Marriage, Divorce, and Asymmetric Information Leora FriedbergSteven SternUniversity of Virginia March 2007

Model U h, U w = 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 V h (U h,U w ) and V w (U h,U w ) Non-negative derivatives Bounds on altruism

Perfect Information With perfect information, the marriage continues iff V h (U h,U w ) + V w (U h,U w ) >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 ε w The probability of a divorce is F w (ε w * )

Partial Information

The husband chooses p * :

An Equilibrium Exists: (monotonicity) (reservation values) ε w *, ε h * (effect of p on res val) (comp statics for p) (information in p) (comp statics for div prob)

Proof sketch Assume (temporarily) that

Proof Sketch And show that And then

Proof Sketch And then And then Schauder fixed point theorem And then comp stats for divorce probs

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[U h ] = [  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 U w  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[U h ] = [  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[U h ]+E[U w ], 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 households surveyed in 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 households surveyed in 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.

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]

Overheard Interviews and Bias

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 = E h [u* w ] =  w, v* w = E w [u* h ] =  h observable happiness depends on observable control variables X i : either  h i = X i  h,  w = X i  w or  h i = X i ,  w = X i  People actually report discrete values: u h, u w, v h, v w 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 UnrestrictedRestricted VariableMaleFemaleOwnSpouse Constant 1.224**1.459**1.383**1.394** (-0.108)(0.091)(0.089)(0.088) Age/ ** (0.015)(0.013) (0.012) White 0.260**0.237**0.243** (0.069)(0.058)(0.055) Black **-0.324**-0.322** (-0.084)(0.071)(0.068)  Race **-0.143* (0.095)(0.086)(0.083) HS Diploma (0.063)(0.054)(0.052) College Degree 0.275**0.185**0.214** (0.042)(0.034)(0.033) ∆Education (0.044)(0.037)(0.036)

Table 4 Estimation Results for Model Without Caring Preferences UnrestrictedRestricted VariableMaleFemaleOwnSpouse t1t **-0.727** (0.020) t2t t3t **0.830** (0.013) t4t **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 (θ h,θ w ) 0.411**0.409** (0.0008)(0.008) Log Likelihood

Table 5 Moments of Predicated Behavior Standard Deviation MeanAcross HouseholdsWithin Households Divorce probablities No caring preferences without divorce data with divorce data Caring preferences Side payments No caring preferences without divorce data with divorce data Caring preferences

Estimation w/ Caring Specify Impose restrictions:

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 HouseholdsWithin Households Divorce probablities No caring preferences without divorce data with divorce data Caring preferences Side payments No caring preferences without divorce data with divorce data Caring preferences

Table 6 Estimation Results With Divorce Data Variable WithWithout Variable WithWithout Caring Own Constant 1.45**0.841** t1t **-0.826** (0.240)(0.013)(0.087)(0.003) Spouse constant 1.469**0.534** t3t **3.702** (0.139)(0.013)(0.2173)(0.086) Age/ ** t4t **5.117** (1.428)(0.001)(0.128)(0.004) White 0.599**-0.126** Var (θh) 1.305**1.476** (0.097)(0.003)(0.548)(0.004) Black 0.471**0.520** Var (θw) 1.618**1.374** (0.197)(0.009)(0.369)(0.007) ∆Race ** Corr (θh,θw) 0.678**0.367** (0.054)(0.002)(0.014)(0.004) HS Diploma ** Φ ** (0.414)(0.002)(0.202) College Degree **-0.099** Φ ** (0.064)(0.002)(0.020) ∆Education 0.111*-0.189** Φ 10 1 (0.071)(0.003) Φ 11 * ** (0.0003) Objective function Φ 20 * ** (0.021)

Specification Tests Kids on divorce – no significant effect Marriage duration on signal noise variance – t-statistic = New kid on signal noise variance – t- statistic = 2.20