# Matching in Closed-Form: Equilibrium, Identification, and Comparative Statics

aa r X i v : . [ ec on . T H ] F e b MATCHING IN CLOSED-FORM:EQUILIBRIUM, IDENTIFICATION, AND COMPARATIVE STATICS

RAICHO BOJILOV § AND ALFRED GALICHON † Abstract.

This paper provides closed-form formulas for a multidimensional two-sidedmatching problem with transferable utility and heterogeneity in tastes. When the matchingsurplus is quadratic, the marginal distributions of the characteristics are normal, and whenthe heterogeneity in tastes is of the continuous logit type, as in Choo and Siow (2006), weshow that the optimal matching distribution is also jointly normal and can be computedin closed form from the model primitives. Conversely, the quadratic surplus function canbe identiﬁed from the optimal matching distribution, also in closed-form. The closed-formformulas make it computationally easy to solve problems with even a very large numberof matches and allow for quantitative predictions about the evolution of the solution asthe technology and the characteristics of the matching populations change.

Keywords : matching, marriage, assignment.

JEL codes : C78, D61, C13.

Date : Date: January 12, 2016. A preliminary version of this paper containing the main results was ﬁrstpresented in September 2012 under the title “Closed-Form Formulas for Multivariate Matching”. We wouldlike to thank Nicholas Yannelis, the Editor, and two anonymous referees, as well as Arnaud Dupuy, BernardSalani´e and seminar participants at CREST and the 2013 Search and Matching Conference in Paris forhelpful comments. Galichon’s research has received funding from the European Research Council under theEuropean Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreements no 313699,and from FiME, Laboratoire de Finance des March´es de l’Energie. Bojilov’s work is supported by a grantof the French National Research Agency (ANR), ”Investissements d’Avenir” (ANR-11-IDEX-0003/LabexEcodec/ANR-11-LABX-0047). Introduction

Many economic problems involve markets with supply and demand restricted to a unit ofan indivisible good. Examples include occupational choice, task and schedule assignment,sorting of CEOs to ﬁrms, venture capital investment, consumer choice of heterogeneousgoods, and the marriage market. Models of discrete choice have been applied often to theempirical analysis of such problems. While having important advantages, they generallydo not incorporate scarcity constraints and equilibrium price eﬀects. Assignment modelsprovide an alternative framework which accommodates these issues. In assignment models,the supply of goods of each type is ﬁxed, and prices adjust at equilibrium so that supply anddemand clear. Yet, there have been just a few general theoretical results in this environmentbeyond the proof of existence, while econometric work has advanced only recently. The pi-oneering work by Choo and Siow [9] proposes an econometric framework for the estimationof assignment models based on the introduction of logit-type unobserved heterogeneity intastes. Their equilibrium approach has a number of advantages: it incorporates unobservedheterogeneity, allows for matching on many dimensions, and easily lends itself to nonpara-metric identiﬁcation. However, the equilibrium quantities in Choo and Siow’s setting aredeﬁned implicitly by a set of nonlinear equations, and explicit, closed-form solutions, evenunder further assumptions on the primitives of the model, have been missing so far. Thepresent contribution addresses this issue and also proposes a simple model with a completeclosed-form characterization of the equilibrium.This paper considers an environment of one-to-one matching with transferable utilitywhen sorting occurs on multiple dimensions in the presence of unobserved heterogeneity.Our goal is to develop a simple and easy to estimate framework that can be used in bothapplied and empirical problems. Thus, we study two related issues: on the one hand, solvingfor the optimal matching distribution, given a surplus technology and the characteristicsof the matching populations; and, on the other hand, ﬁnding the surplus technology, giventhe optimal matching and the characteristics of the two sides of the market. We follow thesetting of Choo and Siow [9] and impose two additional assumptions: quadratic matchingsurplus and normality on the distributions of the characteristics of the matching parties.

ATCHING IN CLOSED-FORM 3

These assumptions play a key role in the derivation of our main result: they are strongbut not uncommon in the related literature. In particular, economists have recognized theusefulness of the quadratic-normal setting and employed it in models of hedonic pricingand matching since Tinbergen [36]. In addition, the introduction of randomness in theform of taste shocks has not only some theoretical appeal, but it is also indispensableeconometrically. As Choo and Siow [9] discuss, such unobserved heterogeneity is crucial foridentiﬁcation in matching models, even when the characteristics of the matching parties arecontinuous.We show that under the model assumptions both the identiﬁcation and equilibrium prob-lems have explicit solutions in closed-form. In the equilibrium problem, the optimal match-ing distribution is a multivariate Gaussian distribution, and we provide explicit expres-sions for the relation between the parameters of the optimal matching distribution and theprimitives of the model. Moreover, we derive closed-form expressions for social welfare,equilibrium transfers, and the payoﬀs of market participants. In the identiﬁcation problem,we derive a closed-form expression for the maximum-likelihood estimator of the surplustechnology. In contrast to estimation methods based on numerical methods, these closed-form formulas make it computationally easy to solve problems that involve a large numberof matches and characterize the associated equilibrium. They also allow for quantitativepredictions about the sorting that will occur and how it evolves as the complementarityon various dimensions changes or as the distribution of characteristics in the populationsvaries.

Literature Review.

This paper is closely related to the literature on identiﬁcationof the matching function when the surplus is unobservable (see [24] for a good survey),following Choo and Siow [9]. Fox [18] proposes a maximum score estimator which relies ona rank-order property. Galichon and Salani´e [21] show that the social welfare in this settinghas a tractable formulation involving the entropy of the joint distribution, from whichidentiﬁcation can be deduced. They ([21], [22]) further introduce a parametric estimatorof the surplus function. Dupuy and Galichon [14] extend the model to the continuous caseand propose a decomposition of the surplus function into indices of mutual attractiveness,in order to best approximate the matching patterns by lower-dimensional models, and

