Lecture 5: Estimation of a One-to-One Transferable Utility Matching Model

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1 Lecture 5: Estimation of a One-to-One Transferable Utility Matching Model Bryan S Graham, UC - Berkeley & NBER September 4, 2015 Consider a single matching market composed of two large populations of, for concreteness, women and men For each woman and man we respectively observe the discretely-valued characteristics W i W = {w 1,,, w K } and X j X = {x 1,, x L } 1 The K types of women and L types of men may encode, for example, different unique combinations of yearsof-schooling and age While K and L are assumed finite, they may be very large in practice Observationally identical women have heterogeneous preferences over different types of men, but are indifferent between men of the same type Specifically female i s utility from matching with male j is given by U ( W i, X j, ε i ) = α ( Wi, X j) + τ ( W i, X j) + ε i ( X j ), where α (w k, x l ) is the systematic utility a type W i = w k women derives from matching with a type X j = x l man, ε i (X j ) = L l=1 1 (Xj = x l ) ε il captures unobserved heterogeneity in women s preferences over alternative types of men, and τ (w k, x l ) is the equilibrium transfer that a type X j = x l man must pay a type W i = w k women in order to match Transfers may be negative and their determination is discussed below Here 1 ( ) denotes the indicator function Since the stochastic component of female match utility, ε i (X j ), varies with male type alone (ie, his specific identify does not matter), women are indifferent amongst observationally identical men A similar restriction on male preferences ensures that the equilibrium transfer, τ (w k, x l ), depends on agent types alone A women may also choose to remained unmatched, or single, in which case her utility is given by U (W i, ε i ) = α (W i ) + ε i0 1 The subscript i denotes a generic random draw from the population of women, while the superscript j one from men See Shapley and Shubik (1971) for the theory of one-to-one matching with transfers 1

2 Table 1: Feasible matchings W\M Single(x 0 ) x 1 x L Single(w 0 ) - q 1 K k=1 r k1 q L K k=1 r kl - w 1 p 1 L l=1 r 1l r 11 r 1L p 1 w K p K L l=1 r Kl r K1 r KL p K - q 1 q L Notes: Let r kl 0 denote the number of k-to-l matches Feasibility of a matching imposes the K + L adding up constraints L l=1 r kl p k for k = 1,, K and K k=1 r kl q l for l = 1,, L Men also have heterogenous preferences Man j s utility from matching with woman i is given by V ( W i, X j, υ j) = β ( W i, X j) τ ( W i, X j) + υ j (W i ), where β (w k, x l ) is the systematic utility a type X j = x l men derives from matching with a type W i = w k woman and υ j (W i ) = K k=1 1 (W i = w k ) υ j k is a heterogenous component of match utility Here τ (w, x) enters with a negative sign as we conceptually imagine men paying women (recall that transfers may be negative) The utility from remaining unmatched is V ( X j, υ j) = β ( X j) + υ j 0 Preference heterogeneity ensures that, for any given transfer function τ (w, x), observationally identical women will match with different types of men If the support of the heterogeneity distribution is rich enough all types of matches will be observed in equilibrium Let ε = (ε i0, ε i1,, ε il ) and υ = ( υ j 0, υ j 1,, υ j K) I assume that the components of these vectors are independently and identically distributed Type I extreme value random variables F ε W (e W = w k ) = F υ X (v X = x l ) = L l=0 K k=0 ( ( exp exp e l σ ε )) ( ( exp exp v )) k σ υ (1) Assumption (1) is slightly more general than that maintained by Choo and Siow (2006a,b) who additionally impose the restriction σ ε = σ υ 2 2 Graham (2011) considers the case where F ε W and F υ X are left nonparametric 2 Bryan S Graham 2015

3 Equilibrium Let α kl = α (w k, x l ), α k0 = α (w k ), β kl = β (w k, x l ), β 0l = β (x l ) and τ kl = τ (w k, x l ) Let θ be a vector of model parameters to be more precisely specified below and τ a KL 1 vector of transfers The total number of type k women is given by p k, that of type l men by q l Let p = (p 1,, p K ) and q = (q 1,, q L ) Denote the probability, given a hypothetical transfer vector τ, that a type k woman matches with a type l man by e D kl (θ; τ) The probability of remaining unmatched is ed k0 (θ, τ) = 1 L l=1 ed kl (θ, τ) Under the Type I extreme value assumption we have for k = 1,, K (McFadden, 1974): e D k0 (θ, τ) = e D kl (θ, τ) = L n=1 exp (σ 1 ε [α kn α k0 + τ kn ]) exp (σε 1 [α kl α k0 + τ kl ]) 1 +, l = 1,, L L n=1 exp (σ 1 ε [α kn α k0 + τ kn ]) Total demand for type l men by type k women is therefore r D k0 (θ, τ, p, q) r D kl (θ, τ, p, q) = p k e D k0 (θ, τ) (2) = p k e D kl (θ, τ), which, after some manipulation, gives ( ) r D σ ε ln kl (θ, τ, p, q) = α rk0 D kl α (θ, τ, p, q) k0 + τ kl (3) For l = 1,, L we get a conditional supply of type l men to each of the k = 1,, K types of women equal to r S 0l (θ, τ, p, q) r S kl (θ, τ, p, q) = q l g S 0l (θ, τ) (4) = q l g S kl (θ, τ), where g S 0l (θ, τ) = g S kl (θ, τ) = ( ]) K m=1 exp συ [β 1 ml β 0l τ ml ( ]) exp συ [β 1 kl β 0l τ kl 1 + ( ]), k = 1,, K, K m=1 exp συ [β 1 ml β 0l τ ml 3 Bryan S Graham 2015

