Identification of Matching Complementarities: A Geometric Viewpoint
IIDENTIFICATION OF MATCHING COMPLEMENTARITIES: AGEOMETRIC VIEWPOINT
ALFRED GALICHON § Abstract.
We provide a geometric formulation of the problem of identification of thematching surplus function and we show how the estimation problem can be solved by theintroduction of a generalized entropy function over the set of matchings.
Keywords : matching, marriage, assignment.
JEL codes : C78, D61, C13. Setting
We consider the Becker model of the marriage market as a bipartite matching game withtransferable utility. Let X and Y be finite sets of “types” of men and women where |X | = d x and |Y| = d y . Assume that the number of men and women is equal, and that the number ofmen of type x (resp. of women of type y ) is p x (resp. q y ). We normalize the total number ofmen and women to one, that is we set (cid:80) x ∈X p x = 1 and (cid:80) y ∈Y q y = 1. Let Φ xy ≥ Date
Advances inEconometrics, vol. 31: Structural Econometric Models , DOI: https://doi.org/10.1108/S0731-9053(2013)0000032005 . a r X i v : . [ ec on . E M ] F e b ALFRED GALICHON § joint surplus (to be split endogenously across the pair) from matching a man of type x anda woman of type y . For the clarity of exposition we do not allow for unmatched individuals.Recall that under transferable utility, in the Shapley-Shubik model, the stable matchingalso maximizes the total surplus (cid:88) x,y µ xy Φ xy over µ ∈ M the set of matchings , defined by M = (cid:40) µ : µ xy ≥ , (cid:88) y µ xy = p x , (cid:88) x µ xy = q y (cid:41) , where µ xy is interpreted as the number of ( x, y ) pairs, which is allowed to be a fractionalnumber.Note that the equations defining M have d x + d y − M is d x d y − d x − d y + 1 = ( d x −
1) ( d y − µ and ˜ µ are in M , then for t ∈ [0 , tµ + (1 − t ) ˜ µ ) is also in M . Finally, M is obviously bounded in R d x d y . Hence: Claim 1.
The set of matchings M is a compact convex set of R d x d y . Identification
One observes a matching ˆ µ ∈ M and one wonders whether ˆ µ is rationalizable, i.e. whetherthere exists some surplus function Φ such that ˆ µ is the optimal matching in the problemwith surplus Φ, that is ˆ µ ∈ arg max µ ∈M (cid:88) x,y µ xy Φ xy . As it is classically the case in revealed preference analysis, some restrictions on Φ areneeded in order to have a meaningful definition. Indeed, the null surplus function Φ xy = 0always trivially rationalizes any matching; similarly, Φ xy = f x + g y which also rationalizes DENTIFICATION OF MATCHING COMPLEMENTARITIES: A GEOMETRIC VIEWPOINT 3 any µ as the value of the total surplus evaluated at µ is (cid:80) x p x f x + (cid:80) y q y g y irrespective of µ ∈ M . Hence in order to have some empirical bite, we need to imposearg max µ ∈M (cid:88) x,y µ xy ˆΦ xy (cid:54) = M . Let S be the set of Φ such that Φ xy does not coincides with f x + g y for some vectors ( f x )and ( g y ). We shall thus seek Φ in S . The following assertion characterizes Φ in dimensiontwo. Claim 2.
Assume d x = d y = 2 . Then S is the set of (Φ xy ) such that Φ + Φ (cid:54) = Φ + Φ . The previous considerations lead to the following definition:
Definition 1. ˆ µ ∈ M is rationalizable if there is ˆΦ ∈ S such that ˆ µ ∈ arg max µ ∈M (cid:88) x,y µ xy ˆΦ xy . (1)Introducing W the indirect surplus function , defined as W (Φ) = max µ ∈M (cid:104) µ, Φ (cid:105) (2)where the product (cid:104) µ, Φ (cid:105) is defined as (cid:104) µ, Φ (cid:105) = (cid:88) xy µ xy Φ xy , (3)condition (1) is equivalent, by the Envelope theorem, toˆ µ ∈ ∂ W (cid:16) ˆΦ (cid:17) where ∂ W (Φ) denotes the subgradient of W at Φ. See the Appendix for some basic resultson convex analysis. In the terminology of convex analysis, W is the support function of set M , a geometric property which we shall develop in the next paragraph.The following remark is obvious. ALFRED GALICHON § Claim 3. W is positive homogenous of degree one, hence for t > , one has W ( t Φ) = t W (Φ) (4) ∂ W ( t Φ) = ∂ W (Φ) . (5)3. Geometry
The following result provides the geometric interpretation of rationalizability. Formula(1) means that for ˆ µ to be rationalizable, it needs to maximize a linear functional over thecompact convex set M . As it is well known, a necessary and sufficient for this to hold isthat ˆ µ should belong to the boundary of M . Theorem 1.