BOJILOV AND GALICHON estimate the number of relevant dimensions on which the sorting eﬀectively occurs. In thecase of matching with non-transferable utility, Menzel [30] provides a characterization ofthe equilibrium in a continuous logit framework, which, as we discuss in the conclusion,could also be handled by our approach. Relative to these papers, we derive a closed-formformula for the maximum likelihood estimator in the normal-quadratic setting, and wepresent comparative statics that investigate the sensitivity of the estimates to changes inthe observed data. Moreover, our results allow for easy counterfactual experiments.A number of papers provide closed-form formulas for the equilibrium matching problemwith quadratic surplus and Gaussian distributions of the populations, in the case when thereis no unobserved heterogeneity. Early examples include mathematically oriented papers byOlkin and Pukensheim [31], as well as Dowson and Landau [13]. More recently, Lindenlaub[27] and Machado [28] work within the same setting to study assortative matching andcomplementarities in multidimensional economic settings. Our paper also connects withthe literature on characterizing matching equilibria and comparative statics in matchingmarkets. This literature includes, for example, contributions by Gretsky, Ostroy, and Zame[25] who analyze of the welfare properies of non-atomic assignment models, and EkelandGalichon, and Henry [16] who investigate the falsiﬁability of incompletely speciﬁed matchingmodels. Moreover, Decker et al. [12] study the equilibrium problem in Choo and Siow’sdiscrete setting and provide a number of comparative statics with a focus on the impact ofa change in the size of matching populations of given characteristics. In relation to theseworks, we ﬁnd closed-form solutions to both the identiﬁcation and equilibrium problemswith heterogeneity. In addition, we derive closed-form formulas for the optimal matchingdistribution, social welfare, transfers, individual utility and ﬁrm proﬁts. Furthermore, weconduct a comparative statics exercise in a continuous multidimensional setting to quantifyhow the characterization varies with changes in the properties of the surplus technologyand the distributional characteristics of the populations.Moreover, our work relates to the literature on hedonic pricing, which includes the fa-mous normal-quadratic model of Tinbergen [36] and the seminal contribution of Rosen [33].Brown and Rosen [3] suggest the use of linear approximations to the equilibrium conditionsfor empirical work and this approach underpins most of the following empirical studies.

ATCHING IN CLOSED-FORM 5

Ekeland, Heckman, and Nesheim [17] investigate the empirical content of various hedonicpricing models, speciﬁcally models with quasi-linear preferences and the normal-quadraticmodel. Chiappori, McCann, and Nesheim [6] establish the link between models of hedonicpricing and matching: they show that models of hedonic pricing with quasi-linear prefer-ences are equivalent to models of stable matching with transferable utility. Furthermorestable matching assignments in the latter correspond to equilibrium assignments in the for-mer and the same equivalence applies for prices. Finally, Ekeland [15] investigates furtherthe existence, uniqueness, and eﬃciency of hedonic pricing models. Given the equivalence inthe environments, our work also contributes to this literature by presenting a multidimen-sional model with closed-form solutions. In contrast, the famous linear-quadratic-normalmodel in Tinbergen [36] yields closed-fom solutions only in the special case when the sur-plus technology and the variances of the characteristics of the two sides in the market arediagonal. As Ekeland, Heckman, and Nesheim [17] point out (p. S70) “eﬀectively, this isa scalar case in which each attribute is priced separately.”The setting in the paper shares common features with several other literatures. In the casewithout unobserved heterogeneity, the solution to this equilibrium problem is well-known inthe applied mathematics literature and solved e.g. in [13]. Our mathematical contributionhere is to move beyond this benchmark and solve the problem with logit heterogeneitywithin a framework of clear economic interpretation. The quadratic-normal setting is usedalso in many-to-many matching, for example Gomes and Pavan [23], and coordinationgames of incomplete information, as in Angeletos and Pavan [1]. In addition, logit-typeheterogeneity is a key component of most choice models. Our use of logit heterogeneity in acontinuous choice setting relates to the spatial choice model of Ben-Akiva and Watanatada[4]. Relatedly, Dagsvik [11] provides a more general analysis of continuous choice, max-stable processes, and the independence of irrelevant alternative.Finally, there is a vast panel of applications of assignment models. In the literature onindustrial organization, Fox and Kim [19] use an extension of Choo and Siow’s model tostudy the formation of supply chains. In the corporate governance literature, Gabaix andLandier [20] and Tervi¨o [37] applied an assignment model to explain the sorting of CEOs andﬁrms and the determinants of the CEO compensation. In the marriage literature, Chiappori,

BOJILOV AND GALICHON

Salani´e and Weiss [8] used a heteroskedastic version of Choo and Siow’s model to estimatethe changes in the returns to education on the US marriage market; Chiappori, Oreﬃce andQuintana-Domeque [7] study the interplay between the sorting on anthropomorphic and onsocioeconomic characteristics. The context of our model is closest to Hsieh, Hurst, Jones,and Klenow [26] who use a variation of the Roy model to study discrimination in the US.This list is necessarily very incomplete.

Organization of the paper . The rest of the paper is organized as follows. Section 2presents the model and states the optimal matching problem. Section 3 starts with the set ofconditions that link the solution of the matching problem with the parameters of the socialsurplus and then derives closed-form expression for the optimal matching, social surplus,wage transfers, individual utilities and proﬁts. Then, it presents the closed-form solution tothe identiﬁcation problem and concludes with asymptotic analysis and comparative staticsof the closed-form solutions. Section 4 concludes.

Notation . Throughout the paper, we denote the transpose of matrix M as M ∗ and theinverse of this transpose as M −∗ . Similarly, the inverse of the square-root of matrix

M, M , is denoted M − . Let the generalized inverse (see [5]) of a non-square matrix M be denoted M + and its transpose, M + ∗ . All distributions are centered at 0. The random variables aredenoted with capital letters, while the corresponding sample points are denoted with smallletter. 2.

Model

We consider a market for heterogeneous goods with demand and supply limited to unity,in which transfers enter additively in the utility functions. We assume away market fric-tions and focus on long-term outcomes and contexts in which frictions play a relativelysmall role. The main advantage of applying one-to-one matching with transferable utility(hedonic pricing) to the analysis of such environments consists in the possibility to studythe interdependence of sorting (assignment) due to equilibrium price eﬀects. Due to itstractability, this setting has been used numerous times to study marriage, CEO pay, ven-ture capital, occupational choice and wage diﬀerentials, etc. In contexts in which market

ATCHING IN CLOSED-FORM 7 frictions are of primary interest, Shimer and Smith [35] provide an alternative approachthat extends the transferable utility framework to environments with both search frictionsand matching. Their focus is on the conditions under which assortative matching may bepreserved in this more complex environment. While appealing because of its realism, thismodel has proven to be operationally challenging in applied and empirical work, partic-ularly in multidimensional environments. Still further, the classical models of search byinspection and by experience provide an alternative when there are search frictions butinterdependence of sorting decisions is not of importance.We present the theoretical model in the context of labor matching between a populationof ﬁrms and a population of workers. We adopt the setting of Dupuy and Galichon [14],extending Choo and Siow [9] to the case when the characteristics of the matching popula-tions are continuous. As it is standard in the search and matching literature, we interpreteach ﬁrm as a unique job position or task. A worker can be employed by only one ﬁrm anda ﬁrm can employ only one worker. The populations are assumed to be of equal size, andwe assume that it is always better to be matched than to remain alone and generate nosurplus. Speciﬁcally, the focus is on markets in which remaining unmatched is empiricallyof little relevance. Examples include the markets for CEOs, sport managers and trainers,or specialists like accountants, lawyers, and PR agents. The model also covers some casesof one-to-many matching in which the proﬁt of each ﬁrm is additive in the characteristicsof the workers, i.e. there are no externalities between the assignment of individuals to thesame ﬁrm.2.1.