4 so that ( ) r S σ υ ln kl (θ, τ, p, q) = β kl β (θ, τ, p, q) 0l τ kl (5) r S 0l The transfer vector, τ, adjusts to equate the KL female demands with the KL male supplies so that in equilibrium kl (θ, τ eq, p, q) = r D kl (θ, τ eq, p, q) = r S kl (θ, τ eq, p, q), k = 1,, K, l = 1,, L, (6) with the eq superscript denoting an equilibrium quantity Let γ kl = α kl + β kl α k0 β 0l σ ε + σ υ, λ = σ υ σ ε + σ υ Imposing (6), adding (3) and (5), exponentiating and rearranging yields kl = ( k0 )1 λ ( 0l )λ exp (γ kl ) (7) where I let kl = kl (θ, τ eq, p, q) to economize on notation See Graham (2013) for a fixed point representation of the matching market equilibrium The set of feasible matchings is depicted in Table 1 Estimation Let i and j index two independent random draws from the equilibrium distribution of matches Let A klmn ij = 1 (W i {w k, w m }) 1 (W j {w k, w m }) 1 (X i {x l, x n }) 1 (X j {x l, x n }) be an indicator for the event that this pair of matches belongs to the klmn sub-allocation That is the two women in the matches are of type w k or w m and the two men are of type x l or x n We can assume that, without loss of generality, that the support points of female and male types are ordered so that w k < w m and x l < x n Now ine S ij = sgn {(W i W j ) (X i X j )} We have that S ij = 1 if the higher type women matches with the higher type man and S ij = 1 in the configuration is anti-assortative If either W i = W j or X i = X j or both, then S ij = 0 and the configuration is neither assortative or anti-assortative Using (7) above 4 Bryan S Graham 2015

5 we have Pr ( S ij = 1 S ij { 1, 1}, A klmn ij = 1 ) = = = kl req mn kl req mn + kn req ml exp (γ kl ) exp (γ mn ) exp (γ kl ) exp (γ mn ) + exp (γ kn ) exp (γ ml ) exp (γ mn γ ml (γ kn γ kl )) 1 + exp (γ mn γ ml (γ kn γ kl )) If we let A ij be the J = ( K 2 )( L 2) vector of sub-allocation indicators and ine φ to be the corresponding vector of complementarity parameters φ klmn = γ mn γ ml (γ kn γ kl ), we have Pr (S ij = 1 S ij { 1, 1}, A ij ) = exp ( A ijφ ) 1 + exp ( A ij φ) In some settings it may be desirable to impose more structure on the complementarity parameters Let γ (w, x) = a (w) + b (x) + e (w, x) η with e (w, x) a low dimensional vector of basis functions and a (w) and b (x) arbitrary If we let M be a matrix composed of rows m klmn = {e (w m, x n ) e (w m, x l ) [e (w k, x n ) e (w k, x l )]}, then we have φ = Mη and hence Pr (S ij = 1 S ij { 1, 1}, A ij ) = exp ( A ijmη ) 1 + exp ( A ij Mη) We can consistently estimate η but choosing ˆη to minimize the U-process: L N (η) = References 2 N (N 1) N S ij { S ija ijmη ln [ 1 + exp ( S ija ijmη )]} i=1 j<i [1] Choo, Eugene and Aloysius Siow (2006a) Who marries whom and why? Journal of Political Economy 114 (1): Bryan S Graham 2015

6 [2] Choo, Eugene and Aloysius Siow (2006b) Estimating a marriage matching model with spillover effects, Demography 43 (3): [3] Graham, Bryan S (2011) Econometric methods for the analysis of assignment problems in the presence of complementarity and social spillovers, Handbook of Social Economics 1B: (J Benhabib, A Bisin, & M Jackson, Eds) Amsterdam: North-Holland [4] Graham, Bryan S (2013) Comparative static and computational methods for an empirical one-to-one transferable utility matching model, Advances in Econometrics 31: [5] McFadden, Daniel (1974) Conditional logit analysis of qualitative choice behavior, Frontiers in Econometrics: (P Zarembka, Ed) Academic Press: New York [6] Shapley, Lloyd S and Martin Shubik (1971) "The assignment game I: The core," International Journal of Game Theory 1 (1): Bryan S Graham 2015