The following three conditions are equivalent:(i) ˆ µ is rationalizable,(ii) ˆ µ lies on M\M int , the boundary of M ,(iii) There is ˆΦ ∈ S such that ˆ µ ∈ ∂ W (cid:16) ˆΦ (cid:17) . (6)This theorem is illustrated in Figure 1. While the equivalence between part (ii), ofgeometric kind and part (iii), of analytic nature follows from standard convex analysis,the insight of this result is to connect this to the economic notion of rationalizability (i),of revealed preference flavour. This result provides a geometric understanding of revealedpreference analysis in matching models with transferable utility. See Echenique et al. (2012).Geometrically, this means that the matchings that are rationalizable lie on the boundaryof M . We give a very simple example of a ˆ µ which is rationalizable. Example 1.
Assume d x = d y = 2 and consider matrix ˆ µ = then any ˆΦ such that ˆΦ + ˆΦ > ˆΦ + ˆΦ rationalizes ˆ µ . DENTIFICATION OF MATCHING COMPLEMENTARITIES: A GEOMETRIC VIEWPOINT 5
Figure 1.
Geometric view of rationalizability. In order for matching µ tobe rationalized by surplus function Φ, µ need to lie on the geometric frontierof M .We now give a very simple example of a ˆ µ which not is rationalizable, i.e. where ˆ µ is inthe strict interior of M . Example 2.
Assume d x = d y = 2 and consider matrix ˆ µ = . . . . . This matrix is equal to . + 0 . . Hence for a production function Φ ,we get (cid:88) xy ˆ µ xy Φ xy = 0 . + Φ ) + 0 . + Φ ) Hence it cannot be rationalized by a production function Φ unless Φ + Φ = Φ + Φ .But in that case, set a = Φ , b = 0 , a = Φ , and b = Φ − Φ , thus Φ ij = a i + b j –which contradicts Φ ∈ S . Therefore ˆ µ cannot be rationalized. ALFRED GALICHON § Example 3.
As another example, consider p ⊗ q defined by ( p ⊗ q ) xy = p x q y . Clearly, p ⊗ q ∈ M ; intuitively this matching corresponds to matching randomly men and women, sothat the characteristics of the partner are independent. This matching cannot be rationalizedas it lies in the strict interior of M . Indeed, p ⊗ q is the barycenter of the full set M . Entropy
In practice, it is almost never the case that a matching ˆ µ observed in the population isrationalizable. This is understandable using the geometric interpretation provided above:the locus of matchings that are rationalizable being the frontier of a convex set, it is “small”with respect to the set of matchings that are not rationalizable, which is the strict interiorof this same convex set.Mathematically speaking, we are looking for a solution Φ ∈ S satisfyingˆ µ ∈ ∂ W (Φ) . (7)If W was “well behaved,” more precisely if W was strictly convex and continuouslydifferentiable, then the gradient ∇W would exist and be invertible with inverse ∇W ∗ ,where W ∗ is the convex conjugate of W . Then relation (7) would imply Φ = ∇W ∗ (ˆ µ ).But W is not strictly convex, so this approach does not work, and in fact relation (7) hasno solution. Geometrically, it is quite clear why. As remarked above, the image of ∂ W isincluded in the frontier of M , hence if ˆ µ does not lie on the geometric frontier of M , thenrelation (7) cannot possibly have a solution.In order to be able to estimate Φ based on the observation of ˆ µ , most of the literaturefollowing the seminal paper of Choo and Siow (2005) introduce heterogeneities in matchingsurpluses. Without trying to be exhaustive, let us mention Fox (2010, 2011), Galichon andSalani´e (2010, 2012), Decker et al. (2012), Chiappori et al. (2012). As argued in Galichonand Salani´e (2012), this consists in essence in introducing a generalized entropy function I ( µ ) which is strictly convex, and which is such that I ( µ ) = + ∞ if µ / ∈ M , DENTIFICATION OF MATCHING COMPLEMENTARITIES: A GEOMETRIC VIEWPOINT 7 such that I is differentiable on M int the interior of M , with, for all µ ∈ M int , ∇I ( µ ) ∈ S , and such ˆΦ is identified by ˆΦ = ∇I (ˆ µ ) . (8)Noting that (8) is the first order condition to the following optimization program W I (Φ) = max µ ∈M (cid:104) µ, Φ (cid:105) − I ( µ ) (9)which, as argued in Galichon and Salani´e (2010, 2012), can be interpreted in some cases asthe social welfare of a matching model with unobserved heterogeneity. Example 4.
