Marriage with Labour Supply Jean-Marc Robin (joint with Nicolas - PowerPoint PPT Presentation

Marriage with Labour Supply Jean-Marc Robin (joint with Nicolas Jacquemet, Paris 1) Sciences-Po & UCL Matching Workshop, Chicago, 4-6 June 2012

INTRODUCTION

Aim • Integrate Chiappori’s (Ecta, 1988, JPE, 1992) collective model within a search-matching framework derived from Shimer-Smith (Ecta, 2000). • Indeed, collective models make the sharing rule depend on matching and market factors (wage ratio, aggregate sex ratio, divorce rules, etc) without providing a formal model. • Long term aim: evaluate the effects of policies targeted at the household level (WFTC, EITC, ...) • On the methodological side, this papers adds to Shimer-Smith by studying the identification and the estimation of two-sided search-matching models from cross-sectional data on wages and hours worked.

Why (random) search? ... instead of perfect-information assignment as recent work by Siow and Chiappori (and coauthors). • More realistic? It takes time to find the right partner. • Naturally yields mismatch. • Easier to deal with continuous characteristics. • Forward looking agents and risk are naturally incorporated. • ...

Literature • Bargaining models. • Manser, Brown (IER, 80), McElroy, Horney (IER, 81), Becker (1981), Lundberg, Pollak (JPE, 93, JEPersp, 96). • On non-unitary models of the household see survey by Chiappori, Donni (2009). • Non-equilibrium search models of match formation. • Ermisch (2003), Gould, Paserman (JUrbE, 2003). • Rich applied, essentially macro literature of search-matching models aiming at explaining time trends (such as declining marriage rate, increasing female college graduation rate) and the role of policy. • Aiyagari, Greenwood, Guner (JPED, 2000), Greenwood, Guner, Knowles (AER, 2000) Caucutt, Guner, John Knowles (RED, 2002), Brien, Lillard, Stern (IER, 2006), Chiappori, Weiss (JEEA, 2006, JoLE, 2007), Chiappori and Oreffice (JPED, 2008), etc.

• Perfect-information match formation and intra-family ressources allocation. • Choo, Seitz and Siow (2008) and Chiappori, Salanie, Weiss (2010). • Theory of search and matching in marriage markets. • Sattinger (IER, 1995), Lu, McAfee (inbook, 1996), Burdett, Coles (QJE, 1997), Shimer and Smith (Ecta, 2000).

THE MODEL

Populations • Only source of heterogeneity is labour productivity or wages. • ℓ m ( x ) , ℓ f ( y ) denote the number of males, females with labour productivity (wage) x , y . ´ ´ • L m = ℓ m ( x ) d x and L f = ℓ f ( y ) d y • u m ( x ) and u f ( y ) are the measures of singles of types x and y . ´ ´ • U m = u m ( x ) d x and U f = u f ( y ) d y • n ( x , y ) is the number of married couples with characteristics ( x , y ) ˜ • N = n ( x , y ) d x d y

Meetings, matching, separations • δ is the (exogenous) divorce rate. • Only singles search. • M ( U m , U f ) is the meeting function (number of meetings per unit of time) • λ m = M ( U m , U f ) and λ f = M ( U m , U f ) are the meeting rates. U m U f • Not all meetings induce marriage. • α ( x , y ) ∈ [ 0 , 1 ] is the (endogenous) probability of marriage ( matching probability).

Steady-state restrictions • In steady-state equilibrium, the number of divorces per unit of time is equal to the number of marriages: u f ( y ) δ n ( x , y ) = u m ( x ) λ m α ( x , y )= λ u m ( x ) u f ( y ) α ( x , y ) U f where λ = M ( U m , U f ) . U m U f • Marginalisation: ˆ ˆ δ n ( x , y ) d y = δ [ ℓ m ( x ) − u m ( x )] = λ u m ( x ) u f ( y ) α ( x , y ) d y • Same for u f ( y )

Linear preferences • Individuals draw utility from consumption ( c ) and leisure ( ℓ ). • Indirect utility flow: v m ( x , xT + t ) = xT + t − A m ( x ) B m ( x ) where T is total time and t is non-labour income. • Leisure follow by Roy’s identity as ℓ m ( x , xT + t ) = A ′ m ( x ) + b ′ m ( x )[ xT + t − A m ( x )] where b m ( x ) = log B m ( x ) and b ′ m denotes derivative.

Time Use for Married Individuals • Marriage allows individuals to benefit from economies of scale and task specialisation. • Home production is H ( p m , p f , x , y ) + z , a function of • time spent in home production by both spouses, p m , p f , • productivity x , y , • a source of noise, z , drawn at the first meeting from a zero-mean distribution denoted G . It aims at capturing all other dimensions of mutual attractiveness but labor market productivity. • Optimal time use p 1 m ( x , y ) , p 1 f ( x , y ) solve C ( x , y ) = max p m , p f { H ( p m , p f , x , y ) − xp m − yp f } .

