matching models of the marriage market: theory and empirical

Matching models of the marriage market: theory andempirical applications

Pierre-André Chiappori Columbia University

Cemmap Masterclass, March 2011

P.A. Chiappori, Columbia () Matching modelsCemmap Masterclass, March 2011 1 /

20

Overview

Session 1: Background: Collective models of household behavior

Session 2: Matching models: theory (Part 1)


Session 4: Matching models: applications

Session 5: Matching models: empirical applications (Part 1)



20

Overview








20

Session 1

Background: Collective models of householdbehavior


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Gains from marriage (what do we want to model?)

Ecclesiastes (4: 9-10) ; "Two are better than one, becausethey have a good reward for their toil. For if they fall, one willlift up the other; but woe to one who is alone and falls and doesnot have another to help. Again, if two lie together, they keepwarm; but how can one keep warm alone?"


20

Gains from marriage (what do we want to model?)

1 Sharing of public (non rival) goods. For instance, both partners canequally enjoy their children, share the same information and use thesame home.

2 Division of labor to exploit comparative advantage and increasingreturns to scale. For instance, one partner works at home and theother works in the market.

3 Home production. For example, coordinating child care, (which is apublic good for the parents).

4 Extending credit and coordination of investment activities. Forexample, one partner works when the other is in school.

5 Risk pooling. For example, one partner works when the other is sickor unemployed.


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Modeling household behavior: the �unitary�framework

Drawbacks:

Contradicts individualism

Weak theoretical foundations

Samuelson: postulates a group utility W [U1, . . . ,US ]; W must beindependent of prices, wages, distribution factors,...Becker �rotten kid�: concludes that the group behaves as a singledecision maker.But: strong assumptions required:

Speci�c decision processTransferable utility and/or production functionIn a sense, assumes away most heterogeneity

Group as a �black box�

Group formation, dissolution,. . .Intragroup allocation ignored (inequality)�Power�issues ignored: income pooling (ex: targeting)


20

Modeling household behavior: the collective approach

Basic ideas:

Di¤erent individuals may have di¤erent preferences

Emphasis put on decision process

Hence: natural interpretation of �power�

Addresses issues of power redistribution: targeting, . . .


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Modeling household behavior: the collective approach

General assumption: Pareto e¢ ciency

Justi�cations:

General principle (�no money left on the table�)Repeated interactionsCan be seen as a benchmark

Note, however, that commitment may be a problem

Basic insight:! powers summarized by Pareto weights


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The collective approach: key concepts

Distribution factors:

Any factor that:- in�uences the decision process

- does not a¤ect preferences or budget sets

Examples: income, wealth ratios; control of land; background familyfactors; sex ratio; divorce laws; targeted bene�ts

Interpretation:

threat points in a bargaining contextsocial weight in a sociological interpretationstrategic position in a market contextetc.


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The collective approach: main questions

Three main questions:

Testability (on demand data)

Identi�ability (recovering preferences and �power�); ! Distinctionidenti�ability/identi�cation

Theoretical underpinning: where do Pareto weights come from?


20

The collective approach: the setting

S agents; consumptions : public X 2 RN , private xs 2 Rn; note that(x1, ..., xS ) may not be observed.

Prices P, p; (group) income y ; intragroup production could beintroduced.

Distribution factors zk , k = 1, . . . ,K! could include individual incomes

Preferences : Us (X , x1, ..., xS ) (most general case)Particular cases:

�egoistic�: Us (X , xs )

�caring�: W shU1 (X , x1) , ...,US (X , xS )

iNote that: an allocation that is e¢ cient for caring is e¢ cient foregoistic as well

TU:Us (X , xs ) = F s

�As�X , x2s , ..., x

ns

�+ x1s b (X )

�P.A. Chiappori, Columbia () Matching models

Cemmap Masterclass, March 2011 12 /20


(Aggregate) market demand:

ξ = (X , x1 + ...+ xS )

as function of π = (P, p), y and possibly z ; budget constraint:

π0ξ = P 0X + p0 (x1 + ...+ xS ) = y

E¢ ciency: for all (p,P, y), there exist µ1, ..., µS withµ1 + ...+ µS = 1 such that (X , x1, . . . , xS ) solves:

maxX ,x1,...,xS

∑s

µsUs under the BC

or the more general version (with production):

maxX ,x1,...,xS

∑s

µsUs (P)

underΦ (X , x1 + ...+ xS ) = ξ and π0ξ = y


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Key remark: in general, µ1, ..., µS depend on prices and incomes (anddistribution factors)

Therefore:

may de�ne household utility

UG (ξ, µ) = maxX ,x1,...,xS

∑s

µsUs s.t. Φ (X , x1 + ...+ xS ) = ξ

but it is price dependent.

Particular case: private goods only. Then e¢ ciency equivalent to theexistence of a sharing rule: there exists ρ = (ρ1, ..., ρS ) with∑ ρs = y such that xs solves

maxUs (xs ) st p0xs = ρs

Questions: testability and identi�cation


20

The collective approach: testability (price variations)

Normalization: y = 1; therefore Slutsky matrix S(π) = (Dπξ)�I � πξT

�Basic result (Browning-Chiappori 1998) :

Proposition

(The SNR(H � 1) condition). If the C 1 function ξ(π) solves problem(P), then the Slutsky matrix S(π) can be decomposed as:

S (π) = Σ (π) + R (π) (1)

where:

the matrix Σ (π) is symmetric and negativethe matrix R (π) is of rank at most H � 1.

Equivalently, there exists a subspace E (π) of dimension at least N + n� Ssuch that the restriction of S (π) to E (π) is symmetric, de�nite negative


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The collective approach: testability (no price variation)

Engel curves: ξ i (y , z1, ..., zk )

De�nition of z-demands: assume there is at least one good j and oneobservable distribution factor, say z1, such that ξ j (y , z) strictlymonotone in z1. Then:

z1 = ζ(y , z�1, ξ j )

where z = (z1, z�1). Then substituting into the demand for good i 6= j :

ξ i (y , z1, z�1) = ξ i [y , ζ(y , z�1, ξ j ), z�1] = θij (y , z�1, ξj ).