Recall the definition ( p ⊗ q ) xy = p x p y , and remember that p ⊗ q is never onthe frontier of M , hence never rationalizable. When ˆ µ is not rationalizable either, one mayconsider the smallest t such that p ⊗ q + t (ˆ µ − p ⊗ q ) is rationalizable. This number existsand is finite because the halfline which starts from p ⊗ q through ˆ µ must cross the frontierof M , which is a convex and compact set. Letting t ∗ be the corresponding value of t , and µ ∗ = p ⊗ q + t ∗ (ˆ µ − p ⊗ q ) , there exists by definition an element ˆΦ ∈ S \ { } such that µ ∗ ∈ ∂ W (cid:16) ˆΦ (cid:17) , where W is as in (2). Note that if ˆ µ is rationalizable, then t ∗ = 1 and µ ∗ = µ . See Figure 2.This construction can be expressed in terms of I . Letting I (ˆ µ ) = − max t ≥ { t : p ⊗ q + t (ˆ µ − p ⊗ q ) ∈ M} if ˆ µ ∈ M (10)= + ∞ else (11) so that I (ˆ µ ) can be formulated as a max-min problem, that is, for ˆ µ ∈ M , I (ˆ µ ) = − max t ≥ min Φ ∈ S { t + W (Φ) − (cid:104) Φ , p ⊗ q + t (ˆ µ − p ⊗ q ) (cid:105)} . Because the objective function is convex in Φ and linear in t , this problem has a saddle-point which will be denoted (Φ ∗ , t ∗ ) . Let µ ∗ = p ⊗ q + t ∗ (ˆ µ − p ⊗ q ) . By optimality withrespect to Φ , µ ∗ ∈ ∂ W (Φ ∗ ) , thus Φ ∗ rationalizes µ ∗ . By the envelope theorem t ∗ Φ ∗ = ∇I (ˆ µ ) , ALFRED GALICHON § thus we take ˆΦ = t ∗ Φ ∗ . and µ ∗ is the matching which is on the halfline which starts from p ⊗ q through ˆ µ andwhich is rationalizable. Figure 2.
Geometric illustration of Example 4. ˆ µ is not rationalizable, butit is associated to some proximate µ ∗ on the boundary of M , which is itselfrationalized by ˆΦ. Example 5.
In the Choo and Siow (2005) model, the surplus function is Φ ij = Φ ( x, y ) + ε iy + η jx where ε iy and η jx are iid extreme value type I random variables. Choo and Siowuse this model nonparametrically identifies Φ . Galichon and Salani´e (2010) show that thismodel leads to the following specification of I : I ( µ ) = (cid:88) xy µ xy log µ xy if µ ∈ M (12)= + ∞ else. DENTIFICATION OF MATCHING COMPLEMENTARITIES: A GEOMETRIC VIEWPOINT 9
Example 6.
