How does the level of income impact the chances of break-up of married couples v

Question

How does the level of income impact the chances of break-up of married couples vs cohabiting couples and why is that? Why are the probabilities of union dissolution in relation to relative income of spouses so differently shaped? When Mr and Mrs Rose got married, their gain from marriage, in that first year, was U_1 = 100. They also had their outside options. Mrs Rose's outside option was U_F = 8 and Mr Rose's was U_M = 12. With time, however, the spark went gradually out of their marriage, so that in every year, their gain from marriage diminished by 10%. Expressed formally, at any time t, U_t = 0.9U_t - 1. The outside options stayed constant. How long will that marriage last before they would get divorced? If each of the spouses gets exactly half of the overall gain from marriage in any given year and this split is fixed and non-renegotiable, when will the couple divorce and who will initiate the divorce? Which of the two subquestions above, a. or b., implies that the Becker-Coase theorem does not apply and how can you tell?

Explanation / Answer

a. Total utility from marriage Ut = 0.9*Ut-1

Utility from other other options = Uf + Um = 8 + 12 = 20

They will get divorced if there tottal ytility falls below this level of other options so

So, they will get divorced if

Ut <= 20

Ut = 0.9Ut-1 is a geometric series

a1=100 , r=910 , an=20 , n=?

To find n we use formula

an=a1r^n1

In this example we have a1=100  ,  r=9/10 ,  an=20. After substituting these values to above formula, we obtain:

an=a1rn

20 =100+(9/10)^n1

log(0.9)^n-1 =log(1/5)

(n-1)log(0.9) = log(1/5)

n-1 = 15.27553

n = 16.27

Thus they will get divorced in 16 years

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