Therefore, the efficiencyloss associated with using a structural model is only × . . . percent.Table 8: Empirical and Predicted Assocation Levels Statistics Γ W,MH,H Γ W,MH,E Γ W,ME,H Γ W,ME,E
Empirical .7439 .2546 .2638 .6468
Predicted .6545 .2017 .2218 .6041Table 9: Difference Between Realized and Estimated Distributions
Kullback-Leibler Divergence ( P i ˆ m i log { ˆ m i /m i } ) .3952 Shannon Entropy of Predicted Distribution ( − P i ˆ m i log { ˆ m i } ) Efficiency Loss (%)
Appendix: Proofs
Proof. of Lemma X := supp ( F ) and Y := supp ( G ) have countably many elements.For given matching distribution M , define vec M as the density vector defined by M . More specifically, therows of vec M represents the densities of ( x, y ) ∈ X × Y implied by M . For P,N concordance improving τ P,N ( x, y, x ′ , y ′ ; 1) , define P,N transfer vector t P,N ( x, y, x ′ , y ′ ) such that (a) the rowassociated with ( x, y ) and ( x ′ , y ′ ) is ; (b) the row associated with ( x ′ , y ) and ( x, y ′ ) is − ; and (c) the restof the rows are . Let T ( P, N ) denote the set of P,N transfer vectors.It suffices to show that the following statement holds:vec M − vec M ′ = X t ∈T ( P,N ) α t t , ∀ α t ≥ ⇔ M (cid:23) P,N M ′ . (8)Define Q · t P,N ( x, y, x ′ , y ′ ) as follows: Q · t P,N ( x, y, x ′ , y ′ ) = Q ( x, y ) + Q ( x ′ , y ′ ) − Q ( x ′ , y ) − Q ( x, y ′ ) . Note that the aggregate output improves with P,N transfers: Q · t P,N ( x, y, x ′ , y ′ )= Q ( x , ..., x K , y , ..., y L ) + Q ( x ′ , ..., x ′ K , y ′ , ..., y ′ L ) − Q ( x , ..., x K , y ′ , ..., y ′ L ) − Q ( x ′ , ..., x ′ K , y , ..., y L )= K P i =1 L P j =1 Q (cid:0) x , ..., x i − , x i , x ′ i +1 , ..., x ′ K , y , ..., y j − , y j , y ′ j +1 , ..., y ′ L (cid:1) + Q (cid:0) x , ..., x i − , x ′ i , x ′ i +1 , ..., x ′ K , y , ..., y j − , y ′ j , y ′ j +1 , ..., y ′ L (cid:1) − Q (cid:0) x , ..., x i − , x i , x ′ i +1 , ..., x ′ K , y , ..., y j − , y ′ j , y ′ j +1 , ..., y ′ L (cid:1) − Q (cid:0) x , ..., x i − , x ′ i , x ′ i +1 , ..., x ′ K , y , ..., y j − , y j , y ′ j +1 , ..., y ′ L (cid:1) ≥ .Consequently, it holds that Q ∈ C ( P, N ) ⇔ Q · t ≥ ∀ t ∈ T ( P, N ) . (9)Equation (8) holds if and only if vec M − vec M ′ belongs to the convex cone C ( P, N ) generated by T ( P, N ) : C ( P, N ) = nP t ∈T ( P,N ) α t t : α t ≥ , ∀ t ∈ T ( P, N ) o .From Equation (9), it follows that C ( P, N ) is the dual cone of C ( P, N ) . Since C ( P, N ) is convex and closed, C ( P, N ) is the dual cone of C ( P, N ) due to Luenberger (1969, p:215), i.e. for { β t } > , it holds that P t ∈T ( P,N ) β t t ∈ C ( P, N ) ⇔ P t ∈T ( P,N ) β t Q · t ≥ ∀ Q ∈ C ( P, N ) . Therefore, M (cid:23) P,N M ′ if and only if vec M − vec M ′ ∈ C ( P, N ) . Claim. (a)
The set of P,N undominated distributions is a subset of weak P,N assortative matching distributions.32 b) The set of P,N dominant distributions and global P,N assortative matching distributions coincide.
