Dynamic Marriage Matching: An Empirical Framework Eugene Choo University of Calgary Matching Problems Conference, June 5th 2012 1/31
Introduction Interested in rationalizing the marriage distribution of ‘who marries � whom’ by age. To allow for dynamics in marriage and marital decisions. � Empirically quantify the marital gains across gender and age. � How important are dynamic considerations in marital decisions? Propose a dynamic version of the Becker-Shapley-Shubik model. � Model rationalizes a new marriage matching function, � µ = G ( m , f ; Π ) where µ is the distribution of new marriages, m and f are the vectors of available single men and women, and Π is a matrix of parameters, . 2/31
Contributions - Dynamic Marriage Matching Function z ij � 1 2 ( βS ) k � µ i + k, 0 µ 0 ,j + k � � µ ij = Π ij m i f j . (1) m i + k f j + k k =0 ( i, j ) denote the ages (or types) of males and females respectively. � µ ij is the number of observed new ( i, j ) matches, � µ i 0 is the number of i men who remained single and µ 0 j is the number of j women who remained single m i and f j are the number of single type i men and j women respectively. � discount factor is β, divorce rate is δ, survival probability S = 1 − δ � z ij = Z − max( i, j ) , measures the maximum length of a marriage. � 3/31
Contributions - Dynamic Marriage Matching Function continues Π ij is the present discounted value of an ( i, j ) match relative to � remaining single for the duration of the match. z ij ( βS ) k [( α ijk + γ ijk ) − ( α i + k, 0 + γ 0 ,j + k )] − 2 κ � Π ij = (2) k =0 α ijk be the k ′ th period marital output accrued to a type i male when � married to a type j female today, similarly γ ijk be the k ′ th period marital output accrued to a type j � female when married to a type i male, α i 0 and γ 0 j are the per-period utilities from remaining single for i � type males and j type females respectively. κ is the geometric sum of Euler’s constants. � 4/31
Empirical Application Use model to analyze the fall in marital gains by age and gender � between 1970 and 1990 in the US Show that dynamic component marital gains is large especially � among the young. Ignoring dynamics severely unstate the drop in marital gains between � 1970 and 1990 especially among young couples. 5/31
Literature Builds on frictionless Becker-Sharpley-Shubik transferable utility � model of marriage Extends ideas in Choo and Siow (2006) and Choo and Siow (2005) � Adopt the dynamic discrete choice framework of Rust (1987) � Growing body of empirical work on marriage matching: � Chiappori, Salanie and Weiss (2011), Chiappori, McCann and Nesheim (2009), Galichon and Salanie (2010), Echenique, Lee, Shum and Yenmez (2011), Ariely, Hortacsu and Hitsch (2006, 2010), Fox (2010). 6/31
The Model - Assumptions State Variables: Single individuals has two state variables: � 1. ( i, j ) denote male and female’s age when single, terminal age is Z. 2. ǫ ig , is a ( Z + 1) vector of i.i.d idiosyncratic payoffs specific to type i male individual, g ( ǫ jG for type j female, G ), unobserved to econometrician. Agents observe ǫ at beginning of period. Stationarity: Single males and females, m i and f j ∀ i, j at each � period taken as given. Actions: a ig ∈ { 0 , 1 , . . . , Z } (or a jG ) denote the action of a single � type i male g (or single type j female G ). If g (or G ) chooses to remain single, a ig = 0 (or a jG = 0) , else if g (or G ) chooses to match with a type k spouse, a ig = k (or a jG = k ) . 7/31
The Model - Assumptions continues Exogenous Parameters: discount factor is β, divorce rate δ, the � survival probability S = 1 − δ. Adopt Dynamic Discrete Choice framework of Rust(1987), � maintain Rust’s Additive Separability (AS) and Conditional Independence (CI). Additive Separability (AS) in utilities � Utility function of a single male g decomposes to v ( a ig , i, ǫ ig ) = v a ( i ) + ǫ iag , similarly utility function of a single female G takes the form, w ( a jG , j, ǫ jG ) = w a ( j ) + ǫ jaG . 8/31
The Model - Assumptions continues Conditional Independence (CI): State transition probability � factorize as h ( ǫ | i ) · F a ( i ′ | i ) P { i ′ , ǫ ′ ig | i, ǫ , a } = h ( ǫ | i ) · R a ( j ′ | j ) . P { j ′ , ǫ ′ jG | j, ǫ , a } = F a ( i ′ | i ) is the transition probability that a type i male g will next � find himself single at age i ′ given his action a at age i . R a ( j ′ | j ) is the transition probability that a type j female G will � next find herself single at age j ′ given her action a. ǫ are i.i.d. Type I Extreme Value random variables. � full commitment, transferable utility setup. � 9/31
