The Marriage Market, Labor Supply and Education ChoicesThe Marriage Market, Labor Supply and...

The Marriage Market, Labor Supply and EducationChoices

P.A. Chiappori, M. Costa Dias, C. Meghir

Columbia University, UCL and IFS, Yale University

Workshop ‘Risk and Family Economics’IFS, June 2014

Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 1

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Gary Becker, 1930-2014


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Education, matching and labor supply

Education as a human capital investment

What are the returns?

Obvious: labor market (higher wages, lower unemployment, bettercarreer,...)As important, but largely ignored: marriage market

Education affects

Marriage probabilitySpouse’s education and incomeTotal surplus generated by marriageIntra-household division of that surplus

Matching models useful to analyze these returns


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Obvious: labor market (higher wages, lower unemployment, bettercarreer,...)

As important, but largely ignored: marriage market

Education affects




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Education affects




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Education affects

Marriage probability

Spouse’s education and incomeTotal surplus generated by marriageIntra-household division of that surplus



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Education affects

Marriage probabilitySpouse’s education and income

Total surplus generated by marriageIntra-household division of that surplus



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Education affects

Marriage probabilitySpouse’s education and incomeTotal surplus generated by marriage

Intra-household division of that surplus



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Education affects




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A Beckerian Paradox

Consider two of Becker’s fundamental intuitions:

Role of specialization

Matching as an effi cient allocation (surplus maximization)... and a paradox:

Effi ciency may require specialization (one spouse specializes indomestic production) ...... but then why should matching be assortative on education?→ In principle, if one spouse does not work, one may expect negativeassortative matchingData: clearly positive assortative, even ‘back then’, even for coupleswhere one spouse does not work

Possible answer (‘domestic capital’): domestic productivity alsoincreases with education


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A Beckerian Paradox




Effi ciency may require specialization (one spouse specializes indomestic production) ...

... but then why should matching be assortative on education?→ In principle, if one spouse does not work, one may expect negativeassortative matchingData: clearly positive assortative, even ‘back then’, even for coupleswhere one spouse does not work



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A Beckerian Paradox




Effi ciency may require specialization (one spouse specializes indomestic production) ...... but then why should matching be assortative on education?→ In principle, if one spouse does not work, one may expect negativeassortative matching

Data: clearly positive assortative, even ‘back then’, even for coupleswhere one spouse does not work



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A Beckerian Paradox




Effi ciency may require specialization (one spouse specializes indomestic production) ...... but then why should matching be assortative on education?→ In principle, if one spouse does not work, one may expect negativeassortative matchingData: clearly positive assortative, even ‘back then’, even for coupleswhere one spouse does not work



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Assortative matching in education

Figure: Wife’s education by H’s Year of Birth, US.


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This model: main ingredient

Two person household

Leisure, private and public consumption; public good domesticallyproduced

Matching based on human capital

Human capital’s role is twofold:

Increase potential wageIncrease domestic productivity

Extension: risk sharing

Basic issue: When do we expect assortative matching?

Note that the model will typically be ITU!


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Increase potential wage

Increase domestic productivity





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Increase potential wageIncrease domestic productivity





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Matching models: three main families

1 Matching under NTU (Gale-Shapley)Idea: no transfer possible between matched partners

2 Matching under TU (Becker-Shapley-Shubik)

Transfers possible without restrictionsTechnology: constant ‘exchange rate’between utilesIn particular: (strong) version of interpersonal comparison of utilities→ requires restrictions on preferences

3 Matching under Imperfectly TU (ITU)

Transfers possibleBut no restriction on preferences→ technology involves variable ‘exchange rate’


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Matching models: three main families

Similarities and differences

All aimed at understanding who is matched with whom

Only the last 2 address how the surplus is divided

Only the third allows for impact on the group’s aggregate behavior


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Formal structure: Common components

Compact, separable metric spaces X ,Y (‘women, men’) with finitemeasures F and G . Note that the spaces may be multidimensional

Spaces X ,Y often ‘completed’to allow for singles:X = X ∪ {∅} , Y = Y ∪ {∅}A matching defines of a measure h on X × Y (or X × Y ) such thatthe marginals of h are F and G