Proof.
Kakutani Fixed Point Theorem:
Let A ⊆ R N be a non-empty, compact and convex set; U : A P ( A ) be a non-empty-valued, convex-valued correspondence with a closed graph. Correspondence U : A P ( A ) has a fixed point.For a given M ∈ M ( F, G ) and P, N , define correspondence U ( vec M ; P, N ) = vec M + P t ∈T ( P,N ) / T ( N,P ) α t t + P t ∈T ( P,N ) ∩T ( N,P ) β t t , ∀ α t , β t ≥ and for some α t > vec M oth. . (10)
It is clear that U ( · ; P, N ) is non-empty-valued, convex-valued, and has a closed graph. Therefore, ∃ M ∗ ∈M ( F, G ) such that vec M ∗ ∈ U ( vec M ∗ , P, N ) by Kakutani fixed point theorem. (a) Let vec M ∗ be a fixed point of the correspondence described in Equation 10.Due to Lemma 1, every P,Nundominated matching distribution corresponds to a fixed point of the correspondence above. This provesthat the set of P,N undominated distributions is non-empty. By definition, any distribution violating weakP,N sorting cannot be a fixed point of the correspondence. (b) ( ⇒ ) Suppose not. Let M ∈ M ( F, G ) be P,N dominant and not globally P,N assortative. Without lossof generality, assume that (1 , ∈ P and M violates (1 , positive sorting by assigning positive mass to ( x, y ′ ) , ( x ′ , y ) such that x > x ′ and y > y ′ . For matching distribution M ′ ∈ M ( F, G ) satisfying positivesorting between the first attributes, we have R QdM ′ > R QdM when Q ( x, y ) = x y . Consequently, M isnot a P,N dominant distribution. ( ⇐ ) Let M ∈ M ( F, G ) satisfy global P,N assortativeness. For any ( x, y ) , ( x ′ , y ′ ) ∈ supp ( M ) , we have ( ♣ )( x i − x ′ i ) (cid:0) y j − y ′ j (cid:1) ≥ for all ( i, j ) ∈ P and ( ♠ ) (cid:0) x p − x ′ p (cid:1) (cid:0) y q − y ′ q (cid:1) ≥ for all ( p, q ) ∈ N . Since every pairof matched couples under M is P,N weak concordant, any matching distribution M ′ ∈ M ( F, G ) satisfies thefollowing condition: vec M = vec M ′ + X t ∈T ( P,N ) α t t , for α t ≥ . Due to Lemma 1, it immediately follows that any matching distribution which satisfies global P,N sorting isP,N dominant.
Claim.
If the set of P,N dominant distributions is non-empty, then it coincides with the set of P,N undomi-nated distributions. 33 roof. of Claim. ( ⇒ ) Trivial. ( ⇐ ) Suppose not. Let M ∈ M ( F, G ) be P,N undominated but not P,N dominant. Consider a P,N dominantdistribution: M ′ ∈ M ( F, G ) . Since M is not P,N dominant and M ′ is, it holds that (a) R QdM ≤ R QdM ′ for all Q ∈ C ( P, N ) ; and (b) there exists output function Q ∈ C + ( P, N ) such that R QdM < R QdM ′ dueto Equation (9). Consequently, M ′ strictly dominates M in P,N modular order. Therefore, M is not a P,Nundominated distribution. Proof. of Proposition 1.
If a matching distribution does not satisfy within-group P,N sorting, then there exist a P,N concordanceimproving transfer which is not N,P concordance improving. In other words, the matching distribution doesnot satisfy weak P,N sorting.
