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

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8. Sharing the gains from marriage 373
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3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2
FIGURE 8.7. Pareto frontier
Because of the public consumption, our simple model exhibits what
Lundberg and Pollack call ‘production dominance’; that is, any single man
and any single woman can do better by marrying. To see why, just note
that a single man with income y chooses Q = cm = y/2 for a utility of y 2 /4,
while a single woman with income y 0 > ¯c achieves a utility that equals y 0 .
Now, by marrying, they achieve an income (y + y 0 ).Ify 0 ≤ y +¯c, hecan
achieve (y +(y 0 − ¯c)) 2 /4 >y 2 /4 while she gets ¯c +(y + y 0 ) /2 >y 0 .If,on
the contrary, y 0 >y+¯c then he can achieve (y 0 − ¯c) y>y 2 /4, while she
remains at y 0 . Therefore, in a frictionless model like this one (and without
non-monetary gains or costs), either all women or all men marry: singles
can only be on one side of the marriage market.
Assortativeness
The Pareto frontier just derived has a particularly tractable form. Indeed,
let us analyze the stability conditions along the lines previously described.
For v ≥ (y+y0 )+¯c
2 ,weget:
∂H (y + y0 ,v)
∂ (y + y0 ) = v − ¯c, ∂H (y + y0 ,v)
= − (2v − (¯c +(y + y
∂v
0 ))) (8.51)
implying that
∂2H (y + y0 ,v)
∂ (y + y0 ) 2 =0 and ∂2H (y + y0 ,v)
∂ (y + y0 =1 (8.52)
) ∂v
u