Matching models of the marriage market: theory and
empirical applications
Pierre-André Chiappori
Columbia University
Cemmap Masterclass, March 2011
P.A. Chiappori, Columbia ()
Matching models
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Overview
Session 1: Background: Collective models of household behavior
Session 2: Matching models: theory (Part 1)
Session 3: Matching models: theory (Part 2)
Session 4: Matching models: applications
Session 5: Matching models: empirical applications (Part 1)
Session 6: Matching models: empirical applications (Part 2)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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Overview
Session 1: Background: Collective models of household behavior
Session 2: Matching models: theory (Part 1)
Session 3: Matching models: theory (Part 2)
Session 4: Matching models: applications
Session 5: Matching models: empirical applications (Part 1)
Session 6: Matching models: empirical applications (Part 2)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
3/
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Session 1
Background: Collective models of household
behavior
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Gains from marriage (what do we want to model?)
Ecclesiastes (4: 9-10) ; "Two are better than one, because
they have a good reward for their toil. For if they fall, one will
lift up the other; but woe to one who is alone and falls and does
not have another to help. Again, if two lie together, they keep
warm; but how can one keep warm alone?"
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Gains from marriage (what do we want to model?)
1
Sharing of public (non rival) goods. For instance, both partners can
equally enjoy their children, share the same information and use the
same home.
2
Division of labor to exploit comparative advantage and increasing
returns to scale. For instance, one partner works at home and the
other works in the market.
3
Home production. For example, coordinating child care, (which is a
public good for the parents).
4
Extending credit and coordination of investment activities. For
example, one partner works when the other is in school.
5
Risk pooling. For example, one partner works when the other is sick
or unemployed.
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Modeling household behavior: the ‘unitary’framework
Drawbacks:
Contradicts individualism
Weak theoretical foundations
Samuelson: postulates a group utility W [U 1 , . . . , U S ]; W must be
independent of prices, wages, distribution factors,...
Becker ‘rotten kid’: concludes that the group behaves as a single
decision maker.
But: strong assumptions required:
Speci…c decision process
Transferable utility and/or production function
In 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)
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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 interactions
Can 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 family
factors; sex ratio; divorce laws; targeted bene…ts
Interpretation:
threat points in a bargaining context
social weight in a sociological interpretation
strategic position in a market context
etc.
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The collective approach: main questions
Three main questions:
Testability (on demand data)
Identi…ability (recovering preferences and ‘power’); ! Distinction
identi…ability/identi…cation
Theoretical underpinning: where do Pareto weights come from?
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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 be
introduced.
Distribution factors zk , k = 1, . . . , K
! could include individual incomes
Preferences : U s (X , x1 , ..., xS ) (most general case)
Particular cases:
‘egoistic’: U s h(X , xs )
i
‘caring’: W s U 1 (X , x1 ) , ..., U S (X , xS )
Note that: an allocation that is e¢ cient for caring is e¢ cient for
egoistic as well
TU:
U s (X , xs ) = F s As X , xs2 , ..., xsn + xs1 b (X )
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The collective approach: the setting
(Aggregate) market demand:
ξ = (X , x1 + ... + xS )
as function of π = (P, p ), y and possibly z; budget constraint:
π 0 ξ = P 0 X + p 0 (x1 + ... + xS ) = y
E¢ ciency: for all (p, P, y ), there exist µ1 , ..., µS with
µ1 + ... + µS = 1 such that (X , x1 , . . . , xS ) solves:
max
∑ µs U s
under the BC
X ,x1 ,...,xS s
or the more general version (with production):
max
∑ µs U s
(P)
X ,x1 ,...,xS s
under
Φ (X , x1 + ... + xS ) = ξ and π 0 ξ = y
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The collective approach: the setting
Key remark: in general, µ1 , ..., µS depend on prices and incomes (and
distribution factors)
Therefore:
may de…ne household utility
U G (ξ, µ) =
∑ µs U s
max
X ,x1 ,...,xS s
s.t. Φ (X , x1 + ... + xS ) = ξ
but it is price dependent.
Particular case: private goods only. Then e¢ ciency equivalent to the
existence of a sharing rule: there exists ρ = (ρ1 , ..., ρS ) with
∑ ρs = y such that xs solves
max U s (xs ) st p 0 xs = ρs
Questions: testability and identi…cation
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The collective approach: testability (price variations)
Normalization: y = 1; therefore Slutsky matrix S (π ) = (Dπ ξ ) I
Basic result (Browning-Chiappori 1998) :
πξ T
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 negative
the matrix R (π ) is of rank at most H
1.
Equivalently, there exists a subspace E (π ) of dimension at least N + n S
such 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 one
observable distribution factor, say z1 , such that ξ j (y , z) strictly
monotone in z1 . Then:
z1 = ζ (y , z
where z = (z1 , z
1 ).
ξ i (y , z1 , z
P.A. Chiappori, Columbia ()
1, ξ
j
)
Then substituting into the demand for good i 6= j:
1)
= ξ i [y , ζ (y , z
1, ξ
j
Matching models
), z
1]
= θ ij (y , z
1, ξ
j
).
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The collective approach: testability (no price variation,
BBC 2009)
Proposition
A given system of demand functions is compatible with collective
rationality 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 that
ξ i (y , z) = Ξi [y , µ(y , z)]
8i
ii) household demand functions satisfy
∂ξ j /∂zk
∂ξ i /∂zk
=
∂ξ i /∂zl
∂ξ j /∂zl
8i, j, k, l
iii) there exists at least one good j such that:
∂θ ij (y , z 1 , ξ j )
=0
∂zk
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8i 6= j and k = 2, .., K
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The collective approach: identi…ability (C-E 2009)
Basic assumptions:
Egoistic preferences
Exclusion restrictions: for each agent s, there exists at elast one
commodity that s does not consume
Basic result (C-E 2009):
Proposition
Generically, the welfare relevant structure can be recovered
Moreover, in the absence of price variations, if all goods are private (or
under separability):
The sharing rule can be identi…ed up to a constant
Individual Engel curves can be recovered
Exclusion not needed!
