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The Econometrics and Some Properties of Separable Matching Models
{ Keywords: sorting, matching, marriage market, gross substitutes.}
{ JEL Classification: D3, J21, J23 and J31.\vskip50pt }
Eugene Choo and Aloysius Siow's (2006) contribution has renewed interest in empirical applications of matching with perfectly transferable utility (TU). Unobserved heterogeneity in joint surplus is a paramount consideration in the specification of these models. Choo and Siow chose a separable multilogit model, which leads to highly tractable formul\ae. But unobserved heterogeneity could originate from variation in tastes, from division of labor within the partners, and other sources. It is therefore important to allow for flexibility in the stochastic specification of the joint surplus. Alfred Galichon and Bernard Salani\'{e} (2016) have explored a general class of models of bilateral matching which sets very few constraints on the distributions of unobserved heterogeneity beyond {\em separability\/} of the joint surplus. These separable models have a nicely convex and (usually) smooth structure that generates very useful econometric and analytic properties.
We start by summarizing our main results concerning identification and inference in separable models of one-to-one matching under TU\footnote{Galichon--Salani\'e (2016) has detailed arguments, along with somehat weaker assumptions than we use here.}. We then show how in models with separable heterogeneity and full support, we can use the implicit function theorem and matrix algebra to get explict formul\ae\ 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 formul\ae\ 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 heterosexual marriage market \`a la Becker. We assume that agents on both sides of the market belong to continuous sets $\mathcal{I}$ and $\mathcal{J}$, which are partitioned into finite sets of types. A man $i\in \mathcal{I}$ has a type $x_i\in \mathcal{X}$ and a woman $j\in\mathcal{J}$ has a type $y_j\in\mathcal{Y}$, where $\mathcal{X}$ and $\mathcal{Y}$ are finite. The mass of men of type $x$ (resp.\ women of type $y$) is $n_x$ (resp.\ $m_y$). The distinction between types and identities is data-driven: while participants on the market are assumed to operate under perfect information, the analyst only observes the types $x$ and $y$. We also assume that joint surplus is {\em separable}: \[ \tilde{\Phi}_{ij}=\Phi_{x_i y_j}+{\Greekmath 0122}_{iy_j}+{\Greekmath 0111}_{jx_i}. \] Separability excludes interactions between unobserved characteristics of $i$ and $j$ conditional on observed types $(x,y)$. As an example, let types describe education, as in Pierre--Andr\'e Chiappori, Salani\'e, and Yoram Weiss (2016). Then separability does allow for unlimited unobserved heterogeneity in the way more-educated men value the education of their partners for instance; it rules out considerations like matching on physical characteristics, which certainly exists but may not be that relevant for the study of some economic questions at least.
We know from Chiappori-Salani\'e-Weiss (2016) and Galichon--Salani\'e (2016) that if $i$ and $j$ match then the man receives utility $ U_{x_iy_j}+{\Greekmath 0122} _{iy_j}$ and the woman receives utility $V_{x_iy_j}+{\Greekmath 0111} _{jx}$, where the terms $\bm{U}=(U_{xy})$ and $\bm{V}=(V_{xy})$ are endogenously determined at equilibrium so that $U_{xy}+V_{xy}=\Phi _{xy}$. A single man $i$ receives utility ${\Greekmath 0122} _{i0}$, while a single woman $j$ receives ${\Greekmath 0111} _{j0}$. The interpretation of this result is simple: the ${\Greekmath 0122}_{iy}$ of man $i$ has the same value for ll women 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 $\mathcal{X}_0=\mathcal{X} \cup \{0\}$ and $\mathcal{Y}_0=\mathcal{Y} \cup \{0\}$. We shall assume that the random vector $\bm{{\Greekmath 0122}}_x =({\Greekmath 0122}_{iy})\in \mathbb{R}^{\mathcal{Y} _{0}}$ is distributed as $\mathbf{P}_{x}$ identically and independently across the population of men $i$ of type $x$; and we introduce $\mathbf{Q}_{y}$ in the same way for women. In this note we will also impose {\em full support}: for each $ x\in \mathcal{X}$, $\mathbf{P}_{x}$ has a nonvanishing density on $\mathbb{R} ^{\mathcal{Y}_{0}}$, and for each $y\in \mathcal{Y}$, $\mathbf{Q}_{y}$ has a nonvanishing density on $\mathbb{R}^{\mathcal{X}_{0}}$.
