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

Since f < O, it follows that e_bε < e_lε and hence

^ebε < ^elε,   ^ ^ε,

which contradicts the budget constraint (5). Therefore inequality (22)
must be true.

Second, we consider an increase in R^f so that e₁ changes to e_bo. Then η
changes to a^oη. Hence the sharing constant decreases if inequality (22) holds
with a and b being replaced by a^o and b^o. Therefore the same method by
which the first part of Lemma 1 has been proven can be applied here. ■

Appendix B: Proof of Proposition 2

a) Sufficiency: Suppose that F(e) is a quadratic function. Then f (e) =
a + be. Hence (17) implies linear relative sharing rules for two investors i and
j. Therefore all absolute rules are also linear.

b) Necessity: Differentiate (15) with respect to ε; this yields

ʃ//ʌ _λde_i dθ
^{f (e}“⁾ * =   ■/

Now suppose a linear absolute sharing rule for every investor: e_iε = a_i +
β_i ε, so that β~ = β_i. Then it follows from (25) for any two investors i and j

^{f (êis)} η. = ^f> ^(êj£'⁾ η^l.' ’   ^ ^ε'

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