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The collective approach: testability (no price variation,BBC 2009)

PropositionA given system of demand functions is compatible with collectiverationality if and only if either K = 1 or it satis�es any of the following,equivalent conditions :i) there exist real valued functions Ξ1, .....,Ξn and µ such thatξ i (y , z) = Ξi [y , µ(y , z)] 8iii) household demand functions satisfy

∂ξ i/∂zk∂ξ i/∂zl

=∂ξ j/∂zk∂ξ j/∂zl

8i , j , k, l

iii) there exists at least one good j such that:

∂θij (y , z�1, ξj )

∂zk= 0 8i 6= j and k = 2, ..,K


20

The collective approach: identi�ability (C-E 2009)

Basic assumptions:

Egoistic preferences

Exclusion restrictions: for each agent s, there exists at elast onecommodity that s does not consume

Basic result (C-E 2009):

PropositionGenerically, the welfare relevant structure can be recovered

Moreover, in the absence of price variations, if all goods are private (orunder separability):

The sharing rule can be identi�ed up to a constant

Individual Engel curves can be recovered

Exclusion not needed!


20

The collective approach: empirical issues

Empirical tests:

various distribution factors (income, wealth ratios; control of land;background family factors; sex ratio; divorce laws; targeted bene�ts;...)various behavior: labor supply (CFL 2002), consumption ofgender-speci�c commodities (BBCL 1994), food consumption(Attanasio Lechene 2009), ...unitary restrictions generally rejected (income pooling)collective restrictions generally not rejected

Sex ratio: link with with the maket for marriage (Becker)


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The collective approach: open issues

Dynamics and commitment

Risk and risk sharing

�Upstream�model: the collective model takes as given:

group compositiondecision process (Pareto weights)

! Questions:

group formation: who marries whom and why (and dissolution)?distribution of powers as an endogenous phenomenon

Basic tools:

bargaining theorymatching models (frictionless)search models (search frictions)


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43

Overview








43

Session 2

Matching models: theory (1)


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Background: The Collective Model

Takes as given:

group compositiondecision process (Pareto weights)

Questions:

group formation: who marries whom?distribution of powers as an endogenous phenomenon

Basic tools:

matching models (frictionless)search models (search frictions)bargaining theory

Here: emphasis on matching models; couples only; mostly TU


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Transferable Utility (TU)

De�nitionA group satis�es TU if there exists monotone transformations of individualutilities such that the Pareto frontier is an hyperplane for all values ofprices and income.

In practice:

Quasi Linear (QL) preferences (but highly unrealistic)�Generalized Quasi Linear (GQL, Bergstrom and Cornes 1981):

us (qs ,Q) = Fs�As�q2s , ..., q

ns ,Q

�+ q1s bs (Q)

�with bs (Q) = b (Q) for all s

Note:

Ordinal propertyRestrictions on heterogeneityBut �acceptably�realistic


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Properties of Transferable Utility (TU)

Unanimity regarding group�s decisions

clear distinction between aggregate behavior and intragroup allocationof power/resources/welfarehere: concentrate of �power�issues

Matching models: ! mathematical structure: optimal transportationmodels


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Optimal Transportation Problems (Monge-Kantorovitch)

General Structure:

Complete, separable metric spaces X ,Y with measures F and GSurplus s (x , y) upper semicontinuousProblem: �nd a measure h on X � Y such that the marginals of h areF and G , and h solves

maxh

ZX�Y

s (x , y) dh (x , y)

Hence: linear programming

Dual problem: dual functions u (x) , v (y) and solve

minu,v

ZXu (x) dF (x) +

ZYv (y) dG (y)

under the constraint

u (x) + v (y) � s (x , y) for all (x , y) 2 X � Y


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General Structure:Complete, separable metric spaces X ,Y with measures F and G

Surplus s (x , y) upper semicontinuousProblem: �nd a measure h on X � Y such that the marginals of h areF and G , and h solves

maxh

ZX�Y

s (x , y) dh (x , y)



minu,v

ZXu (x) dF (x) +

ZYv (y) dG (y)




43


General Structure:Complete, separable metric spaces X ,Y with measures F and GSurplus s (x , y) upper semicontinuous

Problem: �nd a measure h on X � Y such that the marginals of h areF and G , and h solves

maxh

ZX�Y

s (x , y) dh (x , y)



minu,v

ZXu (x) dF (x) +

ZYv (y) dG (y)




43


General Structure:Complete, separable metric spaces X ,Y with measures F and GSurplus s (x , y) upper semicontinuousProblem: �nd a measure h on X � Y such that the marginals of h areF and G , and h solves

maxh

ZX�Y

s (x , y) dh (x , y)



minu,v

ZXu (x) dF (x) +

ZYv (y) dG (y)




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Matching Models under Transferable Utility

Becker-Shapley-Shubik (as opposed to Gale-Shapley)

Same general structure

Complete, separable metric spaces X ,Y with measures F and GSurplus s (x , y) upper semicontinuous

A matching consists of:

a measure h on X � Y such that the marginals of h are F and Gtwo functions u (x) , v (y) (�imputations�) such that

u (x) + v (y) = s (x , y) for all (x , y) 2 Supp (h)

Stability: the matching is stable if:


Interpretation


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Stability: assume

u (x) + v (y) < s (x , y) for all (x , y) 2 X � Y

Interpretation


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Stability: the matching is stable if:


Interpretation: �divorce at will�


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Basic Result (duality)

We have the following:

TheoremA matching is stable if and only if it solves the surplus maximizationproblem

Consequence: existence; generic uniqueness of the measure underadditional conditions

Note that:

sets can be multidimensionalextends to hedonic models.


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Note that:

sets can be multidimensional

extends to hedonic models.


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Note that:

sets can be multidimensionalextends to hedonic models.