Environment.

Each worker has a vector of observable characteristics x known to theeconometrician , where x ∈ R m . Similarly, vector y summarizes the observable character-istics of ﬁrms, where y ∈ R n . We assume that the surplus enjoyed by worker x choosingoccupation y is the sum of three terms: one reﬂecting her intrinsic taste for occupation y ,as predicted by her observable type x ; one reﬂecting the amount of net monetary compen-sation; and one reﬂecting unobserved heterogeneity in tastes for occupation y . In general,it is suﬃcient that the random taste shocks have a known distribution conditional on char-acteristics x . We follow the commonly adopted assumption in the literature and consider BOJILOV AND GALICHON the special case when the taste shocks are independent of x. The formal assumptions of themodel are presented below.

Assumption 1: Surplus.

The surplus of a worker with characteristics x from matchingwith a ﬁrm with characteristics y is x ′ By + τ ( x, y ) + χ ( y ) (2.1)where B is a m × n real matrix, x ′ By is the nonpecuniary amenity of career with ﬁrm y , τ ( x, y ) is the monetary transfer or compensation received by the worker from the ﬁrm, and χ ( y ) is a random utility process drawn by each individual worker and whose distribution ischaracterized in Assumption 3 below. Similarly, the surplus of a ﬁrm with characteristics y from matching with a worker of characteristics x is x ′ Γ y − τ ( x, y ) + ξ ( x ) (2.2)where Γ is a m × n real matrix, x ′ Γ y is the economic value created by worker x at ﬁrm y , and ξ ( x ) is a random productivity process whose distribution is also characterized inAssumption 3 below.Assumption 1 speciﬁes the form of heterogeneity that we investigate, namely heterogene-ity in the preferences over the observable characteristics of one’s partner. As a result of thisassumption, the joint matching surplus is separable in the sense that˜Φ = x ′ Ay + χ ( y ) + ξ ( x )where the m × n real matrix A, A = B + Γ , is called the aﬃnity matrix . The ﬁrst termin the formula for the joint surplus is the observable surplus. The second term allows forunobserved variation in the preference of workers for observed ﬁrm characteristics, andthe third allows for unobserved variation in the preferences of ﬁrms over the observedcharacteristics of workers. In other words, we rule out an idiosyncratic term that representsthe interaction of unobserved ﬁrm and worker characteristics.Thus, Assumption 1 follows Choo and Siow [9] in the way it speciﬁes the ﬁrm andworker surplus from matching, except for the speciﬁc functional form of the surplus imposed ATCHING IN CLOSED-FORM 9 in our setting. The quadratic speciﬁcation is the simplest nontrivial form of the jointsurplus yielding complementarities between any pair of ﬁrm and worker characteristics.This speciﬁcation provides an intuitive and meaningful interpretation of the interactionbetween the characteristics of ﬁrms and workers: namely, A ij is simply the strength of thecomplementarity (positive or negative) between the i -th characteristics of the ﬁrm and the j -th characteristics of the worker. For this reason, it has been widely used in empiricalwork and applied models since Tinbergen [36]. Finally, note that x and y can be of diﬀerentdimension.’ Assumption 2: Unobservable Heterogeneity.

The function χ ( . ) is modelled as anExtreme-value stochastic process χ ( y ) = max k (cid:8) − λ ( y k − y ) + σ χ ik (cid:9) where (cid:0) y k , χ ik (cid:1) are the points of a Poisson process on R n × R of intensity dy × e − χ dχ forworker i , and λ ( z ) = 0 if z = 0, λ ( z ) = + ∞ otherwise. Similarly, ξ ( x ) = max l n − λ ( x l − x ) + σ ξ jl o where (cid:16) x l , ξ jl (cid:17) are the points of a Poisson process on R m × R of intensity dx × e − ξ dξ forﬁrm j .Assumption 2 implies that each worker and each ﬁrm draws an inﬁnite, but discretenumber of “acquaintances” from the opposite side of the market, along with a randomsurplus shock. In essence, this speciﬁcation allows for the extension of the convenient logitprobability function to the analysis of continuous choice problems. Let σ be the sum ofthe scale parameters σ and σ deﬁned in Assumption 2. It captures the total amountof heterogeneity and will play an important role in the sequel. Further details on thecontinuous logit model can be found in [4]. Assumption 3: Observable Heterogeneity . We assume that X has a Gaussiandistribution P = N (0 , Σ X ) and, similarly, that Y has distribution Q = N (0 , Σ Y ). Assumption 3 is arguably the strongest restriction of the model. It imposes normalmarginal distributions for the characteristics of the matching populations. The theoreti-cal literature has long recognized the usefulness of the quadratic-normal setting, startingwith Tinbergen [36]. Following our discussion of comparative statics below, we proposean overidentiﬁcation test that can be used to determine whether the model assumption,and in particular the quadratic-nomal setting, are statistically acceptable. Empirically, thenormality assumption is appropriate in many cases, while this is not the case in others dueto the presence of discrete characteristics, skewness, fat tails, etc. For example, researchershave often found that the normal distribution provides a good approximation to the actualdistribution of IQ tests, test of non-cognitive skills, height and weight. Moreover, most ofthe applied and empirical literature following the seminal contribution of Rosen [33] is alsobased on the quadratic-normal setting. As discussed in Brown and Rosen [3], researchershave interpreted the equilibrium conditions associated with the setting as linear approxima-tions to the actual equilibrium conditions. The same quadratic-normal setting plays the roleof a benchmark case in the identiﬁcation analysis of hedonic models in Ekeland, Heckman,and Nesheim [17]. Finally, quadratic-Gaussian models have a long tradition also in coor-dination games under incomplete information, for example Angeletos and Pavan [1], andin the analysis of many-to-many matching with multidimensional characteristics, includingGomes and Pavan [23], and the references therein.’

Assumption 4: Information.

The ﬁrms and workers know x , y, and the realized tasteshocks at the time they decide with whom to match, but the econometrician observes only x and y .This assumption deﬁnes the information structure of the problem: agents have full infor-mation, the econometrician does not observe preference shocks.2.2. Equilibrium Conditions.

