Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

10. An equilibrium model of marriage, fertility and divorce 445
probability 0.5, sothat
W1 = EW1,0(θ) = 7
2
Y + 1
4
1 5
Y + a =
2 6
(10.22)
If p =0, W1,1(θ) is the highest for all θ, implying that all couples marry,
have children and do not divorce, so that
W1 = EW1,1(θ) =4Y +(q ∗ − c) = 3
6
(10.23)
The calculation of welfare is a bit more complex if p =0.25. Inthiscase,
the maximum is given by W0(θ) if θ ≤ 0 and by W1(θ) if θ ≥ 0. Thus,
W1 =
7
2
Y + 1
4
Y + 1
2
3 a + 2E(θ/θ ≤ 0)
+
2
4Y +(q∗ − c)+2E(θ/θ ≥ 0)
2
= 4
6
(10.24)
These calculations illustrate that ex-antewelfarerisesaswemovetoequilibrium
points with higher p, reflecting the positive externality associated
with an increase in the aggregate number of singles.
10.3 Income uncertainty and ex-post heterogeneity
The simple model assumed perfect symmetry among spouses and that all
individuals have the same incomes which remain fixed over time. We now
allow income to change over time, which creates income heterogeneity expost.
As before, all men and women have the same income, Y ,inthefirst
period of their life. However, with probability λ income in the second period
rises to Y h and with probability 1−λ it declines to Y l . To maintain ex-ante
symmetry, we assume that the incomes of men and women follow this same
process. To simplify, we shall assume now that the quality of the match, θ,
is revealed only at the end of each period. The realized value of θ at the end
of the first period can trigger divorce, while the realized value of θ at the
end of the second period has no behavioral consequences in our two period
model. Since there are gains from marriage, and the commitment is only
for one period, everyone marries in the first period. However, in this case,
changes in incomes as well as changes in the quality of match can trigger
divorce. We continue to assume risk neutrality and joint consumption.
The main difference from the previous model is that at the beginning
of the second period there will be two types of potential mates, rich and
poor. Let α be the expected remarriage rate and π the proportion of high
income individuals among the divorcees, and let y = πY h +(1− π)Y l be
the average income of the divorcees. Then the expected values of being
unattached in the beginning of the second period for each type are
V j (α, π) =Y j + αy, j = l, h. (10.25)