The matching is pure if the support of the measure is included in thegraph of some function φTranslation: matching is pure if y = φ (x) a.e.→ no ‘randomization’


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Formal structure: differences

Defining the problem: populations X ,Y plus

NTU: two funtions u (x , y) , v (x , y)TU: one function s (x , y) (intrapair allocation is endogenous)ITU: Pareto frontier u = P (x , y , v)

Defining the solution

NTU: only the measure h; stability as usualTU: measure h and two functions u (x) , v (y) such that

u (x) + v (y) = s (x , y) for (x , y) ∈ Supp (h)and stability

u (x) + v (y) ≥ s (x , y) for all (x , y)ITU: measure h and two functions u (x) , v (y) such that

u (x) = P (x , y , v (y)) for (x , y) ∈ Supp (h)and stability

u (x) ≥ F (x , y , v (y)) for all (x , y)


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Defining the problem: populations X ,Y plusNTU: two funtions u (x , y) , v (x , y)

TU: one function s (x , y) (intrapair allocation is endogenous)ITU: Pareto frontier u = P (x , y , v)








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Defining the problem: populations X ,Y plusNTU: two funtions u (x , y) , v (x , y)TU: one function s (x , y) (intrapair allocation is endogenous)

ITU: Pareto frontier u = P (x , y , v)








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Defining the problem: populations X ,Y plusNTU: two funtions u (x , y) , v (x , y)TU: one function s (x , y) (intrapair allocation is endogenous)ITU: Pareto frontier u = P (x , y , v)








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Defining the solutionNTU: only the measure h; stability as usual

TU: measure h and two functions u (x) , v (y) such that






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Defining the solutionNTU: only the measure h; stability as usualTU: measure h and two functions u (x) , v (y) such that


u (x) + v (y) ≥ s (x , y) for all (x , y)

ITU: measure h and two functions u (x) , v (y) such that




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Defining the solutionNTU: only the measure h; stability as usualTU: measure h and two functions u (x) , v (y) such that






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Formal structure: differences (cont.)

Characterization:

NTU: existence (Gale-Shapley), uniqueness not guaranteed (latticestructure of the set of stable matchings)ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley),uniqueness not guaranteedTU: highly specific

Stability equivalent to surplus maximizationtherefore: existence easy to establish‘generic’uniqueness

In a nutshell

NTU: intragroup allocation exogenously imposed; transfers are ruledout by assumptionTU and ITU: intragroup allocation endogenous; transfers areparamount and determined (or constrained) by equilibrium conditionsTU: life much easier (GQL → equivalent to surplus maximization) ...... but price to pay: couple’s (aggregate) behavior does not depend on‘powers’, therefore on equilibrium conditions


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Characterization:NTU: existence (Gale-Shapley), uniqueness not guaranteed (latticestructure of the set of stable matchings)

ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley),uniqueness not guaranteedTU: highly specific


In a nutshell



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Characterization:NTU: existence (Gale-Shapley), uniqueness not guaranteed (latticestructure of the set of stable matchings)ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley),uniqueness not guaranteed

TU: highly specific


In a nutshell



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Characterization:NTU: existence (Gale-Shapley), uniqueness not guaranteed (latticestructure of the set of stable matchings)ITU: existence (Kelso-Crawford’s generalization of Gale-Shapley),uniqueness not guaranteedTU: highly specific


In a nutshell



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Stability equivalent to surplus maximization

therefore: existence easy to establish‘generic’uniqueness

In a nutshell



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Stability equivalent to surplus maximizationtherefore: existence easy to establish

‘generic’uniqueness

In a nutshell



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In a nutshell



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In a nutshellNTU: intragroup allocation exogenously imposed; transfers are ruledout by assumption

TU and ITU: intragroup allocation endogenous; transfers areparamount and determined (or constrained) by equilibrium conditionsTU: life much easier (GQL → equivalent to surplus maximization) ...... but price to pay: couple’s (aggregate) behavior does not depend on‘powers’, therefore on equilibrium conditions


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In a nutshellNTU: intragroup allocation exogenously imposed; transfers are ruledout by assumptionTU and ITU: intragroup allocation endogenous; transfers areparamount and determined (or constrained) by equilibrium conditions