43

Hedonic Models

Structure:

Separable metric spaces X, Y, Z; measures F, GUtilities u(x,z) and c(z,y) upper semicontinuous;Problem: �nd a pricing function P(z) such that if:

x solves maxz u(x , z)� P(z)y solves maxz P (z)� c (y , z)

then markets clear

Basic property: equivalent to a matching with

s(x , y) = maxz(u(x , z)� c(y , z))

(see Chiappori-McCann-Neishem 2010)

Note that: n-dimensional; single crossing not needed; covexity notneeded.


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Proof

Start from:

u(x) + v(y) � s (x , y) � u (x , z)� c (y , z) on X � Y � Z ,

hencec (y , z) + v (y) � u (x , z)� u(x) on X � Y � Z

andinfy2Y

fc (y , z) + v (y)g � supx2X

fu (x , z)� u (x)g on Z .

Take any P (z) such that

infy2Y

fc (y , z) + v (y)g � P (z) � supx2X

fu (x , z)� u (x)g on Z .


43

Supermodularity and assortative matching

One-dimensional:

s is supermodular if whenever x � x 0 and y � y 0 then

s (x , y) + s�x 0, y 0

�� s

�x , y 0

�+ s

�x 0, y

�Then stable matching is assortative; indeed, surplus maximization

Interpretation: single crossing (Spence - Mirrlees). Assume that s isC 1 then

s (x , y)� s�x 0, y

�� s

�x , y 0

�� s

�x 0, y 0

�and ∂s/∂x increasing in y ; if s is C 2 then

∂2s∂x∂y

� 0

Of course, similar results with submodularity (∂s/∂x decreasing in y)

In both case, ∂s/∂x monotonic in y ; if strict then injective


43

Supermodularity and assortative matching

Problem: both super- (or sub-) modularity and assortative matchingare typically one-dimensional

Generalization (CMcCN ET 2010):

De�nitionA surplus function s : X �Y �! [0,∞[ is said to be X�twisted if there isa set XL � X0 of zero volume such that ∂x s(x0, y1) is disjoint from∂x s(x0, y2) for all x0 2 X0 n XL and y1 6= y2 in Y .

Then the stable matching is unique and pure

De�nitionThe matching is pure if the measure µ is born by the graph of a function:for almost all x there exists exactly one y such that x matched with y .

! excludes �mixed strategies�


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Counter example (C-McC-N 09)Transportation on a circle


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Lake, students

Lake, students, schools


Problem: allocate children to schoolsso as to minimize total transportation cost

Interpretation: matching with transfers (�tuition�)

Note that: single crossing cannot hold!


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Stable match:


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Application: marriage market


43

Application: marriage markets

Structure:

Men and women, respective income distributions F and G; mass 1 formen, r for womenTU; surplus s(x , y), derived from a collective modelIn general, TU implies s(x , y) supermodularExample (CIW07):

uh = Q�1+ qh

�uw = Q (a+ qw )

then

s (x , y) =(x + y + 1+ a)2

4Note that:

surplus depends on total income: s (x , y) = H (x + y)H convex (due to TU) therefore supermodularity


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Application: marriage markets

Who marries whom? Assortative

Characterization: for any couple (x , y), the number of men withincome larger than x equals the number of women with income largerthan y :

1� F (x) = r (1� G (y))Therefore

x = φ (y) = F�1 [1� r (1� G (y))]

y = ψ (x) = G�1�1� 1

r(1� F (x))

�If r > 1 then y0 = ψ (0) �last married woman�

If r < 1 then x0 = φ (0) �last married man�


43

Matching with imperfectly transferable utility

General case: the Pareto frontier has the form

u = H (x , y , v) (1)

with H(0, 0, v) = 0 for all v . Here, H is decreasing in v and increasing inx and y .Two remarks:

stability can still be de�ned:

u (x) � H (x , y , v (y)) for all (x , y)

but no longer equivalent to surplus maximization


43

Session 3

Matching models: theory (2)


43

Sharing the surplus

Basic result:

Intramatch allocation of welfare is pinned downby the equilibrium conditions


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Sharing the surplus

Stability impliesu (x) = max

y(s (x , y)� v (y))

therefore

u0 (x) =∂s∂x(x ,ψ (x)) and v 0 (y) =

∂s∂y(φ (y) , y)

and

u (x) = k +Z x

0

∂s∂x(t,ψ (t)) dt , v (y) = k 0 +

Z y

0

∂s∂y(φ (s) , s) ds

where k + k 0 given

Pinning down the constant: if r > 1 the �last married�womanreceives no surplus

Hence: endogeneize intrahousehold allocation


43

Sharing the surplus: application

Assume that s (x , y) = H (x + y), H convex. Then

u0 (x) = H 0 (x + ψ (x)) , v 0 (y) = H (φ (y) + y)

Application: �linear shift�: r = 1 and F (t) = G (αt � β) for someα < 1, β > 0

Satis�ed for instance if Lognormal distributions with same σ.P.A. Chiappori, Columbia () Matching models


Example: shifting female income distribution

Then φ (y) = (y + β) /α and ψ (x) = αx � β; from

u0 (x) = H 0 ((α+ 1) x � β) ,

one gets

u (x) = K 0 +1

1+ αH (x + ψ (x))

and

v (y) = K +α

1+ αH (φ (y) + y)

where K +K 0 = 0


43


Consider an upward shift in female income: y becomes ky with k > 1.Then:

same matching patterns

but changes in the redistribution of surplus:

∂vk∂k

=αy

α+ 1H 0 (y + x) +

α

(α+ 1)2H (y + x) and

∂uk∂k

=y

α+ 1H 0 (y + x)� α

(α+ 1)2H (y + x)


43





∂vk∂k

=αy

α+ 1H 0 (y + x)

∂uk∂k

=y

α+ 1H 0 (y + x)


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∂vk∂k

=αy

α+ 1H 0 (y + x) +

α

(α+ 1)2H (y + x) and

∂uk∂k

=y

α+ 1H 0 (y + x)� α

(α+ 1)2H (y + x)


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As above, stability implies that:

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

where the maximum is actually reached for y = ψ (x) (or x = φ (y)).First order conditions imply that

∂H∂y(φ (y) , y , v (y)) + v 0 (y)

∂H∂v(φ (y) , y , v (y)) = 0.

or:

v 0 (y) = �∂H∂y (φ (y) , y , v (y))∂H∂v (φ (y) , y , v (y))

.

and

u0 (x) =∂H∂x(x ,ψ (x) , v (ψ (x)))


43


Second order conditions:

∂

∂y

�∂H∂y(φ (y) , y , v (y)) + v 0 (y)

∂H∂v(φ (y) , y , v (y))

�� 0 8y .