The Model - Utility Functions If male g (or female G ) chooses to marry an age j female (or i male), � v ( a ig = j, i, ǫ ig ) = α i ( j ) − τ ij + ǫ ijg , and w ( a jG = i, j, ǫ jG ) = γ j ( i ) + τ ij + ǫ ajG z ij z ij ( βS ) k α ijk , ( βS ) k γ ijk . � � where α i ( j ) = and γ j ( i ) = k =0 k =0 α ijk (or γ ijk ) be the k ′ th period marital output accrued to a type i male � (or j female) when married to a type j female (or i male) today. If male g (or female G ) chooses to remain single, then � v ( a ig = 0 , i, ǫ ig ) = α i 0 + ǫ i 0 g , and w ( a jG = 0 , j, ǫ jG ) = γ 0 j + ǫ 0 jG 10/31
The Model - Convenient representation Rust’s framework permits Value function to have convenient form, � { ˜ v ia + ǫ iag } V α ( i, ǫ ig ) = max a ∈D { ˜ w aj + ǫ ajG } W γ ( j, ǫ jG ) = max a ∈D where the mean components, ˜ v ij and ˜ w ij are also referred to as the � choice specific value functions for type i males and j females respectively. R i ( j ′ | j ) · W j ′ � w ij ˜ = ( γ j ( i ) + τ ij ) I ( i � = 0) + γ 0 j I ( i = 0) + j ′ F j ( i ′ | i ) · V i ′ . � v ij ˜ = ( α i ( j ) − τ ij ) I ( j � = 0) + α i 0 I ( j = 0) + i ′ V i and W j are the integrated value function (value function where the � unobservable state is integrated out) � � V i = V α ( i, ǫ g ) dH ( ǫ g ) , W j = W γ ( j, ǫ G ) dH ( ǫ G ) . 11/31
The Model - Choice Probabilities Define the conditional choice probability P ij for males and Q ij for � females: � P ij I { j = arg max v ia + ǫ iag ) } h ( d ǫ ) , = a ∈D (˜ � Q ij = I { i = arg max a ∈D ( ˜ w aj + ǫ ajG ) } h ( d ǫ ) . The probabilities have the familiar multinomial logit form, � v ij − ˜ w ij − ˜ exp(˜ v i 0 ) exp( ˜ w i 0 ) P ij = Q ij = , . 1 + � Z 1 + � Z r =1 exp(˜ v ir − ˜ v i 0 ) r =1 exp( ˜ w rj − ˜ w 0 j ) 12/31
The Model - Quasi Demand and Supply Log-odds ratios delivers a system of ( Z × Z ) quasi-demand and � quasi-supply equations respectively. z ij ( βS ) k ln P i + k, 0 � ln P ij − α i ( j ) − α i (0) − τ ij − κ = k =0 z ij ( βS ) k ln Q 0 ,j + k � ln Q ij − γ j ( i ) − γ j (0) + τ ij − κ. = k =0 where κ = cβS (1 − ( βS ) z ij ) / (1 − βS ) , ( c is the Euler’s constant) z ij z ij ( βS ) k α ijk , ( βS ) k γ ijk , � � α i ( j ) = γ j ( i ) = k =0 k =0 z ij z ij ( βS ) k α i + k, 0 , ( βS ) k γ 0 ,j + k . � � α i (0) = and γ j (0) = k =0 k =0 13/31
The Model - Equilibrium A marriage market equilibrium consists of a vector of males, m and females, f across individual type, the vector of marriage µ , and the vector of transfers, τ such that the number of i type men who want to marry j type spouses exactly equals the number of j type women who agree to marry type i men for all combinations of ( i, j ) . That is, for each of the ( Z × Z ) sub-markets, m i P ij = f j Q ij = µ ij � 14/31
The Model - Dynamic Marriage Matching Function Let p ij and q ij denote the maximum likelihood estimators of P ij and � Q ij , that is, p ij = µ ij /m i and q ij = µ ij /f j . z ij � 1 2 ( βS ) k � µ i + k, 0 µ 0 ,j + k � � µ ij = Π ij m i f j m i + k f j + k k =0 k =0 ( βS ) k [( α ijk + γ ijk ) − ( α i + k, 0 + γ 0 ,j + k )] − 2 κ where Π ij = � z ij 15/31
The Model - Dynamic Marriage Matching Function The dynamic marriage matching function also needs to satisfy the � accounting constraints given by, Z � f j ∀ j µ 0 j + µ ij = i =1 Z � m i ∀ i µ i 0 + µ ij = j =1 ≥ 0 ∀ i, j µ 0 j , µ i 0 , µ ij 16/31
Inverse Problem Given a matrix of preferences Π , whose elements are non-negative � and strictly positive population vectors, m and f , does there exist a unique non-negative marital distribution µ that is consistent with Π , that satisfies Dynamic Marriage Matching Function and accounting constraints. Reformulate the model to an I + J system with I + J number of � unmarrieds of each type, µ i 0 and µ 0 j , as unknowns. This reduced system is defined by z ij I � 1 2 ( βS ) k � µ i + k, 0 µ 0 ,j + k � � � m i − µ i 0 = Π ij m i f j (3) m i + k f j + k i =1 k =0 z ij J � 1 2 ( βS ) k � µ i + k, 0 µ 0 ,j + k � � � f j − µ 0 j = Π ij m i f j (4) m i + k f j + k j =1 k =0 17/31
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