Given Assumptions 1 to 4, a matching assignment is adistribution π over the characteristics of ﬁrms and workers ( X, Y ) with marginals for X and Y equal to P and Q respectively. Let M ( P, Q ) be the set of matching assignments,i.e. distributions over (

X, Y ) , such that X ∼ P , and Y ∼ Q . We start by reviewing the ATCHING IN CLOSED-FORM 11 equilibrium conditions that establish a link between the primitives of the model, particu-larly the surplus technology, and the optimal matching, denoted π XY . As usual, workersmaximize utility by solving the following problem:max y x ′ By + τ ( x, y ) + χ ( y )From the properties of the Generalized Extreme Value (GEV) distribution,log π XY ( x, y ) = x ′ By + τ ( x, y ) − a ( x ) σ (2.3)where a ( x ) is a normalization term that depends only on xa ( x ) = − σ log f ( x ) R exp h x ′ By + τ ( x,y ) σ i dy and f ( x ) is the probability density function associated with characteristics X . Similarly, oneobtains another condition that links the optimal matching distribution and the technologyof the ﬁrm x ′ Γ y : log π XY ( x, y ) = x ′ Γ y − τ ( x, y ) − b ( y ) σ (2.4)where b ( y ) = − σ log g ( y ) R exp h x ′ Γ y − τ ( x,y ) σ i dx and g ( y ) is the probability density function associated with characteristics Y . Equilibriumin the market requires that supply equals demand and, therefore, the probabilities in equa-tions (2.3) and (2.4) must be consistent. Using this equilibrium requirement, we solve for τ ( x, y ) and π XY ( x, y ) by combining (2.3) and (2.4) to obtain:log π XY ( x, y ) = x ′ Ay − a ( x ) − b ( y ) σ (2.5) τ ( x, y ) = σ ( x ′ Γ y − b ( y )) − σ ( x ′ By − a ( x )) σ (2.6)where σ = σ + σ . To complete the conditions that characterize the equilibrium, we deﬁnethe feasibility constraints: π XY ∈ M ( P, Q ) . These equilibrium conditions coincide with the conditions that characterize the solutionto the associated social welfare problem. We recall the characterization of the equilibriumgiven by Dupuy and Galichon [14], who show how the problem of estimating the equilibrium allocation, or optimal matching, can be solved by estimating the corresponding social welfareproblem of the planner. Theorem 1 in [14] implies that:(i) At equilibrium, the optimal matching π XY ∈ M ( P, Q ) is a maximizer of the socialwelfare W ( A ) = sup π ∈M ( P,Q ) (cid:0) E π (cid:2) X ′ AY (cid:3) − σ E ( π ) (cid:1) (2.7)where E ( π ) = E π [ln π ( X, Y )] if π is absolutely continuous= + ∞ otherwise.(ii) π XY is solution of (2.7) if and only if it is a solution of A ij = σ ∂ log π Y | X ( y | x ) ∂x i ∂y j π XY ∈ M ( P, Q ) . (2.8)(iii) for any σ >

0, the solution of (2.8) exists and is unique.In particular, part (ii) establishes the equivalence. Diﬀerentiating condition (2.5), weobtain the ﬁrst part of equilibrium conditions (2.8), while the second part is just a restate-ment of the feasibility constraints. The equivalence of the matching problem to a linearoptimization problem helps provide an expression for the social gain from matching, de-ﬁned in part (i) above. This result can be seen as an extension of the Monge-Kantorovichtheory (see Chapter 2 of [38]). It implies that the introduction of unobserved heterogeneitynaturally leads to the classical matching problem with an additional information term thatattracts the optimal solution toward a random matching: the entropy of the joint distri-bution E π XY [ln π XY ( X, Y )] . If σ is large, then optimality requires minimizing the mutualinformation which happens when ﬁrms and workers are matched randomly to each other.On the other hand, if σ is small, optimality requires maximizing the observable surplus.Uniqueness of the solution is a more general result in [14], but in our speciﬁc case it can besimply veriﬁed after solving the problem.Finally, the optimality conditions for the worker (2.3) and the equilibrium condition (2.5)yield an expression for the deterministic part of the utility of a worker of characteristics x ATCHING IN CLOSED-FORM 13 matching with a ﬁrm of characteristics y : U ( x, y ) = σ x ′ Ay − σ b ( y ) + σ a ( x ) σ Similarly, we use (2.4) and (2.5) to obtain the proﬁts of a ﬁrm with characteristics y matchingwith a worker with characteristics x :Π ( x, y ) = σ x ′ Ay − σ a ( x ) + σ b ( y ) σ The expressions a ( x ) and b ( y ) correspond to the Lagrange multipliers associated withthe scarcity constraints for workers of characteristics x and for ﬁrms of characteristics y. Consequently, the contribution of a ( x ) to individual utility captures the intuition thatthe share of the surplus that goes to a particular worker increases in the scarcity of hercharacteristics . Interestingly, the utility also increases in the scale of the taste shocks overthe ﬁrm characteristics. A similar interpretation also holds for proﬁts.3.

Closed-Form Formulas

This section contains our main results. We start by discussing how the quadratic settingand the distributional assumptions give rise to a tractable condition that allows us to ﬁnda closed-form formula for the optimal matching in terms of the aﬃnity matrix A, and theparameters that describe the distributions of X and Y. Then, we use the same condition torecover the expression of the surplus, namely the aﬃnity matrix A, in terms of the observeddata. Denote Σ XY = ( E π XY [ X i Y j ]) ij = E π XY (cid:2) XY ′ (cid:3) (3.1)to be the cross-covariance matrix computed at the optimal π XY solution of (2.8). We willshow that under assumptions 1 to 4 the optimal solution is normal and that (Σ X , Σ Y , Σ XY )completely parameterize the distribution π XY .We consider two related problems. The equilibrium computation problem requires, givena surplus technology, to determine the optimal matching π XY . Finding out how to dothis in a tractable way, possibly in closed form, allows to make quantitative predictionsabout the sorting that will occur on the market and to derive comparative statics. In contrast, the identiﬁcation problem starts with an observed matching that is assumed tobe stable (or equivalently, optimal) and the goal is to determine the underlying surplustechnology. This problem is the inverse problem of the former and is of primary interestto the econometrician. The conditions (2.8), in particular the ﬁrst one, play a crucial rolein our analysis because they establish a link between the cross-covariance matrix and theaﬃnity matrix. Thus, the equilibrium computation problem involves solving the ﬁrst-orderconditions (2.8) for the optimal matching and Σ XY , while the identiﬁcation problem involvessolving the same conditions but for A given Σ XY . Closed-form formulas shall be providedfor these two problems.3.1.

Equilibrium Computation Problem.

The following result provides an explicit so-lution to the equilibrium problem. It requires some educated ‘guesswork’ and consists oftwo parts. In the ﬁrst part, we propose a multivariate normal distribution as a candidatefor the solution, and then we verify that the candidate satisﬁes the equilibrium conditions(2.8). From above, we know that this solution is unique. To the interested readers, we revealhow we arrive at our candidate for the solution in the second part of the proof. Namely,we start with the ﬁrst condition in (2.8) and note that in the case of multivariate normaldistributions it reduces to a quadratic matrix equation in Σ XY , since the cross-derivative ofthe optimal matching is constant. Most such equations do not have closed-form solutionsbut we redeﬁne variables in a way that allows us to solve for the unknown Σ XY in closed-form. While members of the exponential family satisfy some of the equilibrium conditions,we ﬁnd that it is only the multivariate normal distribution that satisfy all of them. Theorem 1.