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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 of
gender-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 composition
decision process (Pareto weights)
! Questions:
group formation: who marries whom and why (and dissolution)?
distribution of powers as an endogenous phenomenon
Basic tools:
bargaining theory
matching models (frictionless)
search models (search frictions)
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Matching models of the marriage market: theory and
empirical applications
Pierre-André Chiappori
Columbia University
Cemmap Masterclass, March 2011
P.A. Chiappori, Columbia ()
Matching models
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Overview
Session 1: Background: Collective models of household behavior
Session 2: Matching models: theory (Part 1)
Session 3: Matching models: theory (Part 2)
Session 4: Matching models: applications
Session 5: Matching models: empirical applications (Part 1)
Session 6: Matching models: empirical applications (Part 2)
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Session 2
Matching models: theory (1)
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Background: The Collective Model
Takes as given:
group composition
decision 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…nition
A group satis…es TU if there exists monotone transformations of individual
utilities such that the Pareto frontier is an hyperplane for all values of
prices and income.
In practice:
Quasi Linear (QL) preferences (but highly unrealistic)
‘Generalized Quasi Linear (GQL, Bergstrom and Cornes 1981):
us (qs , Q ) = Fs As qs2 , ..., qsn , Q + qs1 bs (Q )
with bs (Q ) = b (Q ) for all s
Note:
Ordinal property
Restrictions on heterogeneity
But ‘acceptably’realistic
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Properties of Transferable Utility (TU)
Unanimity regarding group’s decisions
clear distinction between aggregate behavior and intragroup allocation
of power/resources/welfare
here: concentrate of ‘power’issues
Matching models: ! mathematical structure: optimal transportation
models
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Optimal Transportation Problems (Monge-Kantorovitch)
General Structure:
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Optimal Transportation Problems (Monge-Kantorovitch)
General Structure:
Complete, separable metric spaces X , Y with measures F and G
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Optimal Transportation Problems (Monge-Kantorovitch)
General Structure:
Complete, separable metric spaces X , Y with measures F and G
Surplus s (x, y ) upper semicontinuous
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Optimal Transportation Problems (Monge-Kantorovitch)
General Structure:
Complete, separable metric spaces X , Y with measures F and G
Surplus s (x, y ) upper semicontinuous
Problem: …nd a measure h on X Y such that the marginals of h are
F and G , and h solves
max
h
Z
X Y
s (x, y ) dh (x, y )
Hence: linear programming
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Optimal Transportation Problems (Monge-Kantorovitch)
General Structure:
Complete, separable metric spaces X , Y with measures F and G
Surplus s (x, y ) upper semicontinuous
Problem: …nd a measure h on X Y such that the marginals of h are
F and G , and h solves
max
h
Z
X Y
s (x, y ) dh (x, y )
Hence: linear programming
Dual problem: dual functions u (x ) , v (y ) and solve
min
u,v
Z
u (x ) dF (x ) +
X
Z
v (y ) dG (y )
Y
under the constraint
u (x ) + v (y )
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s (x, y ) for all (x, y ) 2 X
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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 G
Surplus s (x, y ) upper semicontinuous
A matching consists of:
a measure h on X Y such that the marginals of h are F and G
two 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:
u (x ) + v (y )
s (x, y ) for all (x, y ) 2 X
Y
Interpretation
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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 G
Surplus s (x, y ) upper semicontinuous
A matching consists of:
a measure h on X Y such that the marginals of h are F and G
two functions u (x ) , v (y ) (‘imputations’) such that
u (x ) + v (y ) = s (x, y ) for all (x, y ) 2 Supp (h)
Stability: assume
u (x ) + v (y ) < s (x, y ) for all (x, y ) 2 X
Y
Interpretation
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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 G
Surplus s (x, y ) upper semicontinuous
A matching consists of:
a measure h on X Y such that the marginals of h are F and G
two 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:
u (x ) + v (y )
s (x, y ) for all (x, y ) 2 X
Y
Interpretation: ‘divorce at will’
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Basic Result (duality)
We have the following:
Theorem
A matching is stable if and only if it solves the surplus maximization
problem
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Basic Result (duality)
We have the following:
Theorem
A matching is stable if and only if it solves the surplus maximization
problem
Consequence: existence; generic uniqueness of the measure under
additional conditions
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Basic Result (duality)
We have the following:
Theorem
A matching is stable if and only if it solves the surplus maximization
problem
Consequence: existence; generic uniqueness of the measure under
additional conditions
Note that:
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Basic Result (duality)
We have the following:
Theorem
A matching is stable if and only if it solves the surplus maximization
problem
Consequence: existence; generic uniqueness of the measure under
additional conditions
Note that:
sets can be multidimensional
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Basic Result (duality)
We have the following:
Theorem
A matching is stable if and only if it solves the surplus maximization
problem
Consequence: existence; generic uniqueness of the measure under
additional conditions
Note that:
sets can be multidimensional
extends to hedonic models.
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Hedonic Models
Structure:
Separable metric spaces X, Y, Z; measures F, G
Utilities 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 ) = max(u (x, z )
z
c (y , z ))
(see Chiappori-McCann-Neishem 2010)
Note that: n-dimensional; single crossing not needed; covexity not
needed.
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Proof
Start from:
u (x ) + v (y )
s (x, y )
u (x, z )
c (y , z )
on X
Y
Z,
hence
c (y , z ) + v (y )
u (x, z )
u (x )
on X
Y
Z
and
inf fc (y , z ) + v (y )g
y 2Y
sup fu (x, z )
u (x )g
on Z .
x 2X
Take any P (z ) such that
inf fc (y , z ) + v (y )g
y 2Y
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P (z )
sup fu (x, z )
u (x )g
on Z .
x 2X
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Supermodularity and assortative matching
One-dimensional:
s is supermodular if whenever x
s (x, y ) + s x 0 , y 0
x 0 and y
y 0 then
s x, y 0 + s x 0 , y
Then stable matching is assortative; indeed, surplus maximization
Interpretation: single crossing (Spence - Mirrlees). Assume that s is
C 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
∂2 s
∂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
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Supermodularity and assortative matching
Problem: both super- (or sub-) modularity and assortative matching
are typically one-dimensional
Generalization (CMcCN ET 2010):
De…nition
A surplus function s : X Y ! [0, ∞[ is said to be X twisted if there is
a 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…nition
The 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
Lake, students
Lake, students, schools
Counter example (C-McC-N 09)
Transportation on a circle
Problem: allocate children to schools
so as to minimize total transportation cost
Interpretation: matching with transfers (‘tuition’)
Note that: single crossing cannot hold!
