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Marriage, Divorce, and Asymmetric Information
Leora Friedberg Steven Stern
University of Virginia University of Virginia
March 2007
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Model
Uh, Uw = utility of husband, wife from being married
h, w = component of U that is observable to spouse
h, w = component of U that is private information
p = side payment (p>0 if the husband makes a side payment to the wife)
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Caring Preferences
• Vh(Uh ,Uw) and Vw(Uh ,Uw)
• Non-negative derivatives
• Bounds on altruism
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Perfect Information
• With perfect information, the marriage continues iff Vh(Uh ,Uw) + Vw(Uh ,Uw) >0
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Perfect information
• If preferences are not caring, marriages continue as long as:
– Suppose spouse j is unhappy (j+j<0)
– Spouse i is willing to pay p to j so that j is happy (j+p+j>0) as long as spouse i remains happy enough (i-p+i>0)
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Perfect Information
• If preferences are caring, then there is a reservation value of εw
• The probability of a divorce is Fw(εw*)
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Partial Information
0),(:
0),(:*
*
*
)(
)(),(),(
pV ww
pV wwwwhhh
hh
www
www
dF
dFppVpV
0),(:
0),(:*
*
*
)(
)(),(),(
pV hh
pV hhwwhhw
ww
hhh
hhh
dF
dFppVpV
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Partial Information
• The husband chooses p*:
]0),(Pr[),(maxarg *** pVpVp wwhhp
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An Equilibrium Exists:
• (monotonicity)
• (reservation values) εw*, εh
*
• (effect of p on res val)
• (comp statics for p)
• (information in p)
• (comp statics for div prob)
0),(
,0),( **
h
hh
w
ww pVpV
0)(
,0)( **
p
p
p
p hw
0 ,0 ,0***
hwh
ppp
hhp *
00Pr ,00Pr **
wh
wh
VV
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Proof sketch
• Assume (temporarily) that
0 , ,0),( *
**
p
pV ww
w
ww
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Proof Sketch
• And show that
• And then
• And then
• And then
• And then
• And then
0*
h
hV
*h
0*
ph
0 ,0 ,0***
hwh
ppp
hhp *
0*
w
wV
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Proof Sketch
• And then
• And then
• And then Schauder fixed point theorem
• And then comp stats for divorce probs
*w
0*
pw
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Partial Information wo/ Caring
• Suppose the husband makes an offer p• As before, they fail to agree (and divorce) if p is
such that:h-p+h< 0 or w+p+w< 0
• Now, this may occur inefficiently:– a higher p could preserve the marriage
– a higher p won’t be offered because the wife is unobservably unhappier than the husband believes
• If p is acceptable, the marriage continues
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Partial Information wo/ Caring
• The husband chooses his offer p* as follows:– he has beliefs about the density f(w) of his wife’s private
information w
– p* maximizes his expected utility from marriage, given those beliefs:
E[Uh] = [h-p+h]*[1-F(-w-p)]
p* solves [h-p+h ]*[f(-w-p)]-[1-F(-w-p)] = 0
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Partial information• p* is bigger if the husband is happier (unobservably or
observably):
dp*/dh> 0, dp*/dh>0
• p* is smaller if the wife is observably happier:
dp*/dw< 0
• The probability that Uw 0 (so the marriage continues after the offer p*) is higher if the husband is observably happier:
Pr[w+p+w 0]/h 0
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Other results
• We can compute utility from marriage, after the side payment
• Expected utility from marriage• Loss in utility (or expected utility) due to
asymmetric information
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Government policy
• Consider adding (or increasing) a divorce cost D• Husband pays D, wife pays (1-)D• Now, p* maximizes the husband’s expected utility
from marriage minus expected divorce costs:
E[Uh] = [h-p+h]*[1-F(-w-(1-)D-p)]
- D*F(-w-(1- )D-p)
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Impact of the divorce cost• Fewer divorces
• p* may rise or fall
• Expected utility from marriage may rise or fall
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An example
• Assume that i iid N(0,1), i = h,w
• Then the optimal payment p( hh) solves:
– we can use this to compute p*, the divorce probability, total expected value E[Uh]+E[Uw], welfare effects
– we can show how they vary with the husband’s happiness h+h and the wife’s observable happiness w
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Empirical analysis
• Data from the National Survey of Families and Households (NSFH)
• The NSFH reports:– each spouse’s happiness in marriage
– each spouse’s beliefs about the other’s happiness
• We can estimate determinants of each spouse’s happiness, the correlation of their happiness
• We can infer the magnitude of side payments
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Selection
• The NSFH sample is a random sample of 13008 households surveyed in 1987.
