The Econometrics and Some Properties of Separable Matching Models
aa r X i v : . [ ec on . E M ] F e b The Econometrics and Some Properties of Separable Matching ModelsAlfred Galichon ∗ Bernard Salani´e † December 28, 2016
Abstract
We present a class of one-to-one matching models with perfectly transferable utility.We discuss identification and inference in these separable models, and we show howtheir comparative statics are readily analyzed.
Keywords : sorting, matching, marriage market, gross substitutes.
JEL Classification : D3, J21, J23 and J31.
Eugene Choo and Aloysius Siow’s (2006) contribution has renewed interest in empiricalapplications of matching with perfectly transferable utility (TU). Unobserved heterogeneityin joint surplus is a paramount consideration in the specification of these models. Chooand Siow chose a separable multilogit model, which leads to highly tractable formulæ. Butunobserved heterogeneity could originate from variation in tastes, from division of laborwithin the partners, and other sources. It is therefore important to allow for flexibility in ∗ Department of Economics, FAS, and Department of Mathematics, Courant Institute, New York Univer-sity. Email: [email protected]. † Department of Economics, Columbia University. Email: [email protected]. separability of the joint surplus.These separable models have a nicely convex and (usually) smooth structure that generatesvery useful econometric and analytic properties.We start by summarizing our main results concerning identification and inference inseparable models of one-to-one matching under TU . We then show how in models withseparable heterogeneity and full support, we can use the implicit function theorem andmatrix algebra to get explict formulæ for any small change in the primitives of the model:arrival or departure of a mass of individuals of a given type, or changes in joint surplus.We illustrate the usefulness of our formulæ on a simple example. In this paper we will call “men” and “women” the agents on both sides of the market,as is traditional; but our results apply more generally than in this implicit heterosexualmarriage market `a la Becker. We assume that agents on both sides of the market belong tocontinuous sets I and J , which are partitioned into finite sets of types. A man i ∈ I has atype x i ∈ X and a woman j ∈ J has a type y j ∈ Y , where X and Y are finite. The mass ofmen of type x (resp. women of type y ) is n x (resp. m y ). The distinction between types andidentities is data-driven: while participants on the market are assumed to operate underperfect information, the analyst only observes the types x and y . We also assume that jointsurplus is separable : ˜Φ ij = Φ x i y j + ε iy j + η jx i . Separability excludes interactions between unobserved characteristics of i and j conditionalon observed types ( x, y ). As an example, let types describe education, as in Pierre–Andr´e Galichon–Salani´e (2016) has detailed arguments, along with somehat weaker assumptions than we usehere. i and j match then the man receives utility U x i y j + ε iy j and the woman receives utility V x i y j + η jx ,where the terms U = ( U xy ) and V = ( V xy ) are endogenously determined at equilibrium sothat U xy + V xy = Φ xy . A single man i receives utility ε i , while a single woman j receives η j . The interpretation of this result is simple: the ε iy of man i has the same value for llwomen of type y , and since there is a continuum of them they will compete for it until the“price” of man i fully incorporates it.We now denote X = X ∪{ } and Y = Y ∪{ } . We shall assume that the random vector ε x = ( ε iy ) ∈ R Y is distributed as P x identically and independently across the populationof men i of type x ; and we introduce Q y in the same way for women. In this note we willalso impose full support : for each x ∈ X , P x has a nonvanishing density on R Y , and foreach y ∈ Y , Q y has a nonvanishing density on R X . When P x and Q y are Gumbel distributions for all x and y , the model boils down to themodel of Choo and Siow (2006). More generally, Galichon and Salani´e (2016) introduce theconvex functions G x ( U ) = E P x (cid:20) max y ∈Y { U xy + ε y , ε } (cid:21) and H y ( V ) = E Q y (cid:20) max x ∈X { V xy + η x , η } (cid:21) , and