Transcript leora5 - University of Virginia

Marriage, Divorce, and
Asymmetric Information
Leora Friedberg
University of Virginia
Steven Stern
University of Virginia
March 2007
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)
Caring Preferences
• Vh(Uh ,Uw) and Vw(Uh ,Uw)
• Non-negative derivatives
• Bounds on altruism
Perfect Information
• With perfect information, the marriage
continues iff Vh(Uh ,Uw) + Vw(Uh ,Uw) >0
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)
Perfect Information
• If preferences are caring, then there is a
reservation value of εw
• The probability of a divorce is Fw(εw*)
Partial Information
Vw* ( w
Vh* ( h


, p) 


, p) 
*
h :Vh (  h , p )  0
*
w :Vw (  w , p )  0
Vw ( h  p   h , w  p   w )dFh ( h )

*
h :Vh (  h , p )  0
dFh ( h )
Vh ( h  p   h ,  w  p   w )dFw ( w )

*
w :Vw (  w , p )  0
dFw ( w )
Partial Information
• The husband chooses p*:
p*  arg max Vh* ( h , p) Pr[Vw* ( w , p)  0]
p
An Equilibrium Exists:
•
•
•
•
•
•
Vw* ( w , p)
Vh* ( h , p)
 0,
0
 w
 h
(monotonicity)
(reservation values) εw*, εh*
(effect of p on res val) p( p)  0, p( p)  0
(comp statics for p) p  0, p  0, p  0
(information in p) p    
(comp statics for div prob)  PrV  0  0,  PrV  0  0
*
w
*
h
*
*
*
h
w
h
*
h
h
*
w
h
*
w
h
Proof sketch
• Assume (temporarily) that
*
Vw* ( w , p)


*
 0,  w , w  0
 w
p
Proof Sketch
•
•
•
•
•
•
Vh*
0
 h
And show that
And then  h*
And then p  0
And then p  0, p


