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

Show that dc2(y)
d(k2−k) = −
ˆ H (k2 −k)
ˆ
H
∈ (0, 1) evaluated at ˆ λ and (ĉ2(y), ĉ2(x)).
ˆ
H (k2−k) = (1 − py) U ′ 2[c2(x)] 2 py U ′ 1[w(y) − c2(y)] R1[w(x) − c2(x)] −
(1 − py) U ′ 2[c2(x)] 2 py U ′ 1[w(y) − c2(y)] R1[w(y) − c2(y)] < 0 (19)
because U ′ (.) > 0, w(y) − c2(y) < w(x) − c2(x) < w(x) − c2(x), 18 and utility functions exhibit
DARA, so that R1[w(y) − c2(y)] > R1[w(x) − c2(x)].
Comparing (19) with (18) , it follows that 0 < ˆ
H
+ ˆ
H (k2−k) . QED
Show that dc2(y)
dy
= −
ˆ
H(y)
ˆ
H
∈ (0, 1) evaluated at ˆ λ and (ĉ2(y), ĉ2(x)).
ˆ
H(y) = (1 − py) U ′ 2[c2(x)] 2 py U ′′
1 [w(y) − c2(y)] =
= − (1 − py) U ′ 2[c2(x)] 2 py U ′ 1[w(y) − c2(y)] R1[w(y) − c2(y)] < 0 (20)
Comparing (20) with (18), it follows that 0 < ˆ
H
+ ˆ
H (k2−k) . QED
Show that dc2(y)
dpy
= −
ˆ
Hpy
ˆ
H
> 0 evaluated at ˆ λ and (ĉ2(y), ĉ2(x)).
ˆ
Hpy = [U2[c2(y)] − U2[c2(x)]] py U ′ 2 [c2(y)] U ′ 1 [w(x) − c2(x)] [R1[w(x) − c2(x)] + R2[c2(x)]]
− (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] U ′ 1 [w(x)−c2(x)]
1−py
because Ri(.) > 0, Ui ′(.) > 0, and c2(y) ≤ c2(x), so that [U2[c2(y)] − U2[c2(x)]] < 0. QED
Show that dc2(y)
dx
= −
ˆ
H(x)
ˆ
H
< 0 and dc2(y)
d¯x
= −
ˆ
H(¯x)
ˆ
H
< 0
< 0 evaluated at ˆ λ and (ĉ2(y), ĉ2(x)).
ˆ
H(x) = − (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] px U ′′
1 [w(x) − c2(x)] > 0.
H(¯x) = − (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] p¯x U ′′
1 [w(¯x) − c2(x)] > 0.
These follow from U ′ (.) > 0 and U ′′ (.) < 0. QED
Show that dc2(y)
d p¯x
= −
px+p¯x
ˆ
H p¯x
px+p¯x
ˆ
< 0 evaluated at
H
ˆ λ and (ĉ2(y), ĉ2(x)).
ˆ H p¯x
px+p¯x
= (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] [U ′ 1 [w(x) − c2(x)] − U ′ 1 [w(¯x) − c2(x)]] > 0
18 The optimal sharing rule gives some insurance to both parties, for both are risk-averse. Appendix B proves this
result under full information. The proof under hidden information runs similarly.
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