Suppose not. Let M be a solution to the planner’s problem for Q ∈ C + ( P, N ) which is not a weaklyP,N assortative matching distribution. Since a swap between pair of couples which violates weak P,N sortingstrictly improves the aggregate output, M cannot be a solution. The existence of a weak P,N assortative distribution immediately follows from the existence of the fixedpoint of the correspondence given in Equation 10.Case 1: P ∪ N = ∅ .Trivial. Every matching distribution M ∈ M ( F, G ) is associated with the same level of aggregate output.Case 2: P ∪ N = ∅ .Let M ∈ M ( F, G ) be a solution to the planner’s problem for Q ∈ C ( P, N ) . Suppose M is not weak P,Nassortative distribution. Choose M ′ ∈ M ( F, G ) such that (i) vec M ′ ∈ U ( vec M ; P, N ) , and (ii) vec M ′ ∈ U ( vec M ′ ; P, N ) . Since M ′ is obtained from M via a sequence of P,N concordance improving transfers, itcannot worsen the level of aggregate output for any Q ∈ C ( P, N ) . Consequently, M ′ is at least as goodas M for any Q ∈ C ( P, N ) . Thus, M ′ also solves the planner’s problem. By construction, M ′ is a P,Nundominated distribution. Thus M ′ is satisfies weak P,N sorting. Let M be a solution to the planner’s problem for Q ∈ C + ( P, N ) that does not satisfy global P,Nsorting. Since the set of P,N dominant distributions coincides with the set of globally P,N assortativematching distributions, and M is not a globally P,N assortative matching distribution. Thus, there exists anN,P concordant pair of matched couples under M . In other words, every globally P,N assortative matchingdistribution strictly dominates M due to Lemma 1. Therefore, M cannot solve the planner’s problem. The set of P,N dominant distributions coincides with the set of globally P,N assortative matchingdistributions. Thus, every globally P,N assortative matching distribution solves the planner’s problem for34ll Q ∈ C ( P, N ) . Proof. of Proposition 2.
Immediate from Lemma 2.
Proof. of Corollary 1.
Immediate from Proposition 2.
Proof. of Proposition 3.
Immediate from Lemma 2.
Proof. of Corollary 2.
Immediate from Definition 14.35 ppendix: Tables
Table 10: Distributions of Women’s Health Status and Education Levels by Survey Year .0082 .0136 .008 .0026 .0015 .0178 .0374 .0237 .0094 .0042 .0772 .0423 .0216 .0227 .0968 .0964 .0754 .0411 .0115 .0541 .0747 .0788 .04422012 EducationHealth 1 2 3 4 51 .0074 .0154 .0083 .0037 .0015 .0161 .0386 .0252 .0098 .0047 .0754 .0419 .0215 .0214 .0945 .0987 .0806 .0434 .0099 .0530 .0702 .0827 .04712014 EducationHealth 1 2 3 4 51 .0079 .0144 .0082 .0031 .0018 .0167 .0374 .0256 .0118 .0064 .0764 .0445 .024 .0185 .0899 .0984 .0836 .0492 .0121 .0471 .0709 .0805 .05092016 EducationHealth 1 2 3 4 51 .0061 .0118 .0078 .0031 .0021 .0139 .0347 .0241 .013 .0067 .0792 .0488 .0291 .0204 .0855 .0970 .0871 .05175 .0122 .05 .0682 .0806 .0503 .0073 .015 .009 .0036 .0015 .0165 .0364 .0241 .0109 .0049 .0971 .0745 .0432 .0224 .0214 .0975 .1017 .0781 .0422 .0123 .0538 .0701 .0774 .04582013 EducationHealth 1 2 3 4 51 .007 .0138 .0085 .0031 .0014 .0171 .0368 .0254 .0106 .0055 .0757 .0432 .0229 .0189 .0921 .0987 .0827 .0477 .0114 .0538 .0708 .0791 .04922015 EducationHealth 1 2 3 4 51 .006 .0133 .0094 .0036 .002 .0151 .0336 .0247 .0122 .0053 .0758 .0459 .0255 .0212 .0914 .0946 .0835 .0485 .0125 .0519 .068 .0816 .05312017 EducationHealth 1 2 3 4 51 .0062 .0121 .0082 .0029 .0024 .0133 .0325 .0258 .0122 .0051 .0809 .0514 .0283 .0172 .0844 .0968 .0914 .0535 .0116 .0501 .0661 .0811 .0526 .0127 .0146 .0084 .0046 .0021 .0191 .0376 .0223 .0113 .0064 .0967 .069 .0441 .0256 .026 .0981 .0862 .0734 .0431 .0158 .061 .0637 .0693 .0492012 EducationHealth 1 2 3 4 51 .0108 .0168 .0081 .004 .0032 .0203 .0367 .0225 .0108 .0069 .0875 .068 .0434 .0275 .0256 .0955 .0878 .0776 .0484 .0148 .0568 .0644 .074 .05022014 EducationHealth 1 2 3 4 51 .0101 .0156 .0077 .0038 .0019 .0191 .0354 .0244 .0114 .0079 .0912 .0707 .0468 .0301 .0235 .0921 .0912 .0783 .0484 .015 .0588 .0613 .0701 .05142016 EducationHealth 1 2 3 4 51 .0087 .0132 .0081 .0038 .0029 .015 .032 .0241 .0124 .0078 .089 .0756 .0511 .0304 .0248 .0894 .0869 .0818 .05285 .0159 .0561 .0619 .0716 .0511 .0108 .015 .0098 .0041 .0027 .0185 .0347 .0227 .0125 .0069 .0942 .067 .0417 .0237 .0262 .0997 .0901 .0742 .0468 .0153 .058 .0621 .0707 .05032013 EducationHealth 1 2 3 4 51 .0115 .0145 .0089 .0029 .0019 .0188 .0344 .0232 .0118 .0077 .068 .0446 .0288 .0246 .0932 .0907 .0811 .0484 .0143 .056 .0629 .0718 .05242015 EducationHealth 1 2 3 4 51 .0088 .0135 .0093 .004 .0018 .0172 .0343 .0242 .0126 .008 .0914 .0702 .0462 .0293 .0257 .0914 .0879 .0777 .0507 .0159 .0577 .0615 .0752 .05172017 EducationHealth 1 2 3 4 51 .0081 .0122 .0082 .0031 .0023 .0152 .0344 .0244 .0141 .0079 .0735 .0504 .0315 .0221 .0859 .0887 .085 .05375 .0126 .0555 .0596 .075 .0511 .0095 .0125 .0071 .0031 .0017 .0196 .0346 .0211 .0106 .0064 .0993 .0692 .0435 .0246 .0286 .0996 .0861 .0737 .0444 .0146 .062 .0661 .0717 .0492012 EducationHealth 1 2 3 4 51 .0089 .0137 .0073 .0037 .0029 .0195 .0352 .0216 .0114 .0066 .0908 .0677 .0418 .0278 .0279 .0973 .0891 .0772 .047 .0143 .0564 .065 .0754 .05192014 EducationHealth 1 2 3 4 51 .0084 .0133 .0075 .0038 .0025 .0196 .035 .0244 .0108 .0081 .0908 .0677 .0453 .028 .0255 .0947 .0911 .0782 .0501 .014 .0594 .0646 .0724 .05112016 EducationHealth 1 2 3 4 51 .0066 .0114 .0074 .0031 .0023 .016 .0326 .023 .0125 .0083 .0873 .0744 .048 .0301 .0252 .0928 .0892 .0827 .0519 .0164 .0557 .0625 .0744 .0524 2011 EducationHealth 1 2 3 4 51 .0078 .0133 .0085 .0045 .0023 .0188 .0349 .0215 .0109 .0068 .0929 .0664 .0429 .0284 .0283 .1014 .0909 .0743 .0461 .0147 .0591 .0644 .0707 .05052013 EducationHealth 1 2 3 4 51 .0081 .0124 .0072 .0036 .0024 .0199 .0332 .0229 .0113 .0081 .0922 .0666 .0428 .0284 .0248 .0948 .0903 .0816 .0487 .0142 .059 .0667 .0729 .05152015 EducationHealth 1 2 3 4 51 .0068 .0129 .0084 .0037 .0024 .017 .0308 .0234 .0125 .0073 .09 .0685 .0453 .0294 .026 .0955 .0891 .079 .0496 .0163 .059 .0637 .0752 .05282017 EducationHealth 1 2 3 4 51 .0061 .0126 .0074 .0034 .0022 .0155 .0321 .0219 .0121 .0072 .0722 .0497 .0313 .0231 .0886 .0926 .0856 .0528 .0146 .0564 .0606 .0769 .053 .0089 .0172 .0101 .004 .0021 .0174 .0386 .023 .0113 .0063 .0783 .0438 .0226 .0225 .0954 .0945 .0743 .0401 .0124 .0558 .074 .075 .04152012 EducationHealth 1 2 3 4 51 .008 .0181 .0104 .0036 .0027 .0157 .0382 .0253 .0121 .0058 .0731 .0443 .0231 .0223 .0912 .0996 .0797 .0421 .0118 .0554 .0693 .0792 .04452014 EducationHealth 1 2 3 4 51 .0078 .0154 .0096 .0039 .0026 .0148 .0392 .0244 .0126 .0072 .0778 .0478 .0264 .019 .086 .0979 .0826 .0479 .013 .049 .0698 .0766 .04822016 EducationHealth 1 2 3 4 51 .0061 .0129 .0103 .0047 .0026 .0121 .0352 .0247 .0135 .0085 .08 .0515 .0314 .0207 .0856 .0938 .086 .04965 .0123 .052 .0675 .077 .0477 .0086 .0172 .0099 .0042 .0025 .0158 .0357 .0251 .0128 .006 .0958 .0748 .0446 .0221 .0218 .0966 .099 .0772 .0424 .0132 .0547 .0705 .0744 .04372013 EducationHealth 1 2 3 4 51 .0078 .0163 .0099 .0035 .0021 .0151 .0362 .0257 .0123 .0067 .0751 .045 .0253 .02 .0911 .1001 .0811 .0458 .012 .0536 .0683 .0767 .04682015 EducationHealth 1 2 3 4 51 .0068 00151 .0091 .004 .0025 .0144 .0035 .0275 .0132 .0062 .0756 .0478 .0279 .0218 .0886 .0914 .0829 .0477 .0124 .0526 .0679 .0791 .05012017 EducationHealth 1 2 3 4 51 .0055 .0133 .009 .0039 .0021 .0126 .0348 .0271 .0135 .0078 .0816 .0539 .0299 .0178 .0816 .0958 .0881 .05225 .0115 .0489 .0642 .0798 .0493 .0074 .0069 .0038 .0024 .0096 .041 .0242 .0111 .0064 .0116 .0287 .1863 .0338 .0177 .0049 .0125 .0402 .2493 .0254 .0027 .0071 .0179 .0287 .20682012 MenWomen 1 2 3 4 51 .013 .0065 .0084 .0053 .0032 .01 .0417 .0227 .0121 .0078 .0114 .0285 .1724 .0357 .0196 .0055 .0136 .0424 .252 .0251 .0031 .0068 .0189 .0297 .20462014 MenWomen 1 2 3 4 51 .0114 .0082 .0084 .0043 .0032 .0098 .0443 .0245 .0126 .0066 .0089 .0269 .1772 .0335 .019 .006 .0122 .0418 .2523 .0272 .003 .0066 .0206 .0307 .20062016 MenWomen 1 2 3 4 51 .011 .0054 .0072 .0046 .0027 .0083 .0421 .024 .0112 .0068 .0102 .023 .1868 .0343 .0194 .005 .0129 .042 .2567 .0252 .0022 .008 .0197 .0289 .2024 2011 MenWomen 1 2 3 4 51 .0129 .0072 .0083 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