G ( U ) = X x ∈X n x G x ( U ) and H ( V ) = X y ∈Y m y H y ( V ) . U minimizes the expression G ( U ) + H ( Φ − U ). Under separability and full support, the functions G and H are strictly convexand twice differentiable, and the first-order conditions characterize the unique equilibrium: ∇ G ( U ) = ∇ H ( Φ − U ) . (3.1)These conditions are easily interpreted. By the Daly-Zachary-Williams theorem, the massof men of type x wishing to match with women of type y ∈ Y given a vector U is U is µ xy = ∂G ( U ) /∂U xy . Similarly, the number of women of type y wishing to match withmen of type x ∈ X is µ xy = ∂H ( V ) /∂V xy . In equilibrium, the two quantities ∇ G ( U ) and ∇ H ( V ) must coincide; and since U + V = Φ , U is determined in equilibrium by (3.1). Alsonote that the expected utility of the average man of type x is u x = G x ( U ) in equilibrium.Galichon–Salani´e (2016, section 5) details several approaches to computing the equilibriumefficiently. The convexity and smoothness of the problem make it very tractable numerically. Convex duality is the key to the approach in Galichon and Salani´e (2016). Remember thatgiven any function f ( a ), its Legendre–Fenchel transform is the function f ∗ such that f ∗ ( b ) = sup a ( a · b − f ( b )) . The function f ∗ may be badly-behaved: it may take infinite values, for instance. But since f ∗ is the supremum of linear functions of b , it is convex. And if f is convex, it is theLegendre–Fenchel transform of f ∗ ; and if f and f ∗ are strictly convex, then b = ∇ f ( a ) iff a = ∇ f ∗ ( b ) . Let us first apply this “convex inversion formula” to the strictly convex function f = G : µ = ∇ G ( U ) iff U = ∇ G ∗ ( µ ) . Given a full specification for the distributions P x , the function G can be computed, and itsLegendre–Fenchel transform too. Feeding the observed matching patterns into U = ∇ G ∗ ( µ )4irectly identifies U . Proceeding in the same way with f ( U ) = H ( Φ − U ) identifies Φ − U = ∇ H ∗ ( µ ); and adding up, Φ = ∇ G ∗ ( µ ) + ∇ H ∗ ( µ ) , which identifies the joint surplus Φ from the (assumed) knowledge of the distributions P x and Q y . This is “conditional unrestricted identification”: the joint surplus is identified withoutany prior restriction if the analyst somehow knows the distribution of unobserved hetero-geneity. If for instance these distributions are only assumed to be known up to scale, thenin order to achieve point identification of the joint surplus the analyst will need to imposerestrictions on it. There is an unavoidable trade off here, which can be alleviated by poolingdata from several markets and assuming some common features across markets .Once identification is achieved, inference is straightforward. It can be based directly onthe equations above, or proceed via maximum likelihood, or by matching moments of somebasis functions. The latter method is based on a linear expansionΦ xy ( λ ) = λ · φ xy where φ is a vector of basis functions . Galichon and Salani´e (2016) show that findingthe parameter vector λ that matches the observed comoments ˆ C = ˆ E φ gives a consistentestimator. The separable structure of the problem naturally generates a number of comparative staticsresults that extend those obtained by Colin Decker et al. (2012) and Bryan Graham (2013)for the Choo and Siow model. Our assumptions on the unobserved heterogeneity yieldenough smoothness and convexity that simple formulæ can be obtained. Chiappori, Salani´e and Weiss (2016) gives an example, with an heteroskedastic version of the Choo andSiow model. For instance, a simple “assortative matching basis function” would be 11( x = y ). G and H are not only twice differentiable and strictly convex: they arealso submodular. The economic interpretation is straightforward. Given differentiability,the submodularity of G requires that ∂G /∂U xy ∂U x ′ y ′ ≤ x ′ , y ′ ) = ( x, y ). But since µ xy = ∂G x ( U ) /∂U xy , this simply says that ∂µ xy /∂U xy ′ ≤