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Counter example (C-McC-N 09)
Transportation on a circle
Stable match:
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Counter example (C-McC-N 09)
Transportation on a circle
Stable match:
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Application: marriage market
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Application: marriage markets
Structure:
Men and women, respective income distributions F and G; mass 1 for
men, r for women
TU; surplus s (x, y ), derived from a collective model
In general, TU implies s (x, y ) supermodular
Example (CIW07):
then
uh
= Q 1 + qh
uw
= Q (a + q w )
(x + y + 1 + a )2
s (x, y ) =
4
Note 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 with
income larger than x equals the number of women with income larger
than y :
1 F (x ) = r (1 G (y ))
Therefore
x
= φ (y ) = F
1
y
= ψ (x ) = G
1
[1
r (1 G (y ))]
1
1
(1 F (x ))
r
If r > 1 then y0 = ψ (0) ’last married woman’
If r < 1 then x0 = φ (0) ’last married man’
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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 in
x 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
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Session 3
Matching models: theory (2)
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Sharing the surplus
Basic result:
Intramatch allocation of welfare is pinned down
by the equilibrium conditions
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Sharing the surplus
Stability implies
u (x ) = max (s (x, y )
y
v (y ))
therefore
u 0 (x ) =
∂s
∂s
(x, ψ (x )) and v 0 (y ) =
( φ (y ) , y )
∂x
∂y
and
u (x ) = k +
Z x
∂s
0
∂x
(t, ψ (t )) dt , v (y ) = k 0 +
Z y
∂s
0
∂y
(φ (s ) , s ) ds
where k + k 0 given
Pinning down the constant: if r > 1 the ‘last married’woman
receives no surplus
Hence: endogeneize intrahousehold allocation
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Sharing the surplus: application
Assume that s (x, y ) = H (x + y ), H convex. Then
u 0 (x ) = H 0 (x + ψ (x )) , v 0 (y ) = H (φ (y ) + y )
Application: ‘linear shift’: r = 1 and F (t ) = G (αt
α < 1, β > 0
β) for some
Satis…ed for instance if Lognormal distributions with same σ.
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Example: shifting female income distribution
Then φ (y ) = (y + β) /α and ψ (x ) = αx
u 0 (x ) = H 0 ((α + 1) x
β; from
β) ,
one gets
u (x ) = K 0 +
1
H (x + ψ (x ))
1+α
v (y ) = K +
α
H ( φ (y ) + y )
1+α
and
where K + K 0 = 0
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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Example: shifting female income distribution
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
=
∂uk
∂k
=
P.A. Chiappori, Columbia ()
αy
α
H (y + x ) and
H 0 (y + x ) +
α+1
( α + 1)2
y
α
H 0 (y + x )
H (y + x )
α+1
( α + 1)2
Matching models
Cemmap Masterclass, March 2011
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43
Example: shifting female income distribution
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
∂uk
∂k
=
=
P.A. Chiappori, Columbia ()
αy
H 0 (y + x )
α+1
y
H 0 (y + x )
α+1
Matching models
Cemmap Masterclass, March 2011
30 /
43
Example: shifting female income distribution
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
=
∂uk
∂k
=
P.A. Chiappori, Columbia ()
αy
α
H (y + x ) and
H 0 (y + x ) +
α+1
( α + 1)2
y
α
H 0 (y + x )
H (y + x )
α+1
( α + 1)2
Matching models
Cemmap Masterclass, March 2011
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43
Matching with imperfectly transferable utility
As above, stability implies that:
u (x ) = max H (x, y , v (y ))
y
where the maximum is actually reached for y = ψ (x ) (or x = φ (y )).
First order conditions imply that
∂H
∂H
(φ (y ) , y , v (y )) + v 0 (y )
(φ (y ) , y , v (y )) = 0.
∂y
∂v
or:
0
v (y ) =
∂H
∂y
∂H
∂v
(φ (y ) , y , v (y ))
(φ (y ) , y , v (y ))
.
and
u 0 (x ) =
P.A. Chiappori, Columbia ()
∂H
(x, ψ (x ) , v (ψ (x )))
∂x
Matching models
Cemmap Masterclass, March 2011
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43
Matching with imperfectly transferable utility
Second order conditions:
∂
∂y
∂H
∂H
(φ (y ) , y , v (y )) + v 0 (y )
(φ (y ) , y , v (y ))
∂y
∂v
0 8y .
Since
F (y , φ (y )) = 0
where
F (y , x ) =
8y
∂H
∂H
(x, y , v (y )) + v 0 (y )
(x, y , v (y )) .
∂y
∂v
we have:
∂F
∂F 0
+
φ (y ) = 0
∂y
∂x
8y ,
which implies that
∂F
∂y
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0 if and only if
Matching models
∂F 0
φ (y )
∂x
0.
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Matching with imperfectly transferable utility
The second order conditions can hence be written as:
∂2 H
∂2 H
(φ (y ) , y , v (y )) + v 0 (y )
(φ (y ) , y , v (y )) φ0 (y )
∂x ∂y
∂x ∂v
0 8y .
(2)
Assortative matching: φ0 (y )
0, which holds if
∂2 H
∂2 H
(φ (y ) , y , v (y )) + v 0 (y )
(φ (y ) , y , v (y ))
∂x ∂y
∂x ∂v
Since v 0 (y )
is that
0 8y .
