ROYAL ECONOMIC SOCIETY CONFERENCE 2015 SPECIAL SESSION Econometrics of Matching Wednesday 1st April 11.00 — 12.30 Venue: University Place Lecture Theatre A (RES 2015) 1 / 1
E STIMATING T RANSFER F RICTIONS IN THE M ARRIAGE M ARKET Alfred Galichon Based on joint works with Arnaud Dupuy, Yu-Wei Hsieh, Scott Kominers, Bernard Salani´ e, and Simon Weber. The Econometric Journal special session RES conference, Manchester, April 1, 2015 T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 1/ 32
Section 1 I NTRODUCTION T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 2/ 32
U TILITY TRANSFERS IN MATCHING MODELS ◮ Transfers of utility (under the form of money or other exchanges): ◮ Are sometimes clearly forbidden (e.g. school choice problems): Nontransferable Utility (NTU) ◮ Are sometimes clearly allowed (e.g. wages in the market for CEOs): Transferable Utility (TU). ◮ However, it is sometimes unclear (e.g. the marriage market). Indeed, there is a tradition to model the marriage market with transfers (Shapley and Shubik; Becker; Choo and Siow) and without transfers (Gale and Shapley; Dagsvik; Hitsch, Hortacsu and Ariely; Menzel). T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 3/ 32
M ATCHING WITH IMPERFECT TRANSFERS ◮ It makes sense to assume that transfers whithin the couple are possible, but not efficient in the sense that the utility transfered by i to j maybe more costly to i that it is beneficial to j : Imperfectly Transferable Utility (ITU). ◮ Also the case when matching is in the presence of nonquasilinear utilities (Kelso and Crawford, Hatfield and Milgrom); of taxes (Jaffe and Kominers); of risk aversion (Legros and Newman, Chade and Eckehout); of investments (Samuelson and Noeldeke). ◮ However, in the case of the marriage matching market, this is an empirical question ◮ Does the answer matter? I will argue that it does; both for econometric analysis and for policy implications. For this we will need a framework for ITU matching. T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 4/ 32
O UTLINE This talk: 1. Introduction 2. Theoretical framework 3. Empirical framework 4. Application T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 5/ 32
Section 2 T HE THEORETICAL FRAMEWORK T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 6/ 32
M ATCHING ◮ Let µ ij ∈ { 0, 1 } be a dummy variable that is equal to 1 if man i and woman j are matched, 0 else. ◮ Hence, µ ij satisfies ∑ µ ij ≤ 1 j ∑ µ ij ≤ 1. i T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 7/ 32
U TILITIES ◮ Assume that if man i and woman j match, then i has utility α ij and j has utility γ ij , pre-transfer. ◮ Utilities of single individuals are normalized to zero. ◮ Assume that transfers t i ← j and t j ← i are decided, so that if matched, i and j enjoy respectively u i = α ij + t i ← j v j = γ ij + t j ← i . ◮ The link between t i ← j (what i receives) and − t j ← i (what j gives out) is subject to a feasibility constraint , which will specify whether matching is TU, NTU, or ITU. T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 8/ 32
F EASIBILITY ◮ If µ ij = 1, then t i ← j and t j ← i must satisfy a feasibility constraint: ◮ In the TU case t i ← j + t j ← i ≤ 0 � ≤ 0 ◮ In the NTU case, max � t i ← j , t j ← i � ≤ 0, where Ψ ij is ◮ More generally, in the ITU case, Ψ ij � t i ← j , t j ← i continuous and nondecreasing in its two variables. ◮ We can rewrite the feasibility constraint as � ≤ 0. � µ ij = 1 = ⇒ Ψ ij u i − α ij , v j − γ ij ◮ Of particular interest is the Exponentially Transferable Utility (ETU) case, when � exp ( a / τ ) + exp ( b / τ ) � Ψ ( a , b ) = τ log 2 where τ > 0 is a transferability parameter. The ETU model interpolates between TU and NTU; indeed, τ → 0 is NTU, while τ → + ∞ is TU. T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 9/ 32
S TABILITY ◮ For any pair i and j , we need to rule out the case that i and j form a blocking pair, i.e. each achieve higher payoffs by rebargaining. This implies � ≥ 0. � ∀ i , j Ψ ij u i − α ij , v j − γ ij ◮ In the TU case, this is the well-known stability conditions u i + v j ≥ α ij + γ ij , � ≥ 0. ◮ while in the NTU case, this reads max � u i − α ij , v j − γ ij T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 10/ 32