TU: life much easier (GQL → equivalent to surplus maximization) ...... but price to pay: couple’s (aggregate) behavior does not depend on‘powers’, therefore on equilibrium conditions


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In a nutshellNTU: intragroup allocation exogenously imposed; transfers are ruledout by assumptionTU and ITU: intragroup allocation endogenous; transfers areparamount and determined (or constrained) by equilibrium conditionsTU: life much easier (GQL → equivalent to surplus maximization) ...... but price to pay: couple’s (aggregate) behavior does not depend on‘powers’, therefore on equilibrium conditions


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Imperfectly transferable utilities

General case:

Transfers possible...

... but the ‘exchange rate’is not constant.

In practice:u (x) = P (x , y , v (y))

with P decreasing in v , usually increasing in x and y .

Stability:u (x) ≥ P (x , y , v (y)) ∀x ∈ X , y ∈ Y

But: no longer equivalent to a maximization (‘total surplus ’notdefined).


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Imperfectly transferable utility: theory

Stabilityu (x) ≥ max

yP (x , y , v (y))

and equality if marriage probability positive. Hence:

u (x) = maxyP (x , y , v (y))

1st O C:

∂P∂y(x , y , v (y)) + v ′ (y)

∂P∂v(x , y , v (y)) = 0

satisfied for x = φ (y)

Knowing φ, if ∂P/∂y > 0, v defined up to a constant by:

v ′ (y) = −∂P∂y (φ (y) , y , v (y))∂P∂v (φ (y) , y , v (y))

> 0


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Stabilityu (x) ≥ max

yP (x , y , v (y))

and equality if marriage probability positive. Hence:

u (x) = maxyP (x , y , v (y))

1st O C:

∂P∂y(x , y , v (y)) + v ′ (y)

∂P∂v(x , y , v (y)) = 0

satisfied for x = φ (y)Knowing φ, if ∂P/∂y > 0, v defined up to a constant by:

v ′ (y) = −∂P∂y (φ (y) , y , v (y))∂P∂v (φ (y) , y , v (y))

> 0


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Assortativity

1st OC:H (y , φ (y)) = 0 ∀y

where

H (y , x) =∂P∂y(x , y , v (y)) + v ′ (y)

∂P∂v(x , y , v (y)) .

therefore∂H∂y+

∂H∂x

φ′ (y) = 0 ∀y ,

2nd OC:∂H∂y≤ 0 ⇔ ∂H

∂xφ′ (y) ≥ 0.

or:(∂2P∂x∂y

(φ (y) , y , v (y)) + v ′ (y)∂2P∂x∂v

(φ (y) , y , v (y)))

φ′ (y) ≥ 0 ∀y .(1)


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Assortative: φ′ (y) ≥ 0 therefore

∂2P∂x∂y

(φ (y) , y , v (y)) + v ′ (y)∂2P∂x∂v

(φ (y) , y , v (y)) ≥ 0 ∀y . (2)

or:

∂2P∂x∂y

(φ (y) , y , v (y))−∂P∂y (φ (y) , y , v (y))∂P∂v (φ (y) , y , v (y))

∂2P∂x∂v

(φ (y) , y , v (y)) ≥ 0 ∀y .

(3)TU case: P (x , y , v (y)) = s (x , y)− v (y), hence ∂2P

∂x∂v = 0 and condition

∂2P∂x∂y

=∂2s

∂x∂y≥ 0


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The model

Two spouses, utilities:

ui (Ci , Li ,Q) = C1−δii Lδi

i + γiQ

→ heterogeneous:

MRS between leisure and private consumption (δi )preference for public good (γi )

Domestic production: Cobb-Douglas:

lnQ = α ln τ1 + αH lnH1 + β ln τ2 + βH lnH2 + θ lnX

where τi : time in home production by spouse i , X : expenditure ingoods for home production, Hi : human capital of i→ main features:

complementarity between time and HC (→ high HC parents shouldinvest more) and between spouse’s times.heterogeneous productivity of domestic timedomestic productivity depends on HC, flexible


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The model



i + γiQ

→ heterogeneous:MRS between leisure and private consumption (δi )

preference for public good (γi )






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The model



i + γiQ

→ heterogeneous:MRS between leisure and private consumption (δi )preference for public good (γi )






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The model



i + γiQ





complementarity between time and HC (→ high HC parents shouldinvest more) and between spouse’s times.

heterogeneous productivity of domestic timedomestic productivity depends on HC, flexible


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The model



i + γiQ





complementarity between time and HC (→ high HC parents shouldinvest more) and between spouse’s times.heterogeneous productivity of domestic time

domestic productivity depends on HC, flexible


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The model



i + γiQ







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The model (cont.)