0: if alternative y ′ becomes moreattractive, alternative y will be less demanded at equilibrium. This is, of course, a grosssubstitutes property.To state our results, we need some more notation: • we define matching ratios by µ xy = µ My | x n x = µ Wx | y m y ; note that P y µ My | x = 1 − µ M | x and P x µ Wx | y = 1 − µ W | y . • we denote T = (cid:0) D G ( U ) + D H ( Φ − U ) (cid:1) − the inverse of the sum of the Hessiansof G and H at the equilibrium U (the sum is invertible since G and H are strictlyconvex.) • We use specific notation for some of its blocks; for instance, we denote T x · , · y the matrix A with elements A tz = T xt,zy . 6 .1 General results for separable models The primitives of the model are θ = ( n , m , Φ ). The equilibrium U is determined by ∇ G ( U ) = ∇ H ( Φ − U ). Taking differentials, for all x and y we have (cid:8) D G ( U ) + D H ( Φ − U ) (cid:9) d U = ∂ H ( Φ − U ) ∂ V ∂ m d m − ∂ G ( U ) ∂ U ∂ n d n + D H ( Φ − U ) d Φ . (5.1)Given strict convexity, the Hessians are negative definite, and the matrix D G ( U ) + D H ( Φ − U ) is invertible. Therefore we can write d U = T R d θ , (5.2)where R d θ denotes the right-hand side of (5.1). Now since both G and H are submodularand strictly convex, D G and D H are Stieltjes matrices , and so is their sum. By aclassical result on Stieltjes matrices (see e.g. Golub and Van Loan 2013, lemma 11.5.1), allentries of T are nonnegative; and any change in θ such that R d θ is a non-negative vectorcan only increase the equilibrium U xy . Moreover, the average welfare of men of type x ∈ X is given by u x = G x ( U ), and du x = X y ∈Y ∂G x ∂U xy dU xy = X y ∈Y µ My | x dU xy ;so that any such change R d θ ≥ n and m yields very simple formulæ : ∂u x ∂n x ′ = µ M ·| x ′ T x · ,x ′ · µ M ·| x ′ ≤ ∂u x ∂m y ′ = µ M ·| x ′ T x · , · y ′ µ W ·| y ′ ≥ . (5.3)The signs of the entries is a direct consequence of the non-negativity of all elements of T ;it was already known, but now we can easily compute the value of these local effects. In That is, they are positive definite with non-positive off-diagonal terms. The online appendix has the detail of these calculations. ∂u x ∂ Φ x ′ y ′ = µ xy ′ T xy ′ , · y ′ ∂ H y ′ ∂V x ′ y ′ ∂V · y ′ . Since H y ′ is strictly convex and is submodular, the vector of second derivatives in thisexpression has one positive term, while all others are non-positive. Given the non-negativityof all elements of T , an increase in any element Φ x ′ y ′ of the joint surplus should reduce(resp. increase) the expected utility of men whom women of type y ′ see as good (resp. bad)substitutes of type x ′ . These effects are larger for the men who are more likely to marrywomen of type y ′ .More generally, for any small change in the primitives of the model, we recover du x = P y µ My | x dU xy from the solution of the system n x X t ∂ G x ∂U xy ∂U xt dU xt + m y X z ∂ H y ∂V xy ∂V zy dU zy = µ xy d log m y n x + m y X z ∂ H y ∂V xy ∂V zy d Φ zy . (5.4)While G x and H y are functions of U , using the Legendre-Fenchel transform we have µ = ∇ G ∗ ( µ ) + H ∗ ( µ ). Hence all of the elements of (5.4) can be computed from the observeddata, given a structure ( Φ , n , m ) . For a drastically simple illustration, suppose that there is only one type of men and one typeof women: | X | = | Y | = 1 . We simplify the notation by dropping the “1” subscripts, so thatΦ denotes Φ for instance. Equilibrium in this model consists in a number of marriages µ ,and associated expected utilities u and v .Now G ( U ) = nE P max( U + ε, ε ). Let us denote ( F P , f P ) the cdf and pdf of ( ε − ε )under P ; and define k P ( t ) = f P ( F − P ( t )). Then G ′ ( U ) = nF P ( U ) and G ′′ ( U ) = nf P ( U ) . Using similar notation for Q , the equilibrium U and the number of marriages µ are givenby µ = nF P ( U ) = mF Q (Φ − U ). Identification is straightforward: given P and Q , solving8hese equations for Φ gives Φ = F − P (cid:16) µn (cid:17) + F − Q (cid:16) µm (cid:17) . Moving to comparative statics, (5.4) becomes dU = T (cid:16) µd log mn + mk Q (cid:16) µm (cid:17) d Φ (cid:17) with T = 1 /S and S = nk P ( µ/n ) + mk Q ( µ/m ). Since du = ( µ/n ) dU , the change in theexpected utilities of the average man follows directly, and so does the change in the numberof marriages since dµ = F P ( U ) dn + nf P ( U ) dU : du = T µn (cid:16) µd log mn + mk Q d Φ (cid:17) (5.5) dµ = T ( µ ( mk Q d log n + nk P d log m ) + nmk P k Q d Φ) . (5.6)Take a small change ( dn, dm ) in the sizes of the populations of men and of women. 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