And then p    
And then V  0
*
h
*
*
h
w
*
h
*
w
 w
h
p*
 0,
0
 h
Proof Sketch
•
•
•
•
And then 
And then p  0
And then Schauder fixed point theorem
And then comp stats for divorce probs
*
w
*
w
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
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
Partial information
• p* is bigger if the husband is happier (unobservably or
observably):
dp*/dh> 0, dp*/dh>0
• p* is smaller if the wife is observably happier:
dp*/dw< 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
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
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)
Impact of the divorce cost
• Fewer divorces
• p* may rise or fall
• Expected utility from marriage may rise or fall
An example
• Assume that i  iid N(0,1), i = h,w
• Then the optimal payment p( hh) 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
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
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.
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.
Explanatory Variables
Mean
Std Dev Definition
Age
38.50
11.70 Age of Husband
White
0.82
0.38 Husband is White
Black
0.10
0.30 Husband is Black
Husband & wife have
dRace
0.03
0.17 different race
Husband has HS
HS Diploma
0.91
0.29 diploma
Husband has College
College
0.32
0.46 Degree
Husband & wife have
different education
dEducation
0.75
0.43 levels
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; 2Worse; 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]
Happiness of Husband if Separate
Perception of
Husband
Much Worse
0.25
Density
0.2
Worse
0.15
Same
0.1
Better
0.05
Much Better
0
Much Worse
Worse
Same
Perception of Wife
Better
Much Better
Happiness of Wife if Separate
Perception of
Wife
Much Worse
0.25
Density
0.2
Worse
0.15
Same
0.1
Better
Much Better
0.05
0
Much Worse
Worse
Same
Better
Perception of Husband
Much Better
Time Spent Preparing Meals
1.2
Cum Distn
1
0.8
Respondent-husband
Respondent-wife
0.6
Spouse-husband
Spouse-wife
0.4
0.2
0
0
10
20
Hours per Week
30
Joint Density, Fairness: Household Chores
0.6
Husband
Density
0.5
Very Unfair to Me
0.4
Unfair to Me
0.3
Fair
0.2
Unfair to Her
0.1
Very Unfair to Her
0
Very Unfair to
Unfair to
Me
Me
Fair
Wife
Unfair to Very
Him Unfair to
Him
Density
Joint Density, Fairness: Market Work
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Husband
Very Unfair to Me
Unfair to Me
Fair
Unfair to Her
Very Unfair to Her
Very Unfair to
Unfair to
Me
Me
Fair
Wife
Unfair to Very
Him Unfair to
Him
Density
Joint Density, Fairness: Spending Money
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Husband
Very Unfair to Me
Unfair to Me
Fair
Unfair to Her
Very Unfair to Her
Very Unfair to
Unfair to
Me
Me
Fair
Wife
Unfair to Very
Him Unfair to
Him
Joint Density, Fairness: Childcare
0.7
Husband
0.6
Very Unfair to Me
Density
0.5
Unfair to Me
0.4
Fair
0.3
Unfair to Her
0.2
Very Unfair to Her
0.1
0
Very Unfair to
Unfair to
Me
Me
Fair
Wife
Unfair to Very
Him Unfair to
Him
Overheard Interviews and Bias
Density
Overheard Variables Disaggregated by Age
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
20s
30s
40s
50s
husband
wife
husband
wife
overheard overheard overheard > overheard >
few minutes few minutes 15 minutes 15 minutes
husband
wife
overheard
most of
overheard
most of
Happiness in Marriage Disaggregated by
Proportion of Interview Overheard by Spouse
2.5
2
none
1.5
1
0.5
0
few min
> 15 min
most
husband
wife
self happy if
separate
husband
wife
spouse happy if
separate
husband
wife
probability of
separation
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 = Xih, w = Xiw 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
Estimation
• Log likelihood of each couple i:
Table 4
Estimation Results for Model Without Caring Preferences
Unrestricted
Variable
Constant
Age/100
White
Black
DRace
HS Diploma