When $\mathbf{P} _{x}$ and $\mathbf{Q}_{y}$ are Gumbel distributions for all $x$ and $y$, the model boils down to the model of Choo and Siow (2006). More generally, Galichon and Salani\'{e} (2016) introduce the convex functions
and
Galichon and Salani\'{e} (2016, Theorem 2) show that $\bm{U}$ minimizes the expression $G\left(\bm{U}\right) +H\left(\bm{\Phi} -\bm{U}\right)$. Under separability and full support, the functions $G$ and $H$ are strictly convex and twice differentiable, and the first-order conditions characterize the unique equilibrium:
These conditions are easily interpreted. By the Daly-Zachary-Williams theorem, the mass of men of type $x$ wishing to match with women of type $y\in \mathcal{Y} $ given a vector $\bm{U}$ is $\bm{U}$ is ${\Greekmath 0116} _{xy}=\partial G\left(\bm{U}\right)/\partial U_{xy}$. Similarly, the number of women of type $y$ wishing to match with men of type $x\in \mathcal{X}$ is ${\Greekmath 0116}_{xy} =\partial H\left(\bm{V}\right)/\partial V_{xy}$. In equilibrium, the two quantities ${\Greekmath 0272} G\left(\bm{U}\right)$ and ${\Greekmath 0272} H\left(\bm{V}\right)$ must coincide; and since $\bm{U}+\bm{V}=\bm{\Phi}$, $\bm{U}$ is determined in equilibrium by (ref). Also note that the expected utility of the average man of type $x$ is $u_x=G_x(\bm{U})$ in equilibrium. Galichon--Salani\'e (2016, section 5) details several approaches to computing the equilibrium efficiently. 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 that given any function $f(a)$, its Legendre--Fenchel transform is the function $f^\ast$ such that \[ f^\ast(b)=\sup_a \left(a\cdot b-f(b)\right). \] The function $f^\ast$ may be badly-behaved: it may take infinite values, for instance. But since $f^\ast$ is the supremum of linear functions of $b$, it is convex. And if $f$ is convex, it is the Legendre--Fenchel transform of $f^\ast$; and if $f$ and $f^\ast$ are strictly convex, then \[ b={\Greekmath 0272} f(a) \; \mbox{ iff } \; a={\Greekmath 0272} f^\ast(b). \] Let us first apply this “convex inversion formula” to the strictly convex function $f=G$: \[ \bm{{\Greekmath 0116}}={\Greekmath 0272} G(\bm{U}) \; \mbox{ iff } \; \bm{U}={\Greekmath 0272} G^\ast(\bm{{\Greekmath 0116}}). \] Given a full specification for the distributions $\mathbf{P}_x$, the function $G$ can be computed, and its Legendre--Fenchel transform too. Feeding the observed matching patterns into $\bm{U}={\Greekmath 0272} G^\ast(\bm{{\Greekmath 0116}})$ directly identifies $\bm{U}$. Proceeding in the same way with $f(\bm{U})=H(\bm{\Phi}-\bm{U})$ identifies $\bm{\Phi}-\bm{U}={\Greekmath 0272} H^\ast(\bm{{\Greekmath 0116}})$; and adding up, \[ \bm{\Phi}={\Greekmath 0272} G^\ast(\bm{{\Greekmath 0116}})+{\Greekmath 0272} H^\ast(\bm{{\Greekmath 0116}}), \] which identifies the joint surplus $\bm{\Phi}$ from the (assumed) knowledge of the distributions $\mathbf{P}_x$ and $\mathbf{Q}_y.$
This is “conditional unrestricted identification”: the joint surplus is identified without any prior restriction if the analyst somehow knows the distribution of unobserved heterogeneity. If for instance these distributions are only assumed to be known up to scale, then in order to achieve point identification of the joint surplus the analyst will need to impose restrictions on it. There is an unavoidable trade off here, which can be alleviated by pooling data from several markets and assuming some common features across markets\footnote{Chiappori, Salani\'e and Weiss (2016) gives an example, with an heteroskedastic version of the Choo and Siow model.}.
Once identification is achieved, inference is straightforward. It can be based directly on the equations above, or proceed via maximum likelihood, or by matching moments of some basis functions. The latter method is based on a linear expansion \[ \Phi_{xy}(\bm{{\Greekmath 0115}})={\Greekmath 0115}\cdot \bm{{\Greekmath 011E}}_{xy} \] where $\bm{{\Greekmath 011E}}$ is a vector of basis functions\footnote{For instance, a simple “assortative matching basis function” would be ${\rm 1\kern-.40em 1}(x=y)$.}. Galichon and Salani\'e (2016) show that finding the parameter vector $\bm{{\Greekmath 0115}}$ that matches the observed {\em comoments\/} $\hat{\bm{C}}=\hat{E} \bm{{\Greekmath 011E}}$ gives a consistent estimator.
The separable structure of the problem naturally generates a number of comparative statics results 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 yield enough smoothness and convexity that simple formul\ae\ can be obtained.