(3)
0, a su¢ cient (although obviously not necessary) condition
∂2 H
(φ (y ) , y , v (y ))
∂x ∂y
0 and
∂2 H
(φ (y ) , y , v (y ))
∂x ∂v
Note: under TU, H (x, y , v (y )) = h (x, y )
P.A. Chiappori, Columbia ()
Matching models
v (y ) and
∂2 H
∂x ∂v
0.
(4)
=0
Cemmap Masterclass, March 2011
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43
Matching with imperfectly transferable utility
u
P’
P
v
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
35 /
43
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 of
men
Male preferences: Cobb-Douglas
um = cm Q
Female preferences: perfect substitutes
uf (cf ) =
∞ if cf < c̄
= cf + Q if cf
c̄
In particular, if a woman is single, her income must be at least c̄; then
her utility equals her income.
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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43
Matching with imperfectly transferable utility: an example
E¢ cient allocations:
max cm Q
under the constraints
cm + cf + Q = x + y ,
uf = cf + Q
U
Remark: at any e¢ cient allocation:
cf = c̄
U ((x + y ) + c̄ ) /2.
Pareto frontier:
um = H ((x + y ) , uf ) = (uf
where uf
(x +y )+c̄
,and
2
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Q = uf
c̄ ) ((x + y )
c̄, cm = (x + y )
Matching models
uf ) ,
(5)
uf .
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43
v4
3
2
1
0
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
5.2
u
Figure: Pareto frontier
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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43
Characterization
∂H (x + y , v )
=v
∂ (x + y )
c̄,
∂H (x + y , v )
=
∂v
(2v
(c̄ + (x + y )))
therefore assortative matching since
∂2 H (x + y , v )
∂ (x + y )2
= 0 and
∂2 H (x + y , v )
=1
∂ (x + y ) ∂v
Moreover:
v 0 (y ) =
P.A. Chiappori, Columbia ()
αv (y ) αc̄
>0
2αv (y ) (α + 1) y (αc̄ + β)
Matching models
Cemmap Masterclass, March 2011
(6)
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43
Utilities
8
6
4
2
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
x
Figure: Husband’s and Wife’s Utilities, Public Consumption and the Husband’s
Private Consumption (β = 0, α = .8)
P.A. Chiappori, Columbia ()
Matching models
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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
P.A. Chiappori, Columbia ()
Matching models
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43
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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43
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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43
Matching models of the marriage market: theory and
empirical applications
Pierre-André Chiappori
Columbia University
Cemmap Masterclass, March 2011
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
1/
18
Overview
Session 1: Background: Collective models of household behavior
Session 2: Matching models: theory (Part 1)
Session 3: Matching models: theory (Part 2)
Session 4: Matching models: applications
Session 5: Matching models: empirical applications (Part 1)
Session 6: Matching models: empirical applications (Part 2)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
Session 4
Matching models: Applications
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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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)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
Motivation: education and marriage, US
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
5/
18
Motivation: education and marriage, US
Education
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
Motivation: education and marriage, US
Education
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
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 intrahousehold
If larger percentage of educated women, a¤ects matching patterns
Cost of not being educated are higher
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
8/
18
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
9/
18
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
The model
Two equally large populations of men and women to be matched.
Individuals live two periods; investment takes place in the …rst period
of life and marriage in the second period; investment in schooling is
lumpy and takes one period.
All agents of the same level of schooling and gender receive the same
wage.
I (i ) and J (j ) are the schooling "class" of man i and woman j (1 if
uneducated, 2 if educated)
Transferable utility; marital surplus if man i marries woman j:
sij = zI (i )J (j ) + θ i + θ j
with
z11 + z22 > z12 + z21
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
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 across
individuals; 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 ) + θ i + θ j
vi + uj
therefore
vi
= max max zI (i )J (j ) + θ i + θ j
uj , 0
uj
= max max zI (i )J (j ) + θ i + θ j
vi , 0
P.A. Chiappori, Columbia ()
j
i
Matching models
Cemmap Masterclass, March 2011
12 /
18
µ
b
No marriage,
no investment
Marriage, no
investment
R + V2 − V1
m
Rm
−V2
−V1
θ
Marriage and
investment
Investment,
no marriage
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
13 /
18
The model
Proposition:
vi
uj
= VI (i ) + θ i
= UJ ( j ) + θ j
with
P.A. Chiappori, Columbia ()
VI
= max (zIJ
UJ )
UJ
= max (zIJ
VI )
J
I
Matching models
Cemmap Masterclass, March 2011
14 /
18
The model
Theorem:
U1 + V1 = z11
U2 + V2 = z22
and either strict assortative matching:
U1 + V2
z21
U2 + V1
z12
or some people ‘marry down’. If more educated men:
U1 + V2 = z21 therefore U2
U1 = z22
z21 , V2
V1 = z21
z11
U1 = z12
z11 , V2
V1 = z22
z21
whereas if more educated women:
U2 + V1 = z12 therefore U2
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
15 /
18
The model
Equilibrium equations:
Example: # married men = # married women
F (V1 ) +
ZV 2
m
G (R + V2
θ ) f ( θ ) d θ = F ( U1 ) +
V1
ZU 2
G ( R w + U2
θ )f ( θ )d θ
U1
Example: # educated married men = # educated married women
F (V1 ) [1
G (R m + V2
V1 )] = F (U1 ) [1
G ( R w + U2
U1 )]
Then calibration
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
Comparative statics
Compare an "old" regime a "new" regime.
