Heterogeneity of Investors and Asset Pricing in a Risk-Value World

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(e_i)/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(de_i∕de₃) _ _ ^{f (}{e⁾ ɪ f ^z(<¾) de_i

de_j           ^—f,_j^(ej)    ^—f'^(ei) ^de

This proves the equivalence of the first two statements in Proposition
3. Substituting de_i∕dej in equation (29) from equation (28) shows that
d ln(de_i∕de₃∙)∕dβj > [_] [<] 0 if and only if (18) holds.                      ■

Appendix D: Proof of Proposition 4

For any HARA function,

Hence, by the last statement of Proposition 3, e_i (e_j■) is strictly convex
[linear] [strictly concave] if and only if

Hence, by equation (17), e_i(e_j■) 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.                       ■

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