Since

F (y , φ (y)) = 0 8ywhere

F (y , x) =∂H∂y(x , y , v (y)) + v 0 (y)

∂H∂v(x , y , v (y)) .

we have:∂F∂y+

∂F∂x

φ0 (y) = 0 8y ,

which implies that

∂F∂y� 0 if and only if

∂F∂x

φ0 (y) � 0.


43


The second order conditions can hence be written as:�∂2H∂x∂y

(φ (y) , y , v (y)) + v 0 (y)∂2H∂x∂v

(φ (y) , y , v (y))�

φ0 (y) � 0 8y .(2)

Assortative matching: φ0 (y) � 0, which holds if

∂2H∂x∂y

(φ (y) , y , v (y)) + v 0 (y)∂2H∂x∂v

(φ (y) , y , v (y)) � 0 8y . (3)

Since v 0 (y) � 0, a su¢ cient (although obviously not necessary) conditionis that

∂2H∂x∂y

(φ (y) , y , v (y)) � 0 and ∂2H∂x∂v

(φ (y) , y , v (y)) � 0. (4)

Note: under TU, H (x , y , v (y)) = h (x , y)� v (y) and ∂2H∂x∂v = 0


43


v

u

P

P’

Figure: The slope conditionP.A. Chiappori, Columbia () Matching models


Matching with imperfectly transferable utility: an example

continuum of males (income x distributed over [a,A], cdf F )

continuum of females (income y distributed over [b,B ] cdf G ).

Linear shift case φ (y) = (y + β) /α and ψ (x) = αx � β

Number of female is almost equal to, but slightly larger than that ofmen

Male preferences: Cobb-Douglas

um = cmQ

Female preferences: perfect substitutes

uf (cf ) = �∞ if cf < c

= cf +Q if cf � c

In particular, if a woman is single, her income must be at least c ; thenher utility equals her income.


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Matching with imperfectly transferable utility: an example

E¢ cient allocations:max cmQ

under the constraints

cm + cf +Q = x + y , uf = cf +Q � U

Remark: at any e¢ cient allocation:

cf = cU � ((x + y) + c) /2.

Pareto frontier:

um = H ((x + y) , uf ) = (uf � c) ((x + y)� uf ) , (5)

where uf � (x+y )+c2 ,and Q = uf � c , cm = (x + y)� uf .


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3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.20

1

2

3

4

u

v

Figure: Pareto frontier


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Characterization

∂H (x + y , v)∂ (x + y)

= v � c , ∂H (x + y , v)∂v

= � (2v � (c + (x + y)))

therefore assortative matching since

∂2H (x + y , v)

∂ (x + y)2= 0 and

∂2H (x + y , v)∂ (x + y) ∂v

= 1

Moreover:

v 0 (y) =αv (y)� αc

2αv (y)� (α+ 1) y � (αc + β)> 0 (6)


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2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

2

4

6

8

x

Utilities

Figure: Husband�s and Wife�s Utilities, Public Consumption and the Husband�sPrivate Consumption (β = 0, α = .8)


43

Summary

matching as tractable way of modeling general equilibrium interactions

main advantage: explicit derivation of the sharing rule

TU: equivalent to linear programming

More general models are feasible


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18

Overview








18

Session 4

Matching models: Applications


18

Applications: applied theory

Various applications:

Calibration: impact of female education on intrahousehold allocation

Abortion and female empowerment (CO JPE 2006)

Children and divorce (CW JoLE 2007)

Male and female demand for higher education (CIW AER 2009)

Dynamics: divorce and impact of divorce laws (CIW 10)


18

Motivation: education and marriage, US


18

Motivation: education and marriage, USEducation


18



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Asymmetric reactions to increasing returns to education

Possible explanations:

1 Gender discrimination smaller for high incomes?2 Ability?3 Intrahousehold e¤ects

Returns to education have two components: market and intrahouseholdIf larger percentage of educated women, a¤ects matching patternsCost of not being educated are higher


18


18

The model

Two equally large populations of men and women to be matched.

Individuals live two periods; investment takes place in the �rst periodof life and marriage in the second period; investment in schooling islumpy and takes one period.

All agents of the same level of schooling and gender receive the samewage.

I (i) and J(j) are the schooling "class" of man i and woman j (1 ifuneducated, 2 if educated)

Transferable utility; marital surplus if man i marries woman j:

sij = zI (i )J (j) + θi + θj

withz11 + z22 > z12 + z21


18

The model

Investment in schooling is associated with idiosyncratic cost (bene�t),µi for men and µj for women.

θ and µ independent from each other and independent acrossindividuals; distributions F (θ) and G (µ).

Shadow price of woman j is uj , shadow price of man i is vi ; stability:

zI (i )J (j) + θi + θj � vi + uj

therefore

vi = max�maxj

�zI (i )J (j) + θi + θj � uj

�, 0�

uj = max�maxi

�zI (i )J (j) + θi + θj � vi

�, 0�


18

θ1V−2V−

b

mR

2 1mR V V+ −

µ

Investment,no marriage

Marriage, noinvestment

Marriage andinvestment

No marriage,no investment


18

The model

Proposition:

vi = VI (i ) + θi

uj = UJ (j) + θj

with

VI = maxJ(zIJ � UJ )

UJ = maxI(zIJ � VI )


18

The model

Theorem:

U1 + V1 = z11U2 + V2 = z22

and either strict assortative matching:

U1 + V2 � z21U2 + V1 � z12

or some people �marry down�. If more educated men:

U1 + V2 = z21 therefore U2 � U1 = z22 � z21,V2 � V1 = z21 � z11

whereas if more educated women:

U2 + V1 = z12 therefore U2 � U1 = z12 � z11,V2 � V1 = z22 � z21


18

The model

Equilibrium equations:

Example: # married men = # married women

F (V1)+

V2ZV1

G (Rm +V2� θ)f (θ)dθ = F (U1)+

U2ZU1

G (Rw +U2� θ)f (θ)dθ,

Example: # educated married men = # educated married women

F (V1) [1� G (Rm + V2 � V1)] = F (U1) [1� G (Rw + U2 � U1)]

Then calibration


18

Comparative statics

Compare an "old" regime a "new" regime.

