The Birth of Insurance Contracts - The Ataturk Institute for Modern ...

loan is optimal, with ˜c2(y) = ˜w(y), then for parameter values in an open neighborhood of ˜ λ,
˜c2(y) + dc2(y) = ˜w(y) + dw(y).
Show that ˆ
H
> 0 evaluated at ˆ λ and (ĉ2(y), ĉ2(x)).
ˆ
0 py U
H
=
′ 2 [c2(y)] (1 − py) U ′ 2 [c2(x)]
py U ′ 2 [c2(y)] h22 0
(1 − py) U ′ 2 [c2(x)]
=
0 h33
because both h22 and h33 are negative.
and
= − [py U ′ 2 [c2(y)]] 2 h33 − [(1 − py) U ′ 2 [c2(x)]] 2 h22 > 0
h22 = py U ′′
1 [w(y) − c2(y)] + η py U ′′
2 [c2(y)] =
= − py R1[w(y) − c2(y)] U ′ 1[w(y) − c2(y)] − η py R2[c2(y)] U ′ 2[c2(y)] =
= − py R1[w(y) − c2(y)] U ′ 1[w(y) − c2(y)] − py R2[c2(y)] U ′ 1[w(y) − c2(y)] =
= − py [R1[w(y) − c2(y)] + R2[c2(y)]] U ′ 1[w(y) − c2(y)] < 0 (15)
h33 = px U ′′
1 [w(x) − c2(x)] + p¯x U ′′
1 [w(¯x) − c2(x)] + η (1 − py)U ′′
2 [c2(x)] =
= − R1[w(x) − c2(x)] U ′ 1[w(x) − c2(x)] − η (1 − py) R2[c2(x)] U ′ 2[c2(x)] =
= − R1[w(x) − c2(x)] U ′ 1[w(x) − c2(x)] − R2[c2(x)] U ′ 1[w(x) − c2(x)] =
= − [R1[w(x) − c2(x)] + R2[c2(x)]] U ′ 1[w(x) − c2(x)] < 0, (16)
where the first equalities are simply the definitions of h22 and h33; the second equalities derive
from applying the definition of absolute risk-aversion coefficient, R(z) = − U ′′ (z)
U ′ (z) , and defining
U ′′
1 [w(x) − c2(x)]
R1[w(x) − c2(x)] = −
U ′ 1 [w(x) − c2(x)] = − px U ′′
1 [w(x) − c2(x)] + p¯x U ′′
1 [w(¯x) − c2(x)]
px U ′ 1 [w(x) − c2(x)] + p¯x U ′ 1 [w(¯x) − c2(x)]
to make the notation more compact; the third equalities follow from, respectively, the second and
third row in (12) with µy = µx = 0; the fourths, from simply operating; and the inequality, from
U ′ (.) > 0 and U ′′ (.) < 0, so that Ri(.) > 0, for i = 1, 2. QED
Note that ˆ
H
can be expressed as
ˆ
H
= [py U ′ 2 [c2(y)]] 2 U ′ 1 [w(x) − c2(x)] R1[w(x) − c2(x)] +
[py U ′ 2 [c2(y)]] 2 U ′ 1 [w(x) − c2(x)] R2[c2(x)] +
[(1 − py) U ′ 2 [c2(x)]] 2 py U ′ 1 [w(y) − c2(y)] R1[w(y) − c2(y)] +
[(1 − py) U ′ 2 [c2(x)]] 2 py U ′ 1 [w(y) − c2(y)] R2[c2(y)].
26
(17)
(18)