E QUILIBRIUM ◮ We are now ready to define an equilibrium in this market. ◮ An outcome ( µ , u , v ) is an equilibrium if ◮ (i) µ ij ∈ { 0, 1 } , ∑ j µ ij ≤ 1 and ∑ i µ ij ≤ 1 � ≥ 0 ◮ (ii) Ψ ij � u i − α ij , v j − γ ij � = 0. ◮ (iii) µ ij = 1 implies Ψ ij � u i − α ij , v j − γ ij T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 11/ 32
Section 3 T HE EMPIRICAL FRAMEWORK T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 12/ 32
M ATCHING ◮ Assume that there are groups, or clusters of men and women who share similar observable characteristics, called types . There are n x men of type x , and m y women of type y . ◮ Let µ xy ≥ 0 be the number of men of type x matched to women of type y . This quantity satisfies ∑ µ xy ≤ n x y ∑ µ xy ≤ m y x ◮ We shall denote µ x 0 and µ 0 y the number of single men of type x and single women of type y . T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 13/ 32
U TILITIES ◮ Assumption 1 : Assume that if man i of type x and woman j of type y match, then α ij = α xy + ε iy γ ij = γ xy + η jx Utilities of single man i and woman j are respectively ε i 0 and η j 0 . Recall that transfers t i ← j and t j ← i are decided, so that if matched, i and j enjoy respectively u i = α xy + ε iy + t i ← j v j = γ xy + η jx + t j ← i ◮ Assumption 2 : there are a large number of invididuals per group and the ε and the η ’s are i.i.d. Gumbel. T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 14/ 32
F EASIBILITY AND STABILITY ◮ Assumption 3 : If i is of type x and j is of type y , then Ψ ij ( a , b ) = Ψ xy ( a , b ) (i.e. we assume that Ψ ij only depends on i and j through their types). ◮ Thus, we can rewrite the feasibility constraint as (for i in x and j in y ) � ≤ 0. � µ ij = 1 = ⇒ Ψ xy u i − α xy − ε iy , v j − γ xy + η jx and stability � ≥ 0. ∀ i ∈ x , j ∈ y , Ψ xy � u i − α xy − ε iy , v j − γ xy + η jx T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 15/ 32
E QUILIBRIUM TRANSFERS Theorem 1 (Galichon, Kominers and Weber). Under Assumptions 1, 2 and 3 above, equilibrium transfers t i ← j and t j ← i only depend on x and y , the observable types of i and j . Hence, let us denote these quantitites by t x ← y and t y ← x . This theorem extends to the general ITU case a result which was known in the TU case (Choo and Siow, Chiappori, Salani´ e and Weiss, Galichon and Salani´ e). T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 16/ 32
M ATCHING AND DISCRETE CHOICE Implication of this theorem: the matching problem now embeds two sets of discrete choice problems. Indeed, man i and woman j (of types x and y ) solve respectively � � max α xy + t x ← y + ε iy , ε i 0 y � � max γ xy + t y ← x + η jx , η j 0 x which are standard discrete choice problems; thus the log-odds ratio formula applies, and ln µ xy = α xy + t x ← y µ x 0 ln µ xy = γ xy + t y ← x µ 0 y But remember that Ψ xy ( t x ← y , t y ← x ) = 0, thus � � ln µ xy − α xy , ln µ xy Ψ xy − γ xy = 0. µ x 0 µ 0 y T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 17/ 32
E QUILIBRIUM CHARACTERIZATION : RESULT Theorem 2 (GKW). Equilibrium in the ITU problem with logit heterogeneities is fully characterized by the set of nonlinear equations in µ xy , µ x 0 and µ 0 y � � ln µ xy − α xy , ln µ xy Ψ xy − γ xy = 0 µ x 0 µ 0 y ∑ µ xy + µ x 0 = n x y ∑ µ xy + µ 0 y = m y x Under very mild conditions on Ψ it exists; under mild conditions on Ψ it is also unique. Galichon and Hsieh extend this result to the NTU case with general stochastic utilities. T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 18/ 32
E QUILIBRIUM CHARACTERIZATION , BIS Note that first equation defines implicitely µ xy as a function of µ x 0 and µ 0 y , which can be written as a matching function � � µ xy = M xy µ x 0 , µ 0 y hence we can restate the previous result as: Theorem 2’ (GKW). Equilibrium in the ITU problem with logit heterogeneities is fully characterized by the set of nonlinear equations in µ x 0 and µ 0 y � + µ x 0 = n x ∑ � µ x 0 , µ 0 y M xy y � + µ 0 y = m y . ∑ � µ x 0 , µ 0 y M xy x T RANSFER FRICTIONS IN THE MARRIAGE MARKET M ANCHESTER , A PRIL 1, 2015 SLIDE 19/ 32
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