In the old regime:lower returns to educationmore time to be spent at home ! specialization

New regime:higher returns to educationless time spent at home ! both genders demand education

Initial sources of asymmetry?statistical discrimination weaker against educated womendi¤erences in innate ability (Becker-Hubbard-Murphy)impact of birth control

Equilibrium mechanism ampli�es the shiftOld regime: some men marry down, marital returns of schooling higherfor menNew regime: just the opposite (�you can�t a¤ord to be a nurse�)


18


18


Pierre-André Chiappori, Columbia University



34

Overview








34

Session 5

Matching models: empirical applications

Part 1: a semi-structural model


34

Motivation: education and marriage, US


34



34

Motivation: education and marriage, USMarital patterns


34



34


Burtless (EER 1999): over 1979-1996:�The changing correlation of husband and wife earnings has tended toreinforce the e¤ect of greater pay disparity.�

Maybe 1/3 of the increase in household-level inequality (Gini) comesfrom rise of single-adult households and 1/6 from increasedassortative matching. Burtless again, US 1979-1996:�The Spearman rank correlation of husband and wife earningsincreased from .012 to .145�


34

Questions

1 Education: Why the asymmetric response between men and women?Possible answer (CIW, AER 2009): �Marital College Premium�! Is there evidence for a MCP?

2 Changes in matching patterns:

change in attitudes toward assortative matching?or only a mechanical consequences of the changes in populationcompositions?

3 Matching models: scope and empirical implementation

Basic tool: frictionless matching

Reference:

Choo and Siow (JPE 2003)Chiappori, Salanié and Weiss 2010


34

Questions



change in attitudes toward assortative matching?

or only a mechanical consequences of the changes in populationcompositions?



Reference:



34

Questions






Reference:



34

Questions






Reference:

Choo and Siow (JPE 2003)

Chiappori, Salanié and Weiss 2010


34

Questions






Reference:



34

Empirical implementation

Basic issue: capturing imperfections (frictions) in the matching process

Various strategies:

Random matching (Shimer-Smith)

Search

Frictionless matching with unobservable components


34


Basic insight: unobserved characteristics (heterogeneity)! Gain g IJij generated by the match i 2 I , j 2 J:

g IJij = ZIJ + εIJij

where I = 0, J = 0 for singles, and εIJij random shock with mean zero.

Therefore: dual variables (ui , vj ) also random (endogenousdistribution)

Stability: constrained by the inequalities

ui + vj � g IJij for any (i , j)

! large number (one inequality per potential couple)


34


Crucial identifying assumption (Choo-Siow 2006)Assumption S (separability): the idiosyncratic component εij isadditively separable:

εIJij = αIJi + βIJj (S)

where E�αIJi�= E

hβIJj

i= 0.

Then:

Theorem

Under S, there exists U IJ and V IJ such that U IJ + V IJ = Z IJ and for anymatch (i 2 I , j 2 J)

ui = U IJ + αIJi

vj = V IJ + βIJj

General characterization: Galichon-Salanie (2011)


34

Galichon-Salanié 2011

Assume separability, let πIJ the proba that i 2 I marries j 2 J; thenSocial surplus is

Σ = 2∑i ,j

πIJZIJ + ε (nI , nJ ,π)

where ε is a �generalized entropy�

When α, β are extreme values, then

ε (nI , nJ ,π) = �∑I

∑J

πIJ lnπIJnI�∑

I∑J

πIJ lnπIJnJ,

Can be used for identi�cation


34


TheoremA NSC for i 2 I being matched with a spouse in J is:

U IJ + αIJi � U I0 + αI0i

U IJ + αIJi � U IK + αIKi for all K

In practice:

take singlehood as a benchmark (interpretation!)assume the αIJi are extreme value distributedthen logit and expected utility:

uI = E�maxJ

�U IJ + αIJi

��= ln

∑JexpU IJ + 1

!= � ln

�aI0�

Problem: can we compare the u across classes?

needed (marital college premium)but relies on a strong assumption (�What is the unit?�)


34





In practice:take singlehood as a benchmark (interpretation!)

assume the αIJi are extreme value distributedthen logit and expected utility:

uI = E�maxJ

�U IJ + αIJi

��= ln

∑JexpU IJ + 1

!= � ln

�aI0�




34





In practice:take singlehood as a benchmark (interpretation!)assume the αIJi are extreme value distributed

then logit and expected utility:

uI = E�maxJ

�U IJ + αIJi

��= ln

∑JexpU IJ + 1

!= � ln

�aI0�




34





In practice:take singlehood as a benchmark (interpretation!)assume the αIJi are extreme value distributedthen logit and expected utility:

uI = E�maxJ

�U IJ + αIJi

��= ln

∑JexpU IJ + 1

!= � ln

�aI0�




34






uI = E�maxJ

�U IJ + αIJi

��= ln

∑JexpU IJ + 1

!= � ln

�aI0�

Problem: can we compare the u across classes?needed (marital college premium)

but relies on a strong assumption (�What is the unit?�)


34






uI = E�maxJ

�U IJ + αIJi

��= ln

∑JexpU IJ + 1

!= � ln

�aI0�

Problem: can we compare the u across classes?needed (marital college premium)but relies on a strong assumption (�What is the unit?�)


34


Extension: heteroskedasticity

εIJij = σiαIJi + µjβ

IJj (1)

then

ui = U IJ + σI αIJi

vj = V IJ + µJ βIJj

and conditions become:

U IJ

σI+ αIJi � U I0

σI+ αI0i

U IJ

σI+ αIJi � U IK

σI+ αIKi for all K


34

Why does heteroskedasticity matter?