Budget constraint

C1 + C2 + w1L1 + w2L2 = Y − (w1τ1 + w2τ2 + pX )

where Y = y + w1 + w2 : total household resources

Wages → two versions:

without uncertaintywi = WiHi

with uncertaintywi = WiHi ei


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Solution: conditional sharing rule (BCM)

Two stage process

Stage 1: decide on Q, therefore on (τ1, τ2,X ) (cost minimization),and allocate Y − (w1τ1 + w2τ2 + pX ) between spousesStage 2: agents each choose private consumption and labor supply

Backward resolution → stage 2:

maxCi ,Li

C 1−δii Lδi

i

st ρi = Ci + wiLi

gives

Li = δiρiwi, Ci = (1− δi ) ρi and

vi = (1− δi )1−δi

(δiwi

)δi

ρi + γiQ

= ∆i (wi )−δi ρi + γiQ


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Two stage processStage 1: decide on Q, therefore on (τ1, τ2,X ) (cost minimization),and allocate Y − (w1τ1 + w2τ2 + pX ) between spouses

Stage 2: agents each choose private consumption and labor supply


maxCi ,Li

C 1−δii Lδi

i

st ρi = Ci + wiLi

gives


vi = (1− δi )1−δi

(δiwi

)δi

ρi + γiQ



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Two stage processStage 1: decide on Q, therefore on (τ1, τ2,X ) (cost minimization),and allocate Y − (w1τ1 + w2τ2 + pX ) between spousesStage 2: agents each choose private consumption and labor supply


maxCi ,Li

C 1−δii Lδi

i

st ρi = Ci + wiLi

gives


vi = (1− δi )1−δi

(δiwi

)δi

ρi + γiQ



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Stage 1: cost minimization

Effi ciency implies cost minimization:

minτ1,τ2,X

w1τ1 + w2τ2 + pX

such that

lnQ = α ln τ1 + αH lnH1 + β ln τ2 + βH lnH2 + θ lnX .

Solution

w1τ1 + w2τ2 + pX = κ′(w α1w

β2 p

θH−αH1 H−βH

2 Q) 1

α+β+θ

where

κ′ = (α+ β+ θ) exp(− (α ln α+ β ln β+ θ ln θ)

α+ β+ θ

)> 0


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Pareto frontier

Equation:vi = ∆i (wi )

−δi ρi + γiQ

gives∆−11 w

δ1 v1 + ∆−12 w

δ2 v2 = s (H1,H2)

with

s (H1,H2) = maxQ

{ρ1 + ρ2 +Q

(γ1∆1w δ1 +

γ2∆2w δ2

)}= max

Q

{Y − (w1τ1 + w2τ2 + pX ) +Q

(γ1∆1w δ1 +

γ2∆2w δ2

)}Note: straight line, but not TU since the slope depends on the spouse’scharacteristics


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Public good

Optimal amount Q solves:

maxQ

[Y − (w1τ1 + w2τ2 + pX ) +Q

(γ1∆1w δ1 +

γ2∆2w δ2

)]or

maxQ

[Y − κ′

(w α1w

β2 p

θH−αH1 H−βH

2 Q) 1

α+β+θ+Q

(γ1∆1w δ1 +

γ2∆2w δ2

)]which gives:

Q = κ′′(

γ1∆1w δ1 +

γ2∆2w δ2

) (α+β+θ)1−(α+β+θ) (

w α1w

β2 p

θH−αH1 H−βH

2

) 1α+β+θ−1

and finally

s (H1,H2) = Y + κ′′(1− κ′

) (γ1∆1w δ1 +

γ2∆2w δ2

) 11−(α+β+θ)