Take a well-known result: in two-sided matching models, the arrival of newcomers on one side of the market hurts all participants on the same side of the market, and benefits all participants on the opposite side of the market. This was proved by Alexander Kelso and Vincent Crawford (1982, Theorem 5) for a many-to-one matching model under a gross substitutes assumption; by David Gale and Marilda Sotomayor (1985, Theorem 2) for the NTU marriage model; and by Gabrielle Demange and Gale (1985, Corollary 3) for a general class of one-to-one models with transfers. But all of these proofs are purely qualitative. With separable models, it is easy to make these results quantitative, and more generally to analyze the effects of small changes in the primitives.
The functions $G$ and $H$ are not only twice differentiable and strictly convex: they are also submodular. The economic interpretation is straightforward. Given differentiability, the submodularity of $G$ requires that $\partial G^{2}/\partial U_{xy}\partial U_{x^\prime y^{\prime }}\leq 0$ for all $(x^\prime,y^{\prime})\neq (x,y)$. But since ${\Greekmath 0116} _{xy}=\partial G_x\left( \bm{U}\right)/\partial U_{xy}$, this simply says that $\partial {\Greekmath 0116} _{xy}/\partial U_{xy^{\prime}}\leq 0$: if alternative $y^{\prime}$ becomes more attractive, alternative $y$ will be less demanded at equilibrium. This is, of course, a gross substitutes property.
To state our results, we need some more notation:
The primitives of the model are $\bm{{\Greekmath 0112}}=(\bm{n},\bm{m},\bm{\Phi})$. The equilibrium $\bm{U}$ is determined by ${\Greekmath 0272} G\left(\bm{U}\right) ={\Greekmath 0272} H\left(\bm{\Phi} -\bm{U}\right)$. Taking differentials, for all $x$ and $y$ we have
Given strict convexity, the Hessians are negative definite, and the matrix $D^{2}G\left(\bm{U}\right) +D^{2}H\left(\bm{\Phi}-\bm{U}\right)$ is invertible. Therefore we can write
where $\bm{R}d\bm{{\Greekmath 0112}}$ denotes the right-hand side of (ref). Now since both $G$ and $H$ are submodular and strictly convex, $D^{2}G$ and $D^{2}H$ are Stieltjes matrices\footnote{That is, they are positive definite with non-positive off-diagonal terms.}, and so is their sum. By a classical result on Stieltjes matrices (see e.g. Golub and Van Loan 2013, lemma 11.5.1), all entries of $\bm{T}$ are nonnegative; and any change in $\bm{{\Greekmath 0112}}$ such that $\bm{R}d\bm{{\Greekmath 0112}}$ is a non-negative vector can only increase the equilibrium $U_{xy}$. Moreover, the average welfare of men of type $x\in \mathcal{X}$ is given by $ u_x=G_{x}\left(\bm{U}\right)$, and \[ du_x=\sum_{y\in \mathcal{Y}}\frac{\partial G_{x}}{\partial U_{xy}}dU_{xy}=\sum_{y\in \mathcal{Y }}{\Greekmath 0116}^M_{y\vert x}dU_{xy}; \] so that any such change $\bm{R}d\bm{{\Greekmath 0112}}\geq 0$ can only increase the average expected utilities of men of any type.
Applying this to small changes in population sizes $\bm{n}$ and $\bm{m}$ yields very simple formul\ae\footnote{The online appendix has the detail of these calculations.}:
The signs of the entries is a direct consequence of the non-negativity of all elements of $\bm{T}$; it was already known, but now we can easily compute the value of these local effects. In addition, it is easy to prove that \[ \frac{\partial u_x}{\partial \Phi_{x^\prime y^\prime}} ={\Greekmath 0116}_{xy^\prime}\bm{T}_{xy^\prime,\cdot y^\prime} \frac{\partial^2 H_{y^\prime}}{\partial V_{x^\prime y^\prime}\partial V_{\cdot y^\prime}}. \] Since $H_{y^\prime}$ is strictly convex and is submodular, the vector of second derivatives in this expression has one positive term, while all others are non-positive. Given the non-negativity of all elements of $\bm{T}$, an increase in any element $\Phi_{x^\prime y^\prime}$ of the joint surplus should reduce (resp.\ increase) the expected utility of men whom women of type $y^\prime$ see as good (resp.\ bad) substitutes of type $x^\prime$. These effects are larger for the men who are more likely to marry women of type $y^\prime$.