• We excluded 6131 households because there was no married couple, 4 because racial information was missing, 796 because the husband was younger than 20 or older than 65, and 1835 because at least one of the dependent variables was missing.
• This left a sample of 4242 married couples.
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Selection
• The NSFH sample is a random sample of 13008 households surveyed in 1987.
• We excluded 6131 households (no married couple), 4 (racial information was missing), 796 (the husband was younger than 20 or older than 65), and 1835 (at least one of the dependent variables was missing).
• This left a sample of 4242 married couples.
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Mean Std Dev Definition
Age 38.50 11.70 Age of HusbandWhite 0.82 0.38 Husband is WhiteBlack 0.10 0.30 Husband is Black
dRace 0.03 0.17Husband & wife have different race
HS Diploma 0.91 0.29Husband has HS diploma
College 0.32 0.46Husband has College Degree
dEducation 0.75 0.43
Husband & wife have different education levels
Explanatory Variables
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Dependent Variable
• Responses by each spouse to the following questions:– Even though it may be very unlikely, think for a
moment about how various areas of your life might be different if you separated. How do you think your overall happiness would change? [1-Much worse; 2-Worse; 3-Same; 4-Better; 5-Much better]
– How about your partner? How do you think his/her overall happiness might be different if you separated? [same measurement scale]
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Happiness of Husband if Separate
0
0.05
0.1
0.15
0.2
0.25
Much Worse Worse Same Better Much Better
Perception of Wife
Dens
ity
Much Worse
Worse
Same
Better
Much Better
Perception of Husband
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Happiness of Wife if Separate
0
0.05
0.1
0.15
0.2
0.25
Much Worse Worse Same Better Much Better
Perception of Husband
Dens
ity
Much Worse
Worse
Same
Better
Much Better
Perception of Wife
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Time Spent Preparing Meals
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30
Hours per Week
Cu
m D
istn Respondent-husband
Respondent-wife
Spouse-husband
Spouse-wife
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Joint Density, Fairness: Household Chores
0
0.1
0.2
0.3
0.4
0.5
0.6
VeryUnfair to
Me
Unfair toMe
Fair Unfair toHim
VeryUnfair to
Him
Wife
Den
sity
Very Unfair to Me
Unfair to Me
Fair
Unfair to Her
Very Unfair to Her
Husband
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Joint Density, Fairness: Market Work
0
0.10.2
0.30.4
0.5
0.60.7
0.8
VeryUnfair to
Me
Unfair toMe
Fair Unfair toHim
VeryUnfair to
Him
Wife
Den
sity
Very Unfair to Me
Unfair to Me
Fair
Unfair to Her
Very Unfair to Her
Husband
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Joint Density, Fairness: Spending Money
0
0.10.2
0.3
0.4
0.50.6
0.7
0.8
VeryUnfair to
Me
Unfair toMe
Fair Unfair toHim
VeryUnfair to
Him
Wife
Den
sity
Very Unfair to Me
Unfair to Me
Fair
Unfair to Her
Very Unfair to Her
Husband
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Joint Density, Fairness: Childcare
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
VeryUnfair to
Me
Unfair toMe
Fair Unfair to Him
VeryUnfair to
Him
Wife
Den
sity
Very Unfair to Me
Unfair to Me
Fair
Unfair to Her
Very Unfair to Her
Husband
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Overheard Interviews and Bias
Overheard Variables Disaggregated by Age