In the old regime:
lower returns to education
more time to be spent at home ! specialization
New regime:
higher returns to education
less time spent at home ! both genders demand education
Initial sources of asymmetry?
statistical discrimination weaker against educated women
di¤erences in innate ability (Becker-Hubbard-Murphy)
impact of birth control
Equilibrium mechanism ampli…es the shift
Old regime: some men marry down, marital returns of schooling higher
for men
New regime: just the opposite (‘you can’t a¤ord to be a nurse’)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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18
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
18 /
18
Matching models of the marriage market: theory and
empirical applications
Pierre-André Chiappori,
Columbia University
Cemmap Masterclass, March 2011
P.A. Chiappori, Columbia ()
Matching models
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Overview
Session 1: Background: Collective models of household behavior
Session 2: Matching models: theory (Part 1)
Session 3: Matching models: theory (Part 2)
Session 4: Matching models: applications
Session 5: Matching models: empirical applications (Part 1)
Session 6: Matching models: empirical applications (Part 2)
P.A. Chiappori, Columbia ()
Matching models
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Session 5
Matching models: empirical applications
Part 1: a semi-structural model
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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Motivation: education and marriage, US
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Motivation: education and marriage, US
Education
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
5/
34
Motivation: education and marriage, US
Education
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Motivation: education and marriage, US
Marital patterns
P.A. Chiappori, Columbia ()
Matching models
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34
Motivation: education and marriage, US
Marital patterns
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Motivation: education and marriage, US
Marital patterns
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Matching models
Cemmap Masterclass, March 2011
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34
Motivation: education and marriage, US
Marital patterns
Burtless (EER 1999): over 1979-1996:
‘The changing correlation of husband and wife earnings has tended to
reinforce the e¤ect of greater pay disparity.’
Maybe 1/3 of the increase in household-level inequality (Gini) comes
from rise of single-adult households and 1/6 from increased
assortative matching. Burtless again, US 1979-1996:
‘The Spearman rank correlation of husband and wife earnings
increased from .012 to .145’
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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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?
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
or only a mechanical consequences of the changes in population
compositions?
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
or only a mechanical consequences of the changes in population
compositions?
3
Matching models: scope and empirical implementation
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
or only a mechanical consequences of the changes in population
compositions?
3
Matching models: scope and empirical implementation
Basic tool: frictionless matching
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
or only a mechanical consequences of the changes in population
compositions?
3
Matching models: scope and empirical implementation
Basic tool: frictionless matching
Reference:
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
or only a mechanical consequences of the changes in population
compositions?
3
Matching models: scope and empirical implementation
Basic tool: frictionless matching
Reference:
Choo and Siow (JPE 2003)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
11 /
34
Questions
1
2
Education: Why the asymmetric response between men and women?
Possible answer (CIW, AER 2009): ‘Marital College Premium’
! Is there evidence for a MCP?
Changes in matching patterns:
change in attitudes toward assortative matching?
or only a mechanical consequences of the changes in population
compositions?
3
Matching models: scope and empirical implementation
Basic tool: frictionless matching
Reference:
Choo and Siow (JPE 2003)
Chiappori, Salanié and Weiss 2010
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Empirical implementation
Basic issue: capturing imperfections (frictions) in the matching process
Various strategies:
Random matching (Shimer-Smith)
Search
Frictionless matching with unobservable components
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
12 /
34
Empirical implementation
Basic insight: unobserved characteristics (heterogeneity)
! Gain gijIJ generated by the match i 2 I , j 2 J:
gijIJ = Z IJ + εIJ
ij
where I = 0, J = 0 for singles, and εIJ
ij random shock with mean zero.
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
13 /
34
Empirical implementation
Basic insight: unobserved characteristics (heterogeneity)
! Gain gijIJ generated by the match i 2 I , j 2 J:
gijIJ = Z IJ + εIJ
ij
where I = 0, J = 0 for singles, and εIJ
ij random shock with mean zero.
Therefore: dual variables (ui , vj ) also random (endogenous
distribution)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
13 /
34
Empirical implementation
Basic insight: unobserved characteristics (heterogeneity)
! Gain gijIJ generated by the match i 2 I , j 2 J:
gijIJ = Z IJ + εIJ
ij
where I = 0, J = 0 for singles, and εIJ
ij random shock with mean zero.
Therefore: dual variables (ui , vj ) also random (endogenous
distribution)
Stability: constrained by the inequalities
ui + vj
gijIJ for any (i, j )
! large number (one inequality per potential couple)
P.A. Chiappori, Columbia ()
Matching models
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34
Empirical implementation
Crucial identifying assumption (Choo-Siow 2006)
Assumption S (separability): the idiosyncratic component εij is
additively separable:
IJ
IJ
εIJ
(S)
ij = αi + βj
h i
where E αIJ
= E βIJ
= 0.
i
j
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Empirical implementation
Crucial identifying assumption (Choo-Siow 2006)
Assumption S (separability): the idiosyncratic component εij is
additively separable:
IJ
IJ
εIJ
(S)
ij = αi + βj
h i
where E αIJ
= E βIJ
= 0.
i
j
Then:
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Empirical implementation
Crucial identifying assumption (Choo-Siow 2006)
Assumption S (separability): the idiosyncratic component εij is
additively separable:
IJ
IJ
εIJ
(S)
ij = αi + βj
h i
where E αIJ
= E βIJ
= 0.
i
j
Then:
Theorem
Under S, there exists U IJ and V IJ such that U IJ + V IJ = Z IJ and for any
match (i 2 I , j 2 J )
ui
vj
P.A. Chiappori, Columbia ()
= U IJ + αIJ
i
IJ
= V + βIJ
j
Matching models
Cemmap Masterclass, March 2011
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34
Empirical implementation
Crucial identifying assumption (Choo-Siow 2006)
Assumption S (separability): the idiosyncratic component εij is
additively separable:
IJ
IJ
εIJ
(S)
ij = αi + βj
h i
where E αIJ
= E βIJ
= 0.
i
j
Then:
Theorem
Under S, there exists U IJ and V IJ such that U IJ + V IJ = Z IJ and for any
match (i 2 I , j 2 J )
ui
vj
= U IJ + αIJ
i
IJ
= V + βIJ
j
General characterization: Galichon-Salanie (2011)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Galichon-Salanié 2011
Assume separability, let π IJ the proba that i 2 I marries j 2 J; then
Social surplus is
Σ = 2 ∑ π IJ Z IJ + ε (nI , nJ , π )
i ,j
where ε is a ’generalized entropy’
When α, β are extreme values, then
ε (nI , nJ , π ) =
∑ ∑ πIJ ln
I
J
π IJ
nI
∑ ∑ πIJ ln
I
J
π IJ
,
nJ
Can be used for identi…cation
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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34
Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
P.A. Chiappori, Columbia ()
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
Matching models
Cemmap Masterclass, March 2011
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34
Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
16 /
34
Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
take singlehood as a benchmark (interpretation!)
