The name is absent

Incremental Risk Vulnerability

f *(w,y,s) = qι∆(yι) [-u'"(Wι) - u"(Wɪ)r^] + q₂∆(y2) [-u'"(W2) - u"(W2)r^]     (16)

Since q ₁∆(yɪ) + q₂∆(y₂) = 0, it follows that (14) can be rewritten as

f*(w,y,s) = qι∆(yɪ) [(-u’"(Wɪ) - u"(Wɪ)f) - (-u’"(W2) - u"(W2)f)]      (17)

Hence

u'"(W2) - u'"(Wɪ) < -f [u"(W2) - u"(Wɪ)]                    (18)

is a sufficient condition for f* > 0, given f.

As shown in equation (10), Γ is a convex combination of r(W_α), r(Wɪ) and r(W₂) with
Wɪ < W_α < W₂, hence f ∈ {r(W)|W ∈ [Wɪ,W₂]}. Hence, a sufficient condition for (18) is
that

u'"(W2) - u'"(Wɪ) < -r(W) [u"(W2) - u"(Wɪ)]                 (19)

for all {Wɪ ≤ W ≤ W₂} as given by the condition of Proposition 1.

Alternatively, suppose that q0 replaces either qɪ or q₂ in the optimal basis. In this case the
above argument remains the same with q_α instead of either qɪ or q₂, in equation (16).

b) By an analogous argument, it can be shown that ∂r(w)/∂s < [=] 0 is equivalent to
u'"(W2) - u"'(Wɪ) > [=] - r(W)[u''(W2) - u''(Wɪ)] ∀ {Wɪ ≤ W ≤ W2} □

Appendix 2

Proof of Proposition 2

We need to show that conditions (4) and (5) are sufficient for r(w) - ^(w) ≥ 0. E_v [∙] denotes
expectations over v. From the definition of the twice derived risk aversion, ^,

u '( w + y + e )    ,          .

---^v ,_z r---_ ( w + y + e )

Ey+e u ' ( w + y + e ) ^v

^ee ^{u /( w} + ^y + ^e ) E Г ^{u /( w} + ^y + ^e ) r ₍ w + _y + e A
Ey+e u ' ( w + y + e ) ^e t Ee u '( w + y + e )             ʃ

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