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'
becomes: ˆ
'
ˆ
' ˆ
'
μu ( μX
) + 2u
( cX ) c 2 u ( X ) (1 ) u ( Xˆ
c
= μλ μ + − λ ) , yielding the following relationship
2
1
1
2
2
between first and second period outcomes:
'
u ( μXˆ
1)
'
u ( μXˆ
)
2
' ˆ
'
(1 − λ)
u ( X ) ( ˆ
2
c uc
cX
1)
= λ +
−
'
2 ( ˆ
'
μ u μX
) μ u ( μXˆ
(4)
)
2
2
When the last term in (4) is zero (the couple has no children) the equation is the same as (1), and so
we can write:
⎡ u ' (μX ˆ
1
) ⎤
⎢
⎣ u ' (μX ˆ ⎥
2
) ⎦
children
⎡
< u' (μX ˆ
1
) ⎤
⎢
⎣ u ' (μX ˆ ⎥
2
) ⎦
no children
(5)
One can see that μ ˆ X 1
> μ ˆ X 2 and thus ˆ X 1
> ˆ X 2 for heterosexual couples with and without children.
Additionally, the difference between first-period and second-period expenditures is higher for
opposite-sex couples with children than for those without. This means that heterosexual couples
with children save less than do those without, as it is found in the literature (Browning and Ejrnaes,
2009; Scholz and Seshadri, 2007; Browning and Lusardi, 1996).
Within this setting, we can now compare same-sex to opposite–sex couples’ savings. For
homosexual couples we have that U = u(
μ X ) + 2uc ( cX ) + 2λu(
μ ) , so that the first-order
2
1 1
X
2
'
condition is ˆ
'
ˆ
'
μ u ( μX
) + 2u
( cX ) c 2 u ( Xˆ
c
= λμ μ ) , with λ = 1 for lesbian couples and λ < 1for
2
1
1
2
gay couples. It follows that ˆ X 1
> ˆ X 2 for lesbian couples with children. Given that lesbian couples
with no children were shown to save half of their income, and that lesbian couples with children
spend more in the first period, we can safely assert that lesbian couples with children save less than
those who do not have children. For gay couples as well the presence of children decreases
'
⎡ u ( μXˆ
savings: ⎢
'
⎣u
( μXˆ
1
2
) ⎤
⎥
)
⎦
children
'
c uc
( cXˆ
= λ −
'
μ u ( μXˆ
1
2
'
) ⎡ u ( μXˆ
< λ = ⎢
'
)
⎣u
( μXˆ
1
2
) ⎤
⎥
)
⎦
no−children
.
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