Galichon and Salani´e (2012) argue that the model of Choo and Siow actuallyextends in the case where the matching surplus function in the presence of heterogeneitiesbetween man i of type x and woman j of type y is Φ ij = Φ ( x, y ) + ε ixy + η jxy , and letting G x ( U ) = E [max y ( U xy + ε ixy )] and H y ( V ) = E [max x ( V xy + η jxy )] be the ex-ante indirectutilities of respectively the man of type x and the woman of type y , and letting G ∗ and H ∗ their respective convex conjugate transforms, that is G ∗ x (cid:0) µ . | x (cid:1) = sup U xy { (cid:88) y µ y | x U xy − G x ( U ) } if (cid:88) y µ y | x = 1= + ∞ else,and H ∗ y (cid:0) µ . | y (cid:1) = sup V xy { (cid:88) x µ x | y V xy − H y ( V ) } if (cid:88) x µ x | y = 1= + ∞ else.Then I ( µ ) is given by I ( µ ) = (cid:88) x p x G ∗ x (cid:0) µ . | x (cid:1) + (cid:88) y q y H ∗ y (cid:0) µ . | y (cid:1) . (13) which coincides with (12) in the case studied by Choo and Siow, hence the term “generalizedentropy”. As an important consequence, this paves the way to the continuous generalizationof the Choo and Siow model. See Dupuy and Galichon (2012), and Bojilov and Galichon(2013). Example 7.
Applying this setting, Galichon and Salani´e (2012, Example 3) assume that X and Y are finite subsets of R , and that ε ixy = e i y while η jxy = f j x where e i and f j aredrawn from U ([0 , distributions. In this case the utility shocks are perfectly correlatedacross alternatives, in sharp contrast with Example 1, where they are independent. Then,letting Q µY | X = x be the conditional quantile of Y conditional on X = x under distribution µ ,one has I ( µ ) = (cid:88) x p x (cid:90) Q µY | X = x ( t ) tdt + (cid:88) y q y (cid:90) Q µX | Y = y ( t ) tdt. §§ Sciences Po Paris, Department of Economics, Address: 28 rue des Saint-P`eres, 75007Paris, France. E-mail: [email protected].
Facts from Convex Analysis
The definitions below are included for completeness and the reader is referred to Ekelandand Temam (1976) for a thorough exposition of the topic.Take any set Y ⊂ R d ; then the convex hull of Y is the set of points in R d that are convexcombinations of points in Y . We usually focus on its closure, the closed convex hull, denoted cch ( Y ).The support function S Y of Y is defined as S Y ( x ) = sup y ∈ Y x · y for any x in Y , where x · y denotes the standard scalar product. It is a convex function,and it is homogeneous of degree one. Moreover, S Y = S cch ( Y ) where cch ( Y ) is the closedconvex hull of Y , and ∂S Y (0) = cch ( Y ).A point in Y is an boundary point if it belongs in the closure of Y , but not in its interior.Now let u be a convex, continuous function defined on R d . Then the gradient ∇ u of u iswell-defined almost everywhere and locally bounded. If u is differentiable at x , then u (cid:0) x (cid:48) (cid:1) ≥ u ( x ) + ∇ u ( x ) · ( x (cid:48) − x )for all x (cid:48) ∈ R d . Moreover, if u is also differentiable at x (cid:48) , then (cid:0) ∇ u ( x ) − ∇ u (cid:0) x (cid:48) (cid:1)(cid:1) · (cid:0) x − x (cid:48) (cid:1) ≥ . When u is not differentiable in x , it is still subdifferentiable in the following sense. Wedefine ∂u ( x ) as ∂u ( x ) = (cid:110) y ∈ R d : ∀ x (cid:48) ∈ R d , u (cid:0) x (cid:48) (cid:1) ≥ u ( x ) + y · ( x (cid:48) − x ) (cid:111) . Then ∂u ( x ) is not empty, and it reduces to a single element if and only if u is differentiableat x ; in that case ∂u ( x ) = {∇ u ( x ) } . DENTIFICATION OF MATCHING COMPLEMENTARITIES: A GEOMETRIC VIEWPOINT 11
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