00.050.1
0.150.2
0.250.3
0.350.4
husband wife husband wife husband wife
overheardfew minutes
overheardfew minutes
overheard >15 minutes
overheard >15 minutes
overheardmost of
overheardmost of
Dens
ity
20s
30s
40s
50s
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Happiness in Marriage Disaggregated by Proportion of Interview Overheard by Spouse
0
0.5
1
1.5
2
2.5
husband wife husband wife husband wife
self happy ifseparate
spouse happy ifseparate
probability ofseparation
none
few min
> 15 min
most
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Estimation wo/ Caring• Dependent variables: each spouse’s utility from marriage
before side payments p each spouse’s happiness: u*h = h+h , u*w = w+w
• We assume the following:each spouse’s belief about the other spouse’s happiness:
v*h = Eh[u*w] = w , v*w = Ew[u*h] = h
observable happiness depends on observable control variables Xi:
either h i = Xih, w = Xiw or h i = Xi, w = Xi
• People actually report discrete values: uh, uw, vh, vw
• We estimate , the variance of (h,w), and the cutoff points determining how happiness u*,v* maps into discrete values u,v
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Estimation
• Log likelihood of each couple i:
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Table 4
Estimation Results for Model Without Caring Preferences
Unrestricted Restricted
Variable Male Female Own Spouse
Constant1.224** 1.459** 1.383** 1.394**
(-0.108) (0.091) (0.089) (0.088)
Age/100.0235** -0.009 0.001
(0.015) (0.013) (0.012)
White0.260** 0.237** 0.243**
(0.069) (0.058) (0.055)
Black-0.314** -0.324** -0.322**
(-0.084) (0.071) (0.068)
Race-0.084 -0.170** -0.143*
(0.095) (0.086) (0.083)
HS Diploma0.077 0.074 0.071
(0.063) (0.054) (0.052)
College Degree0.275** 0.185** 0.214**
(0.042) (0.034) (0.033)
∆Education0.023 -0.041 -0.021
(0.044) (0.037) (0.036)
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Table 4
Estimation Results for Model Without Caring Preferences
Unrestricted Restricted
Variable Male Female Own Spouse
t1
-0.728** -0.727**
(0.020) (0.020)
t2 0.000 0.000
t3
0.831** 0.830**
(0.013) -0.013
t4
2.071** 2.069**
(0.014) (0.012)
Var (θ)1.226** 1.120** 1.225** 1.117**
(0.059) (0.024) (0.020) (0.023)
Corr (θh,θw)0.411** 0.409**
(0.0008) (0.008)
Log Likelihood -20382.3 -20390.9
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Table 5
Moments of Predicated Behavior
Standard Deviation
MeanAcross Households
Within Households
Divorce probablities
No caring preferences
without divorce data 0.287 0.046 0.191
with divorce data 0.233 0.041 0.213
Caring preferences 0.045 0.068 0.180
Side payments
No caring preferences
without divorce data -1.07 0.083 0.714
with divorce data -1.57 0.164 0.832
Caring preferences -1.26 0.764 2.104
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Estimation w/ Caring
• Specify
• Impose restrictions:
bUUb
UUUUV ji
i
i
jij
21
21
2
0
2
021
,
,
221112
221121
,max
,0,0,0,0
VVV
VVVV
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Estimation w/ Caring
• Objective function is log likelihood function with penalty for not matching divorce probabilities in CPS data
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Table 5
Moments of Predicated Behavior
Standard Deviation
MeanAcross Households
Within Households
Divorce probablities
No caring preferences
without divorce data 0.287 0.046 0.191
with divorce data 0.233 0.041 0.213
Caring preferences 0.045 0.068 0.180
Side payments
No caring preferences
without divorce data -1.07 0.083 0.714
with divorce data -1.57 0.164 0.832