Homoskedastic version: marital college premium measured by thedi¤erence uI � uK (I = college, K = high school) and

uI � uK = ln�aK 0

aI0

�Intuition:

people marry if their (idiosyncratic) gain is larger than some threshold.homoskedastic: one-to-one mapping between the mean and thepercentage below the threshold


34

Why does heteroskedasticity matter?

Heteroskedastic version: now

uI � uK = σK ln�aK 0�� σI ln

�aI0�

and the one-to-one mapping is lost!


34

Covariates

αIJi = Xi .ζIJm + αIJi

βIJj = Yj .ζIJf + β

IJj

and

aIJ = Pr (i , characteristics Xi , matched with a female in J)

=exp

�U IJ + Xi .ζ

IJm

�∑K exp

�U IK + Xi .ζ

IKm

�+ exp

�U I0 + Xi .ζ

I0m

�!standard logit


34

Test and identi�cation: cross sectional data

Basic result:

homoskedastic model exactly identi�ed

heteroskedastic model not identi�ed

Argument (Choo-Siow 2006):

aIJ = Pr (i matched with a female in J)

=exp

�U IJ/σI

�∑K exp (U IK/σI ) + 1

therefore one-to-one correspondence between the aIJ (observed) and theU IJ/σI .


34

Test and identi�cation: dynamic data

Idea: structural modelM =�Z IJ , σI , µJ

�holds for di¤erent cohorts

c = 1, ...,T with varying class compositions. Then:

gij ,c = Z IJc + σI αIJi ,c + µJ βIJj ,c

with the identifying assumption:

Z IJc = ζ Ic + ξJc + ZIJ

Interpretation: trend a¤ecting the surplus but not the supermodularity

Z IJc � Z ILc � ZKJc + ZKLc = Z IJ � Z IL � ZKJ + ZKL

Normalizations

ζ I1 = ξJ1 = 0 so that ZIJ = Z IJ1 .

For any c > 1, the ζ Ic and ξJc are only de�ned up to a (common)additive constant, therefore ξ1c = 0 for all c .


34

Test and identi�cation: dynamic dataTest

Start from:

aIJc = Pr (i 2 I matched with a female in J at c) =exp

�U IJc /σI

�1+∑K exp (UKJc /σK )

De�ne pIJc = UIJc /σI , qIJc = V

IJc /µJ , then

pIJc = ln�

aIJc1�∑K aIKc

�and σI pIJc + µJ qIJc = ζ Ic + ξJc + Z

IJ

gives for all (I � 2, J � 2):

p1Jc � p11c = σI�pIJc � pI1c

�+ µJ

�qIJc � q1Jc

�� µ1

�qI1c � q11c

��Z IJ � Z I1 � Z 1J + Z 11

�P.A. Chiappori, Columbia () Matching models


Test and identi�cation: dynamic dataTest

De�ne the vectors:

PIJ =�pIJ1 � pI11 , ..., pIJT � pI1T

�QIJ =

�qIJ1 � q1J1 , ..., qIJT � q1JT

�RIJ =

�p1J1 � p111 , ..., p1JT � p11T

�1 = (1, ..., 1)

Then for each pair (I � 2, J � 2):

RIJ = σI PIJ + µJ QIJ � µ1QI1 ��Z IJ � Z I1 � Z 1J + Z 11

�1

therefore:

RIJ belongs to the subspace generated by�PIJ ,QIJ ,QI1, 1

,

the coe¢ cient of PIJ (resp. QIJ ,resp. QI1) does not depend on J(resp. I , resp. is constant).


34

Test and identi�cation: dynamic dataIdenti�cation

From

RIJ = σI PIJ + µJ QIJ � µ1QI1 ��Z IJ � Z I1 � Z 1J + Z 11

�1

we (over)identify σI , µJ and�Z IJ � Z I1 � Z 1J + Z 11

�For c = 1:

σI pIJ1 + µJ qIJ1 = ZIJ

identi�es Z IJ

ThenσI pI1c + µ1 qI1c = ζ Ic + Z

I1 8I , 1, cidenti�es ζ Ic , and ξJc follows

In practice: minimum distance.


34

Data

American Community Survey, a representative extract of Census. The2008 survey has info on current marriage status, number of marriages,year of current marriage (633,885 currently married couples).

Born between 1943 and 1970 for men, 1945 and 1972

Three education classes: HS drop out, HS graduate, College andabove

Construct 28 �cohorts�; for each cohort, matrix of marriageproportions by classes (plus singles)

Age ! assumption: husband in cohort c marries wife in cohort c + 2


34

Results (1)

Speci�cation tests: very good �t although formally rejected

Estimate the Z IJ s; strongly supermodular

Variances:

σ1 = .085, σ2 = .05, σ3 = .088, µ1 = .093, µ2 = .063, µ3 = .147

More variance for college-educated women


34

Results: trends

Surplus of diagonal matches


34

Results: marital college premium

In principle, marital college premium has several components:

Marriage probabilitySpouse�s (distribution of) educationSurplus generatedDistribution of the surplus

Our estimates for women:

Year born 1945-47 1970-72Marriage Probability HS 94% Coll 89% HS 78% Coll 81%Coll Educated Spouse HS 39% Coll 84% HS 38% Coll 84%Surplus (Coll Husband) HS .258 Coll .473 HS -.10 Coll .286Share (Coll Husband) HS 47% Coll 51% HS 45% Coll 57%


34



Marriage probability

Spouse�s (distribution of) educationSurplus generatedDistribution of the surplus




34



Marriage probabilitySpouse�s (distribution of) education

Surplus generatedDistribution of the surplus




34



Marriage probabilitySpouse�s (distribution of) educationSurplus generated

Distribution of the surplus




34



Marriage probabilitySpouse�s (distribution of) educationSurplus generatedDistribution of the surplus




34


Expected surplus: women


34


Expected surplus: men


34


In summary:


34

Results: implied intrahousehold resource allocation

Shares:


