The Marriage Market, Labor Supply and Education ChoicesThe Marriage Market, Labor Supply and...
Transcript of 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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 2
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
/ 26
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 probability
Spouse’s education and incomeTotal surplus generated by marriageIntra-household division of that surplus
Matching models useful to analyze these returns
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
/ 26
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 income
Total surplus generated by marriageIntra-household division of that surplus
Matching models useful to analyze these returns
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
/ 26
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 marriage
Intra-household division of that surplus
Matching models useful to analyze these returns
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 3
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 4
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 4
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 4
/ 26
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 matching
Data: clearly positive assortative, even ‘back then’, even for coupleswhere one spouse does not work
Possible answer (‘domestic capital’): domestic productivity alsoincreases with education
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 4
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 4
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 4
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Assortative matching in education
Figure: Wife’s education by H’s Year of Birth, US.
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 5
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
/ 26
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 wage
Increase domestic productivity
Extension: risk sharing
Basic issue: When do we expect assortative matching?
Note that the model will typically be ITU!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 6
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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’
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 7
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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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 8
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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’
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 9
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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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
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Formal structure: differences
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)
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
Formal structure: differences
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)
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
Formal structure: differences
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)
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
Formal structure: differences
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)
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
Formal structure: differences
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)
Defining the solutionNTU: only the measure h; stability as usual
TU: 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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
Formal structure: differences
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)
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 (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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
Formal structure: differences
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)
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 (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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 10
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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 guaranteed
TU: 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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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 maximization
therefore: 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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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 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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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 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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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 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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 11
/ 26
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).
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 12
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 13
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 13
/ 26
Imperfectly transferable utility: theory
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 14
/ 26
Imperfectly transferable utility: theory
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 14
/ 26
Imperfectly transferable utility: theory
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 15
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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 time
domestic productivity depends on HC, flexible
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 16
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 17
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 17
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 17
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 17
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 18
/ 26
Solution: conditional sharing rule (BCM)
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
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 18
/ 26
Solution: conditional sharing rule (BCM)
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
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 18
/ 26
Solution: conditional sharing rule (BCM)
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
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 18
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 19
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 20
/ 26
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 21
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 22
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 23
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 23
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 23
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 24
/ 26
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
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 24
/ 26
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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 25
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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)
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 25
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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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
Conclusion
1 Domestic production (especially children) has a potentially crucialimpact on matching patterns
2 Potential conflict between two logics - specialization versus PAM
3 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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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 PAM
With domestic production: TBE!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
/ 26
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!
Chiappori, Costa, Meghir (Columbia University, UCL and IFS, Yale University)Labor Supply and EducationWorkshop ‘Risk and Family Economics’IFS, June 2014 26
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