Assume σ > and suppose that Assumptions 1 to 4 hold. Let ( X, Y ) ∼ π XY be the solution of (2.7) that satisﬁes conditions (2.8). Then:(i) The relation between X and Y takes the following form Y = T X + ǫ (3.2) ATCHING IN CLOSED-FORM 15 where ǫ ∼ N (cid:0) , Σ Y | X (cid:1) is a random vector independent from X , and matrices T and Σ Y | X are given by T = ∆ (∆ A ∗ Σ X A ∆) − / ∆ A ∗ − σ A + Σ − X (3.3)Σ Y | X = σ ∆ (∆ A ∗ Σ X A ∆) − / ∆ − σ A + Σ − X A + ∗ (3.4) where matrix ∆ is deﬁned as ∆ = (cid:18) σ A + Σ − X A + ∗ + Σ Y (cid:19) / . (ii) The optimal matching π XY is the Gaussian distribution N (0 , Σ) where Σ = Σ X Σ XY Σ ∗ XY Σ Y , (3.5) and the cross-covariance matrix of X and Y , namely Σ XY = E π XY [ XY ′ ] is given by Σ XY = Σ X A ∆ (∆ A ∗ Σ X A ∆) − / ∆ − σ A + ∗ . (3.6) Proof.

Under Assumptions 1-4, and as recalled in Section 2 the solution to the matchingassignment problem is unique and characterized by conditions (2.8). Hence it is suﬃcientjust to verify that taking π XY as the p.d.f. of the N (0 , Σ) distribution where Σ satisﬁes(3.5) and (3.6) satisﬁes conditions (2.8) characterizing the optimal assignment.

Veriﬁcation of optimality.

Let π XY be the distribution of ( X, Y ) where X ∼ N (0 , Σ X )and the distribution of Y conditional on X is given by (3.2), for T and Σ Y | X given by (3.3)and (3.4). We verify that π XY satisﬁes the two conditions of characterization (2.8). Indeed,one has ∂ xy log π Y | X ( y | x ) = T ∗ Σ − Y | X = Aσ and var ( Y ) = T Σ X T ∗ + Σ Y | X = (cid:16) ∆ (∆ A ∗ Σ X A ∆) − / ∆ A ∗ − σ A + Σ − X (cid:17) Σ X T ∗ + Σ Y | X = ∆ − σ ∆ (∆ A ∗ Σ X A ∆) − / ∆ + σ A + Σ − X A + ∗ + Σ Y | X = ∆ − σ A + Σ − X A + ∗ = Σ Y . Hence, condition (2.8) is veriﬁed and π XY is optimal for (2.7), QED.Although this proof by veriﬁcation is suﬃcient from a pure mathematical point of view,it is not very didactic as it is not informative about how the formula for Σ XY was obtained.For the convenience of the reader, we shall now ﬂesh out how to solve for Σ XY . Solving for Σ XY . In this paragraph, we look for a necessary condition on Σ so that N (0 , Σ) is a solution of (2.7). The derivation made before Theorem 2 implies the followingrelation to be inverted between A and Σ XY Aσ = (cid:0) Σ Y (Σ XY ) + Σ X − Σ ∗ XY (cid:1) + . Some algebra leads to the following quadratic equation in Σ XY ∗ XY Σ − X Σ XY + σA + Σ − X Σ XY − Σ Y (3.7)Let Ψ = Σ − / X Σ XY (3.8) D = σ A + Σ − / X (3.9) C = − Σ Y (3.10)so that equation (3.7) becomes Ψ ∗ Ψ + 2 D Ψ + C = 0 (3.11)hence 2 D Ψ = Σ Y − Σ ∗ XY Σ − X Σ XY ATCHING IN CLOSED-FORM 17 is the variance-covariance matrix of the residual ǫ of the regression of Y on X Σ Y = T Σ X T ∗ + Σ Y | X (3.12)and T and Σ Y | X are given by expressions: T = Σ ∗ XY Σ − X (3.13)Σ Y | X = Σ Y − Σ ∗ XY Σ − X Σ XY (3.14)Hence, D Ψ is symmetric positive. Equation (3.11) can be rewritten as(Ψ + D ∗ ) ∗ (Ψ + D ∗ ) = DD ∗ − C Thus, we have for the solution that Ψ = U ∆ − D ∗ (3.15)where ∆ = ( DD ∗ − C ) / = (cid:18) σ A + Σ − X A + ∗ + Σ Y (cid:19) / is a positive semi-deﬁnite matrix and U is an orthogonal matrix to be deﬁned. Note that DD ∗ − C = DD ∗ + Σ Y is a positive symmetric matrix so its square root exists and isuniquely deﬁned.We now determine U . Since D Ψ is symmetric positive, DU ∆ = D Ψ + DD ∗ is symmetricpositive, hence ∆ − DU = ∆ − ( DU ∆) ∆ − as ∆ is symmetric and invertible. LetΛ = ∆ − D Since Λ U is symmetric positive, (Λ U ) = Λ U U ∗ Λ ∗ = ΛΛ ∗ , which implies that Λ U =(ΛΛ ∗ ) / . Hence U is determined by U = Λ − (ΛΛ ∗ ) / = D − ∆ (cid:0) ∆ − DD ∗ ∆ − (cid:1) / . Substituting in (3.15) yieldsΨ = U ∆ − D ∗ = D − ∆ (cid:0) ∆ − DD ∗ ∆ − (cid:1) / ∆ − D ∗ and replacing D, Σ XY , T and Σ Y | X by their respective expressions (3.9), (3.8), (3.13) and(3.14) gives Ψ = Σ / X A ∆ (cid:0) ∆ − (cid:0) A + Σ − X A + ∗ (cid:1) ∆ − (cid:1) / ∆ −

12 Σ − / X A + ∗ Σ XY = Σ X A ∆ (∆ A ∗ Σ X A ∆) − / ∆ − σ A + ∗ T = ∆ (∆ A ∗ Σ X A ∆) − / ∆ A ∗ − σ A + Σ − X Σ Y | X = σ ∆ (∆ A ∗ Σ X A ∆) − / ∆ − σ A + Σ − X A + ∗ . Hence it turns out that the optimal matching solution π XY is jointly normal too, a factthat was not a priori obvious, as π XY may have normal marginal distributions for X and Y without necessarily being jointly normal.In what follows, we provide a characterization of the solution. We ﬁrst consider howthe solution evolves both when the unobserved heterogeneity declines to zero and whenit increases to inﬁnity. The following lemma provides these limiting results, recoveringwell-known results in the optimal transportation literature (see e.g. [13]). Lemma 1.