P.A. Chiappori, Columbia ()
Matching models
Cemmap Masterclass, March 2011
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Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
take singlehood as a benchmark (interpretation!)
assume the αIJ
i are extreme value distributed
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Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
take singlehood as a benchmark (interpretation!)
assume the αIJ
i are extreme value distributed
then logit and expected utility:
ū I = E max U IJ + αIJ
i
J
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= ln
∑ exp U IJ + 1
J
Matching models
!
=
ln aI 0
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34
Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
take singlehood as a benchmark (interpretation!)
assume the αIJ
i are extreme value distributed
then logit and expected utility:
ū I = E max U IJ + αIJ
i
J
= ln
∑ exp U IJ + 1
J
!
=
ln aI 0
Problem: can we compare the ū across classes?
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Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
take singlehood as a benchmark (interpretation!)
assume the αIJ
i are extreme value distributed
then logit and expected utility:
ū I = E max U IJ + αIJ
i
J
= ln
∑ exp U IJ + 1
J
!
=
ln aI 0
Problem: can we compare the ū across classes?
needed (marital college premium)
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Empirical implementation
Theorem
A NSC for i 2 I being matched with a spouse in J is:
U IJ + αIJ
i
U I 0 + αIi 0
U IJ + αIJ
i
U IK + αIK
for all K
i
In practice:
take singlehood as a benchmark (interpretation!)
assume the αIJ
i are extreme value distributed
then logit and expected utility:
ū I = E max U IJ + αIJ
i
J
= ln
∑ exp U IJ + 1
J
!
=
ln aI 0
Problem: can we compare the ū across classes?
needed (marital college premium)
but relies on a strong assumption (‘What is the unit?’)
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Empirical implementation
Extension: heteroskedasticity
IJ
IJ
εIJ
ij = σi αi + µj βj
(1)
then
ui
vj
= U IJ + σI αIJ
i
= V IJ + µJ βIJ
j
and conditions become:
U IJ
+ αIJ
i
σI
U IJ
+ αIJ
i
σI
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UI 0
+ αIi 0
σI
U IK
+ αIK
for all K
i
σI
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Why does heteroskedasticity matter?
Homoskedastic version: marital college premium measured by the
di¤erence ū I ū K (I = college, K = high school) and
ū I
ū K = ln
aK 0
aI 0
Intuition:
people marry if their (idiosyncratic) gain is larger than some threshold.
homoskedastic: one-to-one mapping between the mean and the
percentage below the threshold
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Why does heteroskedasticity matter?
Heteroskedastic version: now
ū I
ū K = σK ln aK 0
σI ln aI 0
and the one-to-one mapping is lost!
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Covariates
αIJ
i
IJ
= Xi .ζ IJ
m + α̃i
βIJ
j
= Yj .ζ IJ
f + β̃j
IJ
and
aIJ
= Pr (i, characteristics Xi , matched with a female in J )
=
exp U IJ + Xi .ζ IJ
m
IK
I0
∑K exp U IK + Xi .ζ m + exp U I 0 + Xi .ζ m
!standard logit
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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 the
U IJ /σI .
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Test and identi…cation: dynamic data
Idea: structural model M = Z IJ , σI , µJ holds for di¤erent cohorts
c = 1, ..., T with varying class compositions. Then:
J IJ
gij ,c = ZcIJ + σI αIJ
i ,c + µ βj ,c
with the identifying assumption:
ZcIJ = ζ Ic + ξ Jc + Z IJ
Interpretation: trend a¤ecting the surplus but not the supermodularity
ZcIJ
ZcIL
ZcKJ + ZcKL = Z IJ
Z IL
Z KJ + Z KL
Normalizations
ζ I1 = ξ J1 = 0 so that Z IJ = Z1IJ .
For any c > 1, the ζ Ic and ξ Jc are only de…ned up to a (common)
additive constant, therefore ξ 1c = 0 for all c.
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Test and identi…cation: dynamic data
Test
Start from:
acIJ = Pr (i 2 I matched with a female in J at c ) =
exp UcIJ /σI
1 + ∑K exp (UcKJ /σK )
De…ne pcIJ = UcIJ /σI , qcIJ = VcIJ /µJ , then
pcIJ = ln
gives for all (I
pc1J
1
acIJ
∑K acIK
2, J
2):
pc11 = σI
pcIJ
Z IJ
P.A. Chiappori, Columbia ()
and σI pcIJ + µJ qcIJ = ζ Ic + ξ Jc + Z IJ
pcI 1 + µJ
ZI1
qcIJ
qc1J
µ1 qcI 1
qc11
Z 1J + Z 11
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Test and identi…cation: dynamic data
Test
De…ne the vectors:
PIJ
=
p1IJ
p1I 1 , ..., pTIJ
pTI 1
QIJ
=
q1IJ
q11J , ..., qTIJ
qT1J
RIJ
=
p11J
p111 , ..., pT1J
pT11
1 = (1, ..., 1)
Then for each pair (I
2, J
RIJ = σI PIJ + µJ QIJ
2):
µ 1 QI 1
Z IJ
ZI1
Z 1J + Z 11 1
therefore:
RIJ belongs to the subspace generated by PIJ , QIJ , QI 1 , 1 ,
the coe¢ cient of PIJ (resp. QIJ ,resp. QI 1 ) does not depend on J
(resp. I , resp. is constant).
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Test and identi…cation: dynamic data
Identi…cation
From
RIJ = σI PIJ + µJ QIJ
µ 1 QI 1
we (over)identify σI , µJ and Z IJ
Z IJ
ZI1
ZI1
Z 1J + Z 11 1
Z 1J + Z 11
For c = 1:
σI p1IJ + µJ q1IJ = Z IJ
identi…es Z IJ
Then
σI pcI 1 + µ1 qcI 1 = ζ Ic + Z I 1 8I , 1, c
identi…es ζ Ic , and ξ Jc follows
In practice: minimum distance.