(w α1w

β2 p

θH−αH1 H−βH

2

) −11−(α+β+θ)


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Pareto frontier

∆−11 wδ1 v1 + ∆−12 w

δ2 v2 = s (H1,H2)

therefore

v1 = ∆1w−δ1 s (H1,H2)− ∆1∆−12 w

−δ1 w δ

2 v2= P (H1,H2, v2)

TheoremSuffi cient condition for PAM:

αH − α ≥ 0 and βH − β ≥ 0


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Intuition

Fact1:∂2P

∂H1∂v2= ∆1∆−12 δw−δ−1

1 w δ2 > 0

Transfering utility is relatively cheaper for wealthy people

→ Suffi cient condition for PAM:∂2P

∂H1∂H2≥ 0

Fact 2:∂2P

∂H1∂H2=

∂2Q∂H1∂H2

+ (≥ 0)

→ Suffi cient condition for PAM:∂2Q

∂H1∂H2≥ 0

Fact 3:∂2Q

∂H1∂H2≥ 0 if αH − α ≥ 0 and βH − β ≥ 0


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Risk

Need a cardinalization:

ui (Ci , Li ,Q) =

(C 1−δii Lδi

i + γiQ)η

η

Then

vi =

(∆iw−δ

i ρi + γiQ)η

η

where

Q (w1,w2) = κ′′(

γ1∆1w δ1 +

γ2∆2w δ2

) (α+β+θ)1−(α+β+θ) (

w α1w

β2 p

θH−αH1 H−βH

2

) 1α+β+θ−1


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Risk

Effi cient Risk Sharing:

maxρ1,ρ2

∫ (∆1w−δ

1 ρ1 + γ1Q (w1,w2))ηf (e) de

+µ∫ (

∆2w−δ2 ρ2 + γ2Q (w1,w2)

)ηf (e) de

under

ρ1 + ρ2 = Y − κ′(w α1w

β2 p

θH−αH1 H−βH

2 Q (w1,w2)) 1

α+β+θ

Note that if

u (ρ) =(

∆1w−δ1 ρ+ γ1Q (w1,w2)

)η

then−u′′ (ρ)u′ (ρ)

= − (η − 1)

ρ+ γ1

(∆1w−δ

1

)−1Q (w1,w2)

and preferences are ISHARA, therefore TU (SW 2007)


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Conclusion

1 Domestic production (especially children) has a potentially crucialimpact on matching patterns

2 Potential conflict between two logics - specialization versus PAM3 PAM requires HC to be an important factor in the ‘childrenproduction function’.

4 Risk sharing may reinforce the PAM logic, although the mechanismsare quite complex:

High HC people have higher but risquier income (or wage)Still better to marry a high HC person ...... but what matters is the ‘second cross derivative’→ less clear!Previous work (CCM): without domestic production, still PAMWith domestic production: TBE!


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Conclusion


2 Potential conflict between two logics - specialization versus PAM

3 PAM requires HC to be an important factor in the ‘childrenproduction function’.




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Conclusion






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Conclusion




High HC people have higher but risquier income (or wage)

Still better to marry a high HC person ...... but what matters is the ‘second cross derivative’→ less clear!Previous work (CCM): without domestic production, still PAMWith domestic production: TBE!


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Conclusion




High HC people have higher but risquier income (or wage)Still better to marry a high HC person ...

... but what matters is the ‘second cross derivative’→ less clear!Previous work (CCM): without domestic production, still PAMWith domestic production: TBE!


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Conclusion




High HC people have higher but risquier income (or wage)Still better to marry a high HC person ...... but what matters is the ‘second cross derivative’→ less clear!

Previous work (CCM): without domestic production, still PAMWith domestic production: TBE!


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Conclusion




High HC people have higher but risquier income (or wage)Still better to marry a high HC person ...... but what matters is the ‘second cross derivative’→ less clear!Previous work (CCM): without domestic production, still PAM

With domestic production: TBE!


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Conclusion






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The Marriage Market, Labor Supply and Education ChoicesThe Marriage Market, Labor Supply and...

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