Suppose that Assumptions 1 to 4 hold, and let Σ XY be the cross-covariancematrix under the optimal matching. Then: (i). lim σ → Σ XY = Σ X A Σ / Y (cid:16) Σ / Y A ∗ Σ X A Σ / Y (cid:17) − / Σ / Y (ii). lim σ →∞ Σ XY = 0 . When the importance of unobserved variables increases, the optimal matching ceases todepend on the observable characteristics X and Y, so in the limit the cross-covariance matrixconverges to zero. In the other limit case, when there is no unobserved heterogeneity, theoptimal matching problem becomes a special case of Brenier’s Theorem (see [38], Ch. 2)and there exists an optimal matching map given by Y = T ( A ∗ X ) , where T is the gradientof a convex function which in our setting has the economic interpretation of the derivative ATCHING IN CLOSED-FORM 19 of the worker’s utility. Note thatlim σ → T = T = Σ / Y (cid:16) Σ / Y A ∗ Σ X A Σ / Y (cid:17) − / Σ / Y A ∗ and, consequently, AT is symmetric positive. Thus, the worker’s equilibrium utility , when σ = 0 , is equal, up to an additive constant, to x ′ AT x. In the likely presence of heterogeneity, however, there exists no such optimal map whichcomplicates the theoretical and econometric problems. Nevertheless, expression (3.6) fromTheorem 1, part i shows that it is still possible to establish a ‘regression-style’ relationbetween X and Y in our setting. It turns out that under the optimal matching one candecompose Y into the sum of two independent normally distributed terms: a linear com-bination of X (a projection of Y onto X ) and an ‘error’ whose distributional propertiesdepend on the primitives of the model, particularly the scale parameter of the unobservedheterogeneity. This ‘regression-style’ relation plays a crucial role in the logic of the proofsof both Theorem 1 and Theorem 2.The closed-form solution for the optimal matching also allows us to derive a closed-formexpression for social welfare (up to an irrelevant integration constant) under the optimalmatching distribution: W ( A ) = T r ( A ∗ Σ XY ) − σ (cid:20)

12 ln (cid:0) det (Σ X ) det (cid:0) Σ Y − Σ ∗ XY Σ − X Σ XY (cid:1)(cid:1)(cid:21) where Σ XY is expressed as a function of A by (3.6). The ﬁrst term is equal to the expectedsurplus from the observables and the second one is equal to the entropy term. Usingthe equilibrium condition (2.5) in combination with Schur decomposition of the variance-covariance matrix, we obtain the following expression for the Lagrange multipliers: a ( x ) = σ x ′ Σ − X | Y xb ( y ) = σ y ′ Σ − Y | X y where Σ X | Y and Σ Y | X are the conditional variances of x and y, given the optimal matchingdistribution deﬁned in Theorem 1. Thus, the equilibrium transfer becomes τ ( x, y ) = 1 σ (cid:16) σ (cid:16) x ′ Γ y − σ y ′ Σ − Y | X y (cid:17) − σ (cid:16) x ′ By − σ x ′ Σ − X | Y x (cid:17)(cid:17) which has a very intuitive interpretation. The wage increases in the gains generated bythe ﬁrm from the match and the market scarcity of workers of characteristics x. At thesame time, it decreases in the worker from the match and in the relative scarcity of ﬁrmsof characteristics y. These results imply that the expected utility at the equilibrium beforethe realization of the taste shocks is: U ( x, y ) = σ σ x ′ Ay − σ y ′ Σ − Y | X y + σ x ′ Σ − X | Y x while the associated expected proﬁts areΠ ( x, y ) = σ σ x ′ Ay + σ y ′ Σ − Y | X y − σ x ′ Σ − X | Y x In other words, each side of the market receives a share of the joint surplus that increases inthe heterogeneity of the shocks over the characteristics of the other side of the market. Inaddition, it obtains a fraction of the equilibrium shadow price of its own characteristics and‘pays’ a fraction of the other side’s shadow price. Interestingly, the division of the surplusas the unobserved heterogeneity vanishes depends on the particular path of convergence of( σ , σ ) to (0 , . For example, when σ = σ , which is a standard restriction in modelsof the marriage market, the division coincides with the utility and proﬁts presented afterLemma 1.3.2. Identiﬁcation Problem.

The identiﬁcation problem is simpler. From Theorem 1,(

X, Y ) is jointly normal, so one may regress Y on X and write Y = T X + ǫ (3.16)where ǫ ∼ N (cid:0) , Σ Y | X (cid:1) is independent from X . Consequently, we can express Σ Y = E [ Y Y ′ ]and Σ ∗ XY = E [ Y X ′ ] as Σ Y = T Σ X T ∗ + Σ Y | X Σ ∗ XY = T Σ X and, solving for T and Σ Y | X , we arrive again at expressions (3.13) and (3.14): T = Σ ∗ XY Σ − X Σ Y | X = Σ Y − Σ ∗ XY Σ − X Σ XY ATCHING IN CLOSED-FORM 21

Since the distribution of Y conditional on X, π Y | X ( y | x ) , is also normal, ∂ xy log π Y | X ( y | x ) = T ∗ Σ − Y | X = Σ − X Σ XY (cid:0) Σ Y − Σ ∗ XY Σ − X Σ XY (cid:1) − . Introducing this expression in condition (2.8) implies A = σ∂ xy log π Y | X ( y | x )= σ Σ − X Σ XY (cid:0) Σ Y − Σ ∗ XY Σ − X Σ XY (cid:1) − This expression can be further simpliﬁed to A = σ (cid:0) Σ Y Σ + XY Σ X − Σ ∗ XY (cid:1) + These observations imply a startlingly simple way to recover the model primitives, giventhe optimal matching. Intuitively, given that the observed matching is optimal, one canregress Y on X . The coeﬃcients of X constitute the corresponding entries in the matrix T and the variance of the error term from the regression is simply the conditional varianceof Y given X, Σ Y | X . Then, by the equilibrium conditions, we have that Aσ = T ∗ Σ − Y | X . Torecover the model primitives, we have simply used the theoretical equilibrium conditionsand OLS, so we have also found a closed-form expression for the MLE for A up to a scaleparameter 1 /σ .As Chow and Siow [9] show, unobserved heterogeneity is crucial for identiﬁcation ofmatching models, even when the characteristics of the matching populations are contin-uous. For example, when the aﬃnity matrix A is approximately trivial, identiﬁcation isnot achieved if the econometrician does not allow for unobserved heterogeneity. As in allchoice models, the model is identiﬁed up to a location and scale parameter: the identiﬁedparameter is the rescaled aﬃnity matrix A/σ . Consequently, we normalize the scale ofheterogeneity to one, σ = 1 , and estimate the norm of the aﬃnity matrix. Finally, if thereare data on wage transfer τ ( x, y ), one can recover also B and Γ , as well as the ratio of thescale parameters σ and σ , using condition (2.6) which links compensation and the utilityand proﬁt functions. We formalize the identiﬁcation result for the aﬃnity matrix A in thetheorem below, whose proof is immediate given the proof of Theorem 1. Theorem 2.