34

Conclusion

1 Frictionless matching: an interesting tool for empirical analysis,especially when not interested in frictions

2 Modelling: link with standard discrete choice models3 Identi�cation: dynamic data4 Advantages of a structural model:

Quantify the various e¤ectsSummarize them into a marital college premium

5 Main �nding: predictions of theoretical models con�rmed


34


Pierre-André Chiappori, Columbia University



22

Overview








22

Session 6

Matching models: empirical applications

Part 2: reduced form models of multidimensionalmatching


22

Multidimensional matching

Theoretical models of matching under TU do not require aone-dimensional framework

Still in practice, most (if not all) empirical models are one-dimensional

Two types of approaches1 Multidimensional framework �reduces to�a single-dimensional one !Chiappori Ore¢ ce Quintana-Domeque 2009

2 �True�multidimensional model ! Chiappori Ore¢ ceQuintana-Domeque 2010

Additional dimension: continuous versus discrete characteristics

not important in theorybut crucial empirically....

Here: �True�multidimensional model, one continuous and one discrete trait


22

Multidimensional matching: the �single index�approach

Framework:

Populations of men and women.

Each potential husband characterized by:

observable characteristics Xi =�X 1i , ...,X

Ki

�unobservable characteristic εi 2 RS , centered and independent of X .all characteristics are continuous

Same for women: observable variables Yj =�Y 1j , ...,Y

Lj

�,

unobservable characteristics ηj 2 RS


22


Key assumption:

Assumption I The �attractiveness�of male i (resp. female j) on themarriage market is fully summarized by a one-dimensional indexIi = I

�X 1i , ...,X

Ki , εi

�(resp. Jj = J

�Y 1j , ...,Y

Lj , ηj

�). Moreover, these

indices are separable in�X 1i , ...,X

Ki

�and

�Y 1j , ...,Y

Lj

�respectively; i.e.

Ii = i�I�X 1i , ...,X

Ki

�, εi

�and

Jj = j�J�Y 1j , ...,Y

Lj

�, ηj

�for some mappings i , j from RS+1 to R and I (resp. J) from RK (resp.RL) to R.


22


In particular:

�Iso-index�pro�les:I�X 1i , ...,X

Ki

�= C

where C is a constant

Compensation ! marginal rate of substitution betweencharacteristics n and m:

MRSm,ni =∂I/∂X n

∂I/∂Xm


22


�Fully summarized�?

The joint density dµ�X 1, ...,XK ,Y 1, ...,Y L

�of observables among

married couples has the form:

dµ�X 1, ...,XK ,Y 1, ...,Y L

�= dν

hI�X 1, ...,XK

�, J�Y 1, ...,Y L

�ifor some measure dν on R2. Therefore:

The conditional distribution of�Y 1, ...,Y L

�given

�X 1, ...,XK

�only

depends on the value I�X 1, ...,XK

�(and same for men)

�Iso-attractiveness�: same conditional distribution of spouse�scharacteristics ! �iso-index�

Here: the subindex I , which only depends on observables, is asu¢ cient statistic for the distribution of characteristics of a man�sspouse (and same for women)


22


In particular, one can (ordinally) non parametrically identify attractivenessindices. Indeed, for any characteristic s of the wife:

EhY s j X 1i , ...,XKi

i= φs

hI�X 1i , ...,X

Ki

�ifor some function φs . Then:

�iso-index�or �iso-attractiveness�pro�les:

EhY s j X 1i , ...,XKi

i= C 0

marginal rate of substitution between characteristics n and m:

MRSm,ni =∂I/∂X n

∂I/∂Xm=

∂E�Y s j X 1i , ...,XKi

�/∂X n

∂E�Y s j X 1i , ...,XKi

�/∂Xm

,

Over identifying restrictions: does not depend on s

Works for any moment


22

�True�multidimensional matching: the economics ofmarital smoking

Setting:

Two populations (men and women) of equal size, normalized to one.

Socio-economic status: continuous variables x and y , uniformlydistributed over [0, 1]

Smoking: dichotomic, independent of status; sM and sW proportionsof smokers

Surplus:

Σ = f (x , y) if both spouses do not smoke

Σ = λf (x , y) otherwise, λ < 1

In practicef (x , y) = (x + y)2 /2


22


Basic remark:

The �twisted�condition does not hold.

Woman, index x0, non smoker:

∂xΣ = (x0 + y1) if she marries a non smoker with index y1∂xΣ = λ (x0 + y2) if she marries a smoker with index y2.

For any y2 2h(1�λ)x0

λ , 1i, if y1 = λy2 � (1� λ) x0, then the couples

(x0, y1) and (x0, y2) violate the twisted buyer condition; works for an openset of values x0 - namely x0 2

�0, λ1�λ

�.

Consequence:

The stable matching may not be pure.


22


Particular case: if sM = sW then:

All smoking women marry smoking men, and conversely

All non smoking women marry non smoking men, and conversely

In words:

Even if λ very close to 1, fully discriminated submarkets

But: in practice,sM > sW


22

Main result

Proposition

Assume that λ � sMsM+1

. There exists four values X ,Y ,Y 0 and p, allbetween 0 and 1, such that:

For all x � X, a non-smoking woman with index x is matched withprobability 1 to a non-smoking husband with index

y =1� sW1� sM

x � sM � sW1� sM

� Y

and a smoking woman with index x is matched with probability 1 to asmoking husband with index

y 0 =sWsMx +

sM � sWsM

� Y 0

In particular, smoking men and non smoking women marry �down�,whereas non-smoking men and smoking women marry �up�.


22

PropositionFor x < X, a non-smoking woman with index x is matched:

with probability p, to a smoking husband with index

y 0 =(1� sW ) p + sW

sMx < Y 0

with probability 1� p, to a non-smoking husband with index

y =(1� sW ) (1� p)

1� sMx < Y

Moreover, for any �xed index x of the wife, smoking husbands have ahigher index than non smoking ones - i.e., y 0 > y.