More generally, for any small change in the primitives of the model, we recover $du_x=\sum_y {\Greekmath 0116}^M_{y\vert x} dU_{xy}$ from the solution of the system
While $G_x$ and $H_y$ are functions of $\bm{U}$, using the Legendre-Fenchel transform we have $\bm{{\Greekmath 0116}}={\Greekmath 0272} G^\ast(\bm{{\Greekmath 0116}})+H^\ast(\bm{{\Greekmath 0116}})$. Hence all of the elements of (ref) can be computed from the observed data, given a structure $(\bm{\Phi},\bm{n},\bm{m}).$
For a drastically simple illustration, suppose that there is only one type of men and one type of women: $\lvertX\rvert=\lvertY\rvert=1.$ We simplify the notation by dropping the “1” subscripts, so that $\Phi$ denotes $\Phi_{11}$ for instance. Equilibrium in this model consists in a number of marriages ${\Greekmath 0116}$, and associated expected utilities $u$ and $v$.
Now $G(U)=nE_{\mathbf{P}} \max(U+{\Greekmath 0122},{\Greekmath 0122}_0)$. Let us denote $(F_P, f_P)$ the cdf and pdf of $({\Greekmath 0122}_0-{\Greekmath 0122})$ under $\mathbf{P}$; and define $k_P(t)=f_P(F_P^{-1}(t))$. Then $G^\prime(U)=nF_P(U)$ and $G^{\prime\prime}(U)=nf_P(U).$ Using similar notation for $\mathbf{Q}$, the equilibrium $U$ and the number of marriages ${\Greekmath 0116}$ are given by ${\Greekmath 0116}=nF_P(U)=mF_Q(\Phi-U)$. Identification is straightforward: given $\mathbf{P}$ and $\mathbf{Q}$, solving these equations for $\Phi$ gives \[ \Phi=F_P^{-1}\left(\frac{{\Greekmath 0116}}{n}\right)+F_Q^{-1}\left(\frac{{\Greekmath 0116}}{m}\right). \] Moving to comparative statics, (ref) becomes \[ dU=T\left({\Greekmath 0116} d\log\frac{m}{n}+m k_Q\left(\frac{{\Greekmath 0116}}{m}\right)d\Phi\right) \] with $T=1/S$ and $S=n k_P({\Greekmath 0116}/n)+m k_Q({\Greekmath 0116}/m)$. Since $du=({\Greekmath 0116}/n)dU$, the change in the expected utilities of the average man follows directly, and so does the change in the number of marriages since $d{\Greekmath 0116}=F_P(U) dn+nf_P(U)dU:$
Take a small change $(dn,dm)$ in the sizes of the populations of men and of women. The resulting log-change $d\log{\Greekmath 0116}$ in the number of marriages will be a weighted average of the log-changes in $n$ and in $m$. More interestingly,
{\em the changes in expected utilities of men and women directly reflect the change in the sex ratio $n/m$; and so do the changes in the percentage of singles in each gender.}
If only the joint surplus of each marriage changes, by $d\Phi$, then
{\em the number of marriages ${\Greekmath 0116}$ changes by a fraction $0<s<1$ of $d\Phi/2$.}
Assume moreover $({\Greekmath 0122}_0-{\Greekmath 0122})$ and $({\Greekmath 0111}_0-{\Greekmath 0111})$ have the same distribution, with cdf $F$ and pdf $f$; and let it be symmetric around 0 and log-concave. Then $0<U<\Phi/2$ and $v>u>0$ if the sex ratio is unfavorable to men, $n<m$. Log-convavity gives us $k_P({\Greekmath 0116}/n)<k_Q({\Greekmath 0116}/m)$, so that $nk_P< mk_Q$ and $Tnk_P<1/2<Tmk_Q.$ Therefore
{\em the number of marriages is more elastic to the size of the smaller population.}
\ \\
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Choo, E., and A. Siow (2006): \textquotedblleft Who Marries Whom and Why,\textquotedblright\ Journal of Political Economy, 114, 175--201.
Decker, C., E. Lieb, R. McCann, and B. Stephens (2012): \textquotedblleft Unique Equilibria and Substitution Effects in a Stochastic Model of the Marriage Market,\textquotedblright\ Journal of Economic Theory, 148, 778--792.
Demange, G. and D. Gale (1985): “The Strategy Structure of Two-Sided Matching Markets ”, Econometrica, 53, 873--888.
Gale, D. and M. Sotomayor (1985): “Some Remarks on the Two-sided Matching Model”, {\em Discrete Applied Mathematics}, 11, 223--232.
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Graham, B. (2013): “Comparative static and computational methods for an empirical one-to-one transferable utility matching model”. Structural Econometric Models 31, 153--181.
Kelso, A. and V. Crawford (1982): “Job Matching, Coalition Formation, and Gross Substitutes”, Econometrica, 50, 1483--1504.