Caring preferences -1.26 0.764 2.104
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Table 6Estimation Results With Divorce Data
VariableWith Without
VariableWith Without
Caring Caring Caring Caring
Own Constant1.45** 0.841**
t1
-0.352** -0.826**
(0.240) (0.013) (0.087) (0.003)
Spouse constant1.469** 0.534**
t3
1.284** 3.702**
(0.139) (0.013) (0.2173) (0.086)
Age/1002.027 0.123**
t4
2.419** 5.117**
(1.428) (0.001) (0.128) (0.004)
White0.599** -0.126**
Var (θh)1.305** 1.476**
(0.097) (0.003) (0.548) (0.004)
Black0.471** 0.520**
Var (θw)1.618** 1.374**
(0.197) (0.009) (0.369) (0.007)
∆Race0.038 -0.035**
Corr (θh,θw)0.678** 0.367**
(0.054) (0.002) (0.014) (0.004)
HS Diploma-0.534 -0.264**
Φ01
1.192**
(0.414) (0.002) (0.202)
College Degree-0.238** -0.099**
Φ02
-0.113**
(0.064) (0.002) (0.020)
∆Education0.111* -0.189**
Φ10 1
(0.071) (0.003)
Φ11 * 100
0.014**
(0.0003)
Objective function -78085 -117905Φ20 * 100
-0.090**
(0.021)
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Specification Tests
• Kids on divorce – no significant effect
• Marriage duration on signal noise variance – t-statistic = -10.11
• New kid on signal noise variance – t-statistic = 2.20
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-2
-0.5 1
2.5 4
5.5 7
-2
2
6
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
density
theta(w)
theta(h)
Smoothed Joint Density of Theta
0.03-0.035
0.025-0.03
0.02-0.025
0.015-0.02
0.01-0.015
0.005-0.01
0-0.005
![Page 51: Marriage, Divorce, and Asymmetric Information Leora FriedbergSteven SternUniversity of Virginia March 2007](https://reader036.vdocuments.site/reader036/viewer/2022062516/56649e025503460f94aebf3d/html5/thumbnails/51.jpg)
Indifference Curves
-6
-4
-2
0
2
4
6
8
10
-4 -3 -2 -1 0 1 2 3 4 5 6
u(w)
u(h
)
v = -1
v = 0
v = 1
v = 2
v = 3
![Page 52: Marriage, Divorce, and Asymmetric Information Leora FriedbergSteven SternUniversity of Virginia March 2007](https://reader036.vdocuments.site/reader036/viewer/2022062516/56649e025503460f94aebf3d/html5/thumbnails/52.jpg)
Variation in Divorce Probabilities
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-4 -2 0 2 4 6 8
theta
pro
b
theta(w) = -1.32
theta(w) = 0.99
theta(w) = 1.91
theta(w) = 4.22
theta(h) = -1.12
theta(h) = 1.18
theta(h) = 2.33
theta(h) = 4.05
![Page 53: Marriage, Divorce, and Asymmetric Information Leora FriedbergSteven SternUniversity of Virginia March 2007](https://reader036.vdocuments.site/reader036/viewer/2022062516/56649e025503460f94aebf3d/html5/thumbnails/53.jpg)
Variation in Side Payments
-6
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -2 0 2 4 6 8
theta
sid
e p
aym
ent
theta(w) = -1.32
theta(w) = 0.99
theta(w) = 1.91
theta(w) = 4.22
theta(h) = -1.12
theta(h) = 1.18
theta(h) = 2.33
theta(h) = 4.05
![Page 54: Marriage, Divorce, and Asymmetric Information Leora FriedbergSteven SternUniversity of Virginia March 2007](https://reader036.vdocuments.site/reader036/viewer/2022062516/56649e025503460f94aebf3d/html5/thumbnails/54.jpg)
Welfare Gains [theta(h) = 1.18]
-1
-0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1 1.2
gamma
Gai
n
theta(h) = -1.32
theta(h) = 0.99
theta(h) = 1.91
theta(h) = 4.22
![Page 55: Marriage, Divorce, and Asymmetric Information Leora FriedbergSteven SternUniversity of Virginia March 2007](https://reader036.vdocuments.site/reader036/viewer/2022062516/56649e025503460f94aebf3d/html5/thumbnails/55.jpg)
Efficient and Inefficient Divorce Probabilities
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-1 0 1 2 3 4 5
theta(h)+theta(w)+eps(h)
Pro
bab
ilit
y
efficient
inefficient