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Data
American Community Survey, a representative extract of Census. The
2008 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 and
above
Construct 28 ’cohorts’; for each cohort, matrix of marriage
proportions by classes (plus singles)
Age ! assumption: husband in cohort c marries wife in cohort c + 2
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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
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Results: trends
Surplus of diagonal matches
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34
Results: marital college premium
In principle, marital college premium has several components:
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Results: marital college premium
In principle, marital college premium has several components:
Marriage probability
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Results: marital college premium
In principle, marital college premium has several components:
Marriage probability
Spouse’s (distribution of) education
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Results: marital college premium
In principle, marital college premium has several components:
Marriage probability
Spouse’s (distribution of) education
Surplus generated
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34
Results: marital college premium
In principle, marital college premium has several components:
Marriage probability
Spouse’s (distribution of) education
Surplus generated
Distribution of the surplus
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Results: marital college premium
In principle, marital college premium has several components:
Marriage probability
Spouse’s (distribution of) education
Surplus generated
Distribution of the surplus
Our estimates for women:
Year born
Marriage Probability
Coll Educated Spouse
Surplus (Coll Husband)
Share (Coll Husband)
P.A. Chiappori, Columbia ()
HS
HS
HS
HS
1945-47
94% Coll 89%
39% Coll 84%
.258 Coll .473
47% Coll 51%
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HS
HS
HS
HS
1970-72
78% Coll 81%
38% Coll 84%
-.10 Coll .286
45% Coll 57%
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34
Results: marital college premium
Expected surplus: women
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34
Results: marital college premium
Expected surplus: men
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34
Results: marital college premium
In summary:
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34
Results: implied intrahousehold resource allocation
Shares:
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34
Conclusion
1
Frictionless matching: an interesting tool for empirical analysis,
especially when not interested in frictions
2
Modelling: link with standard discrete choice models
3
Identi…cation: dynamic data
4
Advantages of a structural model:
Quantify the various e¤ects
Summarize them into a marital college premium
5
Main …nding: predictions of theoretical models con…rmed
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34
Matching models of the marriage market: theory and
empirical applications
Pierre-André Chiappori,
Columbia University
Cemmap Masterclass, March 2011
P.A. Chiappori, Columbia ()
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22
Overview
Session 1: Background: Collective models of household behavior
Session 2: Matching models: theory (Part 1)
Session 3: Matching models: theory (Part 2)
Session 4: Matching models: applications
Session 5: Matching models: empirical applications (Part 1)
Session 6: Matching models: empirical applications (Part 2)
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Session 6
Matching models: empirical applications
Part 2: reduced form models of multidimensional
matching
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22
Multidimensional matching
Theoretical models of matching under TU do not require a
one-dimensional framework
Still in practice, most (if not all) empirical models are one-dimensional
Two types of approaches
1
2
Multidimensional framework ‘reduces to’a single-dimensional one !
Chiappori Ore¢ ce Quintana-Domeque 2009
‘True’multidimensional model ! Chiappori Ore¢ ce
Quintana-Domeque 2010
Additional dimension: continuous versus discrete characteristics
not important in theory
but crucial empirically....
Here: ‘True’multidimensional model, one continuous and one discrete trait
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Multidimensional matching: the ‘single index’approach
Framework:
Populations of men and women.
Each potential husband characterized by:
observable characteristics Xi = Xi1 , ..., XiK
unobservable characteristic εi 2 RS , centered and independent of X .
all characteristics are continuous
Same for women: observable variables Yj = Yj1 , ..., YjL ,
unobservable characteristics η j 2 RS
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Multidimensional matching: the ‘single index’approach
Key assumption:
Assumption I The ‘attractiveness’of male i (resp. female j) on the
marriage market is fully summarized by a one-dimensional index
Ii = I Xi1 , ..., XiK , εi (resp. Jj = J Yj1 , ..., YjL , η j ). Moreover, these
indices are separable in Xi1 , ..., XiK and Yj1 , ..., YjL
Ii
respectively; i.e.
= i I Xi1 , ..., XiK , εi
and
Jj
= j J Yj1 , ..., YjL , η j
for some mappings i, j from RS +1 to R and I (resp. J) from RK (resp.
R L ) to R.
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Multidimensional matching: the ‘single index’approach
In particular:
‘Iso-index’pro…les:
I Xi1 , ..., XiK
=C
where C is a constant
Compensation ! marginal rate of substitution between
characteristics n and m:
MRSim,n =
P.A. Chiappori, Columbia ()
∂I /∂X n
∂I /∂X m
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Multidimensional matching: the ‘single index’approach
‘Fully summarized’?
The joint density d µ X 1 , ..., X K , Y 1 , ..., Y L of observables among
married couples has the form:
h
i
d µ X 1 , ..., X K , Y 1 , ..., Y L = d ν I X 1 , ..., X K , J Y 1 , ..., Y L
for some measure d ν on R2 . Therefore:
The conditional distribution of Y 1 , ..., Y L given X 1 , ..., X K only
depends on the value I X 1 , ..., X K (and same for men)
‘Iso-attractiveness’: same conditional distribution of spouse’s
characteristics ! ‘iso-index’
Here: the subindex I , which only depends on observables, is a
su¢ cient statistic for the distribution of characteristics of a man’s
spouse (and same for women)
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Multidimensional matching: the ‘single index’approach
In particular, one can (ordinally) non parametrically identify attractiveness
indices. Indeed, for any characteristic s of the wife:
h
i
h
i
E Y s j Xi1 , ..., XiK = φs I Xi1 , ..., XiK
for some function φs . Then:
‘iso-index’or ‘iso-attractiveness’pro…les:
h
i
E Y s j Xi1 , ..., XiK = C 0
marginal rate of substitution between characteristics n and m:
MRSim,n =
∂E Y s j Xi1 , ..., XiK /∂X n
∂I /∂X n
=
,
∂I /∂X m
∂E Y s j Xi1 , ..., XiK /∂X m
Over identifying restrictions: does not depend on s
Works for any moment
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‘True’multidimensional matching: the economics of
marital smoking
Setting:
Two populations (men and women) of equal size, normalized to one.