College Degree
∆Education
Restricted
Male
Female
Own
Spouse
1.224**
1.459**
1.383**
1.394**
(-0.108)
(0.091)
(0.089)
(0.088)
.0235**
-0.009
0.001
(0.015)
(0.013)
(0.012)
0.260**
0.237**
0.243**
(0.069)
(0.058)
(0.055)
-0.314**
-0.324**
-0.322**
(-0.084)
(0.071)
(0.068)
-0.084
-0.170**
-0.143*
(0.095)
(0.086)
(0.083)
0.077
0.074
0.071
(0.063)
(0.054)
(0.052)
0.275**
0.185**
0.214**
(0.042)
(0.034)
(0.033)
0.023
-0.041
-0.021
(0.044)
(0.037)
(0.036)
Table 4
Estimation Results for Model Without Caring
Preferences
Unrestricted
Variable
t1
t2
t3
t4
Var (θ)
Corr (θh,θw)
Log Likelihood
Male
Female
Restricted
Own
Spouse
-0.728**
-0.727**
(0.020)
(0.020)
0.000
0.000
0.831**
0.830**
(0.013)
-0.013
2.071**
2.069**
(0.014)
(0.012)
1.226**
1.120**
1.225**
1.117**
(0.059)
(0.024)
(0.020)
(0.023)
0.411**
0.409**
(0.0008)
(0.008)
-20382.3
-20390.9
Table 5
Moments of Predicated Behavior
Standard Deviation
Mean
Across 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
0.045
0.068
0.180
Caring preferences
Side payments
No caring preferences
without divorce data
-1.07
0.083
0.714
with divorce data
-1.57
0.164
0.832
-1.26
0.764
2.104
Caring preferences
Estimation w/ Caring
• Specify
2
2 i
V U1 , U 2    ijU1iU 2j
i 0 j 0
 b  U1 , U 2  b
• Impose restrictions:
V1  0, V2  0, V11  0, V22  0,
V12  max V11 , V22 
Estimation w/ Caring
• Objective function is log likelihood function
with penalty for not matching divorce
probabilities in CPS data
Table 5
Moments of Predicated Behavior
Standard Deviation
Mean
Across 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
0.045
0.068
0.180
Caring preferences
Side payments
No caring preferences
without divorce data
-1.07
0.083
0.714
with divorce data
-1.57
0.164
0.832
-1.26
0.764
2.104
Caring preferences
Table 6
Estimation Results With Divorce Data
With
Without
Variable
Caring
1.45**
Caring
0.841**
Own Constant
(0.240)
(0.013)
1.469**
0.534**
Spouse constant
(0.139)
(0.013)
2.027
0.123**
Age/100
(1.428)
(0.001)
0.599**
-0.126**
(0.097)
(0.003)
0.471**
0.520**
(0.197)
(0.009)
0.038
-0.035**
(0.054)
(0.002)
-0.534
-0.264**
(0.414)
(0.002)
-0.238**
-0.099**
(0.064)
(0.002)
0.111*
-0.189**
(0.071)
(0.003)
White
Black
∆Race
HS Diploma
College Degree
∆Education
t1
t3
t4
Var (θh)
Var (θw)
Corr (θh,θw)
Φ01
Φ02
Φ10
Φ11 * 100
Objective function
With
Without
Caring
Caring
Variable
-78085
-117905
Φ20 * 100
-0.352**
-0.826**
(0.087)
(0.003)
1.284**
3.702**
(0.2173)
(0.086)
2.419**
5.117**
(0.128)
(0.004)
1.305**
1.476**
(0.548)
(0.004)
1.618**
1.374**
(0.369)
(0.007)
0.678**
0.367**
(0.014)
(0.004)
1.192**
(0.202)
-0.113**
(0.020)
1
0.014**
(0.0003)
-0.090**
(0.021)
Specification Tests
• Kids on divorce – no significant effect
• Marriage duration on signal noise variance
– t-statistic = -10.11
• New kid on signal noise variance – tstatistic = 2.20
Smoothed Joint Density of Theta
0.035
0.03
0.03-0.035
0.025
0.02-0.025
0.015
0.015-0.02
0.01
0.01-0.015
0.005
0.005-0.01
6
2
theta(w)
7
5.5
4
2.5
1
-0.5
0
-2
density
0.025-0.03
0.02
-2
theta(h)
0-0.005
Indifference Curves
10
8
6
v = -1
u(h)
4
v= 0
2
v= 1
v= 2
0
-4
-3
-2
-1
0
1
-2
-4
-6
u(w)
2
3
4
5
6
v= 3
Variation in Divorce Probabilities
1.2
1
theta(w) = -1.32
0.8
theta(w) = 0.99
theta(w) = 1.91
prob
0.6
theta(w) = 4.22
theta(h) = -1.12
0.4
theta(h) = 1.18
theta(h) = 2.33
0.2
theta(h) = 4.05
0
-4
0
-2
2
-0.2
theta
4
6
8
Variation in Side Payments
4
side payment
3
-4
-2
2
theta(w) = -1.32
1
theta(w) = 0.99
0
theta(w) = 1.91
-1 0
2
4
6
8
theta(w) = 4.22
theta(h) = -1.12
-2
theta(h) = 1.18
-3
theta(h) = 2.33
-4
theta(h) = 4.05
-5
-6
theta
Welfare Gains [theta(h) = 1.18]
2
1.5
Gain
1
theta(h) = -1.32
theta(h) = 0.99
0.5
theta(h) = 1.91
theta(h) = 4.22
0
0
0.2
0.4
0.6
-0.5
-1
gamma
0.8
1
1.2
Efficient and Inefficient Divorce Probabilities
0.9
0.8
0.7
Probability
0.6
0.5
efficient
0.4
inefficient
0.3
0.2
0.1
0
-1
0
1
2
3
theta(h)+theta(w)+eps(h)
4
5