Individual surpluses • W m ( x ) be the value of singlehood (to be derived later). • W m ( v , x ) is the value of a marriage yielding flow utility v to a male x . • Option value equation: rW m ( v , x ) = v + δ [ W m ( x ) − W m ( v , x )] where r is discount rate. • Individual surplus: S m ( v , x ) = W m ( v , x ) − W m ( x ) = v − rW m ( x ) r + δ • Similar definitions for females

Bargaining • Spouses split home production, t m + t f = C ( x , y ) + z , by Nash bargaining. • Transfers t m and t f solve t m , t f S m ( v m ( x , xT + t m ) , x ) β S f ( v f ( y , yT + t f ) , y ) 1 − β max subject to condition t m + t f ≤ C ( x , y ) + z .

Transfers • Define s m ( x ) and s f ( y ) such as the continuation values: rW m = xT + s m − A m rW f = yT + s f − A f and B m B f • The solution for transfers is: t m ( x , y , z ) = s m ( x ) + β [ C ( x , y ) + z − s m ( x ) − s f ( y )] t f ( x , y , z ) = s f ( y ) + ( 1 − β )[ C ( x , y ) + z − s m ( x ) − s f ( y )]

Matching • Singles x and y decide to match if the overall surplus is positive, i.e. s ( x , y

The values for singles • The value of being single, for males, solves the option value equation: rW m ( x ) = v m ( x , xT + C m ( x )) ¨ + λ max { S m ( v m ( x , xT + t m ( x , y , z )) , x ) , 0 } d G ( z ) u f ( y ) d y where C m ( x ) = max p m { H m ( p m , x ) − xp m } is home production for single men. • Equivalently, s m ( x ) = − ( xT − A m ( x )) + B m rW m ( x ) ¨ λβ = max { z + C ( x , y ) − s m ( x ) − s f ( y ) , 0 } d G ( z ) u f ( y ) d y r + δ • A symmetric expression can be derived for females.

Equilibrium The equilibrium is a fixed point ( u m , u f , s m , s f ) of the following system of four functional equations: ℓ m ( x ) ℓ f ( y ) u m ( x ) = ´ and u f ( y ) = ´ 1 + λ 1 + λ u f ( y ) α ( x , y ) d y u m ( x ) α ( x , y ) d x δ δ ˜ C m ( x ) + λβ max { z + C ( x , y ) − s f ( y ) , s m ( x ) } d G ( z ) u f ( y ) d y r + δ s m ( x ) = 1 + λβ r + δ U f ˜ C f ( y ) + λ ( 1 − β ) max { z + C ( x , y ) − s m ( x ) , s f ( y ) } d G ( z ) u m ( x ) d x r + δ s f ( y ) = 1 + λ ( 1 − β ) r + δ U m with ˆ ˆ λ = M ( U m , U f ) U m = u m ( x ) d x , U f = u f ( y ) d y , U m U f α ( x , y ) = 1 − G ( s m ( x ) + s f ( y ) − C ( x , y ))

DATA AND DESCRIPTIVE STATISTICS

Wage distributions Marginals (a) Density (b) CDF 0.1 1 married females female singles 0.09 0.9 married males male singles 0.08 0.8 0.07 0.7 0.06 0.6 density CDF 0.05 0.5 0.04 0.4 0.03 0.3 0.02 0.2 married females female singles 0.01 0.1 married males male singles 0 0 0 5 10 15 20 25 30 35 40 45 1 1.5 2 2.5 3 3.5 4 wage log wage

Wage distributions Joint distribution of ( x , y ) amongst married couples • 25% correlation! (a) 3-D plot (b) Projection on the ( x , y ) plane 0.7 0.6 0.5 0.4 density 0.3 0.2 0.1 0 3.5 3 3 2.5 2.5 2 2 1.5 male log wage female log wage

Hours Nonparametric estimates of mean hours given own wages 200 180 married males → 160 ← single males 140 120 single females → 100 ← married females 80 1 1.5 2 2.5 3 3.5 4 log wage

Hours Nonparametric estimates of mean hours given both spouses’ wages married males 250 200 150 100 40 30 3 20 2.5 2 10 1.5 male log wage female log wage married females 200 150 100 50 40 30 30 25 20 20 15 10 10 5 male log wage female log wage

IDENTIFICATION

Matching probability • The equilibrium flow condition implies that α ( x , y ) = δ n ( x , y ) u m ( x ) u f ( y ) . λ • The matching probability is identified by comparing the distribution of types among married couples to what it should be in absence of sorting. • We display an estimate for the following calibration of δ and λ : • The divorce rate is set to 8% annual, which is consistent to a median marriage duration of about 8 years (Census, 2005). • The meeting rate is not identified in absence of data on datings. We arbitrarily calibrate it so that the meeting rate would be twice a year for single men ( λ m = 1 / 6 ).

Estimation • Exponential growth. • Steeper along x than y (harder to see but married men earn more!). (a) 3-D plot (b) Flat projection 3.6 3.4 0.5 3.2 0.4 3 0.3 male log wage 2.8 0.2 2.6 2.4 0.1 2.2 0 2 3.5 3 3 1.8 2.5 2.5 2 2 1.6 1.5 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 male log wage female log wage female log wage