Suppose that Assumptions 1 to 4 hold and let σ = 1 . Then the aﬃnity matrix A is given by A = Σ − X Σ XY (cid:0) Σ Y − Σ ∗ XY Σ − X Σ XY (cid:1) − (3.17)= (cid:0) Σ Y Σ + XY Σ X − Σ ∗ XY (cid:1) + . Comparative Statics and Asymptotics.

In many problems, it is important tocompute how an increase in the complementarity between two characteristics in the surplusformula aﬀects the optimal matching. Similarly, one may want to ﬁnd how the optimalmatching changes as the variances in the matching populations increase. From an econo-metric point of view, the diﬀerentiation of the estimator of the aﬃnity matrix with respectto the summary statistics will allow us to derive central limit theorems and to compute con-ﬁdence intervals. We use extensively matrix diﬀerentiation and the Kronecker product, forwhich we now give the basic deﬁnitions (a more detailed review is given in the Appendix).As deﬁned in that appendix, T is the operator such that T vec ( M ) = vec ( M ∗ ) . The following theorem summarizes our results on comparative statics.

Theorem 3.

Suppose that Assumptions 1 to 4 hold. Then: (i)

The rate of change of the matching estimator with respect to the covariation betweenthe matching populations (keeping Σ X and Σ Y constant) is ∂A∂ Σ XY = ( A ∗ ⊗ A ) (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) , (3.18) and the rates of change with respect to the variances of the matching populations (keeping Σ XY constant) are ∂A∂ Σ X = − ( A ∗ ⊗ A ) (cid:2) I ⊗ Σ Y Σ + XY (cid:3) , (3.19) ∂A∂ Σ Y = − ( A ∗ ⊗ A ) (cid:2) Σ X Σ + ∗ XY ⊗ I (cid:3) . (3.20) ATCHING IN CLOSED-FORM 23 (ii)

The rate of change of the optimal cross-covariance matrix with respect to the aﬃnitymatrix A (keeping Σ X and Σ Y constant) is ∂ Σ XY ∂A = (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) + (cid:0) A + ∗ ⊗ A + (cid:1) , (3.21) and the rates of change with respect to the variances of the matching populations (keeping A constant) are ∂ Σ XY ∂ Σ X = (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) + (cid:0) I ⊗ Σ Y Σ + XY (cid:1) , (3.22) ∂ Σ XY ∂ Σ Y = (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) + (cid:0) Σ X Σ + ∗ XY ⊗ I (cid:1) . (3.23) Proof of Theorem 3.

We use extensively the properties of the Kronecker product and the vec operator reviewed in the Appendix and presented in much greater details by Magnusand Neudecker [29]. The Kronecker product and the vec operator are linked by the followingformula vec ( AXB ) = ( B ∗ ⊗ A ) vec ( X )This relation is very useful in deriving the following general product rule. Suppose that f : R n → R m × p and g : R n → R p × q . Then f · g : R n → R m × q . Let I n denote the n × n identity matrix. From the relation between the Kronecker product and the vec operator vec ( I m f ( x ) g ( x ) I q ) = (( g ( x )) ∗ ⊗ I m ) vec ( f ( x )) = ( I q ⊗ f ( x )) vec ( g ( x ))Thus, we have the following product rule: D ( f ( x ) g ( x )) = (( g ( x )) ∗ ⊗ I m ) Df ( x ) + ( I q ⊗ f ( x )) Dg ( x ) (3.24)We start with the comparative statics for A. Using the product rule (3.24), the chain rule,and the rules of matrix diﬀerentiation summarized in Fact 3.1, 3.2, and 3.3 of the Appendix,we ﬁnd the derivative of A with respect to Σ XY : ∂A∂ Σ XY = − ( A ∗ ⊗ A ) (cid:2) − (Σ X ⊗ Σ Y ) (cid:0) Σ + ∗ XY ⊗ Σ + XY (cid:1) − T (cid:3) = ( A ∗ ⊗ A ) (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) Similarly, the derivatives with respect to Σ X and Σ Y are: ∂A∂ Σ X = − ( A ∗ ⊗ A ) (cid:2) I ⊗ Σ Y Σ + XY (cid:3) ∂A∂ Σ Y = − ( A ∗ ⊗ A ) (cid:2) Σ X Σ + ∗ XY ⊗ I (cid:3) We can also use equation (3.17) to derive implicitly ∂ Σ XY ∂A , ∂ Σ XY ∂ Σ X , and ∂ Σ XY ∂ Σ Y . By diﬀer-entiation of the two sides with respect to A , one gets − (cid:0) A + ∗ ⊗ A + (cid:1) = − (Σ X ⊗ Σ Y ) (cid:0) Σ + ∗ XY ⊗ Σ + XY (cid:1) ∂ Σ XY ∂A − T ∂ Σ XY ∂A (cid:2) (Σ X ⊗ Σ Y ) (cid:0) Σ + ∗ XY ⊗ Σ + XY (cid:1) + T (cid:3) ∂ Σ XY ∂A = (cid:0) A + ∗ ⊗ A + (cid:1) Solving for ∂ Σ XY ∂A , we obtain ∂ Σ XY ∂A = (cid:2) (Σ X ⊗ Σ Y ) (cid:0) Σ + ∗ XY ⊗ Σ + XY (cid:1) + T (cid:3) + (cid:0) A + ∗ ⊗ A + (cid:1) = (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) − (cid:0) A + ∗ ⊗ A + (cid:1) Similarly, by diﬀerentiation with respect to Σ X and Σ Y , one gets ∂ Σ XY ∂ Σ X = (cid:2) T + (Σ X ⊗ I ) ( I ⊗ Σ Y ) (cid:0) Σ + ∗ XY ⊗ Σ + XY (cid:1)(cid:3) + (cid:0) I ⊗ Σ Y Σ + XY (cid:1) = (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) + (cid:0) I ⊗ Σ Y Σ + XY (cid:1) ∂ Σ XY ∂ Σ Y = (cid:2) ( I ⊗ Σ Y ) (Σ X ⊗ I ) (cid:0) Σ + ∗ XY ⊗ Σ + XY (cid:1) + T (cid:3) + (cid:0) Σ X Σ + ∗ XY ⊗ I (cid:1) = (cid:2)(cid:0) Σ X Σ + ∗ XY ⊗ Σ Y Σ + XY (cid:1) + T (cid:3) + (cid:0) Σ X Σ + ∗ XY ⊗ I (cid:1) . Theorem 3, part i presents the comparative statics of the solution of the identiﬁcationproblem as the observed cross-covariance matrix varies. In addition, we explore how theestimated surplus technology changes as the characteristics of the matching populationschange. These formulas can be use to evaluate the local stability of the estimated surplustechnology . In particular, given the formulas, one can also easily totally diﬀerentiate achange in the surplus technology into a component due to change in X, a component dueto change in Y, and a component due to change in Σ XY . It can also be used to deriveasymptotic properties, using a standard delta method and the expression of the derivatives(3.18)-(3.20). To analyze the precision of our estimates, one can apply the closed-form