Socio-economic status: continuous variables x and y , uniformly
distributed over [0, 1]
Smoking: dichotomic, independent of status; sM and sW proportions
of smokers
Surplus:
Σ = f (x, y ) if both spouses do not smoke
Σ = λf (x, y ) otherwise, λ < 1
In practice
f (x, y ) = (x + y )2 /2
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‘True’multidimensional matching: the economics of
marital smoking
Basic remark:
The ‘twisted’condition does not hold.
Woman, index x0 , non smoker:
∂x Σ = (x0 + y1 ) if she marries a non smoker with index y1
∂x Σ = λ (x0 + y2 ) if she marries a smoker with index y2 .
h
i
(1 λ )x 0
For any y2 2
,
1
, if y1 = λy2 (1 λ) x0 , then the couples
λ
(x0 , y1 ) and (x0 , y2 ) violate the twisted buyer condition; works for an open
set of values x0 - namely x0 2 0, 1 λ λ .
Consequence:
The stable matching may not be pure.
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‘True’multidimensional matching: the economics of
marital smoking
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
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Main result
Proposition
sM
0
Assume that λ
sM +1 . There exists four values X , Y , Y and p, all
between 0 and 1, such that:
For all x X , a non-smoking woman with index x is matched with
probability 1 to a non-smoking husband with index
y=
1
1
sW
x
sM
sM
1
sW
sM
Y
and a smoking woman with index x is matched with probability 1 to a
smoking husband with index
y0 =
sW
sM sW
x+
sM
sM
Y0
In particular, smoking men and non smoking women marry ‘down’,
whereas non-smoking men and smoking women marry ‘up’.
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Proposition
For x < X , a non-smoking woman with index x is matched:
with probability p, to a smoking husband with index
y0 =
with probability 1
(1
sW ) p + sW
x < Y0
sM
p, to a non-smoking husband with index
y=
(1
sW ) ( 1
1 sM
p)
x <Y
Moreover, for any …xed index x of the wife, smoking husbands have a
higher index than non smoking ones - i.e., y 0 > y .
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22
Proposition
For x < X , a smoking woman with index x is matched with
probability 1 to a smoking husband with index:
y0 =
(1
sW ) p + sW
x < Y0
sM
Finally, there are no couples in which the wife smokes and the
husband does not.
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22
NS
S
S
NS
Y’
X
Y
Women
Men
u
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x
Female utility (dashed = non smokers); sM = .25, sW = .2, λ = .8
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22
v 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
y
Male utility (dashed = non smokers); sM = .25, sW = .2, λ = .8
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22
Predictions
Non smoking husbands always marry a non smoking wife
Smoking husbands with a higher index (y 0
smoking wife with probability 1
Y 0 ) marry a high index,
Smoking husbands with a lower index (y 0 < Y 0 ) marry either a
smoking 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
non smoking husband with probability 1
X ) marry a high index,
Non smoking wifes with a lower index (x < X ) marry either a smoking
or a non smoking husband with positive probability; moreover, the
smoking husband is of higher quality than the non smoker.
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22
Empirical tests
In practice, randomness in the matching
Here: 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 he
smokes and she does not (denoted NS-S) - i.e., the ratio of S-NS to
NS-S couples should be smaller than:
r=
sW ( 1
sM ( 1
sM )
sW )
In our sample, sM = .22, sW = .17, r = .72
P.A. Chiappori, Columbia ()
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22
Empirical tests
Among couples with identical smoking habits, assortative matching
on socioeconomic status.
Non-smoking wives married with a smoking husband have a lower
socioeconomic status than those married with a non-smoking husband
Smoking husband married with a non-smoking wife have a lower
socioeconomic status than those married with a smoking wife.
When two (non smoking) women with the same (low) index marry
respectively a smoker and a non smoker, the non smoker should on
average be of lower status than the smoker. In practice, smoking is
negatively correlated with status, especially for men. Still, we predict
that controlling for the wife’s ‘quality’, the smoking habit of the
husband should be less negatively correlated with his status.
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22
Empirical results
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22
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%
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 5:
Sorting by education
Husband’s age 24-34, Wife’s age 22-32.
CPS 1996–2007.
Both Non-Smokers
Spouse’s Education
N
R2
Wife’s
Education
Husband’s
Education
Wife’s
Education
Husband’s
Education
0.633***
(0.013)
0.694***
(0.014)
0.458***
(0.035)
0.442***
(0.034)
8710
0.47
8710
0.47
1150
0.27
1150
0.28
Smoking Husband
&
Non-Smoking Wife
Spouse’s Education
N
R2
Both Smokers
Non-Smoking Husband
&
Smoking Wife
Wife’s
Education
Husband’s
Education
Wife’s
Education
Husband’s
Education
0.600***
(0.031)
0.618***
(0.036)
0.495***
(0.044)
0.497***
(0.041)
1408
0.43
1408
0.44
767
0.34
767
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***
(0.012)
0.473***
(0.027)
0.684***
(0.013)
0.556***
(0.026)
Spouse Smokes
−0.141**
(0.060)
−0.025
(0.086)
−0.209***
(0.074)
0.160**
(0.076)
10118
0.47
1917
0.29
9477
0.46
2558
0.36
N
R2
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
Own Education
(1)
(2)
(3)
(4)
−0.028***
(0.002)
−0.024***
(0.002)
−0.021***
(0.006)
−0.029***
(0.007)
Test of Equality
(p-value)
Spouse’s Education
N
R2
Women with S Men
0.0146
0.0341
--
−0.006**
(0.002)
--
0.014**
(0.006)
10118
0.05
10118
0.05
2558
0.06
2558
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
(0.338)
0.157*
(0.090)
Smoking Husband
0.819**
(0.337)
0.263***
(0.089)
Non-Smoking Wife × Non-Smoking Husband
0.476
(0.331)
0.185**
(0.088)
Smoking Wife × Smoking Husband
−0.659*
(0.351)
−0.204**
(0.093)
Mean of the dependent variable
0.17
0.31
N
12,035
12,035
2
R
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