22

PropositionFor x < X, a smoking woman with index x is matched withprobability 1 to a smoking husband with index:

y 0 =(1� sW ) p + sW

sMx < Y 0

Finally, there are no couples in which the wife smokes and thehusband does not.


22

Women Men

X

Y

Y’

NS NSS S

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

u

Female utility (dashed = non smokers); sM = .25, sW = .2,λ = .8


22

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

y

v

Male utility (dashed = non smokers); sM = .25, sW = .2,λ = .8


22

Predictions

Non smoking husbands always marry a non smoking wife

Smoking husbands with a higher index (y 0 � Y 0) marry a high index,smoking wife with probability 1

Smoking husbands with a lower index (y 0 < Y 0) marry either asmoking or a non smoking wife with positive probability; moreover,both potential wives have the same quality index.

Similarly:

Smoking wifes always marry a smoking husband

Non smoking wifes with a higher index (x � X ) marry a high index,non smoking husband with probability 1

Non smoking wifes with a lower index (x < X ) marry either a smokingor a non smoking husband with positive probability; moreover, thesmoking husband is of higher quality than the non smoker.


22

Empirical tests

In practice, randomness in the matchingHere: not formally modeled, but qualitative predictions:

Mixed couples in which the wife smokes while the husband does not(denoted S-NS) should be less frequent than those in which hesmokes and she does not (denoted NS-S) - i.e., the ratio of S-NS toNS-S couples should be smaller than:

r =sW (1� sM )sM (1� sW )

In our sample, sM = .22, sW = .17, r = .72


22

Empirical tests

Among couples with identical smoking habits, assortative matchingon socioeconomic status.

Non-smoking wives married with a smoking husband have a lowersocioeconomic status than those married with a non-smoking husband

Smoking husband married with a non-smoking wife have a lowersocioeconomic status than those married with a smoking wife.

When two (non smoking) women with the same (low) index marryrespectively a smoker and a non smoker, the non smoker should onaverage be of lower status than the smoker. In practice, smoking isnegatively correlated with status, especially for men. Still, we predictthat controlling for the wife�s �quality�, the smoking habit of thehusband should be less negatively correlated with his status.


22

Empirical results


22

II. Recently married: marital duration ≤ 4 years

Non-Smoking Wife Smoking Wife Non-Smoking Husband

71.66%

6.88%

Smoking Husband

12.17%

9.29%

Table A7: Observed Matching Husband’s age 24-36, Wife’s age 22-34. PSID 1999-2007. Weighed % I. Full sample

Non-Smoking Wife Smoking Wife

Non-Smoking Husband

74.74%

5.03%

Smoking Husband

11.43%

8.80%

Table 5: Sorting by education Husband’s age 24-34, Wife’s age 22-32. CPS 1996–2007. Both Non-Smokers Both Smokers

Wife’s

Education Husband’s Education

Wife’s Education

Husband’s Education

Spouse’s Education 0.633*** 0.694*** 0.458*** 0.442*** (0.013) (0.014) (0.035) (0.034) N 8710 8710 1150 1150 R2 0.47 0.47 0.27 0.28

Smoking Husband &

Non-Smoking Wife

Non-Smoking Husband &

Smoking Wife

Wife’s

Education Husband’s Education

Wife’s Education

Husband’s Education

Spouse’s Education 0.600*** 0.618*** 0.495*** 0.497*** (0.031) (0.036) (0.044) (0.041) N 1408 1408 767 767 R2 0.43 0.44 0.34 0.33 Note: All regressions include: own age, year and state fixed effects. Reference categories: 2007 and District of Columbia. Sampling weights are used. Robust standard errors. *** p-value < 0.01, ** p-value < 0.05, * p-value < 0.1

Table 6: Regression of Education by Smoking Status on Spouse’s Education and Smoking Behavior Husband’s age 24-34, Wife’s age 22-32. CPS 1996–2007. Wife’s Education Husband’s Education Non-Smoker Smoker Non-Smoker Smoker Spouse’s Education 0.630*** 0.473*** 0.684*** 0.556*** (0.012) (0.027) (0.013) (0.026) Spouse Smokes −0.141** −0.025 −0.209*** 0.160** (0.060) (0.086) (0.074) (0.076) N 10118 1917 9477 2558 R2 0.47 0.29 0.46 0.36 Note: All regressions include: own age, year and state fixed effects. Reference categories: 2007 and District of Columbia. Sampling weights are used. Robust standard errors. *** p-value < 0.01, ** p-value < 0.05, * p-value < 0.1

Table 7: Regression of Smoking Status on Education Husband’s age 24-34, Wife’s age 22-32. CPS 1996–2007. Men with NS Women Women with S Men (1) (2) (3) (4) Own Education −0.028*** −0.024*** −0.021*** −0.029*** (0.002) (0.002) (0.006) (0.007)

Test of Equality 0.0146 0.0341 (p-value)

Spouse’s Education -- −0.006** -- 0.014** (0.002) (0.006) N 10118 10118 2558 2558 R2 0.05 0.05 0.06 0.06 Note: All regressions include: own age, year and state fixed effects. Reference categories: 2007 and District of Columbia. Sampling weights are used. Robust standard errors. *** p-value < 0.01, ** p-value < 0.05, * p-value < 0.1

Table 8: Regression of Educational Difference on Smoking Status Men aged 24-34, Women aged 22-32. CPS 1996–2007.

ΔE = (Wife’s Education –

Husband’s Education)I(ΔE > 0)

Smoking Wife 0.341 0.157* (0.338) (0.090) Smoking Husband 0.819** 0.263*** (0.337) (0.089) Non-Smoking Wife × Non-Smoking Husband 0.476 0.185** (0.331) (0.088) Smoking Wife × Smoking Husband −0.659* −0.204** (0.351) (0.093) Mean of the dependent variable 0.17 0.31 N 12,035 12,035 R2 0.02 0.32 Note: All regressions include: wife’s age, husband’s age, year and state fixed effects. Reference categories: 2007 and District of Columbia. Sampling weights are used. Robust standard errors. *** p-value < 0.01, ** p-value < 0.05, * p-value < 0.1

matching models of the marriage market: theory and empirical

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