ATCHING IN CLOSED-FORM 25 formulas again to the asymptotic results in Theorem 2 and Corollary 1 of Dupuy andGalichon [14].Theorem 3, part ii explores how the solution of the equilibrium problem varies with achange in the surplus technology A, and the characteristics of the two sides in the market.Without closed-forms, one can study the eﬀect of a change in the surplus technology or thecharacteristics of the matching populations only numerically which has the disadvantage ofbeing computationally intensive and often diﬃcult to interpret the causes of the changes.In contrast within our model, we can derive exactly how the solutions are going to evolveas the structural parameters change, which of both applied and empirical interest. Sim-ilarly to above, we can decompose the overall change in the equilibrium matching into acomponent due to changes in the matching technology and components due to changes inthe characteristics of the matching populations X and Y. As a consequence, this theoremcan be used to easily evaluate the impact of counterfactual experiments.Finally, one can test the quadratic-normal setting through an easy to construct overi-dentiﬁcation test. Given the observed match data, the econometrician can ﬁnd an estimateof the aﬃnity matrix, b A, and then given b A and the marginals predict the optimal match-ing b π XY . Under the hypothesis that the normal-quadratic setting is correct, the distancebetween b π XY and the empirically observed π XY should be small, where the formal test sta-tistics have asymptotic distributions that can be easily found given the comparative staticsfrom above. 4. Conclusion

This paper adopts the framework of Choo and Siow [9] and Galichon and Salani´e [21]and imposes additional functional form and distributional assumptions that allow us toderive closed-form formulas for the surplus and the optimal matching. In this setting, onemay solve easily problems that in the past have been computationally intensive. Moreover,the results allow for the characterization of the optimal solution and how it changes asthe primitives of the model change. The model in this paper is suited to static matchingproblems with exogenously ﬁxed populations in which all participants are matched. Thus, it covers a number of contexts with potential for future applied work, such as matchingof workers with tasks, scheduling, matching between specialists and ﬁrms or top managersand ﬁrms.While our framework is Transferable Utility (TU), this paper’s main argument may becarried without diﬃculty into some instances of the Nontransferable Utility (NTU) frame-work. Indeed, a recent contribution by Menzel (2014) implies that in the case with nosingles and a particular structure of (logit) heterogeneity, the equilibrium matching in theTU and the NTU cases coincide. Hence, the results of the present paper would apply tothat framework as well.

Appendix: Matrix Differentiation

The Kronecker product and the vectorization operation are extremely useful when itcomes to studying asymptotic properties involving matrices. The idea is that matrices m × n , where m stands for the number of rows and n for the number of columns, canbe seen as mn × R mn , and linear operation on such matrices can be seen ashigher order matrices. To do this, the fundamental tool is the vectorization operation,which vectorizes a matrix by stacking its columns. Introduce τ mn a collection of invertiblemaps from { , ..., m } × { , ..., n } onto { , ..., mn } , such that τ mn ( i, j ) = m ( j −

1) + i . Deﬁnition 1.

For ( M ) a m × n matrix, vec ( M ) is the vector v ∈ R mn such that v τ mn ( i,j ) = M ij . Next, we introduce the transposition tensor T m,n as the mn × mn matrix such that T m,n vec ( M ) = vec ( M ∗ ) . The matrix operator T m,n is a permutation matrix with zeros and a single 1 on eachrow and column. Note that T m,n = T − n,m , so T m,n T n,m vec ( M ) = vec ( M ) . Furthermore, T m,n = T ∗ n,m . The next deﬁnition deals with Kronecker product, which is closely related tovectorization. ATCHING IN CLOSED-FORM 27

Deﬁnition 2.

Let A be a m × p matrix and B an n × q matrix. One deﬁnes the Kroneckerproduct A ⊗ B as the mn × pq matrix such that ( A ⊗ B ) n ( i − k,q ( j − l = A ij B kl . The following fundamental property characterizes the Kronecker product.

Fact 1.

For all q × p matrix X , vec (cid:0) BXA T (cid:1) = ( A ⊗ B ) vec ( X )The following important basic properties follow. Fact 2.

Let A be a m × p matrix and B an n × q matrix. Then:1. (Associativity) ( A ⊗ B ) ⊗ C = A ⊗ ( B ⊗ C ) .

2. (Distributivity) A ⊗ ( B + C ) = A ⊗ B + A ⊗ C.

3. (Multilinearity) For λ and µ scalars, λA ⊗ µB = λµ ( A ⊗ B )

4. For matrices of appropriate size, ( A ⊗ B ) ( C ⊗ D ) = ( AC ) ⊗ ( BD ) .5. ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ .6. If A and B are invertible, ( A ⊗ B ) − = A − ⊗ B − .7. For vectors a and b , a ′ ⊗ b = ba ′ (in particular, aa ′ = a ′ ⊗ b ).8. If A and B are square matrices of respective size m and n , det ( A ⊗ B ) = (det A ) m (det B ) n . T r ( A ⊗ B ) = T r ( A ) T r ( B ) .10. rank ( A ⊗ B ) = rank ( A ) rank ( B ) .11. The singular values of A ⊗ B are the product of the singular values of A and thoseof B . Let f be a smooth map from the space of m × p matrices to the space of n × q matrix.Deﬁne df ( A ) dA as the ( nq ) × ( mp ) matrix such that for an m × p matrix X , vec (cid:18) lim e → f ( A + eX ) − f ( A ) e (cid:19) = df ( A ) dA .vec ( X ) . We use the notation A −∗ for ( A ∗ ) − . Fact 3.

Let A be a m × p matrix and B an n × q matrix. Then:1. d ( AXB ) dX = B ∗ ⊗ A .2. dA ∗ dA = T m,p .3. dA − dA = − (cid:0) A −∗ ⊗ A − (cid:1) .4. dA dA = I ⊗ A + A T ⊗ I

5. For A symmetric, dA / dA = (cid:0) I ⊗ A / + A / ⊗ I (cid:1) − . § Department of Economics, ´Ecole polytechnique. Address: Route de Saclay, 91128Palaiseau, France. E-mail: [email protected] † Economics Department and Courant Institute, New York University and EconomicsDepartment, Sciences Po. Address: Department of Economics. 19 W 4th Street, New York,NY 10012, USA. Email: [email protected]

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