aggregate payoff ε. This result can be restated using the coefficient of the
negative third to the second derivative of the F-function, —f-z(ei)/ʌ'(e). This
coefficient is called the coefficient of absolute prudence of the investor’s risk
function; it is the analogue to Kimball’s coefficient of absolute prudence. Mul-
tiplying equation (28) by —1, taking logs and differentiating with respect to
βj yields
d ln(dei∕de3) _ _ f ({e) ɪ f z(<¾) dei
(29)
dej —f,j(ej) —f'(ei) de
This proves the equivalence of the first two statements in Proposition
3. Substituting dei∕dej in equation (29) from equation (28) shows that
d ln(dei∕de3∙)∕dβj > [_] [<] 0 if and only if (18) holds. ■
Appendix D: Proof of Proposition 4
For any HARA function,
f z(¾)
Ш (¾)]2
/f (ei).
Hence, by the last statement of Proposition 3, ei (ej■) is strictly convex
[linear] [strictly concave] if and only if
fi (ei)
—di
< [_] . :
—½∙
∀ ej.
Hence, by equation (17), ei(ej■) is strictly convex [linear] [strictly concave]
if and only if investor i’s sharing constant is smaller than [equal to] [greater
than] that of investor j. This proves Proposition 4. ■
39
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