Marriage, Divorce, and Asymmetric Information

Leora Friedberg Steven Stern

University of Virginia 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

0),(:

0),(:*

*

*

)(

)(),(),(

pV ww

pV wwwwhhh

hh

www

www

dF

dFppVpV

0),(:

0),(:*

*

*

)(

)(),(),(

pV hh

pV hhwwhhw

ww

hhh

hhh

dF

dFppVpV

Partial Information

• The husband chooses p*:

]0),(Pr[),(maxarg *** pVpVp wwhhp

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

Proof sketch

• Assume (temporarily) that

0 , ,0),( *

**

p

pV ww

w

ww

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

Proof Sketch

• And then

• And then

• And then Schauder fixed point theorem

• And then comp stats for divorce probs

*w

0*

pw

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*/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

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( 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

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.

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

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]

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

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

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

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

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

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

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

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

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

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

Estimation

• Log likelihood of each couple i:

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)

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

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

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

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

  